diff --git a/stable b/stable index a1c2c6a..f4caccc 120000 --- a/stable +++ b/stable @@ -1 +1 @@ -v0.1.1 \ No newline at end of file +v0.1.2 \ No newline at end of file diff --git a/v0.1 b/v0.1 index a1c2c6a..f4caccc 120000 --- a/v0.1 +++ b/v0.1 @@ -1 +1 @@ -v0.1.1 \ No newline at end of file +v0.1.2 \ No newline at end of file diff --git a/v0.1.2/.documenter-siteinfo.json b/v0.1.2/.documenter-siteinfo.json new file mode 100644 index 0000000..b41c5dc --- /dev/null +++ b/v0.1.2/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.10.3","generation_timestamp":"2024-05-23T08:45:53","documenter_version":"1.4.1"}} \ No newline at end of file diff --git a/v0.1.2/api/index.html b/v0.1.2/api/index.html new file mode 100644 index 0000000..7f4f3d0 --- /dev/null +++ b/v0.1.2/api/index.html @@ -0,0 +1,18 @@ + +API · TransitionsInTimeseries.jl

API

Main analysis functions

TransitionsInTimeseries.estimate_changesFunction
estimate_changes(config::ChangesConfig, x [,t]) → result

Estimate possible transitions for input timeseries x using the approach specified in the configuration type config, see ChangesConfig for possibilities. t is the time vector corresponding to x, which defaults to eachindex(x).

Return the output as subtype of ChangesResults. The particular form of the output depends on the config and is described in its docstring. Regardless of type, result can always be given to significant_transitions to deduce which possible transitions are statistically significant using a variety of significance tests.

source

Sliding window

TransitionsInTimeseries.SlidingWindowConfigType
SlidingWindowConfig <: ChangesConfig
+SlidingWindowConfig(indicators, change_metrics; kwargs...)

A configuration that can be given to estimate_changes. It estimates transitions by a sliding window approach:

  1. Estimate the timeseries of an indicator by sliding a window over the input timeseries.
  2. Estimate changes of an indicator by sliding a window of the change metric over the indicator timeseries.

Both indicators and change metrics are generic Julia functions that input an x::AbstractVector and output an s::Real. Any function may be given and see making custom indicators/change metrics in the documentation for more information on possible optimizations.

indicators can be a single function or a tuple of indicators. Similarly, change_metrics can be a tuple or a single function. If tuples, the length of indicators and change_metrics must match. This way the analysis can be efficiently repeated for many indicators and/or change metrics.

The results output corresponding to SlidingWindowConfig is SlidingWindowResults.

Step 1. is skipped if nothing is provided as indicators, in which case the change metrics are estimated directly from input data.

Keyword arguments

  • width_ind::Int=100, stride_ind::Int=1: width and stride given to WindowViewer to compute the indicator from the input timeseries.
  • width_cha::Int=50, stride_cha::Int=1: width and stride given to WindowViewer to compute the change metric timeseries from the indicator timeseries.
  • whichtime = midpoint: The time vector corresponding to the indicators / change metric timeseries is obtained from t in estimate_changes using the keyword whichtime. Options include:
    • last: use the last timepoint of each window
    • midpoint: use the mid timepoint of each time window
    • first: use first timepoint of each window
    In fact, the indicators time vector is computed simply via
    t_indicator = windowmap(whichtime, t; width_ind, stride_ind)
    +t_change = windowmap(whichtime, t_indicator; width_cha, stride_cha)
    so any other function of the time window may be given to extract the time point itself, such as mean or median.
  • T = Float64: Element type of input timeseries to initialize some computations.
source
TransitionsInTimeseries.SlidingWindowResultsType
SlidingWindowResults <: ChangesResults

A struct containing the output of estimate_changes used with SlidingWindowConfig. It can be used for further analysis, visualization, or given to significant_transitions.

It has the following fields that the user may access

  • x: the input timeseries.

  • t: the time vector of the input timeseries.

  • x_indicator, the indicator timeseries (matrix with each column one indicator).

  • t_indicator, the time vector of the indicator timeseries.

  • x_change, the change metric timeseries (matrix with each column one change metric).

  • t_change, the time vector of the change metric timeseries.

  • config::SlidingWindowConfig, what was used for the analysis.

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Segmented window

TransitionsInTimeseries.SegmentedWindowConfigType
SegmentedWindowConfig <: IndicatorsChangeConfig
+SegmentedWindowConfig(indicators, change_metrics, tseg_start, tseg_end; kwargs...)

A configuration that can be given to estimate_changes. It estimates transitions by estimating indicators and changes in user-defined window segments as follows:

  1. For each segment specified, estimate the corresponding indicator timeseries by sliding a window over the input timeseries (within the window segment).
  2. For each segment of the indicator timeseries, estimate a scalar change metric by applying the change metric over the full segment of the indicator timeseries.d

tseg_start, tseg_end are the starts and ends of the window segments (the window segments may overlap, that's okay). indicators, change_metrics are identical as in SlidingWindowConfig.

The results output corresponding to SlidingWindowConfig is SegmentedWindowResults.

Keyword arguments

  • width_ind::Int=100, stride_ind::Int=1, whichtime = midpoint, T = Float64: keywords identical as in SlidingWindowConfig.
  • min_width_cha::Int=typemax(Int): minimal width required to perform the change metric estimation. If a segment is not sufficiently long, the change metric is NaN.
source
TransitionsInTimeseries.SegmentedWindowResultsType
SegmentedWindowResults <: ChangesResults

A struct containing the output of estimate_changes used with SegmentedWindowConfig. It can be used for further analysis, visualization, or given to significant_transitions.

It has the following fields that the user may access

  • x: the input timeseries.

  • t: the time vector of the input timeseries.

  • x_indicator::Vector{Matrix}, with x_indicator[k] the indicator timeseries (matrix with each column one indicator) of the k-th segment.

  • t_indicator::Vector{Vector}, with t_indicator[k] the time vector of the indicator timeseries for the k-th segment.

  • x_change::Matrix, the change metric values with x[k, i] the change metric of the i-th indicator for the k-th segment.

  • t_change, the time vector of the change metric.

  • config::SegmentedWindowConfig, what was used for the analysis.

  • i1::Vector{Int} indices corresponding to start time of each segment.

  • i2::Vector{Int} indices corresponding to end time of each segment.

  • precomp_change_metrics vector containing the precomputed change metrics of each segment.

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Slope change

TransitionsInTimeseries.SlopeChangeConfigType
SlopeChangeConfig <: ChangesConfig
+SlopeChangeConfig(; indicator = nothing, kw...)

A configuration that can be given to estimate_changes. It estimates a change of slope in the timeseries by fitting two connected linear segments to the timeseries, returning the results (i.e., the two-linear fits) as SlopeChangeResults.

Keyword arguments

  • indicator = nothing: if not nothing. Otherwise it should be a function f(x) -> Real. The slope fitting is then done over an indicator of the timeseries, which itself is estimated via a sliding window exactly as in SlidingWindowConfig.
  • width_ind, stride_ind, whichtime: exactly as in SlidingWindowConfig if indicator is not nothing.
source
TransitionsInTimeseries.SlopeChangeResultsType
SlopeChangeResults <: ChangesResults

A struct containing the output of estimate_changes used with SlopeChangeConfig. It can be used for further analysis, visualization, or given to significant_transitions. The only significance type that you can use this with significant_transitions is SlopeChangeSignificance.

It has the following fields that the user may access:

  • x: the input timeseries.
  • t: the time vector of the input timeseries.
  • x_indicator, the indicator timeseries.
  • t_indicator, the time vector of the indicator timeseries.
  • t_change, the time the slope changes.
  • fitparams = a, b, c, d, the fitted linear coefficients, a + b*t before. t_change and c + d*t after t_change.
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Significance testing

TransitionsInTimeseries.significant_transitionsFunction
significant_transitions(res::ChangesResults, signif::Significance)

Estimate significant transtions in res using the method described by signif. Return flags, a Boolean matrix with identical size as the changes stored in res (which typically is stored in the field res.x_change). flags is true wherever a change metric of res is deemed significant.

source
TransitionsInTimeseries.SurrogatesSignificanceType
SurrogatesSignificance <: Significance
+SurrogatesSignificance(; kwargs...)

A configuration struct for significance testing significant_transitions using timeseries surrogates.

Keyword arguments

  • surromethod = RandomFourier(): method to generate surrogates
  • n = 1000: how many surrogates to generate
  • rng = Random.default_rng(): random number generator for the surrogates
  • p = 0.05: threshold for significance of the p-value
  • tail = :both: tail type used, see below

Description

When used with ChangesResults, significance is estimated as follows: n surrogates from the input timeseries are generated using surromethod, which is any Surrogate subtype provided by TimeseriesSurrogates.jl. For each surrogate, the indicator and then change metric timeseries is extracted. The values of the surrogate change metrics form a distribution of values (one at each time point). The value of the original change metric is compared to that of the surrogate distribution and a p-value is extracted according to the specified tail. The p-value is compared with p to claim significance. After using SurrogatesSignificance, you may access the full p-values before thresholding in the field .pvalues (to e.g., threshold with different p).

The p-value is simply the proportion of surrogate change metric values that exceed (for tail = :right) or subseed (tail = :left) the original change metric at each given time point. Use tail = :left if the surrogate data are expected to have higher change metric values. This is the case for statistics that quantify entropy. For statistics that quantify autocorrelation, use tail = :right instead. For anything else, use the default tail = :both. An iterable of tail values can also be given, in which case a specific tail is used for each change metric in ChangesResults.

Note that the raw p-values can be accessed in the field .pvalues, after calling the significant_transitions function with SurrogatesSignificance, in case you wish to obtain a different threshold of the p-values.

source
TransitionsInTimeseries.ThresholdSignificanceType
ThresholdSignificance(threshold::Real; tail = :right) <: Significance

A configuration struct for significance testing in significant_transitions. Significance is estimated by comparing the value of each change metric with the given threshold. Values that exceed the threshold (if tail = :right) or subseed the threshold (if tail = :left) are deemed significant. If tail = :both then either condition is checked.

source
TransitionsInTimeseries.SigmaSignificanceType
SigmaSignificance(; factor = 3.0, tail = :both) <: Significance

A configuration struct for significance testing significant_transitions. When used with ChangesResults, significance is estimated by comparing how many standard deviations (σ) the value exceeds the mean value (μ). Values that exceed (if tail = :right) μ + factor*σ, or subseed (if tail = :left) μ - factor*σ are deemed significant. If tail = :both then either condition is checked.

factor can also be a vector of values, in which case a different value is used for each change metric.

See also QuantileSignificance.

source
TransitionsInTimeseries.QuantileSignificanceType
QuantileSignificance(; p = 0.95, tail = :both) <: Significance

A configuration struct for significance testing significant_transitions. When used with ChangesResults, significance is estimated by comparing the value of each change metric with its p-quantile. Values that exceed the p-quantile (if tail = :right) or subseed the 1-p-quantile (if tail = :left) are deemed significant. If tail = :both then either condition is checked.

QuantileSignficance guarantees that some values will be significant by the very definition of what a quantile is. See also SigmaSignificance that is similar but does not have this guarantee.

source
TransitionsInTimeseries.SlopeChangeSignificanceType
SlopeChangeSignificance(; moe_slope, moe_offset, slope_diff = moe_slope, pvalue = 0.05)

Test whether the result of SlopeChangeResults is statistically significant.

Two tests are done:

  1. Check whether the margin of error of the fitted parameters a, b, c, d of the two linear segments a + b*t, c + d*t is less than the specified margins of error, for a chosen pvalue.
  2. Test that the two slopes b, d have difference greater than slope_diff.

The Boolean & of the above two is the final test.

The margin of error is simply half the size of the confidence interval, also known as radius of the confidence interval.

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Indicators

Note that any Julia function can be an indicator or change metric, so the list here is only just a couple of indicators directly implemented in this package.

Value distribution

Statistics.meanFunction
mean(A::AbstractArray, w::AbstractWeights[, dims::Int])

Compute the weighted mean of array A with weight vector w (of type AbstractWeights). If dim is provided, compute the weighted mean along dimension dims.

Examples

n = 20
+x = rand(n)
+w = rand(n)
+mean(x, weights(w))
source
StatsBase.skewnessFunction
skewness(v, [wv::AbstractWeights], m=mean(v))

Compute the standardized skewness of a real-valued array v, optionally specifying a weighting vector wv and a center m.

source
StatsBase.kurtosisFunction
kurtosis(v, [wv::AbstractWeights], m=mean(v))

Compute the excess kurtosis of a real-valued array v, optionally specifying a weighting vector wv and a center m.

source

Critical Slowing Down

Statistics.varFunction
var(x::AbstractArray, w::AbstractWeights, [dim]; mean=nothing, corrected=false)

Compute the variance of a real-valued array x, optionally over a dimension dim. Observations in x are weighted using weight vector w. The uncorrected (when corrected=false) sample variance is defined as:

\[\frac{1}{\sum{w}} \sum_{i=1}^n {w_i\left({x_i - μ}\right)^2 }\]

where $n$ is the length of the input and $μ$ is the mean. The unbiased estimate (when corrected=true) of the population variance is computed by replacing $\frac{1}{\sum{w}}$ with a factor dependent on the type of weights used:

  • AnalyticWeights: $\frac{1}{\sum w - \sum {w^2} / \sum w}$
  • FrequencyWeights: $\frac{1}{\sum{w} - 1}$
  • ProbabilityWeights: $\frac{n}{(n - 1) \sum w}$ where $n$ equals count(!iszero, w)
  • Weights: ArgumentError (bias correction not supported)
source
var(ce::CovarianceEstimator, x::AbstractVector; mean=nothing)

Compute the variance of the vector x using the estimator ce.

source
TransitionsInTimeseries.ar1_whitenoiseFunction
ar1_whitenoise(x::AbstractVector)

Return the AR1 regression coefficient of a time series x by computing the analytic solution of the least-square parameter estimation under white-noise assumption for the data-generating process:

\[\theta = \sum_{i=2}^{n} x_i x_{i-1} / \sum_{i=2}^{n} x_{i-1}^2\]

source

Spectrum

TransitionsInTimeseries.LowfreqPowerSpectrumType
LowfreqPowerSpectrum(; q_lofreq = 0.1)

Return a PrecomputableFunction containing all the necessary fields to generate a PrecomputedLowfreqPowerSpectrum. The latter can be initialized by precompute:

lfps = precompute( LowfreqPowerSpectrum() )

Keyword arguments:

  • q_lofreq: a number between 0 and 1 that characterises which portion of the

frequency spectrum is considered to be low. For instance, q_lofreq = 0.1 implies that the lowest 10% of frequencies are considered to be the low ones.

source
TransitionsInTimeseries.PrecomputedLowfreqPowerSpectrumType
PrecomputedLowfreqPowerSpectrum

A struct containing all the precomputed fields to efficiently perform repetitive computation of the low-frequency power spectrum (LFPS), a number between 0 and 1 that characterizes the amount of power contained in the low frequencies of the power density spectrum of x. Once lfps::PrecomputedLowfreqPowerSpectrum is initialized, it can be used as a function to obtain the LFPS of x::AbstractVector by:

lfps(x)
source

Nonlinear dynamics

Indicators that come from nonlinear timeseries analysis typically quantify some entropy-like or complexity measure from the timeseries. Thousands (literally) such measures are provided out of the box by the ComplexityMeasures.jl package. Given that any of these may be used as an indicator or change metric, we made the decision to not copy-paste any measure here, as it is easy for the user to use any of them.

For example, using the permutation entropy as an indicator is as simple as doing

using ComplexityMeasures
+est = OrdinalPatterns(; m = 3) # order 3
+# create a function that given timeseries returns permutation entropy
+indicator = x -> entropy_normalized(est, x)

and giving the created indicator to e.g., SlidingWindowConfig.

Change metrics

Slope

TransitionsInTimeseries.kendalltauFunction
kendalltau(x::AbstractVector)

Compute the kendall-τ correlation coefficient of the time series x. kendalltau can be used as a change metric focused on trend.

source
TransitionsInTimeseries.spearmanFunction
spearman(x::AbstractVector)

Compute the spearman correlation coefficient of the time series x. spearman can be used as a change metric focused on trend.

source

Value distribution differences

TransitionsInTimeseries.difference_of_meansFunction
difference_of_means(x::AbstractArray)

Return the absolute difference of the means of the first and second halfs of x. difference_of_means can be used as a change metric focused on value differences. Creating similar statistical differences using other moments instead of mean is trivial. In fact, the source of difference_of_means is just:

# assumes 1-based indexing
+n = length(x)
+x1 = view(x, 1:n÷2)
+x2 = view(x, (n÷2 + 1):n)
+return abs(mean(x1) - mean(x2))

difference_of_means can also sensibly be used for windows of size 2, in which case the change metric timeseries is the same as the abs.(diff(...)) of the indicator timeseries.

source

Make your own indicator/metric!

The only difference between what is an "indicator" and what is a "change metric" is purely conceptual. As far as the code base of TransitionsInTimeseries.jl is concerned, they are both functions f: x::AbstractVector{Real} -> f(x)::Real. As a user you may give any such function for an indicator or change metric.

There are situations where you may optimize such a function based on knowledge of input x type and length, in which case you may use PrecomputableFunction:

Surrogates

For the surrogate generation, you can use any subtype of Surrogate defined in Timeseriessurrogates.jl.

Sliding windows

TransitionsInTimeseries.WindowViewerType
WindowViewer(x; width, stride)

Initialize an iterator that generates views over the given timeseries x based on a window with a given width, incrementing the window views with the given stride. You can use this directly with map, such as map(std, WindowViewer(x, ...)) would give you the moving-window-timeseries of the std of x.

If not given, the keywords width, stride are respectively taken as default_window_width(x) and 1.

source
TransitionsInTimeseries.windowmapFunction
windowmap(f::Function, x::AbstractVector; kwargs...) → mapped_f

A shortcut for first generating a wv = WindowViewer(x; kwargs...) and then applying mapped_f = map(f, wv). If x is accompanied by a time vector t, you probably also want to call this function with t instead of x and with one of mean, midpoint, midvalue as f to obtain a time vector for the mapped_f output.

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Load data

Visualization

TransitionsInTimeseries.plot_indicator_changesFunction
plot_indicator_changes(res) → (fig, axs)

Return fig::Figure and axs::Matrix{Axis}, on which res::ChangesResults has been visualised.

Keyword arguments:

  • colors = default_colors sets the colors of the line plots that are to

be cycled through. The default correspond to the color scheme of Julia Dynamics.

  • `indicatornames = defaultindicatorlabel(res), chametricnames =

defaultchametriclabel(res)sets the labels for the indicators and the change metrics, with the default inferring them from the names ofres.config.indicatorsandres.config.change_metrics`.

  • accent_linewidth = 3 sets the line width for the original signals (the

surrogates have linewidth = 1)

source
TransitionsInTimeseries.plot_significance!Function
plot_significance!(axs, res, signif)

Update the axs::Matrix{Axis} of a figure obtained with plot_indicator_changes(res) with the signif::SurrogatesSignificance.

Keyword arguments:

  • flags = nothing provides the significance flags, for instance obtained by

thresholding the pvalues.

  • nsurro = 20 sets the number of surrogates to plot.
source

Utils

default_window_width
diff --git a/v0.1.2/assets/documenter.js b/v0.1.2/assets/documenter.js new file mode 100644 index 0000000..c6562b5 --- /dev/null +++ b/v0.1.2/assets/documenter.js @@ -0,0 +1,1050 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + debounce(() => { + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); + }); +}); + +$(document).on("click", ".docs-article-toggle-button", function (event) { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 0 : 400; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(animationSpeed); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(animationSpeed); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(querystring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+
+ ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` + + `; + + let initial_search_body = ` +
Type something to get started!
+ `; + + let search_modal_footer = ` + + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/v0.1.2/assets/themes/documenter-dark.css b/v0.1.2/assets/themes/documenter-dark.css new file mode 100644 index 0000000..777353c --- /dev/null +++ b/v0.1.2/assets/themes/documenter-dark.css @@ -0,0 +1,7 @@ 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.pagination-next:focus,html.theme--documenter-dark .pagination-link:focus,html.theme--documenter-dark .pagination-ellipsis:focus,html.theme--documenter-dark .file-cta:focus,html.theme--documenter-dark .file-name:focus,html.theme--documenter-dark .select select:focus,html.theme--documenter-dark .textarea:focus,html.theme--documenter-dark .input:focus,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input:focus,html.theme--documenter-dark .button:focus,html.theme--documenter-dark .is-focused.pagination-previous,html.theme--documenter-dark .is-focused.pagination-next,html.theme--documenter-dark .is-focused.pagination-link,html.theme--documenter-dark .is-focused.pagination-ellipsis,html.theme--documenter-dark .is-focused.file-cta,html.theme--documenter-dark .is-focused.file-name,html.theme--documenter-dark .select select.is-focused,html.theme--documenter-dark .is-focused.textarea,html.theme--documenter-dark .is-focused.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-focused,html.theme--documenter-dark .is-focused.button,html.theme--documenter-dark .pagination-previous:active,html.theme--documenter-dark .pagination-next:active,html.theme--documenter-dark .pagination-link:active,html.theme--documenter-dark .pagination-ellipsis:active,html.theme--documenter-dark .file-cta:active,html.theme--documenter-dark .file-name:active,html.theme--documenter-dark .select select:active,html.theme--documenter-dark .textarea:active,html.theme--documenter-dark .input:active,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input:active,html.theme--documenter-dark .button:active,html.theme--documenter-dark .is-active.pagination-previous,html.theme--documenter-dark .is-active.pagination-next,html.theme--documenter-dark .is-active.pagination-link,html.theme--documenter-dark .is-active.pagination-ellipsis,html.theme--documenter-dark .is-active.file-cta,html.theme--documenter-dark .is-active.file-name,html.theme--documenter-dark .select select.is-active,html.theme--documenter-dark .is-active.textarea,html.theme--documenter-dark .is-active.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-active,html.theme--documenter-dark .is-active.button{outline:none}html.theme--documenter-dark .pagination-previous[disabled],html.theme--documenter-dark .pagination-next[disabled],html.theme--documenter-dark .pagination-link[disabled],html.theme--documenter-dark .pagination-ellipsis[disabled],html.theme--documenter-dark .file-cta[disabled],html.theme--documenter-dark .file-name[disabled],html.theme--documenter-dark .select select[disabled],html.theme--documenter-dark .textarea[disabled],html.theme--documenter-dark .input[disabled],html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input[disabled],html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark fieldset[disabled] .pagination-previous,fieldset[disabled] html.theme--documenter-dark .pagination-next,html.theme--documenter-dark fieldset[disabled] .pagination-next,fieldset[disabled] html.theme--documenter-dark .pagination-link,html.theme--documenter-dark fieldset[disabled] .pagination-link,fieldset[disabled] html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark fieldset[disabled] .pagination-ellipsis,fieldset[disabled] html.theme--documenter-dark .file-cta,html.theme--documenter-dark fieldset[disabled] .file-cta,fieldset[disabled] html.theme--documenter-dark .file-name,html.theme--documenter-dark fieldset[disabled] .file-name,fieldset[disabled] html.theme--documenter-dark .select select,fieldset[disabled] html.theme--documenter-dark .textarea,fieldset[disabled] html.theme--documenter-dark .input,fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar 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html.theme--documenter-dark .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:hover,html.theme--documenter-dark .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus,html.theme--documenter-dark .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus:not(:active),html.theme--documenter-dark .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--documenter-dark .button.is-black:active,html.theme--documenter-dark .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-outlined.is-hovered,html.theme--documenter-dark 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html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:hover,html.theme--documenter-dark .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus,html.theme--documenter-dark .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus:not(:active),html.theme--documenter-dark .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--documenter-dark .button.is-light:active,html.theme--documenter-dark .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark 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.button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;box-shadow:none;color:#f5f5f5}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark 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kbd.button{background-color:#363636;border-color:#363636;box-shadow:none}html.theme--documenter-dark .button.is-dark.is-inverted,html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;color:#363636}html.theme--documenter-dark .button.is-dark.is-inverted:hover,html.theme--documenter-dark .content kbd.button.is-inverted:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-dark.is-inverted[disabled],html.theme--documenter-dark .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#363636}html.theme--documenter-dark .button.is-dark.is-loading::after,html.theme--documenter-dark .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#363636;color:#363636}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#363636;border-color:#363636;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #363636 #363636 !important}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#363636;box-shadow:none;color:#363636}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#363636}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #363636 #363636 !important}html.theme--documenter-dark 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.button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#f4c72f;box-shadow:none;color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f4c72f #f4c72f !important}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning.is-light{background-color:#fefaec;color:#8c6e07}html.theme--documenter-dark .button.is-warning.is-light:hover,html.theme--documenter-dark .button.is-warning.is-light.is-hovered{background-color:#fdf7e0;border-color:transparent;color:#8c6e07}html.theme--documenter-dark .button.is-warning.is-light:active,html.theme--documenter-dark .button.is-warning.is-light.is-active{background-color:#fdf3d3;border-color:transparent;color:#8c6e07}html.theme--documenter-dark .button.is-danger{background-color:#cb3c33;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:hover,html.theme--documenter-dark .button.is-danger.is-hovered{background-color:#c13930;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:focus,html.theme--documenter-dark .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:focus:not(:active),html.theme--documenter-dark .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(203,60,51,0.25)}html.theme--documenter-dark .button.is-danger:active,html.theme--documenter-dark .button.is-danger.is-active{background-color:#b7362e;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger{background-color:#cb3c33;border-color:#cb3c33;box-shadow:none}html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-inverted:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#cb3c33;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-outlined.is-focused{background-color:#cb3c33;border-color:#cb3c33;color:#fff}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #cb3c33 #cb3c33 !important}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#cb3c33;box-shadow:none;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #cb3c33 #cb3c33 !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fbefef;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#f8e6e5;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#f6dcda;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:2px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#fff;border-color:#dbdbdb;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#f5f5f5;border-color:#dbdbdb;color:#7a7a7a;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:2px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#cdc3e3;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#f5f5f5;border-left:5px solid #dbdbdb;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #dbdbdb;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#cdc3e3}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#cdc3e3}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#cdc3e3}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--documenter-dark .icon.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--documenter-dark .icon.is-medium{height:2rem;width:2rem}html.theme--documenter-dark .icon.is-large{height:3rem;width:3rem}html.theme--documenter-dark .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--documenter-dark .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--documenter-dark .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--documenter-dark .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--documenter-dark div.icon-text{display:flex}html.theme--documenter-dark .image,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--documenter-dark .image img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--documenter-dark .image img.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--documenter-dark .image.is-5by4 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--documenter-dark .image.is-4by3 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by3 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.hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem} diff --git a/v0.1.2/assets/themeswap.js b/v0.1.2/assets/themeswap.js new file mode 100644 index 0000000..9f5eebe --- /dev/null +++ b/v0.1.2/assets/themeswap.js @@ -0,0 +1,84 @@ +// Small function to quickly swap out themes. Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } + // If we find a matching theme (and it's not the default), we'll set active to non-null + if (themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if (themename !== theme) disabled.push(ss); + } + var activeTheme = null; + if (active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } + } + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); + } +} +set_theme_from_local_storage(); diff --git a/v0.1.2/assets/warner.js b/v0.1.2/assets/warner.js new file mode 100644 index 0000000..3f6f5d0 --- /dev/null +++ b/v0.1.2/assets/warner.js @@ -0,0 +1,52 @@ +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } + + // Current version is not a version number, so we can't tell if it's the newest version. Abort. + if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { + return; + } + + // Current version is newest version, so no need to add a warning. + if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { + return; + } + + // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. + if (document.body.querySelector('meta[name="robots"]') === null) { + const meta = document.createElement("meta"); + meta.name = "robots"; + meta.content = "noindex"; + + document.getElementsByTagName("head")[0].appendChild(meta); + } + + const div = document.createElement("div"); + div.classList.add("outdated-warning-overlay"); + const closer = document.createElement("button"); + closer.classList.add("outdated-warning-closer", "delete"); + closer.addEventListener("click", function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE; + div.innerHTML = + 'This documentation is not for the latest stable release, but for either the development version or an older release.
Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", maybeAddWarning); +} else { + maybeAddWarning(); +} diff --git a/v0.1.2/devdocs/index.html b/v0.1.2/devdocs/index.html new file mode 100644 index 0000000..76b2658 --- /dev/null +++ b/v0.1.2/devdocs/index.html @@ -0,0 +1,2 @@ + +Developer's documentation · TransitionsInTimeseries.jl

Developer's documentation

This documentation addresses users that would like to contribute to the software by either solving bugs, improving documentation, or adding new methods. All contributions come in the form of Pull Requests, for which we strongly advise to follow good practices in scientific code, which means that they are properly formatted, documented, tested, etc.

New indicators or change metrics

As explained already in e.g., SlidingWindowConfig, new indicators or change metrics are standard Julia functions, so you only need to define such a function (and document it, test it, etc.).

New pipeline for estimating changes

This means to contribute a fundamentally new "pipeline" for estimating/detecting transitions in timeseries. This new pipeline defines what a "transition" means. To add a new pipeline follow these steps:

  1. Define a new subtype of ChangesConfig
  2. Define a new subtype of ChangesResults
  3. Add a method for estimate_changes which accepts the new ChangesConfig subtype you defined and returns the ChangesResults subtype you defined.

And that's it!

Optionally you can further extend:

  • TransitionsInTimeseries.plot_indicator_changes with the new ChangesResults type you defined. Note the plotting functions are in the ext folder of the repository.
  • significant_transitions(res::ChangesResults, signif::Significance) with your new ChangesResults type and as many Significance subtypes you have the capacity to extend for.

New pipeline for estimating significance

Statistically significant changes are "transitions". However, what "significant" means is not universal. There are different ways to test for significance and TransitionsInTimeseries.jl allows various methods.

To add a new pipeline for estimating significance follow these steps:

  1. Define a new subtype of Significance
  2. Extend significant_transitions(res::ChangesResults, signif::Significance) with your new type and as many ChangesResults subtypes as you have capacity for.

And that's it! Unfortunately so far we have found no way to make the significant_transitions code agnostic of the changes result, so you would need to add a method manually for every ChangesResults.

Optionally you can further extend TransitionsInTimeseries.plot_significance! (which you can find in the ext folder) for visualizing the significance results.

diff --git a/v0.1.2/examples/do-events.jl b/v0.1.2/examples/do-events.jl new file mode 100644 index 0000000..f40cd30 --- /dev/null +++ b/v0.1.2/examples/do-events.jl @@ -0,0 +1,329 @@ +#= +# Dansgaard-Oescher events and Critical Slowing Down + +The $\delta^{18}O$ timeseries of the North Greenland Ice Core Project +([NGRIP](https://en.wikipedia.org/wiki/North_Greenland_Ice_Core_Project)) are, +to this date, the best proxy record for the Dansgaard-Oeschger events +([DO-events](https://en.wikipedia.org/wiki/Dansgaard%E2%80%93Oeschger_event)). +DO-events are sudden warming episodes of the North Atlantic, reaching 10 degrees +of regional warming within 100 years. They happened quasi-periodically over the +last glacial cycle due to transitions between strong and weak states of the Atlantic +Meridional Overturning Circulation and might be therefore be the most prominent +examples of abrupt transitions in the field of climate science. We here propose +to hindcast these events by applying the theory of Critical Slowing Down (CSD) +on the NGRIP data, which can be found [here](https://www.iceandclimate.nbi.ku.dk/data/) +in its raw format. This analysis has already been done in [boers-early-warning-2018](@cite) +and we here try to reproduce Figure 2.d-f. + +## Preprocessing NGRIP + +Data pre-processing is not part of TransitionsInTimeseries.jl, +but a step the user has to do before using the package. +To present an example with a complete scientific workflow, +we will showcase typical data pre-processing here, that consist of the following steps: +1. Load the data, reverse and offset it to have time vector = time before 2000 AD. +2. Filter non-unique points in time and sort the data. +3. Regrid the data from uneven to even sampling. + +The time and $\delta^{18}O$ vectors resulting from the $i$-th preprocessing step are +respectively called $t_i$ and $x_i$. The final step consists in obtaining a residual +$r$, i.e. the fluctuations of the system around the attractor, which, within the CSD +theory, is assumed to be tracked. Over this example, it will appear that the +convenience of TransitionsInTimeseries.jl leads the bulk of the code to be written +for plotting and preprocessing. + +### Step 1: +=# + +using DelimitedFiles, Downloads + +function load_ngrip() + tmp = Base.download("https://raw.githubusercontent.com/JuliaDynamics/JuliaDynamics/"* + "master/timeseries/NGRIP.csv") + data, labels = readdlm(tmp, header = true) + return reverse(data[:, 1]) .- 2000, reverse(data[:, 2]) # (time, delta-18-0) vectors +end + +t1, x1 = load_ngrip() + +#= +### Step 2 +=# + +uniqueidx(v) = unique(i -> v[i], eachindex(v)) +function keep_unique(t, x) + unique_idx = uniqueidx(t) + return t[unique_idx], x[unique_idx] +end + +function sort_timeseries!(t, x) + p = sortperm(t) + permute!(t, p) + permute!(x, p) + return nothing +end + +t2, x2 = keep_unique(t1, x1) +sort_timeseries!(t2, x2) + +#= +### Step 3 +=# + +using BSplineKit + +function regrid2evensampling(t, x, dt) + itp = BSplineKit.interpolate(t, x, BSplineOrder(4)) + tspan = (ceil(minimum(t)), floor(maximum(t))) + t_even = collect(tspan[1]:dt:tspan[2]) + x_even = itp.(t_even) + return t_even, x_even +end + +dt = 5.0 # dt = 5 yr as in (Boers 2018) +t3, x3 = regrid2evensampling(t2, x2, dt) + +#= +### Step 4 + +For the final step we drop the indices: +=# + +using DSP + +function chebyshev_filter(t, x, fcutoff) + ii = 10 # Chebyshev filtering requires to prune first points of timeseries. + responsetype = Highpass(fcutoff, fs = 1/dt) + designmethod = Chebyshev1(8, 0.05) + r = filt(digitalfilter(responsetype, designmethod), x) + xtrend = x - r + return t[ii:end], x[ii:end], xtrend[ii:end], r[ii:end] +end + +fcutoff = 0.95 * 0.01 # cutoff ≃ 0.01 yr^-1 as in (Boers 2018) +t, x, xtrend, r = chebyshev_filter(t3, x3, fcutoff) + +#= +Let's now visualize our data in what will become our main figure. +For the segmentation of the DO-events, we rely on the tabulated +data from [rasmussen-stratigraphic-2014](@cite) (which will soon be available as downloadable): +=# + +using CairoMakie, Loess + +function loess_filter(t, x; span = 0.005) + loessmodel = loess(t, x, span = span) + xtrend = Loess.predict(loessmodel, t) + r = x - xtrend + return t, x, xtrend, r +end + +function kyr_xticks(tticks_yr) + tticks_kyr = ["$t" for t in Int.(tticks_yr ./ 1e3)] + return (tticks_yr, tticks_kyr) +end + +function plot_do(traw, xraw, tfilt, xfilt, t, r, t_transitions, xlims, xticks) + fig = Figure(size = (1600, 1200), fontsize = 24) + + ## Original timeseries with transition marked by vertical lines + ax1 = Axis(fig[1, 1], xlabel = L"Time (kyr) $\,$", ylabel = L"$\delta^{18}$O (permil)", + xaxisposition = :top, xticks = xticks) + lines!(ax1, traw, xraw, color = (:gray70, 0.5)) + lines!(ax1, tfilt, xfilt, color = :gray10, linewidth = 3) + vlines!(ax1, t_transitions, color = Cycled(1), linewidth = 3) + + ## Residual timeseries + ax2 = Axis(fig[2, 1], ylabel = L"Residual $\,$", xticks = xticks, + xticksvisible = false, xticklabelsvisible = false) + lines!(ax2, t, r, color = :gray50, linewidth = 1) + + ## Axes for variance and AC1 timeseries + ax3 = Axis(fig[3, 1], ylabel = L"Variance $\,$", xticks = xticks, + xticksvisible = false, xticklabelsvisible = false) + ax4 = Axis(fig[4, 1], xlabel = L"Time (kyr) $\,$", ylabel = L"Lag-1 autocor. $\,$", + xticks = xticks) + + axs = [ax1, ax2, ax3, ax4] + [xlims!(ax, xlims) for ax in axs] + ylims!(axs[1], (-48, -34)) + rowgap!(fig.layout, 10) + return fig, axs +end + +xlims = (-60e3, -10e3) +xticks = kyr_xticks(-60e3:5e3:5e3) +t_rasmussen = -[-60000, 59440, 58280, 55800, 54220, 49280, 46860, 43340, 41460, 40160, + 38220, 35480, 33740, 32500, 28900, 27780, 23340, 14692, 11703] +tloess, _, xloess, rloess = loess_filter(t3, x3) # loess-filtered signal for visualization +fig, axs = plot_do(t3, x3, tloess, xloess, t, r, t_rasmussen, xlims, xticks) +fig + +#= +## Hindcast on NGRIP data + +As one can see... there is not much to see so far. +Residuals are impossible to simply eye-ball and we therefore use +TransitionsInTimeseries.jl to study the evolution, measured by the ridge-regression +slope of the residual's variance and lag-1 autocorrelation (AC1) over time. +In many examples of the literature, including [boers-early-warning-2018](@cite), +the CSD analysis is performed over segments (sometimes only one) of the timeseries, +such that a significance value is obtained for each segment. By using +`SegmentedWindowConfig`, dealing with segments can be easily done in +TransitionsInTimeseries.jl and is demonstrated here: +=# + +using TransitionsInTimeseries, StatsBase +using Random: Xoshiro + +ac1(x) = sum(autocor(x, [1])) # AC1 from StatsBase +indicators = (var, ac1) +change_metrics = (RidgeRegressionSlope(), RidgeRegressionSlope()) +tseg_start = t_rasmussen[1:end-1] .+ 200 +tseg_end = t_rasmussen[2:end] .- 200 +config = SegmentedWindowConfig(indicators, change_metrics, + tseg_start, tseg_end; whichtime = last, width_ind = Int(200÷dt), + min_width_cha = 100) # require >=100 data points to estimate change metric +results = estimate_changes(config, r, t) +signif = SurrogatesSignificance(n = 1000, tail = [:right, :right], rng = Xoshiro(1995)) +flags = significant_transitions(results, signif) + +#= +That's it! We can now visualise our results with a generic function that we will re-use later: +=# + +function plot_segment_analysis!(axs, results, signif) + (; t_indicator, x_indicator) = results + for k in eachindex(t_indicator) # loop over the segments + for i in axes(signif.pvalues, 2) # loop over the indicators + if !isnan(signif.pvalues[k, i]) # plot if segment long enough + ## Plot indicator timeseries and its linear regression + ti, xi = t_indicator[k], x_indicator[k][:, i] + lines!(axs[i+2], ti, xi, color = Cycled(1)) + m, p = ridgematrix(ti, 0.0) * xi + if signif.pvalues[k, i] < 0.05 + lines!(axs[i+2], ti, m .* ti .+ p, color = :gray5, linewidth = 3) + else + lines!(axs[i+2], ti, m .* ti .+ p, color = :gray60, linewidth = 3) + end + end + end + end +end +plot_segment_analysis!(axs, results, signif) +fig + +#= +In [boers-early-warning-2018](@cite), 13/16 and 7/16 true positives are respectively +found for the variance and AC1, with 16 referring to the total number of transitions. +The timeseries actually includes 18 transition but, in +[boers-early-warning-2018](@cite), some segments are considered too small to be analysed. +In contrast, we here respectively find 9/16 true positives +for the variance and 3/16 for AC1. We can track down the discrepancies to be in the +surrogate testing, since the indicator timeseries computed here are almost exactly +similar to those of [boers-early-warning-2018](@cite). This mismatch points +out that packages like TransitionsInTimeseries.jl are desirable for research to be +reproducible, especially since CSD is gaining attention - not only within the +scientific community but also in popular media. + +## CSD: only a necessary condition, only in some cases + +For codimension-1 systems, approaching a fold, Hopf or transcritical bifurcation implies +a widening of the potential $U$, which defines the deterministic term $f = -∇U$ of the +SDE's right-hand-side. In the presence of noise, this leads to CSD, which is therefore +a **necessary condition** for crossing one of these bifurcations - although it is not +always assessable by analysing the timeseries due to practical limitations (e.g. sparse +data subject to large measurement noise). It is nonetheless not given that DO-events, +as many other real-life applications, can be seen as a codimension-1 fold, Hopf or +transcritical bifurcations. Besides this, we emphasise that CSD is **not a sufficient +condition** for assessing a transition being ahead in near future, since a resilience +loss can happen without actually crossing any bifurcation. This can be illustrated on +the present example by performing the same analysis only until few hundred years before +the transition: +=# + +tseg_end = t_rasmussen[2:end] .- 700 # stop analysis 500 years earlier than before +config = SegmentedWindowConfig(indicators, change_metrics, + tseg_start, tseg_end, whichtime = last, width_ind = Int(200÷dt), + min_width_cha = 100) +results = estimate_changes(config, r, t) +signif = SurrogatesSignificance(n = 1000, tail = [:right, :right], rng = Xoshiro(1995)) +flags = significant_transitions(results, signif) +fig, axs = plot_do(t3, x3, tloess, xloess, t, r, t_rasmussen, xlims, xticks) +plot_segment_analysis!(axs, results, signif) +fig + +#= +For the variance and AC1, we here respectively find 6 and 3 positives, although the transitions are still far ahead. +This shows that what CSD captures is a potential widening induced by a shift of the forcing parameter rather than the actual transition. +We therefore believe, as already suggested in some studies, +that "resilience-loss indicators" is a more accurate name than "early-warning signals" when using CSD. + +We draw attention upon the fact that the $\delta^{18}O$ timeseries is noisy and sparsely re-sampled. +Furthermore, interpolating over time introduces a potential bias in the statistics, even if performed on a coarse grid. +The NGRIP data therefore represents an example that should be handled with care - as many others where CSD analysis has been applied on transitions in the field of geoscience. +To contrast with this, we propose to perform the same analysis on synthethic DO data, obtained from an Earth Model of Intermediate Complexity (EMIC). + +!!! warning "Degrees of freedom" + These sources of error come along the usual problem of arbitrarily choosing (1) a filtering method, (2) windowing parameters and (3) appropriate metrics (for instance when the forcing noise is suspected to be correlated). + This leads to a large number of degrees of freedom (DoF). + Although sensible guesses are possible here, checking that results are robust w.r.t. the DoF should be a standard practice. + +!!! info "Future improvement" + Supporting the computations for uneven timeseries is a planned improvement of TransitionsInTimeseries.jl. + This will avoid the need of regridding data on coarse grids and will prevent from introducing any bias. + +## Hindcasting simulated DO-events + +In CLIMBER-X, the EMIC described in [willeit-earth-2022](@cite), DO-like events can be triggered by forcing the North Atlantic with a (white noise) freshwater input. +Simulated DO-like events present the big advantage of being evenly sampled in time and free of measurement noise. +We run this analysis over two exemplary simulation outputs: +=# + +t_transitions = [[1, 1850, 2970, 3970, 5070, 5810, 7050, 8050], + [1, 3500, 4370, 5790, 7200, 8140]] +t_lb = [[300, 500, 300, 600, 300, 500, 500], [1800, 500, 1000, 900, 500]] +tseg_start = [t_transitions[1][1:end-1] + t_lb[1], t_transitions[2][1:end-1] + t_lb[2]] +tseg_end = [t_transitions[1][2:end] .- 50, t_transitions[2][2:end] .- 50] + +figvec = Figure[] + +for j in 1:2 + ## Download the data and perform loess filtering on it + tmp = Base.download("https://raw.githubusercontent.com/JuliaDynamics/JuliaDynamics/" * + "master/timeseries/climberx-do$(j)-omaxa.csv") + data = readdlm(tmp) + tcx, xcx = data[1, 1000:end], data[2, 1000:end] + t, x, xtrend, r = loess_filter(tcx, xcx, span = 0.02) + + ## Initialize figure + xlims = (0, last(tcx)) + xticks = kyr_xticks(xlims[1]:1e3:xlims[2]) + fig, axs = plot_do(tcx, xcx, t, xtrend, t, r, t_transitions[j], extrema(t), xticks) + ylims!(axs[1], (5, 40)) + axs[1].ylabel = L"Max. Atlantic overturning (Sv) $\,$" + + ## Run sliding analysis and update figure with results + dt = mean(diff(tcx)) + config = SegmentedWindowConfig( + indicators, change_metrics, tseg_start[j], tseg_end[j], + whichtime = last, width_ind = Int(200÷dt), min_width_cha = 100) + results = estimate_changes(config, r, t) + signif = SurrogatesSignificance(n = 1_000, tail = [:right, :right], rng = Xoshiro(1995)) + flags = significant_transitions(results, signif) + + plot_segment_analysis!(axs, results, signif) + vlines!(axs[1], t_transitions[j], color = Cycled(1), linewidth = 3) + push!(figvec, fig) +end +figvec[1] + +#= +It here appears that not all transitions are preceeded by a significant increase of variance and AC1, even in the case of clean and evenly sampled time series. Let's check another case: +=# + +figvec[2] + +#= +Same here! Although CLIMBER-X does not represent real DO-events, the above-performed analysis might be hinting at the fact that not all DO transitions can be forecasted with CSD. Nonetheless, performing a CSD analysis can inform on the evolution of a system's resilience. +=# \ No newline at end of file diff --git a/v0.1.2/examples/do-events/3835a9d2.png b/v0.1.2/examples/do-events/3835a9d2.png new file mode 100644 index 0000000..e1cfc2b Binary files /dev/null and b/v0.1.2/examples/do-events/3835a9d2.png differ diff --git a/v0.1.2/examples/do-events/3f28f250.png b/v0.1.2/examples/do-events/3f28f250.png new file mode 100644 index 0000000..75e2c4b Binary files /dev/null and b/v0.1.2/examples/do-events/3f28f250.png differ diff --git a/v0.1.2/examples/do-events/7494cf2f.png b/v0.1.2/examples/do-events/7494cf2f.png new file mode 100644 index 0000000..43103ed Binary files /dev/null and b/v0.1.2/examples/do-events/7494cf2f.png differ diff --git a/v0.1.2/examples/do-events/ba06f092.png b/v0.1.2/examples/do-events/ba06f092.png new file mode 100644 index 0000000..f90d89c Binary files /dev/null and b/v0.1.2/examples/do-events/ba06f092.png differ diff --git a/v0.1.2/examples/do-events/ecdc7c2d.png b/v0.1.2/examples/do-events/ecdc7c2d.png new file mode 100644 index 0000000..a296407 Binary files /dev/null and b/v0.1.2/examples/do-events/ecdc7c2d.png differ diff --git a/v0.1.2/examples/do-events/index.html b/v0.1.2/examples/do-events/index.html new file mode 100644 index 0000000..086c9fa --- /dev/null +++ b/v0.1.2/examples/do-events/index.html @@ -0,0 +1,191 @@ + +Dansgaard-Oescher events and Critical Slowing Down · TransitionsInTimeseries.jl

Dansgaard-Oescher events and Critical Slowing Down

The $\delta^{18}O$ timeseries of the North Greenland Ice Core Project (NGRIP) are, to this date, the best proxy record for the Dansgaard-Oeschger events (DO-events). DO-events are sudden warming episodes of the North Atlantic, reaching 10 degrees of regional warming within 100 years. They happened quasi-periodically over the last glacial cycle due to transitions between strong and weak states of the Atlantic Meridional Overturning Circulation and might be therefore be the most prominent examples of abrupt transitions in the field of climate science. We here propose to hindcast these events by applying the theory of Critical Slowing Down (CSD) on the NGRIP data, which can be found here in its raw format. This analysis has already been done in (Boers, 2018) and we here try to reproduce Figure 2.d-f.

Preprocessing NGRIP

Data pre-processing is not part of TransitionsInTimeseries.jl, but a step the user has to do before using the package. To present an example with a complete scientific workflow, we will showcase typical data pre-processing here, that consist of the following steps:

  1. Load the data, reverse and offset it to have time vector = time before 2000 AD.
  2. Filter non-unique points in time and sort the data.
  3. Regrid the data from uneven to even sampling.

The time and $\delta^{18}O$ vectors resulting from the $i$-th preprocessing step are respectively called $t_i$ and $x_i$. The final step consists in obtaining a residual $r$, i.e. the fluctuations of the system around the attractor, which, within the CSD theory, is assumed to be tracked. Over this example, it will appear that the convenience of TransitionsInTimeseries.jl leads the bulk of the code to be written for plotting and preprocessing.

Step 1:

using DelimitedFiles, Downloads
+
+function load_ngrip()
+    tmp = Base.download("https://raw.githubusercontent.com/JuliaDynamics/JuliaDynamics/"*
+        "master/timeseries/NGRIP.csv")
+    data, labels = readdlm(tmp, header = true)
+    return reverse(data[:, 1]) .- 2000, reverse(data[:, 2]) # (time, delta-18-0) vectors
+end
+
+t1, x1 = load_ngrip()
([-59944.5, -59939.6, -59935.9, -59930.5, -59925.0, -59919.0, -59914.3, -59909.4, -59904.5, -59899.7  …  -27.40000000000009, -27.40000000000009, -27.299999999999955, -27.299999999999955, -27.200000000000045, -27.09999999999991, -27.09999999999991, -27.0, -26.90000000000009, -26.799999999999955], [-43.52, -42.94, -42.13, -40.04, -41.09, -43.57, -42.28, -41.2, -44.07, -44.53  …  -35.29, -35.18, -35.51, -36.03, -36.52, -37.12, -37.41, -37.21, -36.62, -35.52])

Step 2

uniqueidx(v) = unique(i -> v[i], eachindex(v))
+function keep_unique(t, x)
+    unique_idx = uniqueidx(t)
+    return t[unique_idx], x[unique_idx]
+end
+
+function sort_timeseries!(t, x)
+    p = sortperm(t)
+    permute!(t, p)
+    permute!(x, p)
+    return nothing
+end
+
+t2, x2 = keep_unique(t1, x1)
+sort_timeseries!(t2, x2)

Step 3

using BSplineKit
+
+function regrid2evensampling(t, x, dt)
+    itp = BSplineKit.interpolate(t, x, BSplineOrder(4))
+    tspan = (ceil(minimum(t)), floor(maximum(t)))
+    t_even = collect(tspan[1]:dt:tspan[2])
+    x_even = itp.(t_even)
+    return t_even, x_even
+end
+
+dt = 5.0        # dt = 5 yr as in (Boers 2018)
+t3, x3 = regrid2evensampling(t2, x2, dt)
([-59944.0, -59939.0, -59934.0, -59929.0, -59924.0, -59919.0, -59914.0, -59909.0, -59904.0, -59899.0  …  -74.0, -69.0, -64.0, -59.0, -54.0, -49.0, -44.0, -39.0, -34.0, -29.0], [-43.445268010797946, -42.85408835164853, -41.357360665158936, -39.9365468206892, -41.60554051352076, -43.57000000000001, -42.145558689100454, -41.33160823883399, -44.273597725729985, -44.4622612304012  …  -35.12, -36.190000000000005, -35.14, -37.06, -34.93, -34.23, -33.53, -35.09, -34.00000000000001, -34.46])

Step 4

For the final step we drop the indices:

using DSP
+
+function chebyshev_filter(t, x, fcutoff)
+    ii = 10             # Chebyshev filtering requires to prune first points of timeseries.
+    responsetype = Highpass(fcutoff, fs = 1/dt)
+    designmethod = Chebyshev1(8, 0.05)
+    r = filt(digitalfilter(responsetype, designmethod), x)
+    xtrend = x - r
+    return t[ii:end], x[ii:end], xtrend[ii:end], r[ii:end]
+end
+
+fcutoff = 0.95 * 0.01   # cutoff ≃ 0.01 yr^-1 as in (Boers 2018)
+t, x, xtrend, r = chebyshev_filter(t3, x3, fcutoff)
([-59899.0, -59894.0, -59889.0, -59884.0, -59879.0, -59874.0, -59869.0, -59864.0, -59859.0, -59854.0  …  -74.0, -69.0, -64.0, -59.0, -54.0, -49.0, -44.0, -39.0, -34.0, -29.0], [-44.4622612304012, -44.264401428210874, -44.02956767939856, -42.29334237357212, -42.594341674200344, -41.09704877854712, -40.39837540966913, -44.369155046413255, -42.74492506508012, -43.737014520530145  …  -35.12, -36.190000000000005, -35.14, -37.06, -34.93, -34.23, -33.53, -35.09, -34.00000000000001, -34.46], [-43.89287787976461, -48.663435112414156, -51.31169981050155, -51.269094666491796, -49.90310432724566, -47.721430022512344, -44.36050569418247, -43.87556827410048, -43.57802291229893, -41.36870967826094  …  -34.81349114528442, -36.01634453179986, -36.150446802907226, -36.37595639697143, -35.96843653265903, -33.899634243732976, -32.49784036951535, -33.06575312724142, -34.07956188682297, -34.4079080530096], [-0.5693833506365866, 4.399033684203284, 7.282132131102991, 8.975752292919669, 7.308762653045311, 6.624381243965224, 3.9621302845133433, -0.493586772312777, 0.8330978472188112, -2.368304842269206  …  -0.30650885471557926, -0.17365546820014846, 1.0104468029072273, -0.684043603028572, 1.0384365326590237, -0.3303657562670192, -1.0321596304846505, -2.0242468727585807, 0.07956188682296689, -0.05209194699039831])

Let's now visualize our data in what will become our main figure. For the segmentation of the DO-events, we rely on the tabulated data from (Rasmussen et al., 2014) (which will soon be available as downloadable):

using CairoMakie, Loess
+
+function loess_filter(t, x; span = 0.005)
+    loessmodel = loess(t, x, span = span)
+    xtrend = Loess.predict(loessmodel, t)
+    r = x - xtrend
+    return t, x, xtrend, r
+end
+
+function kyr_xticks(tticks_yr)
+    tticks_kyr = ["$t" for t in Int.(tticks_yr ./ 1e3)]
+    return (tticks_yr, tticks_kyr)
+end
+
+function plot_do(traw, xraw, tfilt, xfilt, t, r, t_transitions, xlims, xticks)
+    fig = Figure(size = (1600, 1200), fontsize = 24)
+
+    # Original timeseries with transition marked by vertical lines
+    ax1 = Axis(fig[1, 1], xlabel = L"Time (kyr) $\,$", ylabel = L"$\delta^{18}$O (permil)",
+        xaxisposition = :top, xticks = xticks)
+    lines!(ax1, traw, xraw, color = (:gray70, 0.5))
+    lines!(ax1, tfilt, xfilt, color = :gray10, linewidth = 3)
+    vlines!(ax1, t_transitions, color = Cycled(1), linewidth = 3)
+
+    # Residual timeseries
+    ax2 = Axis(fig[2, 1], ylabel = L"Residual $\,$", xticks = xticks,
+        xticksvisible = false, xticklabelsvisible = false)
+    lines!(ax2, t, r, color = :gray50, linewidth = 1)
+
+    # Axes for variance and AC1 timeseries
+    ax3 = Axis(fig[3, 1], ylabel = L"Variance $\,$", xticks = xticks,
+        xticksvisible = false, xticklabelsvisible = false)
+    ax4 = Axis(fig[4, 1], xlabel = L"Time (kyr) $\,$", ylabel = L"Lag-1 autocor. $\,$",
+        xticks = xticks)
+
+    axs = [ax1, ax2, ax3, ax4]
+    [xlims!(ax, xlims) for ax in axs]
+    ylims!(axs[1], (-48, -34))
+    rowgap!(fig.layout, 10)
+    return fig, axs
+end
+
+xlims = (-60e3, -10e3)
+xticks = kyr_xticks(-60e3:5e3:5e3)
+t_rasmussen = -[-60000, 59440, 58280, 55800, 54220, 49280, 46860, 43340, 41460, 40160,
+                38220, 35480, 33740, 32500, 28900, 27780, 23340, 14692, 11703]
+tloess, _, xloess, rloess = loess_filter(t3, x3)    # loess-filtered signal for visualization
+fig, axs = plot_do(t3, x3, tloess, xloess, t, r, t_rasmussen, xlims, xticks)
+fig
Example block output

Hindcast on NGRIP data

As one can see... there is not much to see so far. Residuals are impossible to simply eye-ball and we therefore use TransitionsInTimeseries.jl to study the evolution, measured by the ridge-regression slope of the residual's variance and lag-1 autocorrelation (AC1) over time. In many examples of the literature, including (Boers, 2018), the CSD analysis is performed over segments (sometimes only one) of the timeseries, such that a significance value is obtained for each segment. By using SegmentedWindowConfig, dealing with segments can be easily done in TransitionsInTimeseries.jl and is demonstrated here:

using TransitionsInTimeseries, StatsBase
+using Random: Xoshiro
+
+ac1(x) = sum(autocor(x, [1]))   # AC1 from StatsBase
+indicators = (var, ac1)
+change_metrics = (RidgeRegressionSlope(), RidgeRegressionSlope())
+tseg_start = t_rasmussen[1:end-1] .+ 200
+tseg_end = t_rasmussen[2:end] .- 200
+config = SegmentedWindowConfig(indicators, change_metrics,
+    tseg_start, tseg_end; whichtime = last, width_ind = Int(200÷dt),
+    min_width_cha = 100)        # require >=100 data points to estimate change metric
+results = estimate_changes(config, r, t)
+signif = SurrogatesSignificance(n = 1000, tail = [:right, :right], rng = Xoshiro(1995))
+flags = significant_transitions(results, signif)
18×2 BitMatrix:
+ 0  0
+ 0  0
+ 0  0
+ 0  0
+ 1  0
+ 1  1
+ 1  1
+ 1  1
+ 0  0
+ 1  0
+ 1  0
+ 0  0
+ 0  0
+ 1  0
+ 0  0
+ 0  0
+ 1  0
+ 1  0

That's it! We can now visualise our results with a generic function that we will re-use later:

function plot_segment_analysis!(axs, results, signif)
+    (; t_indicator, x_indicator) = results
+    for k in eachindex(t_indicator)             # loop over the segments
+        for i in axes(signif.pvalues, 2)        # loop over the indicators
+            if !isnan(signif.pvalues[k, i])     # plot if segment long enough
+                # Plot indicator timeseries and its linear regression
+                ti, xi = t_indicator[k], x_indicator[k][:, i]
+                lines!(axs[i+2], ti, xi, color = Cycled(1))
+                m, p = ridgematrix(ti, 0.0) * xi
+                if signif.pvalues[k, i] < 0.05
+                    lines!(axs[i+2], ti, m .* ti .+ p, color = :gray5, linewidth = 3)
+                else
+                    lines!(axs[i+2], ti, m .* ti .+ p, color = :gray60, linewidth = 3)
+                end
+            end
+        end
+    end
+end
+plot_segment_analysis!(axs, results, signif)
+fig
Example block output

In (Boers, 2018), 13/16 and 7/16 true positives are respectively found for the variance and AC1, with 16 referring to the total number of transitions. The timeseries actually includes 18 transition but, in (Boers, 2018), some segments are considered too small to be analysed. In contrast, we here respectively find 9/16 true positives for the variance and 3/16 for AC1. We can track down the discrepancies to be in the surrogate testing, since the indicator timeseries computed here are almost exactly similar to those of (Boers, 2018). This mismatch points out that packages like TransitionsInTimeseries.jl are desirable for research to be reproducible, especially since CSD is gaining attention - not only within the scientific community but also in popular media.

CSD: only a necessary condition, only in some cases

For codimension-1 systems, approaching a fold, Hopf or transcritical bifurcation implies a widening of the potential $U$, which defines the deterministic term $f = -∇U$ of the SDE's right-hand-side. In the presence of noise, this leads to CSD, which is therefore a necessary condition for crossing one of these bifurcations - although it is not always assessable by analysing the timeseries due to practical limitations (e.g. sparse data subject to large measurement noise). It is nonetheless not given that DO-events, as many other real-life applications, can be seen as a codimension-1 fold, Hopf or transcritical bifurcations. Besides this, we emphasise that CSD is not a sufficient condition for assessing a transition being ahead in near future, since a resilience loss can happen without actually crossing any bifurcation. This can be illustrated on the present example by performing the same analysis only until few hundred years before the transition:

tseg_end = t_rasmussen[2:end] .- 700    # stop analysis 500 years earlier than before
+config = SegmentedWindowConfig(indicators, change_metrics,
+    tseg_start, tseg_end, whichtime = last, width_ind = Int(200÷dt),
+    min_width_cha = 100)
+results = estimate_changes(config, r, t)
+signif = SurrogatesSignificance(n = 1000, tail = [:right, :right], rng = Xoshiro(1995))
+flags = significant_transitions(results, signif)
+fig, axs = plot_do(t3, x3, tloess, xloess, t, r, t_rasmussen, xlims, xticks)
+plot_segment_analysis!(axs, results, signif)
+fig
Example block output

For the variance and AC1, we here respectively find 6 and 3 positives, although the transitions are still far ahead. This shows that what CSD captures is a potential widening induced by a shift of the forcing parameter rather than the actual transition. We therefore believe, as already suggested in some studies, that "resilience-loss indicators" is a more accurate name than "early-warning signals" when using CSD.

We draw attention upon the fact that the $\delta^{18}O$ timeseries is noisy and sparsely re-sampled. Furthermore, interpolating over time introduces a potential bias in the statistics, even if performed on a coarse grid. The NGRIP data therefore represents an example that should be handled with care - as many others where CSD analysis has been applied on transitions in the field of geoscience. To contrast with this, we propose to perform the same analysis on synthethic DO data, obtained from an Earth Model of Intermediate Complexity (EMIC).

Degrees of freedom

These sources of error come along the usual problem of arbitrarily choosing (1) a filtering method, (2) windowing parameters and (3) appropriate metrics (for instance when the forcing noise is suspected to be correlated). This leads to a large number of degrees of freedom (DoF). Although sensible guesses are possible here, checking that results are robust w.r.t. the DoF should be a standard practice.

Future improvement

Supporting the computations for uneven timeseries is a planned improvement of TransitionsInTimeseries.jl. This will avoid the need of regridding data on coarse grids and will prevent from introducing any bias.

Hindcasting simulated DO-events

In CLIMBER-X, the EMIC described in (Willeit et al., 2022), DO-like events can be triggered by forcing the North Atlantic with a (white noise) freshwater input. Simulated DO-like events present the big advantage of being evenly sampled in time and free of measurement noise. We run this analysis over two exemplary simulation outputs:

t_transitions = [[1, 1850, 2970, 3970, 5070, 5810, 7050, 8050],
+                 [1, 3500, 4370, 5790, 7200, 8140]]
+t_lb = [[300, 500, 300, 600, 300, 500, 500], [1800, 500, 1000, 900, 500]]
+tseg_start = [t_transitions[1][1:end-1] + t_lb[1], t_transitions[2][1:end-1] + t_lb[2]]
+tseg_end = [t_transitions[1][2:end] .- 50, t_transitions[2][2:end] .- 50]
+
+figvec = Figure[]
+
+for j in 1:2
+    # Download the data and perform loess filtering on it
+    tmp = Base.download("https://raw.githubusercontent.com/JuliaDynamics/JuliaDynamics/" *
+        "master/timeseries/climberx-do$(j)-omaxa.csv")
+    data = readdlm(tmp)
+    tcx, xcx = data[1, 1000:end], data[2, 1000:end]
+    t, x, xtrend, r = loess_filter(tcx, xcx, span = 0.02)
+
+    # Initialize figure
+    xlims = (0, last(tcx))
+    xticks = kyr_xticks(xlims[1]:1e3:xlims[2])
+    fig, axs = plot_do(tcx, xcx,  t, xtrend, t, r, t_transitions[j], extrema(t), xticks)
+    ylims!(axs[1], (5, 40))
+    axs[1].ylabel = L"Max. Atlantic overturning (Sv) $\,$"
+
+    # Run sliding analysis and update figure with results
+    dt = mean(diff(tcx))
+    config = SegmentedWindowConfig(
+        indicators, change_metrics, tseg_start[j], tseg_end[j],
+        whichtime = last, width_ind = Int(200÷dt), min_width_cha = 100)
+    results = estimate_changes(config, r, t)
+    signif = SurrogatesSignificance(n = 1_000, tail = [:right, :right], rng = Xoshiro(1995))
+    flags = significant_transitions(results, signif)
+
+    plot_segment_analysis!(axs, results, signif)
+    vlines!(axs[1], t_transitions[j], color = Cycled(1), linewidth = 3)
+    push!(figvec, fig)
+end
+figvec[1]
Example block output

It here appears that not all transitions are preceeded by a significant increase of variance and AC1, even in the case of clean and evenly sampled time series. Let's check another case:

figvec[2]
Example block output

Same here! Although CLIMBER-X does not represent real DO-events, the above-performed analysis might be hinting at the fact that not all DO transitions can be forecasted with CSD. Nonetheless, performing a CSD analysis can inform on the evolution of a system's resilience.

diff --git a/v0.1.2/examples/ks_paleojump.jl b/v0.1.2/examples/ks_paleojump.jl new file mode 100644 index 0000000..012e025 --- /dev/null +++ b/v0.1.2/examples/ks_paleojump.jl @@ -0,0 +1,168 @@ +# # Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries + +# The goal of this example is to show how simple it is to re-create an analysis _similar_ +# to what was done in the paper +# "Automatic detection of abrupt transitions in paleoclimate records", +# [Bagniewski2021](@cite). The same analysis was then used to create a database +# of transitions in paleoclimate records in [Bagniewski2023](@cite) +# Using TransitionsInTimeseries.jl and HypothesisTests.jl, +# the analysis becomes a 10-lines-of-code script (for a given timeseries). + +# ## Scientific background + +# The approach of [Bagniewski2021](@cite) is based on the [two-sample +# Kolmogorov Smirnov test](https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test#Two-sample_Kolmogorov%E2%80%93Smirnov_test). +# It tests whether the samples from two datasets or timeseries +# are distributed according to the same cumulative density function or not. +# This can be estimated by comparing the value of the _KS-statistic_ +# versus some threshold that depends on the required confidence. + +# The application of this test for identifying transitions in timeseries is simple: + +# 1. A sliding window analysis is performed in the timesries +# 1. In each window, the KS statistic is estimated between the first half and the second +# half of the timeseries within this window. +# 1. Transitions are defined by when the KS statistic exceeds a particular value +# based on some confidence. The transition occurs in the middle of the window. + +# We should point out that in [Bagniewski2021](@cite) the authors did a +# more detailed analysis: analyzed many different window widths, added a +# conditional clause to exclude transitions that do not exceed a predefined minimum +# "jump" in the data, and also added another conditional clause that +# filtered out transitions that are grouped in time (which is a natural consequence +# of using the Kolmogorov-Smirov test for detecting transitions). +# +# Here we won't do that post processing, mainly because it is rather simple +# to include these additional conditional clauses to filter transitions after they are found. + +# ## Steps for TransitionsInTimeseries.jl + +# Doing this kind of work with TransitionsInTimeseries.jl is so easy you won't even trip! +# This analysis follows the same sliding window approach showcased in our [Tutorial](@ref), +# and it even excludes the "indicator" aspect: the change metric is estimated directly +# from the input data! + +# As such, we really only need to define/do these things before we have finished the analysis: + +# 1. Load the input data (we will use the same example as the NGRIP data of +# [Dansgaard-Oescher events and Critical Slowing Down](@ref) example) +# and set the appropriate time window. +# 3. Define the function that estimates the change metric (i.e., the KS-statistic) +# 3. Perform the sliding window analysis as in the [Tutorial](@ref) with [`estimate_changes`](@ref) +# 4. Estimate the "confident" transitions in the data by comparing the estimated +# KS-statistic with a predefined threshold. + +# ## Load timeseries and window length +# Following the Dansgaard-Oescher events example, we load +# the data after all the processing steps done in that example: + +using DelimitedFiles, CairoMakie + +tmp = Base.download("https://raw.githubusercontent.com/JuliaDynamics/JuliaDynamics/"* + "master/timeseries/NGRIP_processed.csv") +data = readdlm(tmp) +t, xtrend, xresid, xloess = collect.(eachcol(data)) + +fig, ax = lines(t, xtrend; axis = (ylabel = "NGRIP (processed)", xlabel = "time")) +lines!(ax, t, xloess; linewidth = 2) +fig + +# For the window, since we are using a sliding window here, we will be using a +# window of length 500 (which is approximately 1/2 to 1/4 the span between typical +# transitions found by [rasmussen-stratigraphic-2014](@cite)). + +window = 500 + +# ## Defining the change metric function + +# HypothesisTest.jl implements the Kolmogorov-Smirnov test, however here we are interested +# in the value of the test iself (the so-called KS-statistic), rather than a p-value. +# To this end, we define the following function to compute the statistic, +# which also normalizes it as in [Bagniewski2021](@cite). + +using HypothesisTests + +function normalized_KS_statistic(timeseries) + N = length(timeseries) + i = N÷2 + x = view(timeseries, 1:i) + y = view(timeseries, (i+1):N) + kstest = ApproximateTwoSampleKSTest(x, y) + nx = ny = i # length of each timeseries half of total + n = nx*ny/(nx + ny) # written fully for concreteness + D_KS = kstest.δ # can be compared directly with sqrt(-log(α/2)/2) + ## Rescale according to eq. (5) of the paper + rescaled = 1 - ((1 - D_KS)/(1 - sqrt(1/n))) + return rescaled +end + +N = 1000 # the statistic is independent of `N` for large enough `N`! +x = randn(N) +y = 1.8randn(N) .+ 1.0 +z = randn(N) +w = 0.6randn(N) .- 2.0 + +fig, ax = density(x; color = ("black", 0.5), strokewidth = 4.0, label = "reference distribution") +ax.title = "showcase of normalized KS-statistic" +for q in (y, z, w) + D_KS = normalized_KS_statistic(vcat(x, q)) + density!(ax, q; label = "D_KS = $(D_KS)") +end +axislegend(ax) +fig + +# ## Perform the sliding window analysis + +# This is just a straightforward call to [`estimate_changes`](@ref). +# In fact, it is even simpler than the tutorial. Here we skip completely +# the "indicator" estimation step, and we evaluate the change metric directly +# on input data. We do this by simply passing `nothing` as the indicators. + +using TransitionsInTimeseries + +config = SlidingWindowConfig(nothing, normalized_KS_statistic; width_cha = 500) + +results = estimate_changes(config, xtrend, t) + +# Which we can visualize +function visualize_results(results) + fig, ax1 = lines(t, xtrend; axis = (ylabel = "NGRIP (processed)",)) + ax2, = lines(fig[2, 1], results.t_change, vec(results.x_change), axis = (ylabel = "D_KS (normalized)", xlabel = "time")) + linkxaxes!(ax1, ax2) + hidexdecorations!(ax1; grid = false) + xloess_normed = (xloess .- minimum(xloess))./(maximum(xloess) - minimum(xloess)) + lines!(ax2, t, xloess_normed; color = ("gray", 0.5)) + fig +end + +visualize_results(results) + +# By overplotting the (smoothened) NGRIP timeseries and the +# normalized KS-statistic, it already becomes pretty clear +# that the statistic peaks when transitions occur. + +# The same thing happens if we alter the window duration + +config = SlidingWindowConfig(nothing, normalized_KS_statistic; width_cha = 200) +results = estimate_changes(config, xtrend, t) +visualize_results(results) + +# So one can easily obtain extra confidence by varying window +# size as in [Bagniewski2021](@cite). + +# ## Identifying "confident" transitions + +# As this identification here is done via a simple threshold, +# identifying the transitions is a nearly trivial call +# to [`significant_transitions`](@ref) with [`ThresholdSignificance`](@ref) + +signif = ThresholdSignificance(0.5) # or any other threshold +flags = significant_transitions(results, signif) + +fig = visualize_results(results) +axDKS = content(fig[2,1]) +vlines!(axDKS, results.t_change[vec(flags)], color = ("red", 0.25)) +fig + +# We could proceed with a lot of preprocessing as in [Bagniewski2021](@cite) +# but we skip this here for the sake of simplicity. \ No newline at end of file diff --git a/v0.1.2/examples/ks_paleojump/217f75ed.png b/v0.1.2/examples/ks_paleojump/217f75ed.png new file mode 100644 index 0000000..d2eb077 Binary files /dev/null and b/v0.1.2/examples/ks_paleojump/217f75ed.png differ diff --git a/v0.1.2/examples/ks_paleojump/414a766b.png b/v0.1.2/examples/ks_paleojump/414a766b.png new file mode 100644 index 0000000..dd97c25 Binary files /dev/null and b/v0.1.2/examples/ks_paleojump/414a766b.png differ diff --git a/v0.1.2/examples/ks_paleojump/709306e6.png b/v0.1.2/examples/ks_paleojump/709306e6.png new file mode 100644 index 0000000..7ee0264 Binary files /dev/null and b/v0.1.2/examples/ks_paleojump/709306e6.png differ diff --git a/v0.1.2/examples/ks_paleojump/b13cc719.png b/v0.1.2/examples/ks_paleojump/b13cc719.png new file mode 100644 index 0000000..eb7f19b4 Binary files /dev/null and b/v0.1.2/examples/ks_paleojump/b13cc719.png differ diff --git a/v0.1.2/examples/ks_paleojump/ed8344ce.png b/v0.1.2/examples/ks_paleojump/ed8344ce.png new file mode 100644 index 0000000..7fdfa08 Binary files /dev/null and b/v0.1.2/examples/ks_paleojump/ed8344ce.png differ diff --git a/v0.1.2/examples/ks_paleojump/index.html b/v0.1.2/examples/ks_paleojump/index.html new file mode 100644 index 0000000..dd2c58b --- /dev/null +++ b/v0.1.2/examples/ks_paleojump/index.html @@ -0,0 +1,68 @@ + +Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries · TransitionsInTimeseries.jl

Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries

The goal of this example is to show how simple it is to re-create an analysis similar to what was done in the paper "Automatic detection of abrupt transitions in paleoclimate records", (Bagniewski et al., 2021). The same analysis was then used to create a database of transitions in paleoclimate records in (Bagniewski et al., 2023) Using TransitionsInTimeseries.jl and HypothesisTests.jl, the analysis becomes a 10-lines-of-code script (for a given timeseries).

Scientific background

The approach of (Bagniewski et al., 2021) is based on the two-sample Kolmogorov Smirnov test. It tests whether the samples from two datasets or timeseries are distributed according to the same cumulative density function or not. This can be estimated by comparing the value of the KS-statistic versus some threshold that depends on the required confidence.

The application of this test for identifying transitions in timeseries is simple:

  1. A sliding window analysis is performed in the timesries
  2. In each window, the KS statistic is estimated between the first half and the second half of the timeseries within this window.
  3. Transitions are defined by when the KS statistic exceeds a particular value based on some confidence. The transition occurs in the middle of the window.

We should point out that in (Bagniewski et al., 2021) the authors did a more detailed analysis: analyzed many different window widths, added a conditional clause to exclude transitions that do not exceed a predefined minimum "jump" in the data, and also added another conditional clause that filtered out transitions that are grouped in time (which is a natural consequence of using the Kolmogorov-Smirov test for detecting transitions).

Here we won't do that post processing, mainly because it is rather simple to include these additional conditional clauses to filter transitions after they are found.

Steps for TransitionsInTimeseries.jl

Doing this kind of work with TransitionsInTimeseries.jl is so easy you won't even trip! This analysis follows the same sliding window approach showcased in our Tutorial, and it even excludes the "indicator" aspect: the change metric is estimated directly from the input data!

As such, we really only need to define/do these things before we have finished the analysis:

  1. Load the input data (we will use the same example as the NGRIP data of Dansgaard-Oescher events and Critical Slowing Down example) and set the appropriate time window.
  2. Define the function that estimates the change metric (i.e., the KS-statistic)
  3. Perform the sliding window analysis as in the Tutorial with estimate_changes
  4. Estimate the "confident" transitions in the data by comparing the estimated KS-statistic with a predefined threshold.

Load timeseries and window length

Following the Dansgaard-Oescher events example, we load the data after all the processing steps done in that example:

using DelimitedFiles, CairoMakie
+
+tmp = Base.download("https://raw.githubusercontent.com/JuliaDynamics/JuliaDynamics/"*
+    "master/timeseries/NGRIP_processed.csv")
+data = readdlm(tmp)
+t, xtrend, xresid, xloess = collect.(eachcol(data))
+
+fig, ax = lines(t, xtrend; axis = (ylabel = "NGRIP (processed)", xlabel = "time"))
+lines!(ax, t, xloess; linewidth = 2)
+fig
Example block output

For the window, since we are using a sliding window here, we will be using a window of length 500 (which is approximately 1/2 to 1/4 the span between typical transitions found by (Rasmussen et al., 2014)).

window = 500
500

Defining the change metric function

HypothesisTest.jl implements the Kolmogorov-Smirnov test, however here we are interested in the value of the test iself (the so-called KS-statistic), rather than a p-value. To this end, we define the following function to compute the statistic, which also normalizes it as in (Bagniewski et al., 2021).

using HypothesisTests
+
+function normalized_KS_statistic(timeseries)
+    N = length(timeseries)
+    i = N÷2
+    x = view(timeseries, 1:i)
+    y = view(timeseries, (i+1):N)
+    kstest = ApproximateTwoSampleKSTest(x, y)
+    nx = ny = i # length of each timeseries half of total
+    n = nx*ny/(nx + ny) # written fully for concreteness
+    D_KS = kstest.δ # can be compared directly with sqrt(-log(α/2)/2)
+    # Rescale according to eq. (5) of the paper
+    rescaled = 1 - ((1 - D_KS)/(1 - sqrt(1/n)))
+    return rescaled
+end
+
+N = 1000 # the statistic is independent of `N` for large enough `N`!
+x = randn(N)
+y = 1.8randn(N) .+ 1.0
+z = randn(N)
+w = 0.6randn(N) .- 2.0
+
+fig, ax = density(x; color = ("black", 0.5), strokewidth = 4.0, label = "reference distribution")
+ax.title = "showcase of normalized KS-statistic"
+for q in (y, z, w)
+    D_KS = normalized_KS_statistic(vcat(x, q))
+    density!(ax, q; label = "D_KS = $(D_KS)")
+end
+axislegend(ax)
+fig
Example block output

Perform the sliding window analysis

This is just a straightforward call to estimate_changes. In fact, it is even simpler than the tutorial. Here we skip completely the "indicator" estimation step, and we evaluate the change metric directly on input data. We do this by simply passing nothing as the indicators.

using TransitionsInTimeseries
+
+config = SlidingWindowConfig(nothing, normalized_KS_statistic; width_cha = 500)
+
+results = estimate_changes(config, xtrend, t)
SlidingWindowResults
+ input timeseries:              6975-element Vector{Float64}
+ indicators:                    nothing
+ indicator (window, stride):    (100, 1)
+ change metrics:                [:normalized_KS_statistic]
+ change metric (window, stride): (500, 1)
+

Which we can visualize

function visualize_results(results)
+    fig, ax1 = lines(t, xtrend; axis = (ylabel = "NGRIP (processed)",))
+    ax2, = lines(fig[2, 1], results.t_change, vec(results.x_change), axis = (ylabel = "D_KS (normalized)", xlabel = "time"))
+    linkxaxes!(ax1, ax2)
+    hidexdecorations!(ax1; grid = false)
+    xloess_normed = (xloess .- minimum(xloess))./(maximum(xloess) - minimum(xloess))
+    lines!(ax2, t, xloess_normed; color = ("gray", 0.5))
+    fig
+end
+
+visualize_results(results)
Example block output

By overplotting the (smoothened) NGRIP timeseries and the normalized KS-statistic, it already becomes pretty clear that the statistic peaks when transitions occur.

The same thing happens if we alter the window duration

config = SlidingWindowConfig(nothing, normalized_KS_statistic; width_cha = 200)
+results = estimate_changes(config, xtrend, t)
+visualize_results(results)
Example block output

So one can easily obtain extra confidence by varying window size as in (Bagniewski et al., 2021).

Identifying "confident" transitions

As this identification here is done via a simple threshold, identifying the transitions is a nearly trivial call to significant_transitions with ThresholdSignificance

signif = ThresholdSignificance(0.5) # or any other threshold
+flags = significant_transitions(results, signif)
+
+fig = visualize_results(results)
+axDKS = content(fig[2,1])
+vlines!(axDKS, results.t_change[vec(flags)], color = ("red", 0.25))
+fig
Example block output

We could proceed with a lot of preprocessing as in (Bagniewski et al., 2021) but we skip this here for the sake of simplicity.

diff --git a/v0.1.2/examples/logistic.jl b/v0.1.2/examples/logistic.jl new file mode 100644 index 0000000..b9163cd --- /dev/null +++ b/v0.1.2/examples/logistic.jl @@ -0,0 +1,203 @@ +# # Permutation entropy for dynamic regime changes + +# Permutation entropy is used frequently to detect a transition between +# one dynamic regime to another. It is useful when the mean and std. of the +# timeseries values are very similar between the two regimes, which would mean that +# common distribution-based indicators, or common critical-slowing-down based indicators, +# would fail. + +# This example will also explore different ways to test for significance that +# are arguably better suitable in such an application than the [Tutorial](@ref)'s default +# of significance via random Fourier surrogates. + +# ## Logistic map timeseries + +# A simple example of this is transitions from periodic to weakly chaotic to chaotic +# motion in the logistic map. First, let's generate a timeseries of the logistic map + +using DynamicalSystemsBase +using CairoMakie + +## time-dependent logistic map, so that the `r` parameter increases with time +r1 = 3.83 +r2 = 3.86 +N = 2000 +rs = range(r1, r2; length = N) + +function logistic_drifting_rule(u, rs, n) + r = rs[n+1] # time is `n`, starting from 0 + return SVector(r*u[1]*(1 - u[1])) +end + +ds = DeterministicIteratedMap(logistic_drifting_rule, [0.5], rs) +x = trajectory(ds, N-1)[1][:, 1] + +# Plot it, using as time the parameter value (they coincide) +fig, ax = lines(rs, x; linewidth = 0.5) +ax.xlabel = "r (time)" +ax.ylabel = "x" +fig + +# In this example there is a rather obvious transition to strongly +# chaotic motion at r ≈ 3.855. However, there is also a subtle transition +# to weak chaos at r ≈ 3.847. This transition is barely visible in the +# timeseries, and in fact many of the timeseries statistical properties remain identical. + +# ## Using a simpler change metric + +# Now, let's compute various indicators and their changes, +# focusing on the permutation entropy as an indicator. We use +# order 4 here, because we know that to detect changes in a period `m` we would need +# an order ≥ `m+1` permutation entropy. + +using TransitionsInTimeseries, ComplexityMeasures + +function permutation_entropy(m) + est = SymbolicPermutation(; m) # order 3 + indicator = x -> entropy_normalized(est, x) + return indicator +end + +indicators = (var, ar1_whitenoise, permutation_entropy(4)) +indistrings = ("var", "ar1", "pe") + +# In this example there is no critical slowing down; +# instead, there is a sharp transition between periodic and chaotic motion. +# Hence, we shouldn't be using any trend-based change metrics. +# Instead, we will use the most basic change metric, [`difference_of_means`](@ref). +# With this metric it also makes most sense to use as stride half the window width +metric = (difference_of_means, difference_of_means, difference_of_means) + +width_ind = N÷100 +width_cha = 20 +stride_cha = 10 + +config = SlidingWindowConfig(indicators, metric; + width_ind, width_cha, stride_cha, +) + +results = estimate_changes(config, x, rs) + +# Let's now plot the change metrics of the indicators + +function plot_change_metrics() + fig, ax = lines(rs, x; axis = (ylabel = "input",), figure = (size = (600, 600),)) + hidexdecorations!(ax; grid = false) + ## plot all change metrics + for (i, c) in enumerate(eachcol(results.x_change)) + ax, = scatterlines(fig[i+1, 1], results.t_change, c; + axis = (ylabel = indistrings[i],), label = "input" + ) + if i < 3 + hidexdecorations!(ax; grid = false) + else + ax.xlabel = "r (time)" + end + end + return fig +end + +fig = plot_change_metrics() + +# We already see the interesting results we expect: the permutation entropy shows a +# striking change as we go from periodic to weakly chaotic motion at r ≈ 3.847. +# (Remember: the plotted quantity is how much the indicator changes within a time window. +# High values mean large changes.) + +# Due to its construction, permutation entropy will have a spike for periodic data at the +# start of the timeseries, so we can safely ignore the spike at r ≈ 3.83. + +# ## Significance via random Fourier surrogates + +# One way to test for significance would be via the standard way as in the [Tutorial](@ref), +# utilizing surrogate timeseries and [`SurrogatesSignificance`](@ref). + +# Let's do it here for an example, but, we have to be **careful**. +# It is crucial that for permutation entropy we use `:right` as the `tail`, +# because it is expected that the surrogates will have higher +# differences in the permutation entropy timeseries (because, if there is no +# dynamical change, the permutation entropy will stay the same, while in the surrogates +# there are always random fluctuations! + +surromethod = RandomFourier() + +## Define a function because we will re-use later +using Random: Xoshiro +function overplot_surrogate_significance!(fig, surromethod, color = "black") + + signif = SurrogatesSignificance(; + n = 1000, tail = [:both, :both, :right], surromethod, rng = Xoshiro(42), + ) + flags = significant_transitions(results, signif) + + ## and also plot the flags with same color + for (i, indicator) in enumerate(indicators) + ## To make things visually clear, we will also plot some example surrogate + ## timeseries for each indicator and change metric pair + for _ in 1:10 + s = TimeseriesSurrogates.surrogate(x, surromethod) + p = windowmap(indicator, s; width = width_ind) + q = windowmap(metric[i], p; width = width_cha, stride = stride_cha) + lines!(fig[i+1, 1], results.t_change, q; color = (color, 0.2), linewidth = 1) + end + ## Plot the flags as vertical dashed lines + vlines!(fig[i+1, 1], results.t_change[flags[:, i]]; + color = color, linestyle = :dash, linewidth = 3 + ) + end + ## add a title to the figure with how we estimate significance + content(fig[1, 1]).title = "surrogates: "*string(nameof(typeof(surromethod))) +end + +surromethod = RandomFourier() +overplot_surrogate_significance!(fig, surromethod) + +fig + +# ## Different surrogates +# %% #src +# Random Fourier surrogates perserve the power spectrum of the timeseries, but the power spectrum +# is a property integrated over the whole timeseries. It doesn't contain any information +# highlighting the _local_ dynamics or information that preserves the local +# changes of dynamical behavior. + +# A surrogate type that does a better job in preserving local sharp +# changes in the timeseries (and hence provides **stricter** +# surrogate-based significance) is for example `RelativePartialRandomization`. + +# A much better alternative is to use block-shuffled surrogates, which +# preserve the short term local temporal correlation in the timeseries and hence +# also preserve local short term sharp changes in the dynamic behavior. + +surromethod = RelativePartialRandomization(0.25) +fig = plot_change_metrics() +overplot_surrogate_significance!(fig, surromethod, "gray") +fig + +# Our results have improved. In the permutation entropy, we see only two +# transitions detected as significant, which is correct: only two +# real dynamical transitions exist in the data. +# In the other two indicators we also see fewer transitions, but as we +# have already discussed, no results with the other indicators should +# be taken into meaningful consideration, as these indicators +# are simply inappropriate for what we are looking for here. + +# ## Simpler Significance +# %% #src +# Arguably, exactly because we are using the [`difference_of_means`](@ref) as a +# change metric, we may want to be less strict and more simple with our tests for significance. +# Instead of using [`SurrogatesSignificance`](@ref) we may use the simpler and much faster +# [`SigmaSignificance`](@ref), which simply claims significant time points +# whenever a change metric exceeds some pre-defined factor of its timeseries standard deviation. + +fig = plot_change_metrics() +flags = significant_transitions(results, SigmaSignificance(factor = 5.0)) + +## Plot the flags +for (i, indicator) in enumerate(indicators) + vlines!(fig[i+1, 1], results.t_change[flags[:, i]]; + color = Cycled(3), linestyle = :dash, linewidth = 3 + ) +end +content(fig[1, 1]).title = "significance from std" +fig diff --git a/v0.1.2/examples/logistic/32bfb484.png b/v0.1.2/examples/logistic/32bfb484.png new file mode 100644 index 0000000..245f270 Binary files /dev/null and b/v0.1.2/examples/logistic/32bfb484.png differ diff --git a/v0.1.2/examples/logistic/79a77bed.png b/v0.1.2/examples/logistic/79a77bed.png new file mode 100644 index 0000000..8950099 Binary files /dev/null and b/v0.1.2/examples/logistic/79a77bed.png differ diff --git a/v0.1.2/examples/logistic/7dd3de76.png b/v0.1.2/examples/logistic/7dd3de76.png new file mode 100644 index 0000000..9c645e5 Binary files /dev/null and b/v0.1.2/examples/logistic/7dd3de76.png differ diff --git a/v0.1.2/examples/logistic/c2799241.png b/v0.1.2/examples/logistic/c2799241.png new file mode 100644 index 0000000..b721ae4 Binary files /dev/null and b/v0.1.2/examples/logistic/c2799241.png differ diff --git a/v0.1.2/examples/logistic/e3d84558.png b/v0.1.2/examples/logistic/e3d84558.png new file mode 100644 index 0000000..9054648 Binary files /dev/null and b/v0.1.2/examples/logistic/e3d84558.png differ diff --git a/v0.1.2/examples/logistic/index.html b/v0.1.2/examples/logistic/index.html new file mode 100644 index 0000000..7d2043a --- /dev/null +++ b/v0.1.2/examples/logistic/index.html @@ -0,0 +1,128 @@ + +Permutation entropy for dynamic regime changes · TransitionsInTimeseries.jl

Permutation entropy for dynamic regime changes

Permutation entropy is used frequently to detect a transition between one dynamic regime to another. It is useful when the mean and std. of the timeseries values are very similar between the two regimes, which would mean that common distribution-based indicators, or common critical-slowing-down based indicators, would fail.

This example will also explore different ways to test for significance that are arguably better suitable in such an application than the Tutorial's default of significance via random Fourier surrogates.

Logistic map timeseries

A simple example of this is transitions from periodic to weakly chaotic to chaotic motion in the logistic map. First, let's generate a timeseries of the logistic map

using DynamicalSystemsBase
+using CairoMakie
+
+# time-dependent logistic map, so that the `r` parameter increases with time
+r1 = 3.83
+r2 = 3.86
+N = 2000
+rs = range(r1, r2; length = N)
+
+function logistic_drifting_rule(u, rs, n)
+    r = rs[n+1] # time is `n`, starting from 0
+    return SVector(r*u[1]*(1 - u[1]))
+end
+
+ds = DeterministicIteratedMap(logistic_drifting_rule, [0.5], rs)
+x = trajectory(ds, N-1)[1][:, 1]
2000-element Vector{Float64}:
+ 0.5
+ 0.9575
+ 0.15585767321160574
+ 0.5039019545917718
+ 0.9574529422362281
+ 0.15602441103002537
+ 0.5043473227583317
+ 0.9574501255597653
+ 0.1560361152301728
+ 0.5043840906474082
+ ⋮
+ 0.4253904015445437
+ 0.9434836074964187
+ 0.20581843730311683
+ 0.6309301050251992
+ 0.8988117342940414
+ 0.3510584545931264
+ 0.8793611090928393
+ 0.40948549063776396
+ 0.9333718681550522

Plot it, using as time the parameter value (they coincide)

fig, ax = lines(rs, x; linewidth = 0.5)
+ax.xlabel = "r (time)"
+ax.ylabel = "x"
+fig
Example block output

In this example there is a rather obvious transition to strongly chaotic motion at r ≈ 3.855. However, there is also a subtle transition to weak chaos at r ≈ 3.847. This transition is barely visible in the timeseries, and in fact many of the timeseries statistical properties remain identical.

Using a simpler change metric

Now, let's compute various indicators and their changes, focusing on the permutation entropy as an indicator. We use order 4 here, because we know that to detect changes in a period m we would need an order ≥ m+1 permutation entropy.

using TransitionsInTimeseries, ComplexityMeasures
+
+function permutation_entropy(m)
+    est = SymbolicPermutation(; m) # order 3
+    indicator = x -> entropy_normalized(est, x)
+    return indicator
+end
+
+indicators = (var, ar1_whitenoise, permutation_entropy(4))
+indistrings = ("var", "ar1", "pe")
("var", "ar1", "pe")

In this example there is no critical slowing down; instead, there is a sharp transition between periodic and chaotic motion. Hence, we shouldn't be using any trend-based change metrics. Instead, we will use the most basic change metric, difference_of_means. With this metric it also makes most sense to use as stride half the window width

metric = (difference_of_means, difference_of_means, difference_of_means)
+
+width_ind = N÷100
+width_cha = 20
+stride_cha = 10
+
+config = SlidingWindowConfig(indicators, metric;
+    width_ind, width_cha, stride_cha,
+)
+
+results = estimate_changes(config, x, rs)
SlidingWindowResults
+ input timeseries:              2000-element Vector{Float64}
+ indicators:                    [:var, :ar1_whitenoise, Symbol("#1")]
+ indicator (window, stride):    (20, 1)
+ change metrics:                [:difference_of_means, :difference_of_means, :difference_of_means]
+ change metric (window, stride): (20, 10)
+

Let's now plot the change metrics of the indicators

function plot_change_metrics()
+    fig, ax = lines(rs, x; axis = (ylabel = "input",), figure = (size = (600, 600),))
+    hidexdecorations!(ax; grid = false)
+    # plot all change metrics
+    for (i, c) in enumerate(eachcol(results.x_change))
+        ax, = scatterlines(fig[i+1, 1], results.t_change, c;
+            axis = (ylabel = indistrings[i],), label = "input"
+        )
+        if i < 3
+            hidexdecorations!(ax; grid = false)
+        else
+            ax.xlabel = "r (time)"
+        end
+    end
+    return fig
+end
+
+fig = plot_change_metrics()
Example block output

We already see the interesting results we expect: the permutation entropy shows a striking change as we go from periodic to weakly chaotic motion at r ≈ 3.847. (Remember: the plotted quantity is how much the indicator changes within a time window. High values mean large changes.)

Due to its construction, permutation entropy will have a spike for periodic data at the start of the timeseries, so we can safely ignore the spike at r ≈ 3.83.

Significance via random Fourier surrogates

One way to test for significance would be via the standard way as in the Tutorial, utilizing surrogate timeseries and SurrogatesSignificance.

Let's do it here for an example, but, we have to be careful. It is crucial that for permutation entropy we use :right as the tail, because it is expected that the surrogates will have higher differences in the permutation entropy timeseries (because, if there is no dynamical change, the permutation entropy will stay the same, while in the surrogates there are always random fluctuations!

surromethod = RandomFourier()
+
+# Define a function because we will re-use later
+using Random: Xoshiro
+function overplot_surrogate_significance!(fig, surromethod, color = "black")
+
+    signif = SurrogatesSignificance(;
+        n = 1000, tail = [:both, :both, :right], surromethod, rng = Xoshiro(42),
+    )
+    flags = significant_transitions(results, signif)
+
+    # and also plot the flags with same color
+    for (i, indicator) in enumerate(indicators)
+        # To make things visually clear, we will also plot some example surrogate
+        # timeseries for each indicator and change metric pair
+        for _ in 1:10
+            s = TimeseriesSurrogates.surrogate(x, surromethod)
+            p = windowmap(indicator, s; width = width_ind)
+            q = windowmap(metric[i], p; width = width_cha, stride = stride_cha)
+            lines!(fig[i+1, 1], results.t_change, q;  color = (color, 0.2), linewidth = 1)
+        end
+        # Plot the flags as vertical dashed lines
+        vlines!(fig[i+1, 1], results.t_change[flags[:, i]];
+            color = color, linestyle = :dash, linewidth = 3
+        )
+    end
+    # add a title to the figure with how we estimate significance
+    content(fig[1, 1]).title = "surrogates: "*string(nameof(typeof(surromethod)))
+end
+
+surromethod = RandomFourier()
+overplot_surrogate_significance!(fig, surromethod)
+
+fig
Example block output

Different surrogates

Random Fourier surrogates perserve the power spectrum of the timeseries, but the power spectrum is a property integrated over the whole timeseries. It doesn't contain any information highlighting the local dynamics or information that preserves the local changes of dynamical behavior.

A surrogate type that does a better job in preserving local sharp changes in the timeseries (and hence provides stricter surrogate-based significance) is for example RelativePartialRandomization.

A much better alternative is to use block-shuffled surrogates, which preserve the short term local temporal correlation in the timeseries and hence also preserve local short term sharp changes in the dynamic behavior.

surromethod = RelativePartialRandomization(0.25)
+fig = plot_change_metrics()
+overplot_surrogate_significance!(fig, surromethod, "gray")
+fig
Example block output

Our results have improved. In the permutation entropy, we see only two transitions detected as significant, which is correct: only two real dynamical transitions exist in the data. In the other two indicators we also see fewer transitions, but as we have already discussed, no results with the other indicators should be taken into meaningful consideration, as these indicators are simply inappropriate for what we are looking for here.

Simpler Significance

Arguably, exactly because we are using the difference_of_means as a change metric, we may want to be less strict and more simple with our tests for significance. Instead of using SurrogatesSignificance we may use the simpler and much faster SigmaSignificance, which simply claims significant time points whenever a change metric exceeds some pre-defined factor of its timeseries standard deviation.

fig = plot_change_metrics()
+flags = significant_transitions(results, SigmaSignificance(factor = 5.0))
+
+# Plot the flags
+for (i, indicator) in enumerate(indicators)
+    vlines!(fig[i+1, 1], results.t_change[flags[:, i]];
+        color = Cycled(3), linestyle = :dash, linewidth = 3
+    )
+end
+content(fig[1, 1]).title = "significance from std"
+fig
Example block output
diff --git a/v0.1.2/index.html b/v0.1.2/index.html new file mode 100644 index 0000000..1bfb9dd --- /dev/null +++ b/v0.1.2/index.html @@ -0,0 +1,2 @@ + +TransitionsInTimeseries.jl · TransitionsInTimeseries.jl

TransitionsInTimeseries.jl

TransitionsInTimeseries.jl

TransitionsInTimeseriesModule

TransitionsInTimeseries.jl

CI codecov Package Downloads

TransitionsInTimeseries.jl is a free and open-source software to easily analyse transitions within timeseries in a reproducible, performant, extensible and reliable way. In contrast to other existing software with similar target application, TransitionsInTimeseries.jl defines a generic interface for how to find transitions and how to test for significance. Within this interface, it is easy to expand the software in three orthogonal ways:

  1. Provide the analysis pipelines with new indicators, which can be either self-written or imported from other packages. In particular, the latter offers thousands of metrics that can indicate transitions right out of the box.
  2. Add new analysis pipelines for finding transitions.
  3. Add new ways for significance testing.

TransitionsInTimeseries is a registered Julia package and can be installed by running:

] add TransitionsInTimeseries

All further information is provided in the documentation, which you can either find online or build locally by running the docs/make.jl file.

Alternative names for this package could have been: Early Warning Signals / Resilience Indicators / Regime-Shift Identifiers / Change-Point Detectors, or however else you want to call them!

source
Star us on GitHub!

If you have found this package useful, please consider starring it on GitHub. This gives us an accurate lower bound of the (satisfied) user count.

Content

Multi-stable systems can display abrupt transitions between two stability regimes. To predict such transitions in real-world systems solely based on data, mathematical tools have been developed in the last decades. Numerous terminologies have been used for them, such as early warning signals, resilience indicators, regime-shift identifiers, change-point detection and transition indicators. TransitionsInTimeseries.jl sticks to the latter terminology and provides an interface that:

  • Allows a fast computation of common transition indicators with a couple of lines, as demonstrated in the example section.
  • Makes the surrogate analysis to test for significance under the hub.
  • Can be easily extended by any user without touching the source code.
  • Reduces the programming overhead for any researcher willing to benchmark new methods.
  • Eases the reproducibility thanks to a clear syntax, a simple installation and RNG-seeded surrogate generation.
  • Increases trustworthiness thanks to a large test suite.
Similar projects

An R toolbox and a Python library already exist. However, we believe that they are difficult to extend for the user. Furthermore, they do not offer a native performant code, as here allowed by the use of Julia.

Approaches

Over the last decades, research on transition indicators has largely focused on Critical Slowing Down (CSD). CSD is observed when a system approaches a Hopf, a transcritical or a fold bifurcation and consists in a resilience loss of the system. For instance this can be diagnosed by an increase of the variance and the AR1-regression coefficient, as demonstrated in the example section. However, we emphasize that this is one out of many possible approaches for obtaining transition indicators. Recent work has explored new approaches relying on nonlinear dynamics or machine learning. TransitionsInTimeseries.jl is designed to allow these cutting-edge methods and foster the development of new ones.

diff --git a/v0.1.2/objects.inv b/v0.1.2/objects.inv new file mode 100644 index 0000000..2c45354 Binary files /dev/null and b/v0.1.2/objects.inv differ diff --git a/v0.1.2/refs.bib b/v0.1.2/refs.bib new file mode 100644 index 0000000..8cc9690 --- /dev/null +++ b/v0.1.2/refs.bib @@ -0,0 +1,87 @@ + +@article{willeit-earth-2022, + title = {The {Earth} system model {CLIMBER}-{X} v1.0 – {Part} 1: {Climate} model description and validation​​​​​​​​​​​​​​}, + volume = {15}, + issn = {1991-9603}, + shorttitle = {The {Earth} system model {CLIMBER}-{X} v1.0 – {Part} 1}, + url = {https://gmd.copernicus.org/articles/15/5905/2022/}, + doi = {10.5194/gmd-15-5905-2022}, + abstract = {The newly developed fast Earth system model CLIMBER-X is presented. The climate component of CLIMBER-X consists of a 2.5-D semi-empirical statistical–dynamical atmosphere model, a 3-D frictional–geostrophic ocean model, a dynamic–thermodynamic sea ice model and a land surface model. All the model components are discretized on a regular lat–long grid with a horizontal resolution of 5 ◦ × 5 ◦. The model has a throughput of ∼ 10 000 simulation years per day on a single node with 16 CPUs on a high-performance computer and is designed to simulate the evolution of the Earth system on temporal scales ranging from decades to {\textgreater} 100 000 years. A comprehensive evaluation of the model performance for the present day and the historical period shows that CLIMBER-X is capable of realistically reproducing many observed climate characteristics, with results that generally lie within the range of state-of-theart general circulation models. The analysis of model performance is complemented by a thorough assessment of climate feedbacks and model sensitivities to changes in external forcings and boundary conditions. Limitations and applicability of the model are critically discussed. CLIMBER-X also includes a detailed representation of the global carbon cycle and is coupled to an ice sheet model, which will be described in separate papers. CLIMBER-X is available as open-source code and is expected to be a useful tool for studying past climate changes and for the investigation of the long-term future evolution of the climate.}, + language = {en}, + number = {14}, + urldate = {2022-08-11}, + journal = {Geoscientific Model Development}, + author = {Willeit, Matteo and Ganopolski, Andrey and Robinson, Alexander and Edwards, Neil R.}, + month = jul, + year = {2022}, + pages = {5905--5948}, + file = {Willeit et al. - 2022 - The Earth system model CLIMBER-X v1.0 – Part 1 Cl.pdf:/home/jan/Zotero/storage/75NKS975/Willeit et al. - 2022 - The Earth system model CLIMBER-X v1.0 – Part 1 Cl.pdf:application/pdf}, +} + +@article{boers-early-warning-2018, + title = {Early-warning signals for {Dansgaard}-{Oeschger} events in a high-resolution ice core record}, + volume = {9}, + issn = {2041-1723}, + url = {http://www.nature.com/articles/s41467-018-04881-7}, + doi = {10.1038/s41467-018-04881-7}, + language = {en}, + number = {1}, + urldate = {2022-10-28}, + journal = {Nature Communications}, + author = {Boers, Niklas}, + month = dec, + year = {2018}, + pages = {2556}, + file = {Boers - 2018 - Early-warning signals for Dansgaard-Oeschger event.pdf:/home/jan/Zotero/storage/DS97Z9UC/Boers - 2018 - Early-warning signals for Dansgaard-Oeschger event.pdf:application/pdf}, +} + +@article{rasmussen-stratigraphic-2014, + title = {A stratigraphic framework for abrupt climatic changes during the {Last} {Glacial} period based on three synchronized {Greenland} ice-core records: refining and extending the {INTIMATE} event stratigraphy}, + volume = {106}, + issn = {02773791}, + shorttitle = {A stratigraphic framework for abrupt climatic changes during the {Last} {Glacial} period based on three synchronized {Greenland} ice-core records}, + url = {https://linkinghub.elsevier.com/retrieve/pii/S0277379114003485}, + doi = {10.1016/j.quascirev.2014.09.007}, + abstract = {Due to their outstanding resolution and well-constrained chronologies, Greenland ice-core records provide a master record of past climatic changes throughout the Last InterglacialeGlacial cycle in the North Atlantic region. As part of the INTIMATE (INTegration of Ice-core, MArine and TErrestrial records) project, protocols have been proposed to ensure consistent and robust correlation between different records of past climate. A key element of these protocols has been the formal definition and ordinal numbering of the sequence of Greenland Stadials (GS) and Greenland Interstadials (GI) within the most recent glacial period. The GS and GI periods are the Greenland expressions of the characteristic DansgaardeOeschger events that represent cold and warm phases of the North Atlantic region, respectively. We present here a more detailed and extended GS/GI template for the whole of the Last Glacial period. It is based on a synchronization of the NGRIP, GRIP, and GISP2 ice-core records that allows the parallel analysis of all three records on a common time scale. The boundaries of the GS and GI periods are defined based on a combination of stable-oxygen isotope ratios of the ice (d18O, reflecting mainly local temperature) and calcium ion concentrations (reflecting mainly atmospheric dust loading) measured in the ice. The data not only resolve the well-known sequence of DansgaardeOeschger events that were first defined and numbered in the ice-core records more than two decades ago, but also better resolve a number of short-lived climatic oscillations, some defined here for the first time. Using this revised scheme, we propose a consistent approach for discriminating and naming all the significant abrupt climatic events of the Last Glacial period that are represented in the Greenland ice records. The final product constitutes an extended and better resolved Greenland stratotype sequence, against which other proxy records can be compared and correlated. It also provides a more secure basis for investigating the dynamics and fundamental causes of these climatic perturbations.}, + language = {en}, + urldate = {2023-10-04}, + journal = {Quaternary Science Reviews}, + author = {Rasmussen, Sune O. and Bigler, Matthias and Blockley, Simon P. and Blunier, Thomas and Buchardt, Susanne L. and Clausen, Henrik B. and Cvijanovic, Ivana and Dahl-Jensen, Dorthe and Johnsen, Sigfus J. and Fischer, Hubertus and Gkinis, Vasileios and Guillevic, Myriam and Hoek, Wim Z. and Lowe, J. John and Pedro, Joel B. and Popp, Trevor and Seierstad, Inger K. and Steffensen, Jørgen Peder and Svensson, Anders M. and Vallelonga, Paul and Vinther, Bo M. and Walker, Mike J.C. and Wheatley, Joe J. and Winstrup, Mai}, + month = dec, + year = {2014}, + pages = {14--28}, + file = {Rasmussen et al. - 2014 - A stratigraphic framework for abrupt climatic chan.pdf:/home/jan/Zotero/storage/VSUTVWPN/Rasmussen et al. - 2014 - A stratigraphic framework for abrupt climatic chan.pdf:application/pdf}, +} + +@article{Bagniewski2023, +abstract = {Tipping points (TPs) in Earth's climate system have been the subject of increasing interest and concern in recent years, given the risk that anthropogenic forcing could cause abrupt, potentially irreversible, climate transitions. Paleoclimate records are essential for identifying past TPs and for gaining a thorough understanding of the underlying nonlinearities and bifurcation mechanisms. However, the quality, resolution, and reliability of these records can vary, making it important to carefully select the ones that provide the most accurate representation of past climates. Moreover, as paleoclimate time series vary in their origin, time spans, and periodicities, an objective, automated methodology is crucial for identifying and comparing TPs. To address these challenges, we introduce the open-source PaleoJump database, which contains a collection of carefully selected, high-resolution records originating in ice cores, marine sediments, speleothems, terrestrial records, and lake sediments. These records describe climate variability on centennial, millennial and longer time scales and cover all the continents and ocean basins. We provide an overview of their spatial distribution and discuss the gaps in coverage. Our statistical methodology includes an augmented Kolmogorov–Smirnov test and Recurrence Quantification Analysis; it is applied here, for illustration purposes, to selected records in which abrupt transitions are automatically detected and the presence of potential tipping elements is investigated. These transitions are shown in the PaleoJump database along with other essential information about the records, including location, temporal scale and resolution, as well as temporal plots. This open-source database represents, therefore, a valuable resource for researchers investigating TPs in past climates.}, +archivePrefix = {arXiv}, +arxivId = {2206.06832}, +author = {Bagniewski, Witold and Rousseau, Denis Didier and Ghil, Michael}, +doi = {10.1038/s41598-023-30592-1}, +eprint = {2206.06832}, +isbn = {0123456789}, +issn = {20452322}, +journal = {Scientific Reports}, +number = {1}, +pages = {1--18}, +pmid = {36934110}, +publisher = {Nature Publishing Group UK}, +title = {{The PaleoJump database for abrupt transitions in past climates}}, +url = {https://doi.org/10.1038/s41598-023-30592-1}, +volume = {13}, +year = {2023} +} +@article{Bagniewski2021, +abstract = {Bifurcations and tipping points (TPs) are an important part of the Earth system's behavior. These critical points represent thresholds at which small changes in the system's parameters or in the forcing abruptly switch it from one state or type of behavior to another. Current concern with TPs is largely due to the potential of slow anthropogenic forcing to bring about abrupt, and possibly irreversible, change to the physical climate system and impacted ecosystems. Paleoclimate proxy records have been shown to contain abrupt transitions, or "jumps,"which may represent former instances of such dramatic climate change events. These transitions can provide valuable information for identifying critical TPs in current and future climate evolution. Here, we present a robust methodology for detecting abrupt transitions in proxy records that is applied to ice core and speleothem records of the last climate cycle. This methodology is based on the nonparametric Kolmogorov-Smirnov (KS) test for the equality, or not, of the probability distributions associated with two samples drawn from a time series, before and after any potential jump. To improve the detection of abrupt transitions in proxy records, the KS test is augmented by several other criteria and it is compared with recurrence analysis. The augmented KS test results show substantial skill when compared with more subjective criteria for jump detection. This test can also usefully complement recurrence analysis and improve upon certain aspects of its results.}, +author = {Bagniewski, W. and Ghil, M. and Rousseau, D. D.}, +doi = {10.1063/5.0062543}, +issn = {10897682}, +journal = {Chaos}, +number = {11}, +pmid = {34881579}, +publisher = {AIP Publishing LLC}, +title = {{Automatic detection of abrupt transitions in paleoclimate records}}, +volume = {31}, +year = {2021} +} diff --git a/v0.1.2/refs/index.html b/v0.1.2/refs/index.html new file mode 100644 index 0000000..7006762 --- /dev/null +++ b/v0.1.2/refs/index.html @@ -0,0 +1,2 @@ + +References · TransitionsInTimeseries.jl

References

diff --git a/v0.1.2/search_index.js b/v0.1.2/search_index.js new file mode 100644 index 0000000..f9c3908 --- /dev/null +++ b/v0.1.2/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"EditURL = \"tutorial.jl\"","category":"page"},{"location":"tutorial/#Tutorial","page":"Tutorial","title":"Tutorial","text":"","category":"section"},{"location":"tutorial/#workflow","page":"Tutorial","title":"Workflow","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Computing transition indicators consists of the following steps:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Doing any preprocessing of raw data first, such as detrending. This not part of TransitionsInTimeseries.jl and yields the input timeseries.\nEstimating the timeseries of an indicator by sliding a window over the input timeseries.\nComputing the changes of the indicator by sliding a window over its timeseries. Alternatively, the change metric can be estimated over the whole segment, as examplified in the section Segmented windows.\nGenerating many surrogates that preserve important statistical properties of the original timeseries.\nPerforming step 2 and 3 for the surrogate timeseries.\nChecking whether the indicator change timeseries of the real timeseries shows a significant feature (trend, jump or anything else) when compared to the surrogate data.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"These steps are illustrated one by one in the tutorial below, and then summarized in the convenient API that TransitionsInTimeseries.jl exports.","category":"page"},{"location":"tutorial/#example_stepbystep","page":"Tutorial","title":"Tutorial – Educational","text":"","category":"section"},{"location":"tutorial/#Raw-input-data","page":"Tutorial","title":"Raw input data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Let us load data from a bistable nonlinear model subject to noise and to a gradual change of the forcing that leads to a transition. Furthermore, we also load data from a linear model, which is by definition monostable and therefore incapable of transitioning. This is done to control the rate of false positives, a common problem that can emerge when looking for transition indicators. The models are governed by:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"dfracmathrmdx_lmathrmdt = - x_l - 1 + f(t) + n(t) \ndfracmathrmdx_nlmathrmdt = - x_nl^3 + x_nl + f(t) + n(t)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"with x_l the state of the linear model, x_nl the state of the bistable model, f the forcing and n the noise. For f=0 they both display an equilibrium point at x=-1. However, the bistable model also displays a further equilibrium point at x=1. Loading (and visualizing with Makie) such prototypical data to test some indicators can be done by simply running:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using TransitionsInTimeseries, CairoMakie\n\nt, x_linear, x_nlinear = load_linear_vs_doublewell()\nfig, ax = lines(t, x_linear)\nlines!(ax, t, x_nlinear)\nax.title = \"raw data\"\nfig","category":"page"},{"location":"tutorial/#Preprocessing","page":"Tutorial","title":"Preprocessing","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"note: Not part of TransitionsInTimeseries.jl\nAny timeseries preprocessing, such as the de-trending step we do here, is not part of TransitionsInTimeseries.jl and is the responsibility of the researcher.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The nonlinear system clearly displays a transition between two stability regimes. To forecast such transition, we analyze the fluctuations of the timeseries around the attractor, assumed to be tracked. Therefore, a detrending step is needed - here simply obtained by building the difference of the timeseries with lag 1.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"x_l_fluct = diff(x_linear)\nx_nl_fluct = diff(x_nlinear)\ntfluct = t[2:end]\n\nfig, ax = lines(tfluct, x_l_fluct)\nlines!(ax, tfluct, x_nl_fluct .+ 0.05)\nax.title = \"input timeseries\"\nfig","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"At this point, x_l_fluct and x_nl_fluct are considered the input timeseries.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"info: Detrending in Julia\nDetrending can be performed in many ways. A wide range of Julia packages exists to perform smoothing such as Loess.jl or DSP.jl. There the detrending step consists of subtracting the smoothed signal from the original one.","category":"page"},{"location":"tutorial/#Indicator-timeseries","page":"Tutorial","title":"Indicator timeseries","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"We can then compute the values of some \"indicator\" (a Julia function that inputs a timeseries and outputs a number). An indicator should be a quantity that is likely to change if a transition occurs, or is about to occur in the timeseries. We compute indicators by applying a sliding window over the input timeseries, determined by the width and the stride with which it is applied. Here we demonstrate this computation with the AR1-regression coefficient (under white-noise assumption), implemented as ar1_whitenoise:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"indicator = ar1_whitenoise\nindicator_window = (width = 400, stride = 1)\n\n# By mapping `last::Function` over a windowviewer of the time vector,\n# we obtain the last time step of each window.\n# This therefore only uses information from `k-width+1` to `k` at time step `k`.\n# Alternatives: `first::Function`, `midpoint:::Function`.\nt_indicator = windowmap(last, tfluct; indicator_window...)\nindicator_l = windowmap(indicator, x_l_fluct; indicator_window...)\nindicator_nl = windowmap(indicator, x_nl_fluct; indicator_window...)\n\nfig, ax = lines(t_indicator, indicator_l)\nlines!(ax, t_indicator, indicator_nl)\nax.title = \"indicator timeseries\"\nfig","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The lines plotted above are the indicator timeseries.","category":"page"},{"location":"tutorial/#Change-metric-timeseries","page":"Tutorial","title":"Change metric timeseries","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"From here, we process the indicator timeseries to quantify changes in it. This step is in essence the same as before: we apply some function over a sliding window of the indicator timeseries. We call this new timeseries the change metric timeseries. In the example here, the change metric we will employ will be the slope (over a sliding window), calculated via means of a RidgeRegressionSlope:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"change_window = (width = 30, stride = 1)\nridgereg = RidgeRegressionSlope(lambda = 0.0)\nprecompridgereg = precompute(ridgereg, t[1:change_window.width])\n\nt_change = windowmap(last, t_indicator; change_window...)\nchange_l = windowmap(precompridgereg, indicator_l; change_window...)\nchange_nl = windowmap(precompridgereg, indicator_nl; change_window...)\n\nfig, ax = lines(t_change, change_l)\nlines!(ax, t_change, change_nl)\nax.title = \"change metric timeseries\"\nfig","category":"page"},{"location":"tutorial/#Timeseries-surrogates","page":"Tutorial","title":"Timeseries surrogates","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"As expected from Critical Slowing Down, an increase of the AR1-regression coefficient can be observed. Although eyeballing the timeseries might already be suggestive, we want a rigorous framework for testing for significance.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"In TransitionsIdentifiers.jl we perform significance testing using the method of timeseries surrogates and the TimeseriesSurrogates.jl Julia package. This has the added benefits of reproducibility, automation and flexibility in choosing the surrogate generation method. Note that TimeseriesSurrogates is re-exported by TransitionsInTimeseries, so that you don't have to using both of them.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"To illustrate the surrogate, we compare the change metric computed from the bistable timeseries what that computed from a surrogate of the same timeseries.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# Generate Fourier random-phase surrogates\nusing Random: Xoshiro\ns = surrogate(x_nl_fluct, RandomFourier(), Xoshiro(123))\n\nfunction gridfig(nrows, ncols)\n fig = Figure()\n axs = [Axis(fig[i, j], xticklabelsvisible = i == nrows ? true : false)\n for j in 1:ncols, i in 1:nrows]\n rowgap!(fig.layout, 10)\n return fig, axs\nend\nfig, axs = gridfig(2, 1)\nlines!(axs[1], tfluct, x_nl_fluct, color = Cycled(2))\nlines!(axs[1], tfluct, s .- 0.05, color = Cycled(3))\naxs[1].title = \"real signal vs. surrogate(s)\"\n\n# compute and plot indicator and change metric\nindicator_s = windowmap(indicator, s; indicator_window...)\nchange_s = windowmap(precompridgereg, indicator_s; change_window...)\n\nlines!(axs[2], t_change, change_nl, label = \"nonlin\", color = Cycled(2))\nlines!(axs[2], t_change, change_s, label = \"surrogate\", color = Cycled(3))\naxislegend()\naxs[2].title = \"change metric\"\n\n[xlims!(ax, 0, 50) for ax in axs]\nfig","category":"page"},{"location":"tutorial/#Quantifying-significance","page":"Tutorial","title":"Quantifying significance","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"To quantify the significance of the values of the change metric timeseries we perform a standard surrogate test by computing the p-value w.r.t. the change metrics of thousands of surrogates of the input timeseries. A low p-value (typically p<0.05) is commonly considered as significant. To visualize significant trends, we plot the p-value vs. time:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"n_surrogates = 1_000\nfig, axs = gridfig(2, 2)\naxs[1].title = \"linear\"\naxs[2].title = \"nonlinear\"\n\nfor (j, ax, axsig, x) in zip(1:2, axs[1:2], axs[3:4], (x_l_fluct, x_nl_fluct))\n\n orig_change = j == 1 ? change_l : change_nl\n sgen = surrogenerator(x, RandomFourier(), Xoshiro(123))\n pval = zeros(length(change_s))\n\n # Collect all surrogate change metrics\n for i in 1:n_surrogates\n s = sgen()\n indicator_s = windowmap(indicator, s; indicator_window...)\n change_s = windowmap(precompridgereg, indicator_s; change_window...)\n pval += orig_change .< change_s\n end\n\n pval ./= n_surrogates\n lines!(ax, t_change, orig_change) # ; color = Cycled(j)\n lines!(axsig, t_change, pval) # ; color = Cycled(j+2)\nend\n\n[xlims!(ax, 0, 50) for ax in axs]\nfig","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"As expected, the data generated by the nonlinear model displays a significant increase of the AR1-regression coefficient before the transition, which is manifested by a low p-value. In contrast, the data generated by the linear model does not show anything similar.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Performing the step-by-step analysis of transition indicators is possible and might be preferred for users wanting high flexibility. However, this results in a substantial amount of code. We therefore provide convenience functions that wrap this analysis, as shown in the next section.","category":"page"},{"location":"tutorial/#example_fastforward","page":"Tutorial","title":"Tutorial – TransitionsInTimeseries.jl","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"TransitionsInTimeseries.jl wraps this typical workflow into a simple, extendable, and modular API that researchers can use with little effort. In addition, it allows performing the same analysis for several indicators / change metrics in one go.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The interface is simple, and directly parallelizes the Workflow. It is based on the creation of a ChangesConfig, which contains a list of indicators, and corresponding metrics, to use for doing the above analysis. It also specifies what kind of surrogates to generate.","category":"page"},{"location":"tutorial/#Sliding-windows","page":"Tutorial","title":"Sliding windows","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The following blocks illustrate how the above extensive example is re-created in TransitionsInTimeseries.jl","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using TransitionsInTimeseries, CairoMakie\n\nt, x_linear, x_nlinear = load_linear_vs_doublewell()\n\n# input timeseries and time\ninput = x_nl_fluct = diff(x_nlinear)\nt = t[2:end]\n\nfig, ax = lines(t, input)\nax.title = \"input timeseries\"\nfig","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"To perform all of the above analysis we follow a 2-step process.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Step 1, we decide what indicators and change metrics to use in SlidingWindowConfig and apply those via a sliding window to the input timeseries using estimate_changes.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# These indicators are suitable for Critical Slowing Down\nindicators = (var, ar1_whitenoise)\n\n# use the ridge regression slope for both indicators\nchange_metrics = (RidgeRegressionSlope(), RidgeRegressionSlope())\n\n# choices go into a configuration struct\nconfig = SlidingWindowConfig(indicators, change_metrics;\n width_ind = 400, width_cha = 30, whichtime = last)\n\n# choices are processed\nresults = estimate_changes(config, input, t)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"We can conveniently plot the information contained in results by using plot_indicator_changes:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"fig = plot_indicator_changes(results)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Step 2 is to estimate significance using SurrogatesSignificance and the function significant_transitions. Finally, we can conveniently plot the results obtained by updating the figure obtained above with plot_significance!:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"signif = SurrogatesSignificance(n = 1000, tail = [:right, :right])\nflags = significant_transitions(results, signif)\nplot_significance!(fig, results, signif, flags = flags)\nfig","category":"page"},{"location":"tutorial/#segmented_windows","page":"Tutorial","title":"Segmented windows","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The analysis shown so far relies on sliding windows of the change metric. This is particularly convenient for transition detection tasks. Segmented windows for the change metric computation are however preferable when it comes to prediction tasks. By only slightly modifying the syntax used so far, one can perform the same computations on segmented windows, as well as visualise the results conveniently:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"config = SegmentedWindowConfig(indicators, change_metrics,\n t[1:1], t[1200:1200]; whichtime = last, width_ind = 200,\n min_width_cha = 100)\nresults = estimate_changes(config, input, t)\nsignif = SurrogatesSignificance(n = 1000, tail = [:right, :right])\nflags = significant_transitions(results, signif)\nfig = plot_changes_significance(results, signif)","category":"page"},{"location":"api/#API","page":"API","title":"API","text":"","category":"section"},{"location":"api/#Main-analysis-functions","page":"API","title":"Main analysis functions","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"estimate_changes\nChangesConfig\nChangesResults","category":"page"},{"location":"api/#TransitionsInTimeseries.estimate_changes","page":"API","title":"TransitionsInTimeseries.estimate_changes","text":"estimate_changes(config::ChangesConfig, x [,t]) → result\n\nEstimate possible transitions for input timeseries x using the approach specified in the configuration type config, see ChangesConfig for possibilities. t is the time vector corresponding to x, which defaults to eachindex(x).\n\nReturn the output as subtype of ChangesResults. The particular form of the output depends on the config and is described in its docstring. Regardless of type, result can always be given to significant_transitions to deduce which possible transitions are statistically significant using a variety of significance tests.\n\n\n\n\n\n","category":"function"},{"location":"api/#TransitionsInTimeseries.ChangesConfig","page":"API","title":"TransitionsInTimeseries.ChangesConfig","text":"ChangesConfig\n\nSupertype for how \"changes\" in a timeseries are estimated in estimate_changes. \"Changes\" deemed statistically significant in significant_transitions are \"transitions\" in the timeseries.\n\nExisting subtypes of ChangesConfig are:\n\nSlidingWindowConfig.\nSegmentedWindowConfig.\n\n\n\n\n\n","category":"type"},{"location":"api/#TransitionsInTimeseries.ChangesResults","page":"API","title":"TransitionsInTimeseries.ChangesResults","text":"ChangesResults\n\nSupertype used to gather results of estimate_changes. The concrete subtype instances are described in the docstrings of configuration types.\n\n\n\n\n\n","category":"type"},{"location":"api/#Sliding-window","page":"API","title":"Sliding window","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"SlidingWindowConfig\nSlidingWindowResults","category":"page"},{"location":"api/#TransitionsInTimeseries.SlidingWindowConfig","page":"API","title":"TransitionsInTimeseries.SlidingWindowConfig","text":"SlidingWindowConfig <: ChangesConfig\nSlidingWindowConfig(indicators, change_metrics; kwargs...)\n\nA configuration that can be given to estimate_changes. It estimates transitions by a sliding window approach:\n\nEstimate the timeseries of an indicator by sliding a window over the input timeseries.\nEstimate changes of an indicator by sliding a window of the change metric over the indicator timeseries.\n\nBoth indicators and change metrics are generic Julia functions that input an x::AbstractVector and output an s::Real. Any function may be given and see making custom indicators/change metrics in the documentation for more information on possible optimizations.\n\nindicators can be a single function or a tuple of indicators. Similarly, change_metrics can be a tuple or a single function. If tuples, the length of indicators and change_metrics must match. This way the analysis can be efficiently repeated for many indicators and/or change metrics.\n\nThe results output corresponding to SlidingWindowConfig is SlidingWindowResults.\n\nStep 1. is skipped if nothing is provided as indicators, in which case the change metrics are estimated directly from input data.\n\nKeyword arguments\n\nwidth_ind::Int=100, stride_ind::Int=1: width and stride given to WindowViewer to compute the indicator from the input timeseries.\nwidth_cha::Int=50, stride_cha::Int=1: width and stride given to WindowViewer to compute the change metric timeseries from the indicator timeseries.\nwhichtime = midpoint: The time vector corresponding to the indicators / change metric timeseries is obtained from t in estimate_changes using the keyword whichtime. Options include:\nlast: use the last timepoint of each window\nmidpoint: use the mid timepoint of each time window\nfirst: use first timepoint of each window\nIn fact, the indicators time vector is computed simply via\nt_indicator = windowmap(whichtime, t; width_ind, stride_ind)\nt_change = windowmap(whichtime, t_indicator; width_cha, stride_cha)\nso any other function of the time window may be given to extract the time point itself, such as mean or median.\nT = Float64: Element type of input timeseries to initialize some computations.\n\n\n\n\n\n","category":"type"},{"location":"api/#TransitionsInTimeseries.SlidingWindowResults","page":"API","title":"TransitionsInTimeseries.SlidingWindowResults","text":"SlidingWindowResults <: ChangesResults\n\nA struct containing the output of estimate_changes used with SlidingWindowConfig. It can be used for further analysis, visualization, or given to significant_transitions.\n\nIt has the following fields that the user may access\n\nx: the input timeseries.\nt: the time vector of the input timeseries.\nx_indicator, the indicator timeseries (matrix with each column one indicator).\nt_indicator, the time vector of the indicator timeseries.\nx_change, the change metric timeseries (matrix with each column one change metric).\nt_change, the time vector of the change metric timeseries.\nconfig::SlidingWindowConfig, what was used for the analysis.\n\n\n\n\n\n","category":"type"},{"location":"api/#Segmented-window","page":"API","title":"Segmented window","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"SegmentedWindowConfig\nSegmentedWindowResults","category":"page"},{"location":"api/#TransitionsInTimeseries.SegmentedWindowConfig","page":"API","title":"TransitionsInTimeseries.SegmentedWindowConfig","text":"SegmentedWindowConfig <: IndicatorsChangeConfig\nSegmentedWindowConfig(indicators, change_metrics, tseg_start, tseg_end; kwargs...)\n\nA configuration that can be given to estimate_changes. It estimates transitions by estimating indicators and changes in user-defined window segments as follows:\n\nFor each segment specified, estimate the corresponding indicator timeseries by sliding a window over the input timeseries (within the window segment).\nFor each segment of the indicator timeseries, estimate a scalar change metric by applying the change metric over the full segment of the indicator timeseries.d\n\ntseg_start, tseg_end are the starts and ends of the window segments (the window segments may overlap, that's okay). indicators, change_metrics are identical as in SlidingWindowConfig.\n\nThe results output corresponding to SlidingWindowConfig is SegmentedWindowResults.\n\nKeyword arguments\n\nwidth_ind::Int=100, stride_ind::Int=1, whichtime = midpoint, T = Float64: keywords identical as in SlidingWindowConfig.\nmin_width_cha::Int=typemax(Int): minimal width required to perform the change metric estimation. If a segment is not sufficiently long, the change metric is NaN.\n\n\n\n\n\n","category":"type"},{"location":"api/#TransitionsInTimeseries.SegmentedWindowResults","page":"API","title":"TransitionsInTimeseries.SegmentedWindowResults","text":"SegmentedWindowResults <: ChangesResults\n\nA struct containing the output of estimate_changes used with SegmentedWindowConfig. It can be used for further analysis, visualization, or given to significant_transitions.\n\nIt has the following fields that the user may access\n\nx: the input timeseries.\nt: the time vector of the input timeseries.\nx_indicator::Vector{Matrix}, with x_indicator[k] the indicator timeseries (matrix with each column one indicator) of the k-th segment.\nt_indicator::Vector{Vector}, with t_indicator[k] the time vector of the indicator timeseries for the k-th segment.\nx_change::Matrix, the change metric values with x[k, i] the change metric of the i-th indicator for the k-th segment.\nt_change, the time vector of the change metric.\nconfig::SegmentedWindowConfig, what was used for the analysis.\ni1::Vector{Int} indices corresponding to start time of each segment.\ni2::Vector{Int} indices corresponding to end time of each segment.\nprecomp_change_metrics vector containing the precomputed change metrics of each segment.\n\n\n\n\n\n","category":"type"},{"location":"api/#Slope-change","page":"API","title":"Slope change","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"SlopeChangeConfig\nSlopeChangeResults","category":"page"},{"location":"api/#TransitionsInTimeseries.SlopeChangeConfig","page":"API","title":"TransitionsInTimeseries.SlopeChangeConfig","text":"SlopeChangeConfig <: ChangesConfig\nSlopeChangeConfig(; indicator = nothing, kw...)\n\nA configuration that can be given to estimate_changes. It estimates a change of slope in the timeseries by fitting two connected linear segments to the timeseries, returning the results (i.e., the two-linear fits) as SlopeChangeResults.\n\nKeyword arguments\n\nindicator = nothing: if not nothing. Otherwise it should be a function f(x) -> Real. The slope fitting is then done over an indicator of the timeseries, which itself is estimated via a sliding window exactly as in SlidingWindowConfig.\nwidth_ind, stride_ind, whichtime: exactly as in SlidingWindowConfig if indicator is not nothing.\n\n\n\n\n\n","category":"type"},{"location":"api/#TransitionsInTimeseries.SlopeChangeResults","page":"API","title":"TransitionsInTimeseries.SlopeChangeResults","text":"SlopeChangeResults <: ChangesResults\n\nA struct containing the output of estimate_changes used with SlopeChangeConfig. It can be used for further analysis, visualization, or given to significant_transitions. The only significance type that you can use this with significant_transitions is SlopeChangeSignificance.\n\nIt has the following fields that the user may access:\n\nx: the input timeseries.\nt: the time vector of the input timeseries.\nx_indicator, the indicator timeseries.\nt_indicator, the time vector of the indicator timeseries.\nt_change, the time the slope changes.\nfitparams = a, b, c, d, the fitted linear coefficients, a + b*t before. t_change and c + d*t after t_change.\n\n\n\n\n\n","category":"type"},{"location":"api/#Significance-testing","page":"API","title":"Significance testing","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"significant_transitions\nSignificance\nSurrogatesSignificance\nThresholdSignificance\nSigmaSignificance\nQuantileSignificance\nSlopeChangeSignificance","category":"page"},{"location":"api/#TransitionsInTimeseries.significant_transitions","page":"API","title":"TransitionsInTimeseries.significant_transitions","text":"significant_transitions(res::ChangesResults, signif::Significance)\n\nEstimate significant transtions in res using the method described by signif. Return flags, a Boolean matrix with identical size as the changes stored in res (which typically is stored in the field res.x_change). flags is true wherever a change metric of res is deemed significant.\n\n\n\n\n\n","category":"function"},{"location":"api/#TransitionsInTimeseries.Significance","page":"API","title":"TransitionsInTimeseries.Significance","text":"Significance\n\nSupertype used to test for significance in significant_transitions. Changes that are statistically significant are \"transitions\".\n\nValid subtypes are:\n\nSurrogatesSignificance.\nSigmaSignificance.\nQuantileSignificance.\nThresholdSignificance.\n\n\n\n\n\n","category":"type"},{"location":"api/#TransitionsInTimeseries.SurrogatesSignificance","page":"API","title":"TransitionsInTimeseries.SurrogatesSignificance","text":"SurrogatesSignificance <: Significance\nSurrogatesSignificance(; kwargs...)\n\nA configuration struct for significance testing significant_transitions using timeseries surrogates.\n\nKeyword arguments\n\nsurromethod = RandomFourier(): method to generate surrogates\nn = 1000: how many surrogates to generate\nrng = Random.default_rng(): random number generator for the surrogates\np = 0.05: threshold for significance of the p-value\ntail = :both: tail type used, see below\n\nDescription\n\nWhen used with ChangesResults, significance is estimated as follows: n surrogates from the input timeseries are generated using surromethod, which is any Surrogate subtype provided by TimeseriesSurrogates.jl. For each surrogate, the indicator and then change metric timeseries is extracted. The values of the surrogate change metrics form a distribution of values (one at each time point). The value of the original change metric is compared to that of the surrogate distribution and a p-value is extracted according to the specified tail. The p-value is compared with p to claim significance. After using SurrogatesSignificance, you may access the full p-values before thresholding in the field .pvalues (to e.g., threshold with different p).\n\nThe p-value is simply the proportion of surrogate change metric values that exceed (for tail = :right) or subseed (tail = :left) the original change metric at each given time point. Use tail = :left if the surrogate data are expected to have higher change metric values. This is the case for statistics that quantify entropy. For statistics that quantify autocorrelation, use tail = :right instead. For anything else, use the default tail = :both. An iterable of tail values can also be given, in which case a specific tail is used for each change metric in ChangesResults.\n\nNote that the raw p-values can be accessed in the field .pvalues, after calling the significant_transitions function with SurrogatesSignificance, in case you wish to obtain a different threshold of the p-values.\n\n\n\n\n\n","category":"type"},{"location":"api/#TransitionsInTimeseries.ThresholdSignificance","page":"API","title":"TransitionsInTimeseries.ThresholdSignificance","text":"ThresholdSignificance(threshold::Real; tail = :right) <: Significance\n\nA configuration struct for significance testing in significant_transitions. Significance is estimated by comparing the value of each change metric with the given threshold. Values that exceed the threshold (if tail = :right) or subseed the threshold (if tail = :left) are deemed significant. If tail = :both then either condition is checked.\n\n\n\n\n\n","category":"type"},{"location":"api/#TransitionsInTimeseries.SigmaSignificance","page":"API","title":"TransitionsInTimeseries.SigmaSignificance","text":"SigmaSignificance(; factor = 3.0, tail = :both) <: Significance\n\nA configuration struct for significance testing significant_transitions. When used with ChangesResults, significance is estimated by comparing how many standard deviations (σ) the value exceeds the mean value (μ). Values that exceed (if tail = :right) μ + factor*σ, or subseed (if tail = :left) μ - factor*σ are deemed significant. If tail = :both then either condition is checked.\n\nfactor can also be a vector of values, in which case a different value is used for each change metric.\n\nSee also QuantileSignificance.\n\n\n\n\n\n","category":"type"},{"location":"api/#TransitionsInTimeseries.QuantileSignificance","page":"API","title":"TransitionsInTimeseries.QuantileSignificance","text":"QuantileSignificance(; p = 0.95, tail = :both) <: Significance\n\nA configuration struct for significance testing significant_transitions. When used with ChangesResults, significance is estimated by comparing the value of each change metric with its p-quantile. Values that exceed the p-quantile (if tail = :right) or subseed the 1-p-quantile (if tail = :left) are deemed significant. If tail = :both then either condition is checked.\n\nQuantileSignficance guarantees that some values will be significant by the very definition of what a quantile is. See also SigmaSignificance that is similar but does not have this guarantee.\n\n\n\n\n\n","category":"type"},{"location":"api/#TransitionsInTimeseries.SlopeChangeSignificance","page":"API","title":"TransitionsInTimeseries.SlopeChangeSignificance","text":"SlopeChangeSignificance(; moe_slope, moe_offset, slope_diff = moe_slope, pvalue = 0.05)\n\nTest whether the result of SlopeChangeResults is statistically significant.\n\nTwo tests are done:\n\nCheck whether the margin of error of the fitted parameters a, b, c, d of the two linear segments a + b*t, c + d*t is less than the specified margins of error, for a chosen pvalue.\nTest that the two slopes b, d have difference greater than slope_diff.\n\nThe Boolean & of the above two is the final test.\n\nThe margin of error is simply half the size of the confidence interval, also known as radius of the confidence interval.\n\n\n\n\n\n","category":"type"},{"location":"api/#indicators","page":"API","title":"Indicators","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"Note that any Julia function can be an indicator or change metric, so the list here is only just a couple of indicators directly implemented in this package.","category":"page"},{"location":"api/#Value-distribution","page":"API","title":"Value distribution","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"StatsBase.mean\nStatsBase.skewness\nStatsBase.kurtosis","category":"page"},{"location":"api/#Statistics.mean","page":"API","title":"Statistics.mean","text":"mean(A::AbstractArray, w::AbstractWeights[, dims::Int])\n\nCompute the weighted mean of array A with weight vector w (of type AbstractWeights). If dim is provided, compute the weighted mean along dimension dims.\n\nExamples\n\nn = 20\nx = rand(n)\nw = rand(n)\nmean(x, weights(w))\n\n\n\n\n\n","category":"function"},{"location":"api/#StatsBase.skewness","page":"API","title":"StatsBase.skewness","text":"skewness(v, [wv::AbstractWeights], m=mean(v))\n\nCompute the standardized skewness of a real-valued array v, optionally specifying a weighting vector wv and a center m.\n\n\n\n\n\n","category":"function"},{"location":"api/#StatsBase.kurtosis","page":"API","title":"StatsBase.kurtosis","text":"kurtosis(v, [wv::AbstractWeights], m=mean(v))\n\nCompute the excess kurtosis of a real-valued array v, optionally specifying a weighting vector wv and a center m.\n\n\n\n\n\n","category":"function"},{"location":"api/#Critical-Slowing-Down","page":"API","title":"Critical Slowing Down","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"StatsBase.var\nar1_whitenoise","category":"page"},{"location":"api/#Statistics.var","page":"API","title":"Statistics.var","text":"var(x::AbstractArray, w::AbstractWeights, [dim]; mean=nothing, corrected=false)\n\nCompute the variance of a real-valued array x, optionally over a dimension dim. Observations in x are weighted using weight vector w. The uncorrected (when corrected=false) sample variance is defined as:\n\nfrac1sumw sum_i=1^n w_ileft(x_i - μright)^2 \n\nwhere n is the length of the input and μ is the mean. The unbiased estimate (when corrected=true) of the population variance is computed by replacing frac1sumw with a factor dependent on the type of weights used:\n\nAnalyticWeights: frac1sum w - sum w^2 sum w\nFrequencyWeights: frac1sumw - 1\nProbabilityWeights: fracn(n - 1) sum w where n equals count(!iszero, w)\nWeights: ArgumentError (bias correction not supported)\n\n\n\n\n\nvar(ce::CovarianceEstimator, x::AbstractVector; mean=nothing)\n\nCompute the variance of the vector x using the estimator ce.\n\n\n\n\n\n","category":"function"},{"location":"api/#TransitionsInTimeseries.ar1_whitenoise","page":"API","title":"TransitionsInTimeseries.ar1_whitenoise","text":"ar1_whitenoise(x::AbstractVector)\n\nReturn the AR1 regression coefficient of a time series x by computing the analytic solution of the least-square parameter estimation under white-noise assumption for the data-generating process:\n\ntheta = sum_i=2^n x_i x_i-1 sum_i=2^n x_i-1^2\n\n\n\n\n\n","category":"function"},{"location":"api/#Spectrum","page":"API","title":"Spectrum","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"LowfreqPowerSpectrum\nPrecomputedLowfreqPowerSpectrum","category":"page"},{"location":"api/#TransitionsInTimeseries.LowfreqPowerSpectrum","page":"API","title":"TransitionsInTimeseries.LowfreqPowerSpectrum","text":"LowfreqPowerSpectrum(; q_lofreq = 0.1)\n\nReturn a PrecomputableFunction containing all the necessary fields to generate a PrecomputedLowfreqPowerSpectrum. The latter can be initialized by precompute:\n\nlfps = precompute( LowfreqPowerSpectrum() )\n\nKeyword arguments:\n\nq_lofreq: a number between 0 and 1 that characterises which portion of the\n\nfrequency spectrum is considered to be low. For instance, q_lofreq = 0.1 implies that the lowest 10% of frequencies are considered to be the low ones.\n\n\n\n\n\n","category":"type"},{"location":"api/#TransitionsInTimeseries.PrecomputedLowfreqPowerSpectrum","page":"API","title":"TransitionsInTimeseries.PrecomputedLowfreqPowerSpectrum","text":"PrecomputedLowfreqPowerSpectrum\n\nA struct containing all the precomputed fields to efficiently perform repetitive computation of the low-frequency power spectrum (LFPS), a number between 0 and 1 that characterizes the amount of power contained in the low frequencies of the power density spectrum of x. Once lfps::PrecomputedLowfreqPowerSpectrum is initialized, it can be used as a function to obtain the LFPS of x::AbstractVector by:\n\nlfps(x)\n\n\n\n\n\n","category":"type"},{"location":"api/#Nonlinear-dynamics","page":"API","title":"Nonlinear dynamics","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"Indicators that come from nonlinear timeseries analysis typically quantify some entropy-like or complexity measure from the timeseries. Thousands (literally) such measures are provided out of the box by the ComplexityMeasures.jl package. Given that any of these may be used as an indicator or change metric, we made the decision to not copy-paste any measure here, as it is easy for the user to use any of them.","category":"page"},{"location":"api/","page":"API","title":"API","text":"For example, using the permutation entropy as an indicator is as simple as doing","category":"page"},{"location":"api/","page":"API","title":"API","text":"using ComplexityMeasures\nest = OrdinalPatterns(; m = 3) # order 3\n# create a function that given timeseries returns permutation entropy\nindicator = x -> entropy_normalized(est, x)","category":"page"},{"location":"api/","page":"API","title":"API","text":"and giving the created indicator to e.g., SlidingWindowConfig.","category":"page"},{"location":"api/#change_metrics","page":"API","title":"Change metrics","text":"","category":"section"},{"location":"api/#Slope","page":"API","title":"Slope","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"kendalltau\nspearman\nRidgeRegressionSlope\nPrecomputedRidgeRegressionSlope","category":"page"},{"location":"api/#TransitionsInTimeseries.kendalltau","page":"API","title":"TransitionsInTimeseries.kendalltau","text":"kendalltau(x::AbstractVector)\n\nCompute the kendall-τ correlation coefficient of the time series x. kendalltau can be used as a change metric focused on trend.\n\n\n\n\n\n","category":"function"},{"location":"api/#TransitionsInTimeseries.spearman","page":"API","title":"TransitionsInTimeseries.spearman","text":"spearman(x::AbstractVector)\n\nCompute the spearman correlation coefficient of the time series x. spearman can be used as a change metric focused on trend.\n\n\n\n\n\n","category":"function"},{"location":"api/#TransitionsInTimeseries.RidgeRegressionSlope","page":"API","title":"TransitionsInTimeseries.RidgeRegressionSlope","text":"RidgeRegressionSlope(; lambda = 0.0) → rr\n\nReturn a PrecomputableFunction containing all the necessary fields to generate a PrecomputedRidgeRegressionSlope. rr can be used as a change metric focused on trend.\n\nlambda is a regularization constant, usually between 0 and 1.\n\n\n\n\n\n","category":"type"},{"location":"api/#TransitionsInTimeseries.PrecomputedRidgeRegressionSlope","page":"API","title":"TransitionsInTimeseries.PrecomputedRidgeRegressionSlope","text":"PrecomputedRidgeRegressionSlope\n\nA struct containing the precomputed ridge regression matrix. Once rrslope::PrecomputedRidgeRegressionSlope is initialized, it can be used as a function to obtain the ridge regression slope of x::AbstractVector by applying:\n\nrrslope(x)\n\n\n\n\n\n","category":"type"},{"location":"api/#Value-distribution-differences","page":"API","title":"Value distribution differences","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"difference_of_means","category":"page"},{"location":"api/#TransitionsInTimeseries.difference_of_means","page":"API","title":"TransitionsInTimeseries.difference_of_means","text":"difference_of_means(x::AbstractArray)\n\nReturn the absolute difference of the means of the first and second halfs of x. difference_of_means can be used as a change metric focused on value differences. Creating similar statistical differences using other moments instead of mean is trivial. In fact, the source of difference_of_means is just:\n\n# assumes 1-based indexing\nn = length(x)\nx1 = view(x, 1:n÷2)\nx2 = view(x, (n÷2 + 1):n)\nreturn abs(mean(x1) - mean(x2))\n\ndifference_of_means can also sensibly be used for windows of size 2, in which case the change metric timeseries is the same as the abs.(diff(...)) of the indicator timeseries.\n\n\n\n\n\n","category":"function"},{"location":"api/#own_indicator","page":"API","title":"Make your own indicator/metric!","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"The only difference between what is an \"indicator\" and what is a \"change metric\" is purely conceptual. As far as the code base of TransitionsInTimeseries.jl is concerned, they are both functions f: x::AbstractVector{Real} -> f(x)::Real. As a user you may give any such function for an indicator or change metric.","category":"page"},{"location":"api/","page":"API","title":"API","text":"There are situations where you may optimize such a function based on knowledge of input x type and length, in which case you may use PrecomputableFunction:","category":"page"},{"location":"api/","page":"API","title":"API","text":"PrecomputableFunction\nprecompute","category":"page"},{"location":"api/#TransitionsInTimeseries.PrecomputableFunction","page":"API","title":"TransitionsInTimeseries.PrecomputableFunction","text":"PrecomputableFunction\n\nSupertype of structs containing the necessary field to precompute a ::Function by:\n\nprecompute(f::PrecomputableFunction, t)\n\n\n\n\n\n","category":"type"},{"location":"api/#TransitionsInTimeseries.precompute","page":"API","title":"TransitionsInTimeseries.precompute","text":"precompute(f, t)\n\nPrecompute the function f given a time vector t.\n\n\n\n\n\n","category":"function"},{"location":"api/#Surrogates","page":"API","title":"Surrogates","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"For the surrogate generation, you can use any subtype of Surrogate defined in Timeseriessurrogates.jl.","category":"page"},{"location":"api/#Sliding-windows","page":"API","title":"Sliding windows","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"WindowViewer\nwindowmap\nwindowmap!","category":"page"},{"location":"api/#TransitionsInTimeseries.WindowViewer","page":"API","title":"TransitionsInTimeseries.WindowViewer","text":"WindowViewer(x; width, stride)\n\nInitialize an iterator that generates views over the given timeseries x based on a window with a given width, incrementing the window views with the given stride. You can use this directly with map, such as map(std, WindowViewer(x, ...)) would give you the moving-window-timeseries of the std of x.\n\nIf not given, the keywords width, stride are respectively taken as default_window_width(x) and 1.\n\n\n\n\n\n","category":"type"},{"location":"api/#TransitionsInTimeseries.windowmap","page":"API","title":"TransitionsInTimeseries.windowmap","text":"windowmap(f::Function, x::AbstractVector; kwargs...) → mapped_f\n\nA shortcut for first generating a wv = WindowViewer(x; kwargs...) and then applying mapped_f = map(f, wv). If x is accompanied by a time vector t, you probably also want to call this function with t instead of x and with one of mean, midpoint, midvalue as f to obtain a time vector for the mapped_f output.\n\n\n\n\n\n","category":"function"},{"location":"api/#TransitionsInTimeseries.windowmap!","page":"API","title":"TransitionsInTimeseries.windowmap!","text":"windowmap!(f::Function, out, x::AbstractVector; kwargs...)\n\nSame as windowmap, but writes the output in-place in out.\n\n\n\n\n\n","category":"function"},{"location":"api/#Load-data","page":"API","title":"Load data","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"load_linear_vs_doublewell()","category":"page"},{"location":"api/#TransitionsInTimeseries.load_linear_vs_doublewell-Tuple{}","page":"API","title":"TransitionsInTimeseries.load_linear_vs_doublewell","text":"load_linear_vs_doublewell()\n\nLoad prototypical data from a linear and a double-well model to test some indicators.\n\n\n\n\n\n","category":"method"},{"location":"api/#Visualization","page":"API","title":"Visualization","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"plot_indicator_changes\nplot_significance!\nplot_changes_significance","category":"page"},{"location":"api/#TransitionsInTimeseries.plot_indicator_changes","page":"API","title":"TransitionsInTimeseries.plot_indicator_changes","text":"plot_indicator_changes(res) → (fig, axs)\n\nReturn fig::Figure and axs::Matrix{Axis}, on which res::ChangesResults has been visualised.\n\nKeyword arguments:\n\ncolors = default_colors sets the colors of the line plots that are to\n\nbe cycled through. The default correspond to the color scheme of Julia Dynamics.\n\n`indicatornames = defaultindicatorlabel(res), chametricnames =\n\ndefaultchametriclabel(res)sets the labels for the indicators and the change metrics, with the default inferring them from the names ofres.config.indicatorsandres.config.change_metrics`.\n\naccent_linewidth = 3 sets the line width for the original signals (the\n\nsurrogates have linewidth = 1)\n\n\n\n\n\n","category":"function"},{"location":"api/#TransitionsInTimeseries.plot_significance!","page":"API","title":"TransitionsInTimeseries.plot_significance!","text":"plot_significance!(axs, res, signif)\n\nUpdate the axs::Matrix{Axis} of a figure obtained with plot_indicator_changes(res) with the signif::SurrogatesSignificance.\n\nKeyword arguments:\n\nflags = nothing provides the significance flags, for instance obtained by\n\nthresholding the pvalues.\n\nnsurro = 20 sets the number of surrogates to plot.\n\n\n\n\n\n","category":"function"},{"location":"api/#TransitionsInTimeseries.plot_changes_significance","page":"API","title":"TransitionsInTimeseries.plot_changes_significance","text":"plot_changes_significance(res, signif) → (fig, axs)\n\nReturn fig::Figure and axs::Matrix{Axis}, on which res::ChangesResults and signif::SurrogatesSignificance have been visualised. The source code is as simple as:\n\nfig, axs = plot_indicator_changes(res; kwargs...)\nplot_significance!(axs, res, signif; kwargs...)\n\nFor more information, refer to plot_indicator_changes and plot_significance!.\n\n\n\n\n\n","category":"function"},{"location":"api/#Utils","page":"API","title":"Utils","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"default_window_width","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"EditURL = \"do-events.jl\"","category":"page"},{"location":"examples/do-events/#Dansgaard-Oescher-events-and-Critical-Slowing-Down","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"","category":"section"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"The delta^18O timeseries of the North Greenland Ice Core Project (NGRIP) are, to this date, the best proxy record for the Dansgaard-Oeschger events (DO-events). DO-events are sudden warming episodes of the North Atlantic, reaching 10 degrees of regional warming within 100 years. They happened quasi-periodically over the last glacial cycle due to transitions between strong and weak states of the Atlantic Meridional Overturning Circulation and might be therefore be the most prominent examples of abrupt transitions in the field of climate science. We here propose to hindcast these events by applying the theory of Critical Slowing Down (CSD) on the NGRIP data, which can be found here in its raw format. This analysis has already been done in (Boers, 2018) and we here try to reproduce Figure 2.d-f.","category":"page"},{"location":"examples/do-events/#Preprocessing-NGRIP","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Preprocessing NGRIP","text":"","category":"section"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"Data pre-processing is not part of TransitionsInTimeseries.jl, but a step the user has to do before using the package. To present an example with a complete scientific workflow, we will showcase typical data pre-processing here, that consist of the following steps:","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"Load the data, reverse and offset it to have time vector = time before 2000 AD.\nFilter non-unique points in time and sort the data.\nRegrid the data from uneven to even sampling.","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"The time and delta^18O vectors resulting from the i-th preprocessing step are respectively called t_i and x_i. The final step consists in obtaining a residual r, i.e. the fluctuations of the system around the attractor, which, within the CSD theory, is assumed to be tracked. Over this example, it will appear that the convenience of TransitionsInTimeseries.jl leads the bulk of the code to be written for plotting and preprocessing.","category":"page"},{"location":"examples/do-events/#Step-1:","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Step 1:","text":"","category":"section"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"using DelimitedFiles, Downloads\n\nfunction load_ngrip()\n tmp = Base.download(\"https://raw.githubusercontent.com/JuliaDynamics/JuliaDynamics/\"*\n \"master/timeseries/NGRIP.csv\")\n data, labels = readdlm(tmp, header = true)\n return reverse(data[:, 1]) .- 2000, reverse(data[:, 2]) # (time, delta-18-0) vectors\nend\n\nt1, x1 = load_ngrip()","category":"page"},{"location":"examples/do-events/#Step-2","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Step 2","text":"","category":"section"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"uniqueidx(v) = unique(i -> v[i], eachindex(v))\nfunction keep_unique(t, x)\n unique_idx = uniqueidx(t)\n return t[unique_idx], x[unique_idx]\nend\n\nfunction sort_timeseries!(t, x)\n p = sortperm(t)\n permute!(t, p)\n permute!(x, p)\n return nothing\nend\n\nt2, x2 = keep_unique(t1, x1)\nsort_timeseries!(t2, x2)","category":"page"},{"location":"examples/do-events/#Step-3","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Step 3","text":"","category":"section"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"using BSplineKit\n\nfunction regrid2evensampling(t, x, dt)\n itp = BSplineKit.interpolate(t, x, BSplineOrder(4))\n tspan = (ceil(minimum(t)), floor(maximum(t)))\n t_even = collect(tspan[1]:dt:tspan[2])\n x_even = itp.(t_even)\n return t_even, x_even\nend\n\ndt = 5.0 # dt = 5 yr as in (Boers 2018)\nt3, x3 = regrid2evensampling(t2, x2, dt)","category":"page"},{"location":"examples/do-events/#Step-4","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Step 4","text":"","category":"section"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"For the final step we drop the indices:","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"using DSP\n\nfunction chebyshev_filter(t, x, fcutoff)\n ii = 10 # Chebyshev filtering requires to prune first points of timeseries.\n responsetype = Highpass(fcutoff, fs = 1/dt)\n designmethod = Chebyshev1(8, 0.05)\n r = filt(digitalfilter(responsetype, designmethod), x)\n xtrend = x - r\n return t[ii:end], x[ii:end], xtrend[ii:end], r[ii:end]\nend\n\nfcutoff = 0.95 * 0.01 # cutoff ≃ 0.01 yr^-1 as in (Boers 2018)\nt, x, xtrend, r = chebyshev_filter(t3, x3, fcutoff)","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"Let's now visualize our data in what will become our main figure. For the segmentation of the DO-events, we rely on the tabulated data from (Rasmussen et al., 2014) (which will soon be available as downloadable):","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"using CairoMakie, Loess\n\nfunction loess_filter(t, x; span = 0.005)\n loessmodel = loess(t, x, span = span)\n xtrend = Loess.predict(loessmodel, t)\n r = x - xtrend\n return t, x, xtrend, r\nend\n\nfunction kyr_xticks(tticks_yr)\n tticks_kyr = [\"$t\" for t in Int.(tticks_yr ./ 1e3)]\n return (tticks_yr, tticks_kyr)\nend\n\nfunction plot_do(traw, xraw, tfilt, xfilt, t, r, t_transitions, xlims, xticks)\n fig = Figure(size = (1600, 1200), fontsize = 24)\n\n # Original timeseries with transition marked by vertical lines\n ax1 = Axis(fig[1, 1], xlabel = L\"Time (kyr) $\\,$\", ylabel = L\"$\\delta^{18}$O (permil)\",\n xaxisposition = :top, xticks = xticks)\n lines!(ax1, traw, xraw, color = (:gray70, 0.5))\n lines!(ax1, tfilt, xfilt, color = :gray10, linewidth = 3)\n vlines!(ax1, t_transitions, color = Cycled(1), linewidth = 3)\n\n # Residual timeseries\n ax2 = Axis(fig[2, 1], ylabel = L\"Residual $\\,$\", xticks = xticks,\n xticksvisible = false, xticklabelsvisible = false)\n lines!(ax2, t, r, color = :gray50, linewidth = 1)\n\n # Axes for variance and AC1 timeseries\n ax3 = Axis(fig[3, 1], ylabel = L\"Variance $\\,$\", xticks = xticks,\n xticksvisible = false, xticklabelsvisible = false)\n ax4 = Axis(fig[4, 1], xlabel = L\"Time (kyr) $\\,$\", ylabel = L\"Lag-1 autocor. $\\,$\",\n xticks = xticks)\n\n axs = [ax1, ax2, ax3, ax4]\n [xlims!(ax, xlims) for ax in axs]\n ylims!(axs[1], (-48, -34))\n rowgap!(fig.layout, 10)\n return fig, axs\nend\n\nxlims = (-60e3, -10e3)\nxticks = kyr_xticks(-60e3:5e3:5e3)\nt_rasmussen = -[-60000, 59440, 58280, 55800, 54220, 49280, 46860, 43340, 41460, 40160,\n 38220, 35480, 33740, 32500, 28900, 27780, 23340, 14692, 11703]\ntloess, _, xloess, rloess = loess_filter(t3, x3) # loess-filtered signal for visualization\nfig, axs = plot_do(t3, x3, tloess, xloess, t, r, t_rasmussen, xlims, xticks)\nfig","category":"page"},{"location":"examples/do-events/#Hindcast-on-NGRIP-data","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Hindcast on NGRIP data","text":"","category":"section"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"As one can see... there is not much to see so far. Residuals are impossible to simply eye-ball and we therefore use TransitionsInTimeseries.jl to study the evolution, measured by the ridge-regression slope of the residual's variance and lag-1 autocorrelation (AC1) over time. In many examples of the literature, including (Boers, 2018), the CSD analysis is performed over segments (sometimes only one) of the timeseries, such that a significance value is obtained for each segment. By using SegmentedWindowConfig, dealing with segments can be easily done in TransitionsInTimeseries.jl and is demonstrated here:","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"using TransitionsInTimeseries, StatsBase\nusing Random: Xoshiro\n\nac1(x) = sum(autocor(x, [1])) # AC1 from StatsBase\nindicators = (var, ac1)\nchange_metrics = (RidgeRegressionSlope(), RidgeRegressionSlope())\ntseg_start = t_rasmussen[1:end-1] .+ 200\ntseg_end = t_rasmussen[2:end] .- 200\nconfig = SegmentedWindowConfig(indicators, change_metrics,\n tseg_start, tseg_end; whichtime = last, width_ind = Int(200÷dt),\n min_width_cha = 100) # require >=100 data points to estimate change metric\nresults = estimate_changes(config, r, t)\nsignif = SurrogatesSignificance(n = 1000, tail = [:right, :right], rng = Xoshiro(1995))\nflags = significant_transitions(results, signif)","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"That's it! We can now visualise our results with a generic function that we will re-use later:","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"function plot_segment_analysis!(axs, results, signif)\n (; t_indicator, x_indicator) = results\n for k in eachindex(t_indicator) # loop over the segments\n for i in axes(signif.pvalues, 2) # loop over the indicators\n if !isnan(signif.pvalues[k, i]) # plot if segment long enough\n # Plot indicator timeseries and its linear regression\n ti, xi = t_indicator[k], x_indicator[k][:, i]\n lines!(axs[i+2], ti, xi, color = Cycled(1))\n m, p = ridgematrix(ti, 0.0) * xi\n if signif.pvalues[k, i] < 0.05\n lines!(axs[i+2], ti, m .* ti .+ p, color = :gray5, linewidth = 3)\n else\n lines!(axs[i+2], ti, m .* ti .+ p, color = :gray60, linewidth = 3)\n end\n end\n end\n end\nend\nplot_segment_analysis!(axs, results, signif)\nfig","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"In (Boers, 2018), 13/16 and 7/16 true positives are respectively found for the variance and AC1, with 16 referring to the total number of transitions. The timeseries actually includes 18 transition but, in (Boers, 2018), some segments are considered too small to be analysed. In contrast, we here respectively find 9/16 true positives for the variance and 3/16 for AC1. We can track down the discrepancies to be in the surrogate testing, since the indicator timeseries computed here are almost exactly similar to those of (Boers, 2018). This mismatch points out that packages like TransitionsInTimeseries.jl are desirable for research to be reproducible, especially since CSD is gaining attention - not only within the scientific community but also in popular media.","category":"page"},{"location":"examples/do-events/#CSD:-only-a-necessary-condition,-only-in-some-cases","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"CSD: only a necessary condition, only in some cases","text":"","category":"section"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"For codimension-1 systems, approaching a fold, Hopf or transcritical bifurcation implies a widening of the potential U, which defines the deterministic term f = -U of the SDE's right-hand-side. In the presence of noise, this leads to CSD, which is therefore a necessary condition for crossing one of these bifurcations - although it is not always assessable by analysing the timeseries due to practical limitations (e.g. sparse data subject to large measurement noise). It is nonetheless not given that DO-events, as many other real-life applications, can be seen as a codimension-1 fold, Hopf or transcritical bifurcations. Besides this, we emphasise that CSD is not a sufficient condition for assessing a transition being ahead in near future, since a resilience loss can happen without actually crossing any bifurcation. This can be illustrated on the present example by performing the same analysis only until few hundred years before the transition:","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"tseg_end = t_rasmussen[2:end] .- 700 # stop analysis 500 years earlier than before\nconfig = SegmentedWindowConfig(indicators, change_metrics,\n tseg_start, tseg_end, whichtime = last, width_ind = Int(200÷dt),\n min_width_cha = 100)\nresults = estimate_changes(config, r, t)\nsignif = SurrogatesSignificance(n = 1000, tail = [:right, :right], rng = Xoshiro(1995))\nflags = significant_transitions(results, signif)\nfig, axs = plot_do(t3, x3, tloess, xloess, t, r, t_rasmussen, xlims, xticks)\nplot_segment_analysis!(axs, results, signif)\nfig","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"For the variance and AC1, we here respectively find 6 and 3 positives, although the transitions are still far ahead. This shows that what CSD captures is a potential widening induced by a shift of the forcing parameter rather than the actual transition. We therefore believe, as already suggested in some studies, that \"resilience-loss indicators\" is a more accurate name than \"early-warning signals\" when using CSD.","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"We draw attention upon the fact that the delta^18O timeseries is noisy and sparsely re-sampled. Furthermore, interpolating over time introduces a potential bias in the statistics, even if performed on a coarse grid. The NGRIP data therefore represents an example that should be handled with care - as many others where CSD analysis has been applied on transitions in the field of geoscience. To contrast with this, we propose to perform the same analysis on synthethic DO data, obtained from an Earth Model of Intermediate Complexity (EMIC).","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"warning: Degrees of freedom\nThese sources of error come along the usual problem of arbitrarily choosing (1) a filtering method, (2) windowing parameters and (3) appropriate metrics (for instance when the forcing noise is suspected to be correlated). This leads to a large number of degrees of freedom (DoF). Although sensible guesses are possible here, checking that results are robust w.r.t. the DoF should be a standard practice.","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"info: Future improvement\nSupporting the computations for uneven timeseries is a planned improvement of TransitionsInTimeseries.jl. This will avoid the need of regridding data on coarse grids and will prevent from introducing any bias.","category":"page"},{"location":"examples/do-events/#Hindcasting-simulated-DO-events","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Hindcasting simulated DO-events","text":"","category":"section"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"In CLIMBER-X, the EMIC described in (Willeit et al., 2022), DO-like events can be triggered by forcing the North Atlantic with a (white noise) freshwater input. Simulated DO-like events present the big advantage of being evenly sampled in time and free of measurement noise. We run this analysis over two exemplary simulation outputs:","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"t_transitions = [[1, 1850, 2970, 3970, 5070, 5810, 7050, 8050],\n [1, 3500, 4370, 5790, 7200, 8140]]\nt_lb = [[300, 500, 300, 600, 300, 500, 500], [1800, 500, 1000, 900, 500]]\ntseg_start = [t_transitions[1][1:end-1] + t_lb[1], t_transitions[2][1:end-1] + t_lb[2]]\ntseg_end = [t_transitions[1][2:end] .- 50, t_transitions[2][2:end] .- 50]\n\nfigvec = Figure[]\n\nfor j in 1:2\n # Download the data and perform loess filtering on it\n tmp = Base.download(\"https://raw.githubusercontent.com/JuliaDynamics/JuliaDynamics/\" *\n \"master/timeseries/climberx-do$(j)-omaxa.csv\")\n data = readdlm(tmp)\n tcx, xcx = data[1, 1000:end], data[2, 1000:end]\n t, x, xtrend, r = loess_filter(tcx, xcx, span = 0.02)\n\n # Initialize figure\n xlims = (0, last(tcx))\n xticks = kyr_xticks(xlims[1]:1e3:xlims[2])\n fig, axs = plot_do(tcx, xcx, t, xtrend, t, r, t_transitions[j], extrema(t), xticks)\n ylims!(axs[1], (5, 40))\n axs[1].ylabel = L\"Max. Atlantic overturning (Sv) $\\,$\"\n\n # Run sliding analysis and update figure with results\n dt = mean(diff(tcx))\n config = SegmentedWindowConfig(\n indicators, change_metrics, tseg_start[j], tseg_end[j],\n whichtime = last, width_ind = Int(200÷dt), min_width_cha = 100)\n results = estimate_changes(config, r, t)\n signif = SurrogatesSignificance(n = 1_000, tail = [:right, :right], rng = Xoshiro(1995))\n flags = significant_transitions(results, signif)\n\n plot_segment_analysis!(axs, results, signif)\n vlines!(axs[1], t_transitions[j], color = Cycled(1), linewidth = 3)\n push!(figvec, fig)\nend\nfigvec[1]","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"It here appears that not all transitions are preceeded by a significant increase of variance and AC1, even in the case of clean and evenly sampled time series. Let's check another case:","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"figvec[2]","category":"page"},{"location":"examples/do-events/","page":"Dansgaard-Oescher events and Critical Slowing Down","title":"Dansgaard-Oescher events and Critical Slowing Down","text":"Same here! Although CLIMBER-X does not represent real DO-events, the above-performed analysis might be hinting at the fact that not all DO transitions can be forecasted with CSD. Nonetheless, performing a CSD analysis can inform on the evolution of a system's resilience.","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"EditURL = \"ks_paleojump.jl\"","category":"page"},{"location":"examples/ks_paleojump/#Kolmogorov-Smirnov-test-for-detecting-transitions-in-paleoclimate-timeseries","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"","category":"section"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"The goal of this example is to show how simple it is to re-create an analysis similar to what was done in the paper \"Automatic detection of abrupt transitions in paleoclimate records\", (Bagniewski et al., 2021). The same analysis was then used to create a database of transitions in paleoclimate records in (Bagniewski et al., 2023) Using TransitionsInTimeseries.jl and HypothesisTests.jl, the analysis becomes a 10-lines-of-code script (for a given timeseries).","category":"page"},{"location":"examples/ks_paleojump/#Scientific-background","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Scientific background","text":"","category":"section"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"The approach of (Bagniewski et al., 2021) is based on the two-sample Kolmogorov Smirnov test. It tests whether the samples from two datasets or timeseries are distributed according to the same cumulative density function or not. This can be estimated by comparing the value of the KS-statistic versus some threshold that depends on the required confidence.","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"The application of this test for identifying transitions in timeseries is simple:","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"A sliding window analysis is performed in the timesries\nIn each window, the KS statistic is estimated between the first half and the second half of the timeseries within this window.\nTransitions are defined by when the KS statistic exceeds a particular value based on some confidence. The transition occurs in the middle of the window.","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"We should point out that in (Bagniewski et al., 2021) the authors did a more detailed analysis: analyzed many different window widths, added a conditional clause to exclude transitions that do not exceed a predefined minimum \"jump\" in the data, and also added another conditional clause that filtered out transitions that are grouped in time (which is a natural consequence of using the Kolmogorov-Smirov test for detecting transitions).","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"Here we won't do that post processing, mainly because it is rather simple to include these additional conditional clauses to filter transitions after they are found.","category":"page"},{"location":"examples/ks_paleojump/#Steps-for-TransitionsInTimeseries.jl","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Steps for TransitionsInTimeseries.jl","text":"","category":"section"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"Doing this kind of work with TransitionsInTimeseries.jl is so easy you won't even trip! This analysis follows the same sliding window approach showcased in our Tutorial, and it even excludes the \"indicator\" aspect: the change metric is estimated directly from the input data!","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"As such, we really only need to define/do these things before we have finished the analysis:","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"Load the input data (we will use the same example as the NGRIP data of Dansgaard-Oescher events and Critical Slowing Down example) and set the appropriate time window.\nDefine the function that estimates the change metric (i.e., the KS-statistic)\nPerform the sliding window analysis as in the Tutorial with estimate_changes\nEstimate the \"confident\" transitions in the data by comparing the estimated KS-statistic with a predefined threshold.","category":"page"},{"location":"examples/ks_paleojump/#Load-timeseries-and-window-length","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Load timeseries and window length","text":"","category":"section"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"Following the Dansgaard-Oescher events example, we load the data after all the processing steps done in that example:","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"using DelimitedFiles, CairoMakie\n\ntmp = Base.download(\"https://raw.githubusercontent.com/JuliaDynamics/JuliaDynamics/\"*\n \"master/timeseries/NGRIP_processed.csv\")\ndata = readdlm(tmp)\nt, xtrend, xresid, xloess = collect.(eachcol(data))\n\nfig, ax = lines(t, xtrend; axis = (ylabel = \"NGRIP (processed)\", xlabel = \"time\"))\nlines!(ax, t, xloess; linewidth = 2)\nfig","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"For the window, since we are using a sliding window here, we will be using a window of length 500 (which is approximately 1/2 to 1/4 the span between typical transitions found by (Rasmussen et al., 2014)).","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"window = 500","category":"page"},{"location":"examples/ks_paleojump/#Defining-the-change-metric-function","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Defining the change metric function","text":"","category":"section"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"HypothesisTest.jl implements the Kolmogorov-Smirnov test, however here we are interested in the value of the test iself (the so-called KS-statistic), rather than a p-value. To this end, we define the following function to compute the statistic, which also normalizes it as in (Bagniewski et al., 2021).","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"using HypothesisTests\n\nfunction normalized_KS_statistic(timeseries)\n N = length(timeseries)\n i = N÷2\n x = view(timeseries, 1:i)\n y = view(timeseries, (i+1):N)\n kstest = ApproximateTwoSampleKSTest(x, y)\n nx = ny = i # length of each timeseries half of total\n n = nx*ny/(nx + ny) # written fully for concreteness\n D_KS = kstest.δ # can be compared directly with sqrt(-log(α/2)/2)\n # Rescale according to eq. (5) of the paper\n rescaled = 1 - ((1 - D_KS)/(1 - sqrt(1/n)))\n return rescaled\nend\n\nN = 1000 # the statistic is independent of `N` for large enough `N`!\nx = randn(N)\ny = 1.8randn(N) .+ 1.0\nz = randn(N)\nw = 0.6randn(N) .- 2.0\n\nfig, ax = density(x; color = (\"black\", 0.5), strokewidth = 4.0, label = \"reference distribution\")\nax.title = \"showcase of normalized KS-statistic\"\nfor q in (y, z, w)\n D_KS = normalized_KS_statistic(vcat(x, q))\n density!(ax, q; label = \"D_KS = $(D_KS)\")\nend\naxislegend(ax)\nfig","category":"page"},{"location":"examples/ks_paleojump/#Perform-the-sliding-window-analysis","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Perform the sliding window analysis","text":"","category":"section"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"This is just a straightforward call to estimate_changes. In fact, it is even simpler than the tutorial. Here we skip completely the \"indicator\" estimation step, and we evaluate the change metric directly on input data. We do this by simply passing nothing as the indicators.","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"using TransitionsInTimeseries\n\nconfig = SlidingWindowConfig(nothing, normalized_KS_statistic; width_cha = 500)\n\nresults = estimate_changes(config, xtrend, t)","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"Which we can visualize","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"function visualize_results(results)\n fig, ax1 = lines(t, xtrend; axis = (ylabel = \"NGRIP (processed)\",))\n ax2, = lines(fig[2, 1], results.t_change, vec(results.x_change), axis = (ylabel = \"D_KS (normalized)\", xlabel = \"time\"))\n linkxaxes!(ax1, ax2)\n hidexdecorations!(ax1; grid = false)\n xloess_normed = (xloess .- minimum(xloess))./(maximum(xloess) - minimum(xloess))\n lines!(ax2, t, xloess_normed; color = (\"gray\", 0.5))\n fig\nend\n\nvisualize_results(results)","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"By overplotting the (smoothened) NGRIP timeseries and the normalized KS-statistic, it already becomes pretty clear that the statistic peaks when transitions occur.","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"The same thing happens if we alter the window duration","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"config = SlidingWindowConfig(nothing, normalized_KS_statistic; width_cha = 200)\nresults = estimate_changes(config, xtrend, t)\nvisualize_results(results)","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"So one can easily obtain extra confidence by varying window size as in (Bagniewski et al., 2021).","category":"page"},{"location":"examples/ks_paleojump/#Identifying-\"confident\"-transitions","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Identifying \"confident\" transitions","text":"","category":"section"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"As this identification here is done via a simple threshold, identifying the transitions is a nearly trivial call to significant_transitions with ThresholdSignificance","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"signif = ThresholdSignificance(0.5) # or any other threshold\nflags = significant_transitions(results, signif)\n\nfig = visualize_results(results)\naxDKS = content(fig[2,1])\nvlines!(axDKS, results.t_change[vec(flags)], color = (\"red\", 0.25))\nfig","category":"page"},{"location":"examples/ks_paleojump/","page":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","title":"Kolmogorov-Smirnov test for detecting transitions in paleoclimate timeseries","text":"We could proceed with a lot of preprocessing as in (Bagniewski et al., 2021) but we skip this here for the sake of simplicity.","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"EditURL = \"logistic.jl\"","category":"page"},{"location":"examples/logistic/#Permutation-entropy-for-dynamic-regime-changes","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"","category":"section"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"Permutation entropy is used frequently to detect a transition between one dynamic regime to another. It is useful when the mean and std. of the timeseries values are very similar between the two regimes, which would mean that common distribution-based indicators, or common critical-slowing-down based indicators, would fail.","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"This example will also explore different ways to test for significance that are arguably better suitable in such an application than the Tutorial's default of significance via random Fourier surrogates.","category":"page"},{"location":"examples/logistic/#Logistic-map-timeseries","page":"Permutation entropy for dynamic regime changes","title":"Logistic map timeseries","text":"","category":"section"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"A simple example of this is transitions from periodic to weakly chaotic to chaotic motion in the logistic map. First, let's generate a timeseries of the logistic map","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"using DynamicalSystemsBase\nusing CairoMakie\n\n# time-dependent logistic map, so that the `r` parameter increases with time\nr1 = 3.83\nr2 = 3.86\nN = 2000\nrs = range(r1, r2; length = N)\n\nfunction logistic_drifting_rule(u, rs, n)\n r = rs[n+1] # time is `n`, starting from 0\n return SVector(r*u[1]*(1 - u[1]))\nend\n\nds = DeterministicIteratedMap(logistic_drifting_rule, [0.5], rs)\nx = trajectory(ds, N-1)[1][:, 1]","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"Plot it, using as time the parameter value (they coincide)","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"fig, ax = lines(rs, x; linewidth = 0.5)\nax.xlabel = \"r (time)\"\nax.ylabel = \"x\"\nfig","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"In this example there is a rather obvious transition to strongly chaotic motion at r ≈ 3.855. However, there is also a subtle transition to weak chaos at r ≈ 3.847. This transition is barely visible in the timeseries, and in fact many of the timeseries statistical properties remain identical.","category":"page"},{"location":"examples/logistic/#Using-a-simpler-change-metric","page":"Permutation entropy for dynamic regime changes","title":"Using a simpler change metric","text":"","category":"section"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"Now, let's compute various indicators and their changes, focusing on the permutation entropy as an indicator. We use order 4 here, because we know that to detect changes in a period m we would need an order ≥ m+1 permutation entropy.","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"using TransitionsInTimeseries, ComplexityMeasures\n\nfunction permutation_entropy(m)\n est = SymbolicPermutation(; m) # order 3\n indicator = x -> entropy_normalized(est, x)\n return indicator\nend\n\nindicators = (var, ar1_whitenoise, permutation_entropy(4))\nindistrings = (\"var\", \"ar1\", \"pe\")","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"In this example there is no critical slowing down; instead, there is a sharp transition between periodic and chaotic motion. Hence, we shouldn't be using any trend-based change metrics. Instead, we will use the most basic change metric, difference_of_means. With this metric it also makes most sense to use as stride half the window width","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"metric = (difference_of_means, difference_of_means, difference_of_means)\n\nwidth_ind = N÷100\nwidth_cha = 20\nstride_cha = 10\n\nconfig = SlidingWindowConfig(indicators, metric;\n width_ind, width_cha, stride_cha,\n)\n\nresults = estimate_changes(config, x, rs)","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"Let's now plot the change metrics of the indicators","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"function plot_change_metrics()\n fig, ax = lines(rs, x; axis = (ylabel = \"input\",), figure = (size = (600, 600),))\n hidexdecorations!(ax; grid = false)\n # plot all change metrics\n for (i, c) in enumerate(eachcol(results.x_change))\n ax, = scatterlines(fig[i+1, 1], results.t_change, c;\n axis = (ylabel = indistrings[i],), label = \"input\"\n )\n if i < 3\n hidexdecorations!(ax; grid = false)\n else\n ax.xlabel = \"r (time)\"\n end\n end\n return fig\nend\n\nfig = plot_change_metrics()","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"We already see the interesting results we expect: the permutation entropy shows a striking change as we go from periodic to weakly chaotic motion at r ≈ 3.847. (Remember: the plotted quantity is how much the indicator changes within a time window. High values mean large changes.)","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"Due to its construction, permutation entropy will have a spike for periodic data at the start of the timeseries, so we can safely ignore the spike at r ≈ 3.83.","category":"page"},{"location":"examples/logistic/#Significance-via-random-Fourier-surrogates","page":"Permutation entropy for dynamic regime changes","title":"Significance via random Fourier surrogates","text":"","category":"section"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"One way to test for significance would be via the standard way as in the Tutorial, utilizing surrogate timeseries and SurrogatesSignificance.","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"Let's do it here for an example, but, we have to be careful. It is crucial that for permutation entropy we use :right as the tail, because it is expected that the surrogates will have higher differences in the permutation entropy timeseries (because, if there is no dynamical change, the permutation entropy will stay the same, while in the surrogates there are always random fluctuations!","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"surromethod = RandomFourier()\n\n# Define a function because we will re-use later\nusing Random: Xoshiro\nfunction overplot_surrogate_significance!(fig, surromethod, color = \"black\")\n\n signif = SurrogatesSignificance(;\n n = 1000, tail = [:both, :both, :right], surromethod, rng = Xoshiro(42),\n )\n flags = significant_transitions(results, signif)\n\n # and also plot the flags with same color\n for (i, indicator) in enumerate(indicators)\n # To make things visually clear, we will also plot some example surrogate\n # timeseries for each indicator and change metric pair\n for _ in 1:10\n s = TimeseriesSurrogates.surrogate(x, surromethod)\n p = windowmap(indicator, s; width = width_ind)\n q = windowmap(metric[i], p; width = width_cha, stride = stride_cha)\n lines!(fig[i+1, 1], results.t_change, q; color = (color, 0.2), linewidth = 1)\n end\n # Plot the flags as vertical dashed lines\n vlines!(fig[i+1, 1], results.t_change[flags[:, i]];\n color = color, linestyle = :dash, linewidth = 3\n )\n end\n # add a title to the figure with how we estimate significance\n content(fig[1, 1]).title = \"surrogates: \"*string(nameof(typeof(surromethod)))\nend\n\nsurromethod = RandomFourier()\noverplot_surrogate_significance!(fig, surromethod)\n\nfig","category":"page"},{"location":"examples/logistic/#Different-surrogates","page":"Permutation entropy for dynamic regime changes","title":"Different surrogates","text":"","category":"section"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"Random Fourier surrogates perserve the power spectrum of the timeseries, but the power spectrum is a property integrated over the whole timeseries. It doesn't contain any information highlighting the local dynamics or information that preserves the local changes of dynamical behavior.","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"A surrogate type that does a better job in preserving local sharp changes in the timeseries (and hence provides stricter surrogate-based significance) is for example RelativePartialRandomization.","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"A much better alternative is to use block-shuffled surrogates, which preserve the short term local temporal correlation in the timeseries and hence also preserve local short term sharp changes in the dynamic behavior.","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"surromethod = RelativePartialRandomization(0.25)\nfig = plot_change_metrics()\noverplot_surrogate_significance!(fig, surromethod, \"gray\")\nfig","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"Our results have improved. In the permutation entropy, we see only two transitions detected as significant, which is correct: only two real dynamical transitions exist in the data. In the other two indicators we also see fewer transitions, but as we have already discussed, no results with the other indicators should be taken into meaningful consideration, as these indicators are simply inappropriate for what we are looking for here.","category":"page"},{"location":"examples/logistic/#Simpler-Significance","page":"Permutation entropy for dynamic regime changes","title":"Simpler Significance","text":"","category":"section"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"Arguably, exactly because we are using the difference_of_means as a change metric, we may want to be less strict and more simple with our tests for significance. Instead of using SurrogatesSignificance we may use the simpler and much faster SigmaSignificance, which simply claims significant time points whenever a change metric exceeds some pre-defined factor of its timeseries standard deviation.","category":"page"},{"location":"examples/logistic/","page":"Permutation entropy for dynamic regime changes","title":"Permutation entropy for dynamic regime changes","text":"fig = plot_change_metrics()\nflags = significant_transitions(results, SigmaSignificance(factor = 5.0))\n\n# Plot the flags\nfor (i, indicator) in enumerate(indicators)\n vlines!(fig[i+1, 1], results.t_change[flags[:, i]];\n color = Cycled(3), linestyle = :dash, linewidth = 3\n )\nend\ncontent(fig[1, 1]).title = \"significance from std\"\nfig","category":"page"},{"location":"#TransitionsInTimeseries.jl","page":"TransitionsInTimeseries.jl","title":"TransitionsInTimeseries.jl","text":"","category":"section"},{"location":"","page":"TransitionsInTimeseries.jl","title":"TransitionsInTimeseries.jl","text":"(Image: TransitionsInTimeseries.jl)","category":"page"},{"location":"","page":"TransitionsInTimeseries.jl","title":"TransitionsInTimeseries.jl","text":"TransitionsInTimeseries","category":"page"},{"location":"#TransitionsInTimeseries","page":"TransitionsInTimeseries.jl","title":"TransitionsInTimeseries","text":"TransitionsInTimeseries.jl\n\n(Image: ) (Image: ) (Image: CI) (Image: codecov) (Image: Package Downloads)\n\nTransitionsInTimeseries.jl is a free and open-source software to easily analyse transitions within timeseries in a reproducible, performant, extensible and reliable way. In contrast to other existing software with similar target application, TransitionsInTimeseries.jl defines a generic interface for how to find transitions and how to test for significance. Within this interface, it is easy to expand the software in three orthogonal ways:\n\nProvide the analysis pipelines with new indicators, which can be either self-written or imported from other packages. In particular, the latter offers thousands of metrics that can indicate transitions right out of the box.\nAdd new analysis pipelines for finding transitions.\nAdd new ways for significance testing.\n\nTransitionsInTimeseries is a registered Julia package and can be installed by running:\n\n] add TransitionsInTimeseries\n\nAll further information is provided in the documentation, which you can either find online or build locally by running the docs/make.jl file.\n\nAlternative names for this package could have been: Early Warning Signals / Resilience Indicators / Regime-Shift Identifiers / Change-Point Detectors, or however else you want to call them!\n\n\n\n\n\n","category":"module"},{"location":"","page":"TransitionsInTimeseries.jl","title":"TransitionsInTimeseries.jl","text":"info: Star us on GitHub!\nIf you have found this package useful, please consider starring it on GitHub. This gives us an accurate lower bound of the (satisfied) user count.","category":"page"},{"location":"#content","page":"TransitionsInTimeseries.jl","title":"Content","text":"","category":"section"},{"location":"","page":"TransitionsInTimeseries.jl","title":"TransitionsInTimeseries.jl","text":"Multi-stable systems can display abrupt transitions between two stability regimes. To predict such transitions in real-world systems solely based on data, mathematical tools have been developed in the last decades. Numerous terminologies have been used for them, such as early warning signals, resilience indicators, regime-shift identifiers, change-point detection and transition indicators. TransitionsInTimeseries.jl sticks to the latter terminology and provides an interface that:","category":"page"},{"location":"","page":"TransitionsInTimeseries.jl","title":"TransitionsInTimeseries.jl","text":"Allows a fast computation of common transition indicators with a couple of lines, as demonstrated in the example section.\nMakes the surrogate analysis to test for significance under the hub.\nCan be easily extended by any user without touching the source code.\nReduces the programming overhead for any researcher willing to benchmark new methods.\nEases the reproducibility thanks to a clear syntax, a simple installation and RNG-seeded surrogate generation.\nIncreases trustworthiness thanks to a large test suite.","category":"page"},{"location":"","page":"TransitionsInTimeseries.jl","title":"TransitionsInTimeseries.jl","text":"info: Similar projects\nAn R toolbox and a Python library already exist. However, we believe that they are difficult to extend for the user. Furthermore, they do not offer a native performant code, as here allowed by the use of Julia.","category":"page"},{"location":"#approaches","page":"TransitionsInTimeseries.jl","title":"Approaches","text":"","category":"section"},{"location":"","page":"TransitionsInTimeseries.jl","title":"TransitionsInTimeseries.jl","text":"Over the last decades, research on transition indicators has largely focused on Critical Slowing Down (CSD). CSD is observed when a system approaches a Hopf, a transcritical or a fold bifurcation and consists in a resilience loss of the system. For instance this can be diagnosed by an increase of the variance and the AR1-regression coefficient, as demonstrated in the example section. However, we emphasize that this is one out of many possible approaches for obtaining transition indicators. Recent work has explored new approaches relying on nonlinear dynamics or machine learning. TransitionsInTimeseries.jl is designed to allow these cutting-edge methods and foster the development of new ones.","category":"page"},{"location":"devdocs/#Developer's-documentation","page":"Developer's documentation","title":"Developer's documentation","text":"","category":"section"},{"location":"devdocs/","page":"Developer's documentation","title":"Developer's documentation","text":"This documentation addresses users that would like to contribute to the software by either solving bugs, improving documentation, or adding new methods. All contributions come in the form of Pull Requests, for which we strongly advise to follow good practices in scientific code, which means that they are properly formatted, documented, tested, etc.","category":"page"},{"location":"devdocs/#New-indicators-or-change-metrics","page":"Developer's documentation","title":"New indicators or change metrics","text":"","category":"section"},{"location":"devdocs/","page":"Developer's documentation","title":"Developer's documentation","text":"As explained already in e.g., SlidingWindowConfig, new indicators or change metrics are standard Julia functions, so you only need to define such a function (and document it, test it, etc.).","category":"page"},{"location":"devdocs/#New-pipeline-for-estimating-changes","page":"Developer's documentation","title":"New pipeline for estimating changes","text":"","category":"section"},{"location":"devdocs/","page":"Developer's documentation","title":"Developer's documentation","text":"This means to contribute a fundamentally new \"pipeline\" for estimating/detecting transitions in timeseries. This new pipeline defines what a \"transition\" means. To add a new pipeline follow these steps:","category":"page"},{"location":"devdocs/","page":"Developer's documentation","title":"Developer's documentation","text":"Define a new subtype of ChangesConfig\nDefine a new subtype of ChangesResults\nAdd a method for estimate_changes which accepts the new ChangesConfig subtype you defined and returns the ChangesResults subtype you defined.","category":"page"},{"location":"devdocs/","page":"Developer's documentation","title":"Developer's documentation","text":"And that's it!","category":"page"},{"location":"devdocs/","page":"Developer's documentation","title":"Developer's documentation","text":"Optionally you can further extend:","category":"page"},{"location":"devdocs/","page":"Developer's documentation","title":"Developer's documentation","text":"TransitionsInTimeseries.plot_indicator_changes with the new ChangesResults type you defined. Note the plotting functions are in the ext folder of the repository.\nsignificant_transitions(res::ChangesResults, signif::Significance) with your new ChangesResults type and as many Significance subtypes you have the capacity to extend for.","category":"page"},{"location":"devdocs/#New-pipeline-for-estimating-significance","page":"Developer's documentation","title":"New pipeline for estimating significance","text":"","category":"section"},{"location":"devdocs/","page":"Developer's documentation","title":"Developer's documentation","text":"Statistically significant changes are \"transitions\". However, what \"significant\" means is not universal. There are different ways to test for significance and TransitionsInTimeseries.jl allows various methods.","category":"page"},{"location":"devdocs/","page":"Developer's documentation","title":"Developer's documentation","text":"To add a new pipeline for estimating significance follow these steps:","category":"page"},{"location":"devdocs/","page":"Developer's documentation","title":"Developer's documentation","text":"Define a new subtype of Significance\nExtend significant_transitions(res::ChangesResults, signif::Significance) with your new type and as many ChangesResults subtypes as you have capacity for.","category":"page"},{"location":"devdocs/","page":"Developer's documentation","title":"Developer's documentation","text":"And that's it! Unfortunately so far we have found no way to make the significant_transitions code agnostic of the changes result, so you would need to add a method manually for every ChangesResults.","category":"page"},{"location":"devdocs/","page":"Developer's documentation","title":"Developer's documentation","text":"Optionally you can further extend TransitionsInTimeseries.plot_significance! (which you can find in the ext folder) for visualizing the significance results.","category":"page"},{"location":"refs/#References","page":"References","title":"References","text":"","category":"section"},{"location":"refs/","page":"References","title":"References","text":"Bagniewski, W.; Ghil, M. and Rousseau, D. D. (2021). Automatic detection of abrupt transitions in paleoclimate records. Chaos 31.\n\n\n\nBagniewski, W.; Rousseau, D. D. and Ghil, M. (2023). The PaleoJump database for abrupt transitions in past climates. Scientific Reports 13, 1–18, arXiv:2206.06832.\n\n\n\nBoers, N. (2018). Early-warning signals for Dansgaard-Oeschger events in a high-resolution ice core record. Nature Communications 9, 2556. Accessed on Oct 28, 2022.\n\n\n\nRasmussen, S. O.; Bigler, M.; Blockley, S. P.; Blunier, T.; Buchardt, S. L.; Clausen, H. B.; Cvijanovic, I.; Dahl-Jensen, D.; Johnsen, S. J.; Fischer, H.; Gkinis, V.; Guillevic, M.; Hoek, W. Z.; Lowe, J. J.; Pedro, J. B.; Popp, T.; Seierstad, I. K.; Steffensen, J. P.; Svensson, A. M.; Vallelonga, P.; Vinther, B. M.; Walker, M. J.; Wheatley, J. J. and Winstrup, M. (2014). A stratigraphic framework for abrupt climatic changes during the Last Glacial period based on three synchronized Greenland ice-core records: refining and extending the INTIMATE event stratigraphy. Quaternary Science Reviews 106, 14–28. Accessed on Oct 4, 2023.\n\n\n\nWilleit, M.; Ganopolski, A.; Robinson, A. and Edwards, N. R. (2022). The Earth system model CLIMBER-X v1.0 – Part 1: Climate model description and validation​​​​​​​​​​​​​​. Geoscientific Model Development 15, 5905–5948. Accessed on Aug 11, 2022.\n\n\n\n","category":"page"}] +} diff --git a/v0.1.2/siteinfo.js b/v0.1.2/siteinfo.js new file mode 100644 index 0000000..9cdef15 --- /dev/null +++ b/v0.1.2/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v0.1.2"; diff --git a/v0.1.2/tutorial.jl b/v0.1.2/tutorial.jl new file mode 100644 index 0000000..25a60a9 --- /dev/null +++ b/v0.1.2/tutorial.jl @@ -0,0 +1,263 @@ +#= + +# Tutorial + +## [Workflow] (@id workflow) + +Computing transition indicators consists of the following steps: + +1. Doing any preprocessing of raw data first, such as detrending. _This not part of TransitionsInTimeseries.jl_ and yields the **input timeseries**. +2. Estimating the timeseries of an indicator by sliding a window over the input timeseries. +3. Computing the changes of the indicator by sliding a window over its timeseries. Alternatively, the change metric can be estimated over the whole segment, as examplified in the section [Segmented windows](@ref segmented_windows). +4. Generating many surrogates that preserve important statistical properties of the original timeseries. +5. Performing step 2 and 3 for the surrogate timeseries. +6. Checking whether the indicator change timeseries of the real timeseries shows a significant feature (trend, jump or anything else) when compared to the surrogate data. + +These steps are illustrated one by one in the tutorial below, and then summarized in the convenient API that TransitionsInTimeseries.jl exports. + +## [Tutorial -- Educational] (@id example_stepbystep) + +### Raw input data + +Let us load data from a bistable nonlinear model subject to noise and to a gradual change of the forcing that leads to a transition. Furthermore, we also load data from a linear model, which is by definition monostable and therefore incapable of transitioning. This is done to control the rate of false positives, a common problem that can emerge when looking for transition indicators. The models are governed by: + +```math +\dfrac{\mathrm{d}x_{l}}{\mathrm{d}t} = - x_{l} - 1 + f(t) + n(t) \\ +\dfrac{\mathrm{d}x_{nl}}{\mathrm{d}t} = - x_{nl}^3 + x_{nl} + f(t) + n(t) +``` + +with $x_{l}$ the state of the linear model, $x_{nl}$ the state of the bistable model, $f$ the forcing and $n$ the noise. For $f=0$ they both display an equilibrium point at $x=-1$. However, the bistable model also displays a further equilibrium point at $x=1$. Loading (and visualizing with [Makie](https://docs.makie.org/stable/)) such prototypical data to test some indicators can be done by simply running: +=# + +using TransitionsInTimeseries, CairoMakie + +t, x_linear, x_nlinear = load_linear_vs_doublewell() +fig, ax = lines(t, x_linear) +lines!(ax, t, x_nlinear) +ax.title = "raw data" +fig + +#= +### Preprocessing + +!!! note "Not part of TransitionsInTimeseries.jl" + Any timeseries preprocessing, such as the de-trending step we do here, + is not part of TransitionsInTimeseries.jl and is the responsibility of the researcher. + + +The nonlinear system clearly displays a transition between two stability regimes. To forecast such transition, we analyze the fluctuations of the timeseries around the attractor, assumed to be tracked. Therefore, a detrending step is needed - here simply obtained by building the difference of the timeseries with lag 1. +=# + +x_l_fluct = diff(x_linear) +x_nl_fluct = diff(x_nlinear) +tfluct = t[2:end] + +fig, ax = lines(tfluct, x_l_fluct) +lines!(ax, tfluct, x_nl_fluct .+ 0.05) +ax.title = "input timeseries" +fig + +#= +At this point, `x_l_fluct` and `x_nl_fluct` are considered the **input timeseries**. + +!!! info "Detrending in Julia" + Detrending can be performed in many ways. A wide range of Julia packages exists to perform smoothing such as [Loess.jl](https://github.com/JuliaStats/Loess.jl) or [DSP.jl](https://docs.juliadsp.org/latest/contents/). There the detrending step consists of subtracting the smoothed signal from the original one. + +### Indicator timeseries + +We can then compute the values of some "indicator" (a Julia function that inputs a timeseries and outputs a number). An indicator should be a quantity that is likely to change if a transition occurs, or is about to occur in the timeseries. We compute indicators by applying a sliding window over the **input timeseries**, determined by the width and the stride with which it is applied. Here we demonstrate this computation with the AR1-regression coefficient (under white-noise assumption), implemented as [`ar1_whitenoise`](@ref): +=# + +indicator = ar1_whitenoise +indicator_window = (width = 400, stride = 1) + +## By mapping `last::Function` over a windowviewer of the time vector, +## we obtain the last time step of each window. +## This therefore only uses information from `k-width+1` to `k` at time step `k`. +## Alternatives: `first::Function`, `midpoint:::Function`. +t_indicator = windowmap(last, tfluct; indicator_window...) +indicator_l = windowmap(indicator, x_l_fluct; indicator_window...) +indicator_nl = windowmap(indicator, x_nl_fluct; indicator_window...) + +fig, ax = lines(t_indicator, indicator_l) +lines!(ax, t_indicator, indicator_nl) +ax.title = "indicator timeseries" +fig + +#= +The lines plotted above are the **indicator timeseries**. + +### Change metric timeseries + +From here, we process the **indicator timeseries** to quantify changes in it. This step is in essence the same as before: we apply some function over a sliding window of the indicator timeseries. We call this new timeseries the **change metric timeseries**. In the example here, the change metric we will employ will be the slope (over a sliding window), calculated via means of a [`RidgeRegressionSlope`](@ref): +=# + +change_window = (width = 30, stride = 1) +ridgereg = RidgeRegressionSlope(lambda = 0.0) +precompridgereg = precompute(ridgereg, t[1:change_window.width]) + +t_change = windowmap(last, t_indicator; change_window...) +change_l = windowmap(precompridgereg, indicator_l; change_window...) +change_nl = windowmap(precompridgereg, indicator_nl; change_window...) + +fig, ax = lines(t_change, change_l) +lines!(ax, t_change, change_nl) +ax.title = "change metric timeseries" +fig + +#= +### Timeseries surrogates + +As expected from [Critical Slowing Down](@ref approaches), an increase of the AR1-regression coefficient can be observed. Although eyeballing the timeseries might already be suggestive, we want a rigorous framework for testing for significance. + +In TransitionsIdentifiers.jl we perform significance testing using the method of timeseries surrogates and the [TimeseriesSurrogates.jl](https://github.com/JuliaDynamics/TimeseriesSurrogates.jl) Julia package. This has the added benefits of reproducibility, automation and flexibility in choosing the surrogate generation method. Note that `TimeseriesSurrogates` is re-exported by `TransitionsInTimeseries`, so that you don't have to `using` both of them. + +To illustrate the surrogate, we compare the change metric computed from the bistable timeseries what that computed from a surrogate of the same timeseries. +=# + +## Generate Fourier random-phase surrogates +using Random: Xoshiro +s = surrogate(x_nl_fluct, RandomFourier(), Xoshiro(123)) + +function gridfig(nrows, ncols) + fig = Figure() + axs = [Axis(fig[i, j], xticklabelsvisible = i == nrows ? true : false) + for j in 1:ncols, i in 1:nrows] + rowgap!(fig.layout, 10) + return fig, axs +end +fig, axs = gridfig(2, 1) +lines!(axs[1], tfluct, x_nl_fluct, color = Cycled(2)) +lines!(axs[1], tfluct, s .- 0.05, color = Cycled(3)) +axs[1].title = "real signal vs. surrogate(s)" + +## compute and plot indicator and change metric +indicator_s = windowmap(indicator, s; indicator_window...) +change_s = windowmap(precompridgereg, indicator_s; change_window...) + +lines!(axs[2], t_change, change_nl, label = "nonlin", color = Cycled(2)) +lines!(axs[2], t_change, change_s, label = "surrogate", color = Cycled(3)) +axislegend() +axs[2].title = "change metric" + +[xlims!(ax, 0, 50) for ax in axs] +fig + +#= +### Quantifying significance + +To quantify the significance of the values of the **change metric timeseries** we perform a standard surrogate test by computing the [p-value](https://en.wikipedia.org/wiki/P-value) w.r.t. the change metrics of thousands of surrogates of the input timeseries. A low p-value (typically `p<0.05`) is commonly considered as significant. To visualize significant trends, we plot the p-value vs. time: +=# + +n_surrogates = 1_000 +fig, axs = gridfig(2, 2) +axs[1].title = "linear" +axs[2].title = "nonlinear" + +for (j, ax, axsig, x) in zip(1:2, axs[1:2], axs[3:4], (x_l_fluct, x_nl_fluct)) + + orig_change = j == 1 ? change_l : change_nl + sgen = surrogenerator(x, RandomFourier(), Xoshiro(123)) + pval = zeros(length(change_s)) + + ## Collect all surrogate change metrics + for i in 1:n_surrogates + s = sgen() + indicator_s = windowmap(indicator, s; indicator_window...) + change_s = windowmap(precompridgereg, indicator_s; change_window...) + pval += orig_change .< change_s + end + + pval ./= n_surrogates + lines!(ax, t_change, orig_change) # ; color = Cycled(j) + lines!(axsig, t_change, pval) # ; color = Cycled(j+2) +end + +[xlims!(ax, 0, 50) for ax in axs] +fig + +#= +As expected, the data generated by the nonlinear model displays a significant increase of the AR1-regression coefficient before the transition, which is manifested by a low p-value. In contrast, the data generated by the linear model does not show anything similar. + +Performing the step-by-step analysis of transition indicators is possible and might be preferred for users wanting high flexibility. However, this results in a substantial amount of code. We therefore provide convenience functions that wrap this analysis, as shown in the next section. + +## [Tutorial -- TransitionsInTimeseries.jl] (@id example_fastforward) + +TransitionsInTimeseries.jl wraps this typical workflow into a simple, extendable, and modular API that researchers can use with little effort. In addition, it allows performing the same analysis for several indicators / change metrics in one go. + +The interface is simple, and directly parallelizes the [Workflow](@ref workflow). It is based on the creation of a [`ChangesConfig`](@ref), which contains a list of indicators, and corresponding metrics, to use for doing the above analysis. It also specifies what kind of surrogates to generate. + +### Sliding windows + +The following blocks illustrate how the above extensive example is re-created in TransitionsInTimeseries.jl +=# + +using TransitionsInTimeseries, CairoMakie + +t, x_linear, x_nlinear = load_linear_vs_doublewell() + +## input timeseries and time +input = x_nl_fluct = diff(x_nlinear) +t = t[2:end] + +fig, ax = lines(t, input) +ax.title = "input timeseries" +fig + +#= +To perform all of the above analysis we follow a 2-step process. + +Step 1, we decide what indicators and change metrics to use in [`SlidingWindowConfig`](@ref) and apply those via +a sliding window to the input timeseries using [`estimate_changes`](@ref). +=# + +## These indicators are suitable for Critical Slowing Down +indicators = (var, ar1_whitenoise) + +## use the ridge regression slope for both indicators +change_metrics = (RidgeRegressionSlope(), RidgeRegressionSlope()) + +## choices go into a configuration struct +config = SlidingWindowConfig(indicators, change_metrics; + width_ind = 400, width_cha = 30, whichtime = last) + +## choices are processed +results = estimate_changes(config, input, t) + +#= +We can conveniently plot the information contained in `results` by using +`plot_indicator_changes`: +=# + +fig = plot_indicator_changes(results) + +#= +Step 2 is to estimate significance using [`SurrogatesSignificance`](@ref) +and the function [`significant_transitions`](@ref). Finally, we can +conveniently plot the results obtained by updating the figure obtained +above with `plot_significance!`: +=# + +signif = SurrogatesSignificance(n = 1000, tail = [:right, :right]) +flags = significant_transitions(results, signif) +plot_significance!(fig, results, signif, flags = flags) +fig + +#= +### [Segmented windows] (@id segmented_windows) + +The analysis shown so far relies on sliding windows of the change metric. +This is particularly convenient for transition detection tasks. Segmented +windows for the change metric computation are however preferable when it +comes to prediction tasks. By only slightly modifying the syntax used so far, +one can perform the same computations on segmented windows, as well as +visualise the results conveniently: +=# + +config = SegmentedWindowConfig(indicators, change_metrics, + t[1:1], t[1200:1200]; whichtime = last, width_ind = 200, + min_width_cha = 100) +results = estimate_changes(config, input, t) +signif = SurrogatesSignificance(n = 1000, tail = [:right, :right]) +flags = significant_transitions(results, signif) +fig = plot_changes_significance(results, signif) diff --git a/v0.1.2/tutorial/03a9efaf.png b/v0.1.2/tutorial/03a9efaf.png new file mode 100644 index 0000000..7168447 Binary files /dev/null and b/v0.1.2/tutorial/03a9efaf.png differ diff --git a/v0.1.2/tutorial/10fd46e0.png b/v0.1.2/tutorial/10fd46e0.png new file mode 100644 index 0000000..e078d5d Binary files /dev/null and b/v0.1.2/tutorial/10fd46e0.png differ diff --git a/v0.1.2/tutorial/1b3f597e.png b/v0.1.2/tutorial/1b3f597e.png new file mode 100644 index 0000000..3afb8f3 Binary files /dev/null and b/v0.1.2/tutorial/1b3f597e.png differ diff --git a/v0.1.2/tutorial/6ccd1579.png b/v0.1.2/tutorial/6ccd1579.png new file mode 100644 index 0000000..b74c7ca Binary files /dev/null and b/v0.1.2/tutorial/6ccd1579.png differ diff --git a/v0.1.2/tutorial/8621b213.png b/v0.1.2/tutorial/8621b213.png new file mode 100644 index 0000000..d2e0a91 Binary files /dev/null and b/v0.1.2/tutorial/8621b213.png differ diff --git a/v0.1.2/tutorial/89e43fa2.png b/v0.1.2/tutorial/89e43fa2.png new file mode 100644 index 0000000..6e07024 Binary files /dev/null and b/v0.1.2/tutorial/89e43fa2.png differ diff --git a/v0.1.2/tutorial/a534bebb.png b/v0.1.2/tutorial/a534bebb.png new file mode 100644 index 0000000..fc614c3 Binary files /dev/null and b/v0.1.2/tutorial/a534bebb.png differ diff --git a/v0.1.2/tutorial/adbb8d4a.png b/v0.1.2/tutorial/adbb8d4a.png new file mode 100644 index 0000000..af02a8a Binary files /dev/null and b/v0.1.2/tutorial/adbb8d4a.png differ diff --git a/v0.1.2/tutorial/d811dce9.png b/v0.1.2/tutorial/d811dce9.png new file mode 100644 index 0000000..5058e28 Binary files /dev/null and b/v0.1.2/tutorial/d811dce9.png differ diff --git a/v0.1.2/tutorial/f28a0721.png b/v0.1.2/tutorial/f28a0721.png new file mode 100644 index 0000000..f2f422f Binary files /dev/null and b/v0.1.2/tutorial/f28a0721.png differ diff --git a/v0.1.2/tutorial/index.html b/v0.1.2/tutorial/index.html new file mode 100644 index 0000000..5124256 --- /dev/null +++ b/v0.1.2/tutorial/index.html @@ -0,0 +1,128 @@ + +Tutorial · TransitionsInTimeseries.jl

Tutorial

Workflow

Computing transition indicators consists of the following steps:

  1. Doing any preprocessing of raw data first, such as detrending. This not part of TransitionsInTimeseries.jl and yields the input timeseries.
  2. Estimating the timeseries of an indicator by sliding a window over the input timeseries.
  3. Computing the changes of the indicator by sliding a window over its timeseries. Alternatively, the change metric can be estimated over the whole segment, as examplified in the section Segmented windows.
  4. Generating many surrogates that preserve important statistical properties of the original timeseries.
  5. Performing step 2 and 3 for the surrogate timeseries.
  6. Checking whether the indicator change timeseries of the real timeseries shows a significant feature (trend, jump or anything else) when compared to the surrogate data.

These steps are illustrated one by one in the tutorial below, and then summarized in the convenient API that TransitionsInTimeseries.jl exports.

Tutorial – Educational

Raw input data

Let us load data from a bistable nonlinear model subject to noise and to a gradual change of the forcing that leads to a transition. Furthermore, we also load data from a linear model, which is by definition monostable and therefore incapable of transitioning. This is done to control the rate of false positives, a common problem that can emerge when looking for transition indicators. The models are governed by:

\[\dfrac{\mathrm{d}x_{l}}{\mathrm{d}t} = - x_{l} - 1 + f(t) + n(t) \\ +\dfrac{\mathrm{d}x_{nl}}{\mathrm{d}t} = - x_{nl}^3 + x_{nl} + f(t) + n(t)\]

with $x_{l}$ the state of the linear model, $x_{nl}$ the state of the bistable model, $f$ the forcing and $n$ the noise. For $f=0$ they both display an equilibrium point at $x=-1$. However, the bistable model also displays a further equilibrium point at $x=1$. Loading (and visualizing with Makie) such prototypical data to test some indicators can be done by simply running:

using TransitionsInTimeseries, CairoMakie
+
+t, x_linear, x_nlinear = load_linear_vs_doublewell()
+fig, ax = lines(t, x_linear)
+lines!(ax, t, x_nlinear)
+ax.title = "raw data"
+fig
Example block output

Preprocessing

Not part of TransitionsInTimeseries.jl

Any timeseries preprocessing, such as the de-trending step we do here, is not part of TransitionsInTimeseries.jl and is the responsibility of the researcher.

The nonlinear system clearly displays a transition between two stability regimes. To forecast such transition, we analyze the fluctuations of the timeseries around the attractor, assumed to be tracked. Therefore, a detrending step is needed - here simply obtained by building the difference of the timeseries with lag 1.

x_l_fluct = diff(x_linear)
+x_nl_fluct = diff(x_nlinear)
+tfluct = t[2:end]
+
+fig, ax = lines(tfluct, x_l_fluct)
+lines!(ax, tfluct, x_nl_fluct .+ 0.05)
+ax.title = "input timeseries"
+fig
Example block output

At this point, x_l_fluct and x_nl_fluct are considered the input timeseries.

Detrending in Julia

Detrending can be performed in many ways. A wide range of Julia packages exists to perform smoothing such as Loess.jl or DSP.jl. There the detrending step consists of subtracting the smoothed signal from the original one.

Indicator timeseries

We can then compute the values of some "indicator" (a Julia function that inputs a timeseries and outputs a number). An indicator should be a quantity that is likely to change if a transition occurs, or is about to occur in the timeseries. We compute indicators by applying a sliding window over the input timeseries, determined by the width and the stride with which it is applied. Here we demonstrate this computation with the AR1-regression coefficient (under white-noise assumption), implemented as ar1_whitenoise:

indicator = ar1_whitenoise
+indicator_window = (width = 400, stride = 1)
+
+# By mapping `last::Function` over a windowviewer of the time vector,
+# we obtain the last time step of each window.
+# This therefore only uses information from `k-width+1` to `k` at time step `k`.
+# Alternatives: `first::Function`, `midpoint:::Function`.
+t_indicator = windowmap(last, tfluct; indicator_window...)
+indicator_l = windowmap(indicator, x_l_fluct; indicator_window...)
+indicator_nl = windowmap(indicator, x_nl_fluct; indicator_window...)
+
+fig, ax = lines(t_indicator, indicator_l)
+lines!(ax, t_indicator, indicator_nl)
+ax.title = "indicator timeseries"
+fig
Example block output

The lines plotted above are the indicator timeseries.

Change metric timeseries

From here, we process the indicator timeseries to quantify changes in it. This step is in essence the same as before: we apply some function over a sliding window of the indicator timeseries. We call this new timeseries the change metric timeseries. In the example here, the change metric we will employ will be the slope (over a sliding window), calculated via means of a RidgeRegressionSlope:

change_window = (width = 30, stride = 1)
+ridgereg = RidgeRegressionSlope(lambda = 0.0)
+precompridgereg = precompute(ridgereg, t[1:change_window.width])
+
+t_change = windowmap(last, t_indicator; change_window...)
+change_l = windowmap(precompridgereg, indicator_l; change_window...)
+change_nl = windowmap(precompridgereg, indicator_nl; change_window...)
+
+fig, ax = lines(t_change, change_l)
+lines!(ax, t_change, change_nl)
+ax.title = "change metric timeseries"
+fig
Example block output

Timeseries surrogates

As expected from Critical Slowing Down, an increase of the AR1-regression coefficient can be observed. Although eyeballing the timeseries might already be suggestive, we want a rigorous framework for testing for significance.

In TransitionsIdentifiers.jl we perform significance testing using the method of timeseries surrogates and the TimeseriesSurrogates.jl Julia package. This has the added benefits of reproducibility, automation and flexibility in choosing the surrogate generation method. Note that TimeseriesSurrogates is re-exported by TransitionsInTimeseries, so that you don't have to using both of them.

To illustrate the surrogate, we compare the change metric computed from the bistable timeseries what that computed from a surrogate of the same timeseries.

# Generate Fourier random-phase surrogates
+using Random: Xoshiro
+s = surrogate(x_nl_fluct, RandomFourier(), Xoshiro(123))
+
+function gridfig(nrows, ncols)
+    fig = Figure()
+    axs = [Axis(fig[i, j], xticklabelsvisible = i == nrows ? true : false)
+        for j in 1:ncols, i in 1:nrows]
+    rowgap!(fig.layout, 10)
+    return fig, axs
+end
+fig, axs = gridfig(2, 1)
+lines!(axs[1], tfluct, x_nl_fluct, color = Cycled(2))
+lines!(axs[1], tfluct, s .- 0.05, color = Cycled(3))
+axs[1].title = "real signal vs. surrogate(s)"
+
+# compute and plot indicator and change metric
+indicator_s = windowmap(indicator, s; indicator_window...)
+change_s = windowmap(precompridgereg, indicator_s; change_window...)
+
+lines!(axs[2], t_change, change_nl, label = "nonlin", color = Cycled(2))
+lines!(axs[2], t_change, change_s, label = "surrogate", color = Cycled(3))
+axislegend()
+axs[2].title = "change metric"
+
+[xlims!(ax, 0, 50) for ax in axs]
+fig
Example block output

Quantifying significance

To quantify the significance of the values of the change metric timeseries we perform a standard surrogate test by computing the p-value w.r.t. the change metrics of thousands of surrogates of the input timeseries. A low p-value (typically p<0.05) is commonly considered as significant. To visualize significant trends, we plot the p-value vs. time:

n_surrogates = 1_000
+fig, axs = gridfig(2, 2)
+axs[1].title = "linear"
+axs[2].title = "nonlinear"
+
+for (j, ax, axsig, x) in zip(1:2, axs[1:2], axs[3:4], (x_l_fluct, x_nl_fluct))
+
+    orig_change = j == 1 ? change_l : change_nl
+    sgen = surrogenerator(x, RandomFourier(), Xoshiro(123))
+    pval = zeros(length(change_s))
+
+    # Collect all surrogate change metrics
+    for i in 1:n_surrogates
+        s = sgen()
+        indicator_s = windowmap(indicator, s; indicator_window...)
+        change_s = windowmap(precompridgereg, indicator_s; change_window...)
+        pval += orig_change .< change_s
+    end
+
+    pval ./= n_surrogates
+    lines!(ax, t_change, orig_change)   # ; color = Cycled(j)
+    lines!(axsig, t_change, pval)       # ; color = Cycled(j+2)
+end
+
+[xlims!(ax, 0, 50) for ax in axs]
+fig
Example block output

As expected, the data generated by the nonlinear model displays a significant increase of the AR1-regression coefficient before the transition, which is manifested by a low p-value. In contrast, the data generated by the linear model does not show anything similar.

Performing the step-by-step analysis of transition indicators is possible and might be preferred for users wanting high flexibility. However, this results in a substantial amount of code. We therefore provide convenience functions that wrap this analysis, as shown in the next section.

Tutorial – TransitionsInTimeseries.jl

TransitionsInTimeseries.jl wraps this typical workflow into a simple, extendable, and modular API that researchers can use with little effort. In addition, it allows performing the same analysis for several indicators / change metrics in one go.

The interface is simple, and directly parallelizes the Workflow. It is based on the creation of a ChangesConfig, which contains a list of indicators, and corresponding metrics, to use for doing the above analysis. It also specifies what kind of surrogates to generate.

Sliding windows

The following blocks illustrate how the above extensive example is re-created in TransitionsInTimeseries.jl

using TransitionsInTimeseries, CairoMakie
+
+t, x_linear, x_nlinear = load_linear_vs_doublewell()
+
+# input timeseries and time
+input = x_nl_fluct = diff(x_nlinear)
+t = t[2:end]
+
+fig, ax = lines(t, input)
+ax.title = "input timeseries"
+fig
Example block output

To perform all of the above analysis we follow a 2-step process.

Step 1, we decide what indicators and change metrics to use in SlidingWindowConfig and apply those via a sliding window to the input timeseries using estimate_changes.

# These indicators are suitable for Critical Slowing Down
+indicators = (var, ar1_whitenoise)
+
+# use the ridge regression slope for both indicators
+change_metrics = (RidgeRegressionSlope(), RidgeRegressionSlope())
+
+# choices go into a configuration struct
+config = SlidingWindowConfig(indicators, change_metrics;
+    width_ind = 400, width_cha = 30, whichtime = last)
+
+# choices are processed
+results = estimate_changes(config, input, t)
SlidingWindowResults
+ input timeseries:              2500-element Vector{Float64}
+ indicators:                    [:var, :ar1_whitenoise]
+ indicator (window, stride):    (400, 1)
+ change metrics:                [:PrecomputedRidgeRegressionSlope, :PrecomputedRidgeRegressionSlope]
+ change metric (window, stride): (30, 1)
+

We can conveniently plot the information contained in results by using plot_indicator_changes:

fig = plot_indicator_changes(results)
Example block output

Step 2 is to estimate significance using SurrogatesSignificance and the function significant_transitions. Finally, we can conveniently plot the results obtained by updating the figure obtained above with plot_significance!:

signif = SurrogatesSignificance(n = 1000, tail = [:right, :right])
+flags = significant_transitions(results, signif)
+plot_significance!(fig, results, signif, flags = flags)
+fig
Example block output

Segmented windows

The analysis shown so far relies on sliding windows of the change metric. This is particularly convenient for transition detection tasks. Segmented windows for the change metric computation are however preferable when it comes to prediction tasks. By only slightly modifying the syntax used so far, one can perform the same computations on segmented windows, as well as visualise the results conveniently:

config = SegmentedWindowConfig(indicators, change_metrics,
+    t[1:1], t[1200:1200]; whichtime = last, width_ind = 200,
+    min_width_cha = 100)
+results = estimate_changes(config, input, t)
+signif = SurrogatesSignificance(n = 1000, tail = [:right, :right])
+flags = significant_transitions(results, signif)
+fig = plot_changes_significance(results, signif)
Example block output
diff --git a/versions.js b/versions.js index 7d3080b..d7dd1de 100644 --- a/versions.js +++ b/versions.js @@ -3,5 +3,5 @@ var DOC_VERSIONS = [ "v0.1", "dev", ]; -var DOCUMENTER_NEWEST = "v0.1.1"; +var DOCUMENTER_NEWEST = "v0.1.2"; var DOCUMENTER_STABLE = "stable";