Generalized Additive Models

Overview

  • What is a GAM?
  • What is smoothing?
  • How do GAMs work?
  • Fitting GAMs using dsm

What is a GAM?

"gam"

  1. Collective noun used to refer to a group of whales, or rarely also of porpoises; a pod.
  2. (by extension) A social gathering of whalers (whaling ships).

(via Natalie Kelly, AAD. Seen in Moby Dick.)

Generalized Additive Models

  • Generalized: many response distributions
  • Additive: terms add together
  • Models: well, it's a model…

What does a model look like?

  • Count \( n_j \) distributed according to some count distribution
  • Model as sum of terms

plot of chunk sumterms

Mathematically...

Taking the previous example…

\[ n_j = \color{red}{A_j}\color{blue}{\hat{p}_j} \color{green}{\exp}\left[\color{grey}{ \beta_0 + s(\text{y}_j) + s(\text{Depth}_j)} \right] + \epsilon_j \]

where \( \epsilon_j \sim N(0, \sigma^2) \), \( \quad n_j\sim \) count distribution

  • \( \color{red}{\text{area of segment - offset}} \)
  • \( \color{blue}{\text{probability of detection in segment}} \)
  • \( \color{green}{\text{link function}} \)
  • \( \color{grey}{\text{model terms}} \)

Response

\[ \color{red}{n_j} = A_j\hat{p}_j \exp\left[ \beta_0 + s(\text{y}_j) + s(\text{Depth}_j) \right] + \epsilon_j \]
where \( \epsilon_j \sim N(0, \sigma^2) \), \( \quad \color{red}{n_j\sim \text{count distribution}} \)

Count distributions