Annotated Equations

A quick reference for the models introduced in the R Tutorial.

Non-linear responses: GAM

A Generalized Additive Model (GAM) replaces the linear covariate effect with a smooth function, allowing a non-linear response.

\[ y_t = \mathrm{Pois}( \underbrace{\color{#d53f8c}{\lambda_t}}_{\textstyle\color{#d53f8c}{\text{rate parameter}}} ) \]

\[ \underbrace{\color{#d69e2e}{\log(\lambda_t)}}_{\textstyle\color{#d69e2e}{\text{log link}}} = \underbrace{\color{#2b6cb0}{c}}_{\textstyle\color{#2b6cb0}{\text{constant}}} \;+\; \underbrace{\color{#2f855a}{f(x_{1,t})}}_{\textstyle\color{#2f855a}{\text{smooth covariate effect}}} \;+\; \underbrace{\color{#c05621}{\beta_2 y_{t-1}}}_{\textstyle\color{#c05621}{\text{AR: lag-1}}} \]

• \(y_t\) — the observed count at time \(t\)
• \(\lambda_t\) — the Poisson rate parameter
• \(f(x_{1,t})\) — a smooth function of the covariate, estimated from the data rather than constrained to a straight line; s(mintemp) in the model formula
• \(c\), \(\beta_2 y_{t-1}\) — the same constant and AR terms as before

Note: \(f\) is estimated on the log scale, so view the response on the scale of abundance with plot_predictions().