From 395afa2390790f9a1e9eba92f527d5ba5a49b6fc Mon Sep 17 00:00:00 2001 From: Thibaud Coroller <14895385+tcoroller@users.noreply.github.com> Date: Wed, 3 Jul 2024 11:05:18 -0400 Subject: [PATCH 1/8] Create evaluation_metrics_survival_model.md (#39) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit * Create evaluation_metrics_survival_model.md * edits * fixed MathJax issue * locked scipy version * added dollar for Latex --------- Co-authored-by: Mélodie Monod --- dev/environment.yml | 2 + docs/AUTHORS.md | 1 - docs/conf.py | 2 + docs/index.rst | 1 + docs/notebooks/survival.md | 264 +++++++++++++++++++++++++++++++++++++ 5 files changed, 269 insertions(+), 1 deletion(-) create mode 100644 docs/notebooks/survival.md diff --git a/dev/environment.yml b/dev/environment.yml index 51f6ce2..51ba6c2 100644 --- a/dev/environment.yml +++ b/dev/environment.yml @@ -24,3 +24,5 @@ dependencies: - nbsphinx=0.9.3 - ipython=8.20.0 - twine=4.0.2 + - scipy=1.12.0 + - sphinx-math-dollar=1.2.1 diff --git a/docs/AUTHORS.md b/docs/AUTHORS.md index 98b27bb..51327c4 100644 --- a/docs/AUTHORS.md +++ b/docs/AUTHORS.md @@ -8,4 +8,3 @@ Contributors * David Ohlssen `(contributor)` * Berkman Sahiner `(contributor)` * Nicholas Petrick `(contributor)` - diff --git a/docs/conf.py b/docs/conf.py index 350f69c..50ad7a9 100644 --- a/docs/conf.py +++ b/docs/conf.py @@ -28,8 +28,10 @@ "nbsphinx", "sphinx.ext.viewcode", "sphinxcontrib.bibtex", + "sphinx_math_dollar", ] + # templates_path = ['_templates'] exclude_patterns = ["_build", "Thumbs.db", ".DS_Store"] diff --git a/docs/index.rst b/docs/index.rst index 20c62a9..155613b 100644 --- a/docs/index.rst +++ b/docs/index.rst @@ -16,6 +16,7 @@ Welcome to TorchSurv's documentation! :maxdepth: 2 :caption: Tutorials: + notebooks/survival notebooks/introduction notebooks/momentum diff --git a/docs/notebooks/survival.md b/docs/notebooks/survival.md new file mode 100644 index 0000000..f317942 --- /dev/null +++ b/docs/notebooks/survival.md @@ -0,0 +1,264 @@ +# A statistical introduction + + + +* **Author**: Mélodie Monod + +## Introduction + +The evaluation metrics for assessing the predictive performance of a model depend greatly on the type of response variable. For instance, mean squared error (MSE) or mean absolute error (MAE) are commonly used for continuous response data. In contrast, for binary data, metrics such as true positive rate (TPR) and true negative rate (TNR) are utilized. This document presents evaluation metrics specifically designed for survival data, where the response variable is the time to an event. + +## Evaluation metrics under binary response + +To understand the evaluation metrics for time-to-event data, it is helpful to start with a review of the evaluation metrics used for binary outcomes, as the former are extensions of the latter. + +Assume we have a binary response $Y_i \in \{0,1\}$, for any individual $i$. The model is a probabilistic classifier that outputs a score $\pi_i \in [0,1]$, which is an estimate of the probability $p(Y_i = 1)$. + +The predicted response for individual $i$, denoted $\hat{Y}_i$, is obtained by comparing the score to a threshold $c$, + +$$ +\hat{Y}_i = +\begin{cases} + 1, \text{ if } \pi_i > c \\ + 0, \text{ if } \pi_i \leq c. \\ +\end{cases} +$$ + +Two key quantities are defined to evaluate the accuracy of the score: sensitivity and specificity, which are defined as + +$$ +\text{Sensitivity}(c) = p(\hat{Y}_i = 1 | Y_i = 1) = p(\pi_i > c | Y_i = 1),\\ +\text{Specificity}(c) = p(\hat{Y}_i = 0 | Y_i = 0) = p(\pi_i \leq c | Y_i = 0). +$$ + +Note that the sensitivity is also referred to as the True Positive Rate (TPR), and the specificity is also known as the True Negative Rate (TNR), which equals 1 − False Positive Rate (FPR). + +To visualize the predictive performance of the probabilistic classifier, the Receiver Operating Characteristic (ROC) curve plots the FPR on the x-axis and the TPR on the y-axis for all values of $c$. The Area Under the ROC Curve (AUC) summarizes the predictive performance across all values of $c$. It is given by + +$$ +\text{AUC} = \int_0^1 TPR(FPR(c)) dFPR(c). +$$ + +It can be shown that the AUC is equal to $p(\pi_i > \pi_j|Y_i = 1, Y_j = 0)$. This is the probability that, for a comparable pair, the individual without the event has a lower score than the individual with the event. This probability is also referred to as the C-index (denoted by C). In the binary context, the AUC is equal to the C-index. + +
+ Proof of the AUC's probabilistic interpretation +
+ +Let +$$ +y(c) = \text{TPR}(c) = p(\pi_i > c | Y_i = 1),\\ +x(c) = \text{FPR}(c) = p(\pi_i > c | Y_i = 0). +$$ + +Let us denote $f_1(c) = p(\pi_i = c | Y_i = 1)$ and $F_1(c) = p(\pi_i \leq c | Y_i = 1)$ and similarly $f_0(c) = p(\pi_i = c | Y_i = 0)$ and $F_0(c) = p(\pi_i \leq c | Y_i = 0)$. Notice that $y(c) = 1 - F_1(c)$ and $x(c) = 1 - F_0(c)$. + +The AUC is defined as +$$ +\text{AUC} = \int_0^1 y(x(c)) dx(c). +$$ + +We use a change of variable $c = x(c)$. We have $\frac{dx}{dc}(c) = x'(c)$ and the limits become $x^{-1}(1) = - \infty$ and $x^{-1}(0) = \infty$. Therefore + +$$ +\begin{align*} +\text{AUC} &= \int_{\infty}^{-\infty} y(c) x'(c) dc \\ +&= \int_{\infty}^{-\infty} (1- F_1(c)) (-f_0(c)) dc \\ +&= \int_{-\infty}^\infty (1- F_1(c)) f_0(c) dc \\ +&= \int_{-\infty}^\infty p(\pi_i > c | Y_i = 1) p(\pi_j = c | Y_j = 0) dc \\ +&= p(\pi_i > \pi_j | Y_i = 1, Y_j = 0) \\ +\end{align*} +$$ +
+ +## Evaluation metrics under time-to-event response + +Let's assume that individual $i$ has a time-to-event $X_i$ and a time-to-censoring $D_i$. We observe either the time to event or censoring, $T_i = \min(X_i, D_i)$, along with the event indicator $\delta_i = 1(X_i \leq D_i)$. A probabilistic survival model outputs a score $q_i(t) \in \mathbb{R}$, which estimates the event risk at time $t$ for individual $i$. + + +### Extension of sensitivity and specificity for time-to-event response + +A time-to-event outcome can be viewed as a time-varying binary outcome using the counting process representation: + +$$ +N_i(t) = 1(X_i \leq t). +$$ + + +The risk score chosen can either predicts prevalence or incidence (Heagerty and Zheng, 2005). For example, the cumulative hazard can be seen as a measure of prevalence because it measures the cumulative risk, whereas the instantaneous hazard can be seen as a measure of incidence as it measures the risk of an event in a very short, infinitesimally small time. Consequently, there are two different types of sensitivity depending on the chosen risk score. + +#### Cumulative sensitivity + +If the predicted classification of $i$ is based on a risk score measuring prevalence, it is defined as: + +$$ +\hat{N}_i(t) = +\begin{cases} + 1, \text{ if } q_i(t) > c \\ + 0, \text{ if } q_i(t) \leq c. \\ +\end{cases} +$$ + +In this case, the sensitivity is referred to as cumulative sensitivity. It is the probability that a model correctly predicts that individual $i$ experiences an event before or at time $t$ and is defined as: + +$$ +\text{sensitivity}^{\mathbb{C}}(c,t) = p(\hat{N}_i(t) = 1 | {N}_i(t) =1 ) = p(q_i(t) > c| X_i \leq t ) +$$ + +#### Incident sensitivity + +If the prediction classification of $i$ is based on a risk score measuring incidence, it is defined as: + +$$ +d\hat{N}_i(t) = +\begin{cases} + 1, \text{ if } q_i(t) > c \\ + 0, \text{ if } q_i(t) \leq c. \\ +\end{cases} +$$ + +In this case, the sensitivity is referred to as incident sensitivity. It is the probability that a model correctly predicts that individual $i$ experiences an event at time $t$ and is defined as: + +$$ +\text{sensitivity}^{\mathbb{I}}(c,t) = p(d\hat{N}_i(t) = 1 | d{N}_i(t) =1 ) = p(q_i(t) > c| X_i = t ) +$$ + +#### Dynamic specificity + +Extending the concept from the binary response setting, specificity in the time-to-event response context is called dynamic specificity. It is the probability that a model correctly predicts that individual $i$ does not experience an event before or at time $t$. It is defined as: + +$$ +\text{specificity}^{\mathbb{D}}(c,t) = p(\hat{N}_i(t) = 0 | {N}_i(t) =0 ) = p(q_i(t) \leq c| X_i > t ) +$$ + +Note that there is no distinction between prevalence and incidence in the context of specificity. This is because if we condition on $dN_i(t) = 0$, it implies either the individual had an event before $t$ and is thus excluded, or the individual has an event after $t$ which is equivalent to conditioning on $N_i(t) = 0$. + +### Extension of the AUC for time-to-event response + +#### Cumulative/dynamic AUC +If the risk score measures prevalence, a cumulative/dynamic ROC curve can be defined on the cumulative sensitivity against 1 - dynamic specificity. The area under the cumulative/dynamic ROC curve is called the cumulative/dynamic AUC (AUC C/D). It represents the probability that the model correctly identifies which of two comparable samples experiences an event before or at time $t$ based on their predicted risk scores. It is defined as: + +$$ +\begin{align*} +AUC^{\mathbb{C}/\mathbb{D}}(t) &= p(q_i(t)>q_j(t)|N_i(t) = 1, N_j(t) = 0) \\ +&= p(q_i(t)>q_j(t)|X_i \leq t, X_j >t) +\end{align*} +$$ + +The proof of the probabilitic interpretation of $AUC^{\mathbb{C}/\mathbb{D}}$ is similar to that in the binary response context. + +#### Incident/dynamic AUC +If the risk score measures incidence, a incident/dynamic ROC curve can be defined on the incident sensitivity against 1 - dynamic specificity. The area under the incident/dynamic ROC curve is called the incident/dynamic AUC (AUC I/D). It represents the probability that the model correctly identifies which of two comparable samples experiences an event at time $t$ based on their predicted risk scores. It is defined as: + +$$ +\begin{align*} +\text{AUC}^{\mathbb{I}/\mathbb{D}}(t) &= p(q_i(t)>q_j(t)|dN_i(t) = 1, N_j(t) = 0) \\ +& = p(q_i(t)>q_j(t)|X_i = t, X_j >t) +\end{align*} +$$ + +The proof of the probabilitic interpretation of $AUC^{\mathbb{I}/\mathbb{D}}$ is similar to that in the binary response context. + +### Extension of the C-index for time-to-event response + +The C-index is the probability that a model correctly predict which of two comparable samples experiences an event first based on their predicted risk scores. + +$$ +C = p(q_i(X_i) > q_j(X_i) | X_i < X_j) +$$ + +It can be shown that the C-index is related to the incidence/dynamic AUC, specifically $C = \int_t \text{AUC}^{\mathbb{I}/\mathbb{D}}(t) g(t) dt$ where $g(t) = 2f(t)S(t)$, $f(t)$ is the time-to-event densitity and $S(t)$ is the survival function. + +
+ Proof of the relation between the C-index and the AUC I/D +
+ + +$$ +\begin{align} +C &= p(q_i(X_i) > q_j(X_i) | X_i < X_j) \\ +&= \frac{p(q_i(X_i) > q_j(X_i) , X_i < X_j)}{p(X_i < X_j)} \\ +&= 2 p(q_i(X_i) > q_j(X_i) , X_i < X_j) \\ +&= 2 \int_t p(q_i(X_i) > q_j(X_i) , X_i = t, X_j > t) dt \\ +&= 2 \int_t p(q_i(X_i) > q_j(X_i) | X_i = t, X_j > t) p(X_i = t, X_j > t) dt \\ +&= 2 \int_t \text{AUC}^{\mathbb{I}/\mathbb{D}}(t) p(X_i = t) p(X_j > t) dt \\ +&= 2 \int_t \text{AUC}^{\mathbb{I}/\mathbb{D}}(t) f(t) S(t) dt \\ +\end{align} +$$ +where in the third line we used the fact that $p(X_i < X_j) = 1/2$ by independence, and in the sixth line, we again relied on the independence of the response. +
+ + + +## Discussion + +### How to estimate the AUC and C-index? +In other documentation, we presented the [estimators for the AUC]((../_autosummary/torchsurv.metrics.auc.html)) and the [estimators for the C-index](../_autosummary/torchsurv.metrics.cindex.html). In particular, these estimator needs to account for censored time-to-event data. + +### What risk score should be used? +As explained previously, the risk score can either measure incidence or prevalence. An obvious candidate for measuring incidence is the instantaneous hazard, while the cumulative hazard is suitable for measuring prevalence. The choice of risk score depends on the question of interest and what the model should be good at predicting. For example, if the question of interest is how many deaths occur by time $t$, the risk score chosen should measure prevalence, and the evaluation metrics should be the AUC C/D. + +### What evaluation metric should be used? + +Choosing between AUC I/D and AUC C/D depends on the type of risk score, as previously explained. However, the question remains whether to use the AUC or the C-index. The AUCs are time-dependent and provide a measure of predictive performance at specific time points. In contrast, the concordance index (C-index) offers a global assessment of a fitted survival model over the entire observational period. It is recommended to use AUC instead of the concordance index for time-dependent predictions (e.g., 10-year mortality), as AUC is proper in this context, while the concordance index is not (Blanche et al., 2018). + +Below, we summarize the pros and cons of the C-index, AUC I/D, and AUC C/D (Lambert and Chevret, 2016). + +| | C-index | AUC cumulative/dynamic| AUC incident/dynamic| +|----------|----------|----------|----------| +| Pros | Global measure that does not depend on time | More clinically relevant, where one wishes to predict individuals who will have failed by time $t$.
Proper for time-dependent prediction. | Integral has probabilistic interpretation of the C-index.
Natural companion for hazard models.
Proper for time-dependent prediction. | +| Cons | Not proper for time-dependent prediction (e.g., 10-years mortality). | Integral does not have a nice probabilistic interpretation.
Requires a cumulative risk score (e.g., 1-survival function or cumulative hazard). | Require the exact event times to be recorded.
All subjects that failed before $t$ are ignored.| + +### What are the limits of the AUC and the C-index? + +There are three main limits of the AUC and C-index (Hartman et al., 2023): + +* **Ties in predicted risk score (C-index)**: +A model that assigns the same risk score to individuals with extremely different survival experiences is not performing well, but the C-Index does not detect this inadequacy. +On the other hand, a model that assigns the same risk score to individuals with very similar survival experiences may be appropriately capturing the similarities in underlying risk. +While these two scenarios have very different interpretations, they are treated as the same in the C-Index calculation, since each of these pairs contributes a score of 0.5. + +* **Time dependence (AUC and C-index)**: +Time-to-event outcome is dichotomized at each time point, which potentially generates many comparable pairs that are difficult to discriminate and are not clinically meaningful. +For example, if a individual experiences the event of interest on day t of the study, and another individual, with similar underlying risk, experiences the event shortly afterwards on day t + 1, then these two individuals would be deemed comparable according to the time-dependent C-Index definition. + +* **Discrimination (AUC and C-index)**: +The concordance index disregards the actual values of predicted risk scores – it is a ranking metric – and is unable to tell anything about calibration. + + +## The Brier-Score as an alternative to the AUC and C-index + +### Brier-score and Integrated Brier Score + +Suppose that the probabilistic survival model can output for any individual $i$ an estimate of the survival function $P(X_i > t)$, denoted by $\xi_i(t) \in [0,1]$. The Time-dependent Brier Score (BS) and integrated Brier Score (Graf et al., 1999) are defined as + +$$ +\text{BS}(t) = \mathbb{E}\big[\big(1(X > t) - \xi(t)\big)^2\big], +$$ + +$$ +\text{IBS} = \int_0^{t_{\text{max}}} \text{BS}(t) dW(t), +$$ +where $W(t) = t / t_{\text{max}}$ and $t_{\text{max}}$ is the time-point until which we want to restrict the average. + +### Discrimination (AUC, C-index) vs. calibration (Brier score) measure + +The Brier score assesses both calibration and discrimination and serves as an alternative to the C-index and AUC. However, it is limited to models capable of estimating a survival function, thereby excluding models like the Cox proportional hazards model. Below, we outline the strengths and weaknesses of the C-index and AUC compared to the Brier score. + +| | C-index and AUC | Brier-Score | +|----------|----------|----------| +| Pros | Extremely popular.
Simple to explain to clinical audience.
User-friendly scale (from 0.5 to 1).|Reflects both calibration and discrimination.
Can support the conclusions from a graphical calibration curve. | +| Cons | Already discussed. | Only applicable for models that can estimate a survival function.
Inadequate for very rare (or very frequent) events.
Less interpretable by clinicians.| + +## Conclusion in practice + +A meta-analysis by Zhou et al. (2022) recorded the proportion of evaluation metrics for survival models used from 2010 to 2021 in prominent journals such as "Annals of Statistics," "Biometrika," "Journal of the American Statistical Association," "Journal of the Royal Statistical Society, Series B," "Statistics in Medicine," "Artificial Intelligence in Medicine," and "Lifetime Data Analysis." This analysis indicates that the use of the C-index has been steadily increasing and has recently become a dominant predictive measure. In 2021, the C-index was used in more than 75% of the readings. + + +## Reference + * Heagerty, P. J., & Zheng, Y. (2005). Survival Model Predictive Accuracy and ROC Curves. In Biometrics (Vol. 61, Issue 1, pp. 92–105). Oxford University Press (OUP). https://doi.org/10.1111/j.0006-341x.2005.030814.x + * Lambert, J., & Chevret, S. (2016). Summary measure of discrimination in survival models based on cumulative/dynamic time-dependent ROC curves. In Statistical Methods in Medical Research (Vol. 25, Issue 5, pp. 2088–2102). SAGE Publications + * Hartman, N., Kim, S., He, K., & Kalbfleisch, J. D. (2023). Pitfalls of the concordance index for survival outcomes. In Statistics in Medicine (Vol. 42, Issue 13, pp. 2179–2190). Wiley. https://doi.org/10.1002/sim.9717 + * Graf, E., Schmoor, C., Sauerbrei, W., & Schumacher, M. (1999). Assessment and comparison of prognostic classification schemes for survival data. In Statistics in Medicine (Vol. 18, Issues 17–18, pp. 2529–2545). Wiley. https://doi.org/10.1002/(sici)1097-0258(19990915/30)18:17/18<2529::aid-sim274>3.0.co;2-5 +* Hanpu Zhou; Hong Wang; Sizheng Wang; Yi Zou (2022). SurvMetrics: An R package for Predictive Evaluation Metrics in Survival Analysis. R Journal . Dec2022, Vol. 14 Issue 4, p1-12. 12p. +* Paul Blanche, Michael W Kattan, and Thomas A Gerds. The c-index is not proper for the evaluation of t-year predicted risks. Biostatistics, 20(2):347–357, February 2018. From 09d0655d7aadb1720dd450fe4cf22d1d10685f75 Mon Sep 17 00:00:00 2001 From: Thibaud Coroller <14895385+tcoroller@users.noreply.github.com> Date: Wed, 3 Jul 2024 14:37:29 -0400 Subject: [PATCH 2/8] fixed warning and code QC (#41) --- src/torchsurv/metrics/auc.py | 6 ------ src/torchsurv/metrics/brier_score.py | 27 +++++++++++++-------------- src/torchsurv/metrics/cindex.py | 8 ++++++++ tests/test_cox.py | 4 ---- tests/test_kaplan_meier.py | 7 +++---- tests/test_momentum.py | 1 - tests/utils.py | 9 --------- 7 files changed, 24 insertions(+), 38 deletions(-) diff --git a/src/torchsurv/metrics/auc.py b/src/torchsurv/metrics/auc.py index 14eb256..b5a6a98 100644 --- a/src/torchsurv/metrics/auc.py +++ b/src/torchsurv/metrics/auc.py @@ -1192,7 +1192,6 @@ def _find_torch_unique_indices( def _validate_auc_inputs( estimate, time, auc_type, new_time, weight, weight_new_time ): - # check new_time and weight are provided, weight_new_time should be provided if all([new_time is not None, weight is not None, weight_new_time is None]): raise ValueError( @@ -1222,12 +1221,10 @@ def _update_auc_new_time( weight: torch.tensor, weight_new_time: torch.tensor, ) -> torch.tensor: - # update new time if ( new_time is not None ): # if new_time are specified: ensure it has the correct format - # ensure that new_time are float if isinstance(new_time, int): new_time = torch.tensor([new_time]).float() @@ -1237,7 +1234,6 @@ def _update_auc_new_time( new_time = new_time.unsqueeze(0) else: # else: find new_time - # if new_time are not specified, use unique event time mask = event & (time < torch.max(time)) new_time, inverse_indices, counts = torch.unique( @@ -1261,7 +1257,6 @@ def _update_auc_new_time( def _update_auc_estimate( estimate: torch.tensor, new_time: torch.tensor ) -> torch.tensor: - # squeeze estimate if shape = (n_samples, 1) if estimate.ndim == 2 and estimate.shape[1] == 1: estimate = estimate.squeeze(1) @@ -1281,7 +1276,6 @@ def _update_auc_weight( weight: torch.tensor, weight_new_time: torch.tensor, ) -> torch.tensor: - # if weight was not specified, weight of 1 if weight is None: weight = torch.ones_like(time) diff --git a/src/torchsurv/metrics/brier_score.py b/src/torchsurv/metrics/brier_score.py index fd3f7a1..e50ebe1 100644 --- a/src/torchsurv/metrics/brier_score.py +++ b/src/torchsurv/metrics/brier_score.py @@ -171,10 +171,13 @@ def __call__( ) # update inputs as required - estimate, new_time, weight, weight_new_time = ( - BrierScore._update_brier_score_new_time( - estimate, time, new_time, weight, weight_new_time - ) + ( + estimate, + new_time, + weight, + weight_new_time, + ) = BrierScore._update_brier_score_new_time( + estimate, time, new_time, weight, weight_new_time ) weight, weight_new_time = BrierScore._update_brier_score_weight( time, new_time, weight, weight_new_time @@ -190,14 +193,14 @@ def __call__( # Calculating the residuals for each subject and time point residuals = torch.zeros_like(estimate) - for i, t in enumerate(new_time): - est = estimate[:, i] - is_case = ((time <= t) & (event)).int() - is_control = (time > t).int() + for index, new_time_i in enumerate(new_time): + est = estimate[:, index] + is_case = ((time <= new_time_i) & (event)).int() + is_control = (time > new_time_i).int() - residuals[:, i] = ( + residuals[:, index] = ( torch.square(est) * is_case * weight - + torch.square(1.0 - est) * is_control * weight_new_time[i] + + torch.square(1.0 - est) * is_control * weight_new_time[index] ) # Calculating the brier scores at each time point @@ -827,7 +830,6 @@ def _validate_brier_score_inputs( weight: torch.tensor, weight_new_time: torch.tensor, ) -> torch.tensor: - # check new_time and weight are provided, weight_new_time should be provided if all([new_time is not None, weight is not None, weight_new_time is None]): raise ValueError( @@ -859,7 +861,6 @@ def _update_brier_score_new_time( weight: torch.tensor, weight_new_time: torch.tensor, ) -> torch.tensor: - # check format of new_time if ( new_time is not None @@ -871,7 +872,6 @@ def _update_brier_score_new_time( new_time = new_time.unsqueeze(0) else: # else: find new_time - # if new_time are not specified, use unique time new_time, inverse_indices, counts = torch.unique( time, sorted=True, return_inverse=True, return_counts=True @@ -896,7 +896,6 @@ def _update_brier_score_weight( weight: torch.tensor, weight_new_time: torch.tensor, ) -> torch.tensor: - # if weight was not specified, weight of 1 if weight is None: weight = torch.ones_like(time) diff --git a/src/torchsurv/metrics/cindex.py b/src/torchsurv/metrics/cindex.py index 51d552d..9276a07 100644 --- a/src/torchsurv/metrics/cindex.py +++ b/src/torchsurv/metrics/cindex.py @@ -1,5 +1,6 @@ import copy import sys +import warnings from typing import Optional, Tuple import torch @@ -639,6 +640,13 @@ def _compare_noether(self, other): cindex1_se = self._concordance_index_se() cindex2_se = other._concordance_index_se() + # Suppress the specific warning + warnings.filterwarnings( + "ignore", + message="Metric `SpearmanCorrcoef` will save all targets and predictions in the buffer. For large datasets, this may lead to large memory footprint.", + category=UserWarning, + ) + # compute spearman correlation between risk prediction corr = regression.SpearmanCorrCoef()( self.estimate.reshape(-1), other.estimate.reshape(-1) diff --git a/tests/test_cox.py b/tests/test_cox.py index b8cdbfa..5766403 100644 --- a/tests/test_cox.py +++ b/tests/test_cox.py @@ -57,9 +57,7 @@ def test_boolean_y(self): def test_log_likelihood_without_ties(self): """test cox partial log likelihood without ties on lung and gbsg datasets""" for benchmark_cox_loglik in benchmark_cox_logliks: - if benchmark_cox_loglik["no_ties"][0] == True: - log_lik = -cox( torch.tensor( benchmark_cox_loglik["log_hazard"], dtype=torch.float32 @@ -82,9 +80,7 @@ def test_log_likelihood_without_ties(self): def test_log_likelihood_with_ties(self): """test Efron and Breslow's approximation of cox partial log likelihood with ties on lung and gbsg data""" for benchmark_cox_loglik in benchmark_cox_logliks: - if benchmark_cox_loglik["no_ties"][0] == False: - # efron approximation of partial log likelihood log_lik_efron = -cox( torch.tensor( diff --git a/tests/test_kaplan_meier.py b/tests/test_kaplan_meier.py index a7d555c..cfff910 100644 --- a/tests/test_kaplan_meier.py +++ b/tests/test_kaplan_meier.py @@ -27,7 +27,6 @@ def test_kaplan_meier_survival_distribution_real_data(self): """test Kaplan Meier survival distribution estimate on lung and gbsg datasets""" for benchmark_kaplan_meier in benchmark_kaplan_meiers: - event = torch.tensor(benchmark_kaplan_meier["status"]).bool() time = torch.tensor(benchmark_kaplan_meier["time"], dtype=torch.float32) new_time = torch.tensor( @@ -209,9 +208,9 @@ def test_kaplan_meier_prediction_error_raised(self): for batch in batch_container.batches: (train_time, train_event, test_time, *_) = batch - train_event[-1] = ( - False # if last event is censoring, the last KM is > 0 and it cannot predict beyond this time - ) + train_event[ + -1 + ] = False # if last event is censoring, the last KM is > 0 and it cannot predict beyond this time km = KaplanMeierEstimator() km(train_event, train_time, censoring_dist=False) diff --git a/tests/test_momentum.py b/tests/test_momentum.py index 6e3bdc7..7ee295f 100644 --- a/tests/test_momentum.py +++ b/tests/test_momentum.py @@ -13,7 +13,6 @@ class TestMometum(unittest.TestCase): - def test_momentum_weibull(self): model = Momentum( backbone=nn.Sequential(nn.Linear(8, 2)), # Weibull expect two ouputs diff --git a/tests/utils.py b/tests/utils.py index 3058d02..b0d21d8 100644 --- a/tests/utils.py +++ b/tests/utils.py @@ -279,7 +279,6 @@ def get_input_array(self) -> Tuple[np.array, np.array, np.array, np.array]: ) def _generate_input(self): - # random maximum time in observational period tmax = torch.randint(5, 500, (1,)).item() @@ -297,7 +296,6 @@ def _generate_input(self): self._generate_new_time() def _generate_data(self, tmax: int, n_train: int, n_test: int): - # time-to-event or censoring in train train_time = torch.randint(1, tmax + 1, (n_train,)).float() @@ -340,7 +338,6 @@ def _generate_data(self, tmax: int, n_train: int, n_test: int): def _enforce_conditions_data( self, time: torch.tensor, event: torch.tensor, dataset_type: str ) -> Tuple[torch.tensor, torch.tensor]: - # if test max time should be greater than train max time if dataset_type == "test": if self.test_max_time_gt_train_max_time: @@ -395,7 +392,6 @@ def _enforce_conditions_data( return time, event def _generate_estimate(self): - # random risk score for observations in test estimate = torch.randn(len(self.test_event)) @@ -403,7 +399,6 @@ def _generate_estimate(self): self.estimate = self._enforce_conditions_estimate(estimate) def _enforce_conditions_estimate(self, estimate: torch.tensor) -> torch.tensor: - # if there should be ties in risk score associated to patients with event if self.ties_score_events: estimate[torch.where(self.test_event == 1.0)[0][0]] = estimate[ @@ -425,7 +420,6 @@ def _enforce_conditions_estimate(self, estimate: torch.tensor) -> torch.tensor: return estimate def _generate_new_time(self): - if torch.all(self.test_event == False): # if all patients are censored in test, no evaluation time new_time = torch.tensor([]) @@ -447,7 +441,6 @@ def _generate_new_time(self): self.new_time = self._enforce_conditions_time(new_time) def _enforce_conditions_time(self, new_time: torch.tensor) -> torch.tensor: - # if the test max time should be included in evaluation time if self.test_max_time_in_new_time: new_time = torch.cat( @@ -457,7 +450,6 @@ def _enforce_conditions_time(self, new_time: torch.tensor) -> torch.tensor: return new_time def _evaluate_conditions(self): - # are there ties in event times self.has_train_ties_time_event = self._has_ties( self.train_time[self.train_event == 1] @@ -614,7 +606,6 @@ def generate_batches(self, n_batch: int, flags_to_set: list): n_batch = len(flags_to_set) for i in range(n_batch): - if i >= len(flags_to_set): # simulate data without flag self.generate_one_batch() From 1360a011a3722763149ca68b836262340c6e2d11 Mon Sep 17 00:00:00 2001 From: Thibaud Coroller <14895385+tcoroller@users.noreply.github.com> Date: Wed, 3 Jul 2024 15:31:30 -0400 Subject: [PATCH 3/8] Update README.md --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index d6d7c4a..0220da0 100644 --- a/README.md +++ b/README.md @@ -193,7 +193,7 @@ The table below compares the functionalities of `TorchSurv` with those of [deepsurv](https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/s12874-018-0482-1). While several libraries offer survival modelling functionalities, no existing library provides the flexibility to use a custom PyTorch-based neural networks to define the survival model parameters. -The outputs of both the log-likelihood functions and the evaluation metrics functions have undergone thorough comparison with benchmarks generated using Python packages and R packages. The comparisons are summarised in the [Related packages summary](https://opensource.nibr.com/torchsurv/benchmarks.html). +The outputs of both the log-likelihood functions and the evaluation metrics functions have **undergone thorough comparison with benchmarks generated** using `Python` and `R` packages. The comparisons (at time of publication) are summarised in the [Related packages summary](https://opensource.nibr.com/torchsurv/benchmarks.html). ![Survival analysis libraries in Python](docs/source/table_python_benchmark.png) ![Survival analysis libraries in Python](docs/source/table_python_benchmark_legend.png) From 993cba8114561d56d6d3073a8aec9b00d51acd4e Mon Sep 17 00:00:00 2001 From: corolth1 Date: Wed, 3 Jul 2024 15:37:46 -0400 Subject: [PATCH 4/8] R comp --- README.md | 7 ++++++- docs/source/table_r_benchmark.png | Bin 0 -> 272430 bytes 2 files changed, 6 insertions(+), 1 deletion(-) create mode 100644 docs/source/table_r_benchmark.png diff --git a/README.md b/README.md index 0220da0..48ee8f3 100644 --- a/README.md +++ b/README.md @@ -198,6 +198,11 @@ The outputs of both the log-likelihood functions and the evaluation metrics func ![Survival analysis libraries in Python](docs/source/table_python_benchmark.png) ![Survival analysis libraries in Python](docs/source/table_python_benchmark_legend.png) +Survival analysis libraries in R. For obtaining the evaluation metrics, packages `survival`, `riskRegression`, `SurvMetrics` and `pec` require the fitted model object as input (a specific object format) and `RisksetROC` imposes a smoothing method. Packages `timeROC`, `riskRegression` and pec force the user to choose a form for subject-specific +weights (e.g., inverse probability of censoring weighting (IPCW)). Packages `survcomp` and `SurvivalROC` do not implement the general AUC but the censoring-adjusted AUC estimator proposed by Heagerty et al. (2000). + +![Survival analysis libraries in R](docs/source/table_r_benchmark.png) + ## Contributing We value contributions from the community to enhance and improve this project. If you'd like to contribute, please consider the following: @@ -231,4 +236,4 @@ If you use this project in academic work or publications, we appreciate citing i primaryClass={cs.LG}, doi={https://doi.org/10.48550/arXiv.2404.10761} } -``` +``` \ No newline at end of file diff --git a/docs/source/table_r_benchmark.png b/docs/source/table_r_benchmark.png new file mode 100644 index 0000000000000000000000000000000000000000..d863fbd16254e2a78f9f496e2b9976f7092a67b7 GIT binary patch literal 272430 zcmdqJbyOVN(>96)3lM@6+y)3fxRc;cAR!PucyMD{$!mpt_p!Ac5}=%}wyVPIg;rKQA_VPKF`U|}b7z zi$xxX36~ul(!Gov_6+ysdqn&9A}#Oe!FcVy7D`I(LgaqFU+vk#~`(Cx?@B} zvYrz06z1pKVM?6B!M96#AIrJB-s`MVKQb9Xs6nCebSt$D{jV1N66vK3Hy$!G*Ss`7 z{V)*1M)Wgj9=2+TCF_~QK} zKzj^His4c3FLl{8at#Bdr&GmNpQIoV3*+ zr_}U}%qf@|$6so9>$G73@6=WLG9%$8@!>Qnrczy6Pbj9L91r-2@HrcME_J0%4Qa-6 zp%pBzHwmfKiR7G8t3&b^3ERmt*}&P|KALJo)HeY=j7wQAh@{Qf*Hkaizrr)<1`DW? z6F3{$*clJfSu5hNqqqc1t8*aQ_SJBJGA-)98kt&}qBr-ar}JZ39StmsOTBE$!c9lg 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0000000..40b1b79 --- /dev/null +++ b/paper/paper.md @@ -0,0 +1,107 @@ +--- +title: '\texttt{TorchSurv}: A Lightweight Package for Deep Survival Analysis' +tags: + - Python + - radiomics + - medical imaging +authors: + - name: Alex Zwanenburg + orcid: 0000-0002-0342-9545 + affiliation: "1, 2" + - name: Steffen Löck + orcid: 0000-0002-7017-3738 + affiliation: "2, 3, 4" +affiliations: + - name: "National Center for Tumor Diseases Dresden (NCT/UCC), Germany: German Cancer Research Center (DKFZ), Heidelberg, Germany; Faculty of Medicine and University Hospital Carl Gustav Carus, TUD Dresden University of Technology, Dresden, Germany; Helmholtz-Zentrum Dresden-Rossendorf (HZDR), Dresden, Germany" + index: 1 + - name: OncoRay – National Center for Radiation Research in Oncology, Faculty of Medicine and University Hospital Carl + Gustav Carus, TUD Dresden University of Technology, Helmholtz-Zentrum Dresden-Rossendorf, Dresden, Germany + index: 2 + - name: German Cancer Consortium (DKTK), Partner Site Dresden, and German Cancer Research Center (DKFZ), Heidelberg, Germany + index: 3 + - name: Department of Radiotherapy and Radiation Oncology, Faculty of Medicine and University Hospital Carl Gustav Carus, TUD Dresden University of Technology, Dresden, Germany + index: 4 +date: 7 February 2024 +bibliography: paper.bib +--- + +# Summary + +The forces on stars, galaxies, and dark matter under external gravitational +fields lead to the dynamical evolution of structures in the universe. The orbits +of these bodies are therefore key to understanding the formation, history, and +future state of galaxies. The field of "galactic dynamics," which aims to model +the gravitating components of galaxies to study their structure and evolution, +is now well-established, commonly taught, and frequently used in astronomy. +Aside from toy problems and demonstrations, the majority of problems require +efficient numerical tools, many of which require the same base code (e.g., for +performing numerical orbit integration). + +# Statement of need + +`Gala` is an Astropy-affiliated Python package for galactic dynamics. Python +enables wrapping low-level languages (e.g., C) for speed without losing +flexibility or ease-of-use in the user-interface. The API for `Gala` was +designed to provide a class-based and user-friendly interface to fast (C or +Cython-optimized) implementations of common operations such as gravitational +potential and force evaluation, orbit integration, dynamical transformations, +and chaos indicators for nonlinear dynamics. `Gala` also relies heavily on and +interfaces well with the implementations of physical units and astronomical +coordinate systems in the `Astropy` package [@astropy] (`astropy.units` and +`astropy.coordinates`). + +`Gala` was designed to be used by both astronomical researchers and by +students in courses on gravitational dynamics or astronomy. It has already been +used in a number of scientific publications [@Pearson:2017] and has also been +used in graduate courses on Galactic dynamics to, e.g., provide interactive +visualizations of textbook material [@Binney:2008]. The combination of speed, +design, and support for Astropy functionality in `Gala` will enable exciting +scientific explorations of forthcoming data releases from the *Gaia* mission +[@gaia] by students and experts alike. + +# Mathematics + +Single dollars ($) are required for inline mathematics e.g. $f(x) = e^{\pi/x}$ + +Double dollars make self-standing equations: + +$$\Theta(x) = \left\{\begin{array}{l} +0\textrm{ if } x < 0\cr +1\textrm{ else} +\end{array}\right.$$ + +You can also use plain \LaTeX for equations +\begin{equation}\label{eq:fourier} +\hat f(\omega) = \int_{-\infty}^{\infty} f(x) e^{i\omega x} dx +\end{equation} +and refer to \autoref{eq:fourier} from text. + +# Citations + +Citations to entries in paper.bib should be in +[rMarkdown](http://rmarkdown.rstudio.com/authoring_bibliographies_and_citations.html) +format. + +If you want to cite a software repository URL (e.g. something on GitHub without a preferred +citation) then you can do it with the example BibTeX entry below for @fidgit. + +For a quick reference, the following citation commands can be used: +- `@author:2001` -> "Author et al. (2001)" +- `[@author:2001]` -> "(Author et al., 2001)" +- `[@author1:2001; @author2:2001]` -> "(Author1 et al., 2001; Author2 et al., 2002)" + +# Figures + +Figures can be included like this: +![Caption for example figure.\label{fig:example}](figure.png) +and referenced from text using \autoref{fig:example}. + +Figure sizes can be customized by adding an optional second parameter: +![Caption for example figure.](figure.png){ width=20% } + +# Acknowledgements + +We acknowledge contributions from Brigitta Sipocz, Syrtis Major, and Semyeong +Oh, and support from Kathryn Johnston during the genesis of this project. + +# References From 95d70f388ddf1d82e4d77930c429e1a18375c156 Mon Sep 17 00:00:00 2001 From: melodiemonod Date: Thu, 18 Jul 2024 17:36:48 +0100 Subject: [PATCH 6/8] template paper --- paper/paper.bib | 59 +++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 59 insertions(+) create mode 100644 paper/paper.bib diff --git a/paper/paper.bib b/paper/paper.bib new file mode 100644 index 0000000..72e3977 --- /dev/null +++ b/paper/paper.bib @@ -0,0 +1,59 @@ +@article{Pearson:2017, + url = {http://adsabs.harvard.edu/abs/2017arXiv170304627P}, + Archiveprefix = {arXiv}, + Author = {{Pearson}, S. and {Price-Whelan}, A.~M. and {Johnston}, K.~V.}, + Eprint = {1703.04627}, + Journal = {ArXiv e-prints}, + Keywords = {Astrophysics - Astrophysics of Galaxies}, + Month = mar, + Title = {{Gaps in Globular Cluster Streams: Pal 5 and the Galactic Bar}}, + Year = 2017 +} + +@book{Binney:2008, + url = {http://adsabs.harvard.edu/abs/2008gady.book.....B}, + Author = {{Binney}, J. and {Tremaine}, S.}, + Booktitle = {Galactic Dynamics: Second Edition, by James Binney and Scott Tremaine.~ISBN 978-0-691-13026-2 (HB).~Published by Princeton University Press, Princeton, NJ USA, 2008.}, + Publisher = {Princeton University Press}, + Title = {{Galactic Dynamics: Second Edition}}, + Year = 2008 +} + +@article{gaia, + author = {{Gaia Collaboration}}, + title = "{The Gaia mission}", + journal = {Astronomy and Astrophysics}, + archivePrefix = "arXiv", + eprint = {1609.04153}, + primaryClass = "astro-ph.IM", + keywords = {space vehicles: instruments, Galaxy: structure, astrometry, parallaxes, proper motions, telescopes}, + year = 2016, + month = nov, + volume = 595, + doi = {10.1051/0004-6361/201629272}, + url = {http://adsabs.harvard.edu/abs/2016A%26A...595A...1G}, +} + +@article{astropy, + author = {{Astropy Collaboration}}, + title = "{Astropy: A community Python package for astronomy}", + journal = {Astronomy and Astrophysics}, + archivePrefix = "arXiv", + eprint = {1307.6212}, + primaryClass = "astro-ph.IM", + keywords = {methods: data analysis, methods: miscellaneous, virtual observatory tools}, + year = 2013, + month = oct, + volume = 558, + doi = {10.1051/0004-6361/201322068}, + url = {http://adsabs.harvard.edu/abs/2013A%26A...558A..33A} +} + +@misc{fidgit, + author = {A. M. Smith and K. Thaney and M. Hahnel}, + title = {Fidgit: An ungodly union of GitHub and Figshare}, + year = {2020}, + publisher = {GitHub}, + journal = {GitHub repository}, + url = {https://github.com/arfon/fidgit} +} \ No newline at end of file From 8a9292d7f9727db0d8a8f0d7d50d2747df853036 Mon Sep 17 00:00:00 2001 From: melodiemonod Date: Thu, 18 Jul 2024 17:39:15 +0100 Subject: [PATCH 7/8] Revert "template paper" This reverts commit 95d70f388ddf1d82e4d77930c429e1a18375c156. --- paper/paper.bib | 59 ------------------------------------------------- 1 file changed, 59 deletions(-) delete mode 100644 paper/paper.bib diff --git a/paper/paper.bib b/paper/paper.bib deleted file mode 100644 index 72e3977..0000000 --- a/paper/paper.bib +++ /dev/null @@ -1,59 +0,0 @@ -@article{Pearson:2017, - url = {http://adsabs.harvard.edu/abs/2017arXiv170304627P}, - Archiveprefix = {arXiv}, - Author = {{Pearson}, S. and {Price-Whelan}, A.~M. and {Johnston}, K.~V.}, - Eprint = {1703.04627}, - Journal = {ArXiv e-prints}, - Keywords = {Astrophysics - Astrophysics of Galaxies}, - Month = mar, - Title = {{Gaps in Globular Cluster Streams: Pal 5 and the Galactic Bar}}, - Year = 2017 -} - -@book{Binney:2008, - url = {http://adsabs.harvard.edu/abs/2008gady.book.....B}, - Author = {{Binney}, J. and {Tremaine}, S.}, - Booktitle = {Galactic Dynamics: Second Edition, by James Binney and Scott Tremaine.~ISBN 978-0-691-13026-2 (HB).~Published by Princeton University Press, Princeton, NJ USA, 2008.}, - Publisher = {Princeton University Press}, - Title = {{Galactic Dynamics: Second Edition}}, - Year = 2008 -} - -@article{gaia, - author = {{Gaia Collaboration}}, - title = "{The Gaia mission}", - journal = {Astronomy and Astrophysics}, - archivePrefix = "arXiv", - eprint = {1609.04153}, - primaryClass = "astro-ph.IM", - keywords = {space vehicles: instruments, Galaxy: structure, astrometry, parallaxes, proper motions, telescopes}, - year = 2016, - month = nov, - volume = 595, - doi = {10.1051/0004-6361/201629272}, - url = {http://adsabs.harvard.edu/abs/2016A%26A...595A...1G}, -} - -@article{astropy, - author = {{Astropy Collaboration}}, - title = "{Astropy: A community Python package for astronomy}", - journal = {Astronomy and Astrophysics}, - archivePrefix = "arXiv", - eprint = {1307.6212}, - primaryClass = "astro-ph.IM", - keywords = {methods: data analysis, methods: miscellaneous, virtual observatory tools}, - year = 2013, - month = oct, - volume = 558, - doi = {10.1051/0004-6361/201322068}, - url = {http://adsabs.harvard.edu/abs/2013A%26A...558A..33A} -} - -@misc{fidgit, - author = {A. M. Smith and K. Thaney and M. Hahnel}, - title = {Fidgit: An ungodly union of GitHub and Figshare}, - year = {2020}, - publisher = {GitHub}, - journal = {GitHub repository}, - url = {https://github.com/arfon/fidgit} -} \ No newline at end of file From fa1d306da463327e7e17f6bea3c119fb13f01395 Mon Sep 17 00:00:00 2001 From: melodiemonod Date: Thu, 18 Jul 2024 17:40:07 +0100 Subject: [PATCH 8/8] Revert "template paper" This reverts commit dc59caa6fa3c0b283262c2e5ba19a963ec5572b4. --- paper/paper.md | 107 ------------------------------------------------- 1 file changed, 107 deletions(-) delete mode 100644 paper/paper.md diff --git a/paper/paper.md b/paper/paper.md deleted file mode 100644 index 40b1b79..0000000 --- a/paper/paper.md +++ /dev/null @@ -1,107 +0,0 @@ ---- -title: '\texttt{TorchSurv}: A Lightweight Package for Deep Survival Analysis' -tags: - - Python - - radiomics - - medical imaging -authors: - - name: Alex Zwanenburg - orcid: 0000-0002-0342-9545 - affiliation: "1, 2" - - name: Steffen Löck - orcid: 0000-0002-7017-3738 - affiliation: "2, 3, 4" -affiliations: - - name: "National Center for Tumor Diseases Dresden (NCT/UCC), Germany: German Cancer Research Center (DKFZ), Heidelberg, Germany; Faculty of Medicine and University Hospital Carl Gustav Carus, TUD Dresden University of Technology, Dresden, Germany; Helmholtz-Zentrum Dresden-Rossendorf (HZDR), Dresden, Germany" - index: 1 - - name: OncoRay – National Center for Radiation Research in Oncology, Faculty of Medicine and University Hospital Carl - Gustav Carus, TUD Dresden University of Technology, Helmholtz-Zentrum Dresden-Rossendorf, Dresden, Germany - index: 2 - - name: German Cancer Consortium (DKTK), Partner Site Dresden, and German Cancer Research Center (DKFZ), Heidelberg, Germany - index: 3 - - name: Department of Radiotherapy and Radiation Oncology, Faculty of Medicine and University Hospital Carl Gustav Carus, TUD Dresden University of Technology, Dresden, Germany - index: 4 -date: 7 February 2024 -bibliography: paper.bib ---- - -# Summary - -The forces on stars, galaxies, and dark matter under external gravitational -fields lead to the dynamical evolution of structures in the universe. The orbits -of these bodies are therefore key to understanding the formation, history, and -future state of galaxies. The field of "galactic dynamics," which aims to model -the gravitating components of galaxies to study their structure and evolution, -is now well-established, commonly taught, and frequently used in astronomy. -Aside from toy problems and demonstrations, the majority of problems require -efficient numerical tools, many of which require the same base code (e.g., for -performing numerical orbit integration). - -# Statement of need - -`Gala` is an Astropy-affiliated Python package for galactic dynamics. Python -enables wrapping low-level languages (e.g., C) for speed without losing -flexibility or ease-of-use in the user-interface. The API for `Gala` was -designed to provide a class-based and user-friendly interface to fast (C or -Cython-optimized) implementations of common operations such as gravitational -potential and force evaluation, orbit integration, dynamical transformations, -and chaos indicators for nonlinear dynamics. `Gala` also relies heavily on and -interfaces well with the implementations of physical units and astronomical -coordinate systems in the `Astropy` package [@astropy] (`astropy.units` and -`astropy.coordinates`). - -`Gala` was designed to be used by both astronomical researchers and by -students in courses on gravitational dynamics or astronomy. It has already been -used in a number of scientific publications [@Pearson:2017] and has also been -used in graduate courses on Galactic dynamics to, e.g., provide interactive -visualizations of textbook material [@Binney:2008]. The combination of speed, -design, and support for Astropy functionality in `Gala` will enable exciting -scientific explorations of forthcoming data releases from the *Gaia* mission -[@gaia] by students and experts alike. - -# Mathematics - -Single dollars ($) are required for inline mathematics e.g. $f(x) = e^{\pi/x}$ - -Double dollars make self-standing equations: - -$$\Theta(x) = \left\{\begin{array}{l} -0\textrm{ if } x < 0\cr -1\textrm{ else} -\end{array}\right.$$ - -You can also use plain \LaTeX for equations -\begin{equation}\label{eq:fourier} -\hat f(\omega) = \int_{-\infty}^{\infty} f(x) e^{i\omega x} dx -\end{equation} -and refer to \autoref{eq:fourier} from text. - -# Citations - -Citations to entries in paper.bib should be in -[rMarkdown](http://rmarkdown.rstudio.com/authoring_bibliographies_and_citations.html) -format. - -If you want to cite a software repository URL (e.g. something on GitHub without a preferred -citation) then you can do it with the example BibTeX entry below for @fidgit. - -For a quick reference, the following citation commands can be used: -- `@author:2001` -> "Author et al. (2001)" -- `[@author:2001]` -> "(Author et al., 2001)" -- `[@author1:2001; @author2:2001]` -> "(Author1 et al., 2001; Author2 et al., 2002)" - -# Figures - -Figures can be included like this: -![Caption for example figure.\label{fig:example}](figure.png) -and referenced from text using \autoref{fig:example}. - -Figure sizes can be customized by adding an optional second parameter: -![Caption for example figure.](figure.png){ width=20% } - -# Acknowledgements - -We acknowledge contributions from Brigitta Sipocz, Syrtis Major, and Semyeong -Oh, and support from Kathryn Johnston during the genesis of this project. - -# References