Generalized Additive Models

We can combine the notion of additive models with generalized linear models, to derive the notion of generalized additive models. In other words, the purpose of generalized additive models is to maximize the quality of prediction of a dependent variable Y from various distributions, by estimating unspecific (non-parametric) functions of the predictor variables which are “connected” to the dependent variable via a link function.

Additive Models
The methods described in this section represent a generalization of multiple regression (which is a special case ofgeneral linear models). Specifically, in linear regression, a linear least-squares fit is computed for a set of predictor or X variables, to predict a dependent Y variable. The well known linear regression equation with m predictors, to predict a dependent variable Y, can be stated as:

Y = b0 + b1*X1 + … + bm*Xm

Where Y stands for the (predicted values of the) dependent variable, X1through Xm represent the m values for the predictor variables, and b0, and b1 through bm are the regression coefficients estimated by multiple regression.

A generalization of the multiple regression model would be to maintain the additive nature of the model, but to replace the simple terms of the linear equation bi*Xi with fi(Xi) where fi is a non-parametric function of the predictor Xi. In other words, instead of a single coefficient for each variable (additive term) in the model, in additive models an unspecified (non-parametric) function is estimated for each predictor, to achieve the best prediction of the dependent variable values.

Generalized Linear Models

To summarize the basic idea, the generalized linear model differs from the general linear model (of which multiple regression is a special case) in two major respects: First, the distribution of the dependent or response variable can be (explicitly) non-normal, and does not have to be continuous, e.g., it can be binomial; second, the dependent variable values are predicted from a linear combination of predictor variables, which are “connected” to the dependent variable via a link function. The general linear model for a single dependent variable can be considered a special case of the generalized linear model: In the general linear model the dependent variable values are expected to follow the normal distribution, and the link function is a simple identity function (i.e., the linear combination of values for the predictor variables is not transformed).

To illustrate, in the general linear model a response variable Y is linearly associated with values on the X variables while the relationship in the generalized linear model is assumed to be

Y = g(b0 + b1*X1 + … + bm*Xm)
where g(…) is a function. Formally, the inverse function of g(…), say gi(…), is called the link function; so that:

gi(muY) = b0 + b1*X1 + … + bm*Xm
where mu-Y stands for the expected value of Y.