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<!DOCTYPE html>
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<head>
<title>What are GAMs?</title>
<meta charset="utf-8" />
<meta name="author" content="Eric Pedersen (with material heavily borrowed from David Miller)" />
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class: inverse, middle, left, my-title-slide, title-slide
# What are GAMs?
### Eric Pedersen (with material heavily borrowed from David Miller)
### May 31th, 2021
---
## Overview
- A very quick refresher on GLMs
- What is a GAM?
- How do GAMs work? (*Roughly*)
- What is smoothing?
- Fitting and plotting simple models
---
# A (very fast) refresher on GLMs
---
## What is a Generalized Linear model (GLM)?
Models that look like:
`\(y_i \sim Some\ distribution(\mu_i, \sigma_i)\)`
`\(link(\mu_i) = Intercept + \beta_1\cdot x_{1i} + \beta_2\cdot x_{2i} + \ldots\)`
---
## What is a Generalized Linear model (GLM)?
Models that look like:
`\(y_i \sim Some\ distribution(\mu_i, \sigma_i)\)`
`\(link(\mu_i) = Intercept + \beta_1\cdot x_{1i} + \beta_1\cdot x_{2i} + \ldots\)`
<br />
The average value of the response, `\(\mu_i\)`, assumed to be a linear combination of the covariates, `\(x_{ji}\)`, with an offset
---
## What is a Generalized Linear model (GLM)?
Models that look like:
`\(y_i \sim Some\ distribution(\mu_i, \sigma_i)\)`
`\(link(\mu_i) = Intercept + \beta_1\cdot x_{1i} + \beta_1\cdot x_{2i} + \ldots\)`
<br />
The model is fit (not really...) by maximizing the log-likelihood:
`\(\text{maximize} \sum_{i=1}^n logLik (Some\ distribution(y_i))\)`
`\(\text{ with respect to } Intercept, \ \beta_1,\ \beta_2, \ ...\)`
---
## With normally distributed data (for continuous unbounded data):
`\(y_i = Normal(\mu_i , \sigma_i)\)`
`\(Identity(\mu_i) = Intercept + \beta_1\cdot x_{1i} + \beta_1\cdot x_{2i} + \ldots\)`
<!-- -->
---
## With Poisson-distributed data (for count data):
`\(y_i = Poisson(\mu_i)\)`
`\(\text{ln}(\mu_i) = Intercept + \beta_1\cdot x_{1i} + \beta_1\cdot x_{2i} + \ldots\)`
<!-- -->
---
# Why bother with anything more complicated?
---
## Is this linear?
<!-- -->
---
## Is this linear? Maybe?
```r
lm(y ~ x1, data=dat)
```
```
## `geom_smooth()` using formula 'y ~ x'
```
<!-- -->
---
# What is a GAM?
The Generalized additive model assumes that `\(link(\mu_i)\)` is the sum of some *nonlinear* functions of the covariates
`\(y_i \sim Some\ distribution(\mu_i, \sigma_i)\)`
`\(link(\mu_i) = Intercept + f_1(x_{1i}) + f_2(x_{2i}) + \ldots\)`
<br />
--
But it is much easier to fit *linear* functions than nonlinear functions, so GAMs use a trick:
1. Transform each predictor variable into several new variables, called basis functions
2. Create nonlinear functions as linear sums of those basis functions
---
<!-- -->
<!-- -->
---
<!-- -->
---
#This means that writing a GAM in code is as simple as:
```r
mod <- gam(y~s(x,k=10),data=dat)
```
---
# You've seen basis functions before:
```r
glm(y ~ I(x) + I(x^2) + I(x^3) +...)
```
--
Polynomials are one type of basis function!
--
... But not a good one.
<!-- -->
---
# One of the most common types of smoother are cubic splines
(We won't get into the details about how these are defined)
```r
glm(y ~ ns(x,df = 4))
```
--
But even cubic splines can overfit:
<!-- -->
---
# How do we prevent overfitting?
The second key part of fitting GAMs: penalizing overly wiggly functions
We want functions that fit our data well, but do not overfit: that is, ones that are not too *wiggly*.
--
Remember from before:
`\(\text{maximize} \sum_{i=1}^n logLik (y_i)\)`
`\(\text{ with respect to } Intercept, \ \beta_1,\ \beta_2, \ ...\)`
---
# How do we prevent overfitting?
The second key part of fitting GAMs: penalizing overly wiggly functions
We want functions that fit our data well, but do not overfit: that is, ones that are not too *wiggly*.
We can modify this to add a *penalty* on the size of the model parameters:
`\(\text{maximize} \sum_{i=1}^n logLik (y_i) - \lambda\cdot \mathbf{\beta}'\mathbf{S}\mathbf{\beta}\)`
`\(= \text{maximize} \sum_{i=1}^n logLik (y_i) - \lambda\cdot \sum_{a=1}^{k}\sum_{b=1}^k \beta_a\cdot\beta_b\cdot P_{a,b}\)`
`\(\text{ with respect to } Intercept, \ \beta_1,\ \beta_2, \ ...\)`
---
# How do we prevent overfitting?
`\(\sum_{i=1}^n logLik (y_i) - \lambda\cdot \mathbf{\beta}'\mathbf{S}\mathbf{\beta}\)`
<br />
--
The penalty `\(\lambda\)` trades off between how well the model fits the observed data ( `\(\sum_{i=1}^n logLik (y_i)\)` ), and how wiggly the fitted function is ( `\(\mathbf{\beta}'\mathbf{S}\mathbf{\beta}\)` ).
<br />
--
The matrix `\(\mathbf{S}\)` measures how wiggly different function shapes are. Each type of smoother has its own penalty matrix; `mgcv` handles this.
<br />
--
Some combinations of parameters correspond to a penalty value of zero; these combinations are called the *null space* of the smoother
---
# For instance, for smoothing splines:
We can create a penalty matrix that penalizes the squared second derivative:
`\(\int_{x_1}^{x_n} [f^{\prime\prime}]^2 dx = \boldsymbol{\beta}^{\mathsf{T}}\mathbf{S}\boldsymbol{\beta}\)`
--
<!-- -->
---
<!-- -->
<!-- -->
---
# You've also (probably) already seen penalties before:
Single-level random effects are another type of smoother!
<!-- -->
---
# You've also (probably) already seen penalties before:
Single-level random effects are another type of smoother!
You've probably seen random effects written like:
`\(y_{i,j} =\alpha + \beta_j + \epsilon_i\)`, `\(\beta_i \sim Normal(0, \sigma_{\beta}^2)\)`
--
* The basis functions are the different levels of the discrete variable: `\(f_j(x_i)=1\)` if `\(x_i\)` is in group `\(j\)`, `\(f_j(x_i)=0\)` if not
--
* The `\(\beta_j\)` terms are the parameters the basis functions are being scaled by
--
* The variance of the random effect is equal to `\(1/\lambda\)`, so the `\(S\)` matrix for a random effect is just a diagonal matrix with `\(1/\sigma^2\)` on the diagonal
---
# How are the `\(\lambda\)` penalties fit?
* By default, `mgcv` uses Generalized Cross-Validation, but this tends to work well only with really large data sets
--
* We will use restricted maximum likelihood (REML) throughout this workshop for fitting GAMs
--
* This is the same REML you may have used when fitting random effects models; again, smoothers in GAMs are basically a random effect in a different hat
---
# To review:
* GAMs are like GLMs: they use link functions and likelihoods to model different types of data
--
* GAMs use linear combinations of basis functions to create nonlinear functions to predict data
--
* GAMs use penalty parameters, `\(\lambda\)`, to prevent overfitting the data; this trades off between how wiggly the function is and how well it fits the data
(measured by the likelihood)
--
* the penalty matrix for a given smooth, `\(\textbf{S}\)`, encodes how the shape of the function translates into the total size of the penalty
--
# But enough lecture; on to the live coding!
</textarea>
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