Generalized Additive Models (GAMs)

Traditional linear regression models assume a linear relationship between predictors and the response variable. However, many real-world phenomena exhibit non-linear, complex relationships. GAMs address this limitation by allowing for flexible modeling of these relationships through the use of smoothing functions. This makes GAMs a valuable tool for capturing patterns in data that linear models might miss.

A generalized additive model (GAM) is a generalized linear model in which the linear response variable depends linearly on unknown smooth functions of some predictor variables, and interest focuses on inference about these smooth functions.

Basic Components of a GAM

• Linear Predictors: GAMs include linear predictors, similar to traditional linear regression modelling, but they also incorporate additional components.
• Smooth Functions: GAMs employ smooth functions to capture non-linear relationships. These functions are typically spline functions or other types of smooth curves.
• Link Function: Like generalized linear models (GLMs), GAMs use a link function to relate the expected value of the response variable to the linear predictor.
• Additive Structure: GAMs are additive models, meaning that the contribution of each smooth function is additive, allowing for the modelling of complex relationships as a sum of simpler components.

Understanding GAMs

Its been known that any multivariate function could be represented as sums and compositions of univariate functions.

But they require highly complicated functions and thus are not suitable for modelling approaches. Therefore, GAMs dropped the outer sum and made sure the function belongs to simpler class.

where ???? is a smooth monotonic function. Writing g for the inverse of ????, this is traditionally written as

When this function is approximating the expectation of some observed quantity, it could be written as

This is the standard formulation of a GAM.

A versatile and effective statistical modeling method called a generalized additive model (GAM) expands the scope of linear regression to include non-linear interactions between variables. Generalized additive models (GAMs) are very helpful when analyzing complicated data that displays non-linear patterns, such as time series, and spatial data, or when the connections between predictors and the response variable are difficult to describe by straightforward linear functions. We’ll look at the basics of GAMs in this guide and show you how to use them in the R Programming Language.