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<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>Introduction to Generalized Additive Models with R and mgcv</title>
<meta charset="utf-8" />
<meta name="author" content="Gavin Simpson" />
<link href="libs/remark-css/default.css" rel="stylesheet" />
<link rel="stylesheet" href="https://stackpath.bootstrapcdn.com/bootstrap/4.3.1/css/bootstrap.min.css" type="text/css" />
<link rel="stylesheet" href="slides.css" type="text/css" />
</head>
<body>
<textarea id="source">
class: inverse, middle, left, my-title-slide, title-slide
# Introduction to Generalized Additive Models with R and mgcv
### Gavin Simpson
### 1000–1230 CST (1600–1830 UTC) July 30th, 2020
---
class: inverse middle center big-subsection
# Welcome
???
I'm live streaming today from Treaty 4 lands. These are the territories of the nêhiyawak (nay-hi-yuh-wuk, Cree), Anihšināpēk (uh-nish-i-naa-payk, Saulteaux), Dakota, Lakota, Nakoda, and the homeland of the Métis/Michif Nation. Today, these lands continue to be the shared territory of many diverse peoples.
---
# Logistics
## Slides
Slidedeck: [bit.ly/gam-webinar](https://bit.ly/gam-webinar)
Sources: [bit.ly/gam-webinar-git](https://bit.ly/gam-webinar-git)
Direct download a ZIP of everything: [bit.ly/gam-webinar-zip](https://bit.ly/gam-webinar-zip)
Unpack the zip & remember where you put it
## Q & A
Add questions to Google Doc: [bit.ly/gam-webinar-qa](https://bit.ly/gam-webinar-qa)
## Recording
Livestream will be recorded &mdash; [youtu.be/sgw4cu8hrZM](https://youtu.be/sgw4cu8hrZM)
---
# Donation
In lieu of not charging for this webinar, if you are able to, financially, please make a donation to the University of Regina's **Student Emergency Fund**
<https://giving.uregina.ca/student-emergency-fund>
---
# Today's topics
* What are GAMs?
* How to fit GAMs in R with **mgcv**
* Model Checking
* Model Diagnostics
* Examples
---
class: inverse middle center subsection
# Motivating example
---
# HadCRUT4 time series
<!-- -->
???
Hadley Centre NH temperature record ensemble
How would you model the trend in these data?
---
# Linear Models
`$$y_i \sim \mathcal{N}(\mu_i, \sigma^2)$$`
`$$\mu_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + \cdots + \beta_j x_{ji}$$`
Assumptions
1. linear effects of covariates are good approximation of the true effects
2. conditional on the values of covariates, `\(y_i | \mathbf{X} \sim \mathcal{N}(0, \sigma^2)\)`
3. this implies all observations have the same *variance*
4. `\(y_i | \mathbf{X}\)` are *independent*
An **additive** model address the first of these
---
class: inverse center middle subsection
# Why bother with anything more complex?
---
# Is this linear?
<!-- -->
---
# Polynomials perhaps&hellip;
<!-- -->
---
# Polynomials perhaps&hellip;
We can keep on adding ever more powers of `\(\boldsymbol{x}\)` to the model &mdash; model selection problem
**Runge phenomenon** &mdash; oscillations at the edges of an interval &mdash; means simply moving to higher-order polynomials doesn't always improve accuracy
---
class: inverse middle center subsection
# GAMs offer a solution
---
# HadCRUT data set
```r
library('readr')
library('dplyr')
URL <- "https://bit.ly/hadcrutv4"
gtemp <- read_delim(URL, delim = ' ', col_types = 'nnnnnnnnnnnn', col_names = FALSE) %>%
select(num_range('X', 1:2)) %>% setNames(nm = c('Year', 'Temperature'))
```
[File format](https://www.metoffice.gov.uk/hadobs/hadcrut4/data/current/series_format.html)
---
# HadCRUT data set
```r
gtemp
```
```
## # A tibble: 171 x 2
## Year Temperature
## <dbl> <dbl>
## 1 1850 -0.336
## 2 1851 -0.159
## 3 1852 -0.107
## 4 1853 -0.177
## 5 1854 -0.071
## 6 1855 -0.19
## 7 1856 -0.378
## 8 1857 -0.405
## 9 1858 -0.4
## 10 1859 -0.215
## # … with 161 more rows
```
---
# Fitting a GAM
```r
library('mgcv')
m <- gam(Temperature ~ s(Year), data = gtemp, method = 'REML')
summary(m)
```
.smaller[
```
##
## Family: gaussian
## Link function: identity
##
## Formula:
## Temperature ~ s(Year)
##
## Parametric coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.025327 0.009847 -2.572 0.011 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Approximate significance of smooth terms:
## edf Ref.df F p-value
## s(Year) 7.784 8.62 138.5 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## R-sq.(adj) = 0.875 Deviance explained = 88.1%
## -REML = -89.311 Scale est. = 0.016581 n = 171
```
]
---
# Fitted GAM
<!-- -->
---
class: inverse middle center big-subsection
# GAMs
---
# Generalized Additive Models
<br />
.references[Source: [GAMs in R by Noam Ross](https://noamross.github.io/gams-in-r-course/)]
???
GAMs are an intermediate-complexity model
* can learn from data without needing to be informed by the user
* remain interpretable because we can visualize the fitted features
---
# How is a GAM different?
In LM we model the mean of data as a sum of linear terms:
`$$y_i = \beta_0 +\sum_j \color{red}{ \beta_j x_{ji}} +\epsilon_i$$`
A GAM is a sum of _smooth functions_ or _smooths_
`$$y_i = \beta_0 + \sum_j \color{red}{s_j(x_{ji})} + \epsilon_i$$`
where `\(\epsilon_i \sim N(0, \sigma^2)\)`, `\(y_i \sim \text{Normal}\)` (for now)
Call the above equation the **linear predictor** in both cases
---
# Fitting a GAM in R
```r
model <- gam(y ~ s(x1) + s(x2) + te(x3, x4), # formuala describing model
data = my_data_frame, # your data
method = 'REML', # or 'ML'
family = gaussian) # or something more exotic
```
`s()` terms are smooths of one or more variables
`te()` terms are the smooth equivalent of *main effects + interactions*
---
# How did `gam()` *know*?
<!-- -->
---
class: inverse
background-image: url('./resources/rob-potter-398564.jpg')
background-size: contain
# What magic is this?
.footnote[
<a style="background-color:black;color:white;text-decoration:none;padding:4px 6px;font-family:-apple-system, BlinkMacSystemFont, &quot;San Francisco&quot;, &quot;Helvetica Neue&quot;, Helvetica, Ubuntu, Roboto, Noto, &quot;Segoe UI&quot;, Arial, sans-serif;font-size:12px;font-weight:bold;line-height:1.2;display:inline-block;border-radius:3px;" href="https://unsplash.com/@robpotter?utm_medium=referral&amp;utm_campaign=photographer-credit&amp;utm_content=creditBadge" target="_blank" rel="noopener noreferrer" title="Download free do whatever you want high-resolution photos from Rob Potter"><span style="display:inline-block;padding:2px 3px;"><svg xmlns="http://www.w3.org/2000/svg" style="height:12px;width:auto;position:relative;vertical-align:middle;top:-1px;fill:white;" viewBox="0 0 32 32"><title></title><path d="M20.8 18.1c0 2.7-2.2 4.8-4.8 4.8s-4.8-2.1-4.8-4.8c0-2.7 2.2-4.8 4.8-4.8 2.7.1 4.8 2.2 4.8 4.8zm11.2-7.4v14.9c0 2.3-1.9 4.3-4.3 4.3h-23.4c-2.4 0-4.3-1.9-4.3-4.3v-15c0-2.3 1.9-4.3 4.3-4.3h3.7l.8-2.3c.4-1.1 1.7-2 2.9-2h8.6c1.2 0 2.5.9 2.9 2l.8 2.4h3.7c2.4 0 4.3 1.9 4.3 4.3zm-8.6 7.5c0-4.1-3.3-7.5-7.5-7.5-4.1 0-7.5 3.4-7.5 7.5s3.3 7.5 7.5 7.5c4.2-.1 7.5-3.4 7.5-7.5z"></path></svg></span><span style="display:inline-block;padding:2px 3px;">Rob Potter</span></a>
]
---
class: inverse
background-image: url('resources/wiggly-things.png')
background-size: contain
???
---
# Wiggly things
.center[]
???
GAMs use splines to represent the non-linear relationships between covariates, here `x`, and the response variable on the `y` axis.
---
# Basis expansions
In the polynomial models we used a polynomial basis expansion of `\(\boldsymbol{x}\)`
* `\(\boldsymbol{x}^0 = \boldsymbol{1}\)` &mdash; the model constant term
* `\(\boldsymbol{x}^1 = \boldsymbol{x}\)` &mdash; linear term
* `\(\boldsymbol{x}^2\)`
* `\(\boldsymbol{x}^3\)`
* &hellip;
---
# Splines
Splines are *functions* composed of simpler functions
Simpler functions are *basis functions* & the set of basis functions is a *basis*
When we model using splines, each basis function `\(b_k\)` has a coefficient `\(\beta_k\)`
Resultant spline is a the sum of these weighted basis functions, evaluated at the values of `\(x\)`
`$$s(x) = \sum_{k = 1}^K \beta_k b_k(x)$$`
---
# Splines formed from basis functions
<!-- -->
???
Splines are built up from basis functions
Here I'm showing a cubic regression spline basis with 10 knots/functions
We weight each basis function to get a spline. Here all the basisi functions have the same weight so they would fit a horizontal line
---
# Weight basis functions &#8680; spline
.center[]
???
But if we choose different weights we get more wiggly spline
Each of the splines I showed you earlier are all generated from the same basis functions but using different weights
---
# How do GAMs learn from data?
<!-- -->
???
How does this help us learn from data?
Here I'm showing a simulated data set, where the data are drawn from the orange functions, with noise. We want to learn the orange function from the data
---
# Maximise penalised log-likelihood &#8680; &beta;
.center[]
???
Fitting a GAM involves finding the weights for the basis functions that produce a spline that fits the data best, subject to some constraints
---
class: inverse middle center subsection
# Avoid overfitting our sample
---
class: inverse center middle large-subsection
# How wiggly?
$$
\int_{\mathbb{R}} [f^{\prime\prime}]^2 dx = \boldsymbol{\beta}^{\mathsf{T}}\mathbf{S}\boldsymbol{\beta}
$$
---
class: inverse center middle large-subsection
# Penalised fit
$$
\mathcal{L}_p(\boldsymbol{\beta}) = \mathcal{L}(\boldsymbol{\beta}) - \frac{1}{2} \lambda\boldsymbol{\beta}^{\mathsf{T}}\mathbf{S}\boldsymbol{\beta}
$$
---
# Wiggliness
`$$\int_{\mathbb{R}} [f^{\prime\prime}]^2 dx = \boldsymbol{\beta}^{\mathsf{T}}\mathbf{S}\boldsymbol{\beta} = \large{W}$$`
(Wiggliness is 100% the right mathy word)
We penalize wiggliness to avoid overfitting
---
# Making wiggliness matter
`\(W\)` measures **wiggliness**
(log) likelihood measures closeness to the data
We use a **smoothing parameter** `\(\lambda\)` to define the trade-off, to find
the spline coefficients `\(B_k\)` that maximize the **penalized** log-likelihood
`$$\mathcal{L}_p = \log(\text{Likelihood}) - \lambda W$$`
---
# HadCRUT4 time series
<!-- -->
---
# Picking the right wiggliness
.pull-left[
Two ways to think about how to optimize `\(\lambda\)`:
* Predictive: Minimize out-of-sample error
* Bayesian: Put priors on our basis coefficients
]
.pull-right[
Many methods: AIC, Mallow's `\(C_p\)`, GCV, ML, REML
* **Practically**, use **REML**, because of numerical stability
* Hence `gam(..., method='REML')`
]
.center[
]
---
# Maximum allowed wiggliness
We set **basis complexity** or "size" `\(k\)`
This is _maximum wigglyness_, can be thought of as number of small functions that make up a curve
Once smoothing is applied, curves have fewer **effective degrees of freedom (EDF)**
EDF < `\(k\)`
---
# Maximum allowed wiggliness
`\(k\)` must be *large enough*, the `\(\lambda\)` penalty does the rest
*Large enough* &mdash; space of functions representable by the basis includes the true function or a close approximation to the tru function
Bigger `\(k\)` increases computational cost
In **mgcv**, default `\(k\)` values are arbitrary &mdash; after choosing the model terms, this is the key user choice
**Must be checked!** &mdash; `gam.check()`
---
# GAM summary so far
1. GAMs give us a framework to model flexible nonlinear relationships
2. Use little functions (**basis functions**) to make big functions (**smooths**)
3. Use a **penalty** to trade off wiggliness/generality
4. Need to make sure your smooths are **wiggly enough**
---
class: middle center inverse subsection
# Example
---
# Portugese larks
.row[
.col-7[
```r
library('gamair')
data(bird)
bird <- transform(bird,
crestlark = factor(crestlark),
linnet = factor(linnet),
e = x / 1000,
n = y / 1000)
head(bird)
```
```
## QUADRICULA TET crestlark linnet x y e n
## 13705 NG56 E <NA> <NA> 551000 4669000 551 4669
## 13710 NG56 J <NA> <NA> 553000 4669000 553 4669
## 13715 NG56 P <NA> <NA> 555000 4669000 555 4669
## 13720 NG56 U <NA> <NA> 557000 4669000 557 4669
## 13725 NG56 Z <NA> <NA> 559000 4669000 559 4669
## 13880 NG66 E <NA> <NA> 561000 4669000 561 4669
```
]
.col-5[
<!-- -->
]
]
---
# Portugese larks &mdash; binomial GAM
.row[
.col-6[
```r
crest <- gam(crestlark ~ s(e, n, k = 100),
data = bird,
family = binomial,
method = 'REML')
```
`\(s(e, n)\)` indicated by `s(e, n)` in the formula
Isotropic thin plate spline
`k` sets size of basis dimension; upper limit on EDF
Smoothness parameters estimated via REML
]
.col-6[
.smaller[
```r
summary(crest)
```
```
##
## Family: binomial
## Link function: logit
##
## Formula:
## crestlark ~ s(e, n, k = 100)
##
## Parametric coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -2.24184 0.07785 -28.8 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Approximate significance of smooth terms:
## edf Ref.df Chi.sq p-value
## s(e,n) 74.04 86.46 857.3 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## R-sq.(adj) = 0.234 Deviance explained = 25.9%
## -REML = 2499.8 Scale est. = 1 n = 6457
```
]
]
]
---
# Portugese larks &mdash; binomial GAM
Model checking with binary data is a pain &mdash; residuals look weird
Alternatively we can aggregate data at the `QUADRICULA` level & fit a binomial count model
.smaller[
```r
## convert back to numeric
bird <- transform(bird,
crestlark = as.numeric(as.character(crestlark)),
linnet = as.numeric(as.character(linnet)))
## some variables to help aggregation
bird <- transform(bird, tet.n = rep(1, nrow(bird)),
N = rep(1, nrow(bird)), stringsAsFactors = FALSE)
## set to NA if not surveyed
bird$N[is.na(as.vector(bird$crestlark))] <- NA
## aggregate
bird2 <- aggregate(data.matrix(bird), by = list(bird$QUADRICULA),
FUN = sum, na.rm = TRUE)
## scale by Quads aggregated
bird2 <- transform(bird2, e = e / tet.n, n = n / tet.n)
## fit binomial GAM
crest2 <- gam(cbind(crestlark, N - crestlark) ~ s(e, n, k = 100),
data = bird2, family = binomial, method = 'REML')
```
]
---
# Model checking
.pull-left[
.smaller[
```r
crest3 <- gam(cbind(crestlark, N - crestlark) ~
s(e, n, k = 100),
data = bird2, family = quasibinomial,
method = 'REML')
```
]
Model residuals don't look too bad
Bands of points due to integers
Some overdispersion &mdash; &phi; = 2.32
]
.pull-right[
```r
ggplot(data.frame(Fitted = fitted(crest2),
Resid = resid(crest2)),
aes(Fitted, Resid)) + geom_point()
```
<!-- -->
]
---
class: inverse middle center subsection
# A cornucopia of smooths
---
# A cornucopia of smooths
The type of smoother is controlled by the `bs` argument (think *basis*)
The default is a low-rank thin plate spline `bs = 'tp'`
Many others available
.row[
.col-6[
* Cubic splines `bs = 'cr'`
* P splines `bs = 'ps'`
* Cyclic splines `bs = 'cc'` or `bs = 'cp'`
* Adaptive splines `bs = 'ad'`
* Random effect `bs = 're'`
* Factor smooths `bs = 'fs'`
]
.col-6[
* Duchon splines `bs = 'ds'`
* Spline on the sphere `bs = 'sos'`
* MRFs `bs = 'mrf'`
* Soap-film smooth `bs = 'so'`
* Gaussian process `bs = 'gp'`
]
]
---
# A bestiary of conditional distributions
A GAM is just a fancy GLM
Simon Wood & colleagues (2016) have extended the *mgcv* methods to some non-exponential family distributions
.row[
.col-6[
* `binomial()`
* `poisson()`
* `Gamma()`
* `inverse.gaussian()`
* `nb()`
* `tw()`
* `mvn()`
* `multinom()`
]
.col-6[
* `betar()`
* `scat()`
* `gaulss()`
* `ziplss()`
* `twlss()`
* `cox.ph()`
* `gamals()`
* `ocat()`
]
]
---
# Smooth interactions
Two ways to fit smooth interactions
1. Bivariate (or higher order) thin plate splines
* `s(x, z, bs = 'tp')`
* Isotropic; single smoothness parameter for the smooth
* Sensitive to scales of `x` and `z`
2. Tensor product smooths
* Separate marginal basis for each smooth, separate smoothness parameters
* Invariant to scales of `x` and `z`
* Use for interactions when variables are in different units
* `te(x, z)`
---
# Tensor product smooths
There are multiple ways to build tensor products in *mgcv*
1. `te(x, z)`
2. `t2(x, z)`
3. `s(x) + s(z) + ti(x, z)`
`te()` is the most general form but not usable in `gamm4::gamm4()` or *brms*
`t2()` is an alternative implementation that does work in `gamm4::gamm4()` or *brms*
`ti()` fits pure smooth interactions; where the main effects of `x` and `z` have been removed from the basis
---
# Factor smooth interactions
Two ways for factor smooth interactions
1. `by` variable smooths
* entirely separate smooth function for each level of the factor
* each has it's own smoothness parameter
* centred (no group means) so include factor as a fixed effect
* `y ~ f + s(x, by = f)`
2. `bs = 'fs'` basis
* smooth function for each level of the function
* share a common smoothness parameter
* fully penalized; include group means
* closer to random effects
* `y ~ s(x, f, bs = 'fs')`
---
# Random effects
When fitted with REML or ML, smooths can be viewed as just fancy random effects
Inverse is true too; random effects can be viewed as smooths
If you have simple random effects you can fit those in `gam()` and `bam()` without needing the more complex GAMM functions `gamm()` or `gamm4::gamm4()`
These two models are equivalent
```r
m_nlme <- lme(travel ~ 1, data = Rail, ~ 1 | Rail, method = "REML")
m_gam <- gam(travel ~ s(Rail, bs = "re"), data = Rail, method = "REML")
```
---
# Random effects
The random effect basis `bs = 're'` is not as computationally efficient as *nlme* or *lme4* for fitting
* complex random effects terms, or
* random effects with many levels
Instead see `gamm()` and `gamm4::gamm4()`
* `gamm()` fits using `lme()`
* `gamm4::gamm4()` fits using `lmer()` or `glmer()`
For non Gaussian models use `gamm4::gamm4()`
---
class: inverse center middle subsection
# Model checking
---
# Model checking
So you have a GAM:
- How do you know you have the right degrees of freedom? `gam.check()`
- Diagnosing model issues: `gam.check()` part 2
---
# GAMs are models too
How accurate your predictions will be depends on how good the model is
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---
class: inverse center middle subsection
# How do we test how well our model fits?
---
# Simulated data
```r
set.seed(2)
n <- 400
x1 <- rnorm(n)
x2 <- rnorm(n)
y_val <- 1 + 2*cos(pi*x1) + 2/(1+exp(-5*(x2)))
y_norm <- y_val + rnorm(n, 0, 0.5)
y_negbinom <- rnbinom(n, mu = exp(y_val),size=10)
y_binom <- rbinom(n,1,prob = exp(y_val)/(1+exp(y_val)))
```
---
# Simulated data
<!-- -->
---
class: inverse middle center subsection
# gam.check() part 1: do you have the right functional form?
---
# How well does the model fit?
- Many choices: k, family, type of smoother, &hellip;
- How do we assess how well our model fits?
---
# Basis size *k*
- Set `k` per term
- e.g. `s(x, k=10)` or `s(x, y, k=100)`
- Penalty removes "extra" wigglyness
- *up to a point!*
- (But computation is slower with bigger `k`)
---
# Checking basis size
```r
norm_model_1 <- gam(y_norm~s(x1, k = 4) + s(x2, k = 4), method = 'REML')
gam.check(norm_model_1)
```
```
##
## Method: REML Optimizer: outer newton
## full convergence after 8 iterations.
## Gradient range [-0.0003467788,0.0005154578]
## (score 736.9402 & scale 2.252304).
## Hessian positive definite, eigenvalue range [0.000346021,198.5041].
## Model rank = 7 / 7
##
## Basis dimension (k) checking results. Low p-value (k-index<1) may
## indicate that k is too low, especially if edf is close to k'.
##
## k' edf k-index p-value
## s(x1) 3.00 1.00 0.13 <2e-16 ***
## s(x2) 3.00 2.91 1.04 0.83
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
---
# Checking basis size
```r
norm_model_2 <- gam(y_norm ~ s(x1, k = 12) + s(x2, k = 4), method = 'REML')
gam.check(norm_model_2)
```
```
##
## Method: REML Optimizer: outer newton
## full convergence after 11 iterations.
## Gradient range [-5.658609e-06,5.392657e-06]
## (score 345.3111 & scale 0.2706205).
## Hessian positive definite, eigenvalue range [0.967727,198.6299].
## Model rank = 15 / 15
##
## Basis dimension (k) checking results. Low p-value (k-index<1) may
## indicate that k is too low, especially if edf is close to k'.
##
## k' edf k-index p-value
## s(x1) 11.00 10.84 0.99 0.38
## s(x2) 3.00 2.98 0.86 0.01 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
---
# Checking basis size
```r
norm_model_3 <- gam(y_norm ~ s(x1, k = 12) + s(x2, k = 12),method = 'REML')
gam.check(norm_model_3)
```
```
##
## Method: REML Optimizer: outer newton
## full convergence after 8 iterations.
## Gradient range [-1.136192e-08,6.812328e-13]
## (score 334.2084 & scale 0.2485446).
## Hessian positive definite, eigenvalue range [2.812271,198.6868].
## Model rank = 23 / 23
##
## Basis dimension (k) checking results. Low p-value (k-index<1) may
## indicate that k is too low, especially if edf is close to k'.
##
## k' edf k-index p-value
## s(x1) 11.00 10.85 0.98 0.31
## s(x2) 11.00 7.95 0.95 0.15
```
---
# Checking basis size
<!-- -->
---
class: inverse middle center subsection
# Model diagnostics
---
class: inverse middle center subsection
# Using gam.check() part 2: visual checks
---
# gam.check() plots