About Generalized Linear Model

The Generalized Linear Model (GLM) includes and extends the class of linear models which address and accommodate some restrictive assumptions of the linear models.

Linear models make a set of restrictive assumptions, most importantly, that the target (dependent variable y) is normally distributed conditioned on the value of predictors with a constant variance regardless of the predicted response value. The advantage of linear models and their restrictions include computational simplicity, an interpretable model form, and the ability to compute certain diagnostic information about the quality of the fit.

GLM relaxes these restrictions, which are often violated in practice. For example, binary (yes/no or 0/1) responses do not have same variance across classes. Furthermore, the sum of terms in a linear model typically can have very large ranges encompassing very negative and very positive values. For the binary response example, we would like the response to be a probability in the range [0,1].

GLM accommodates responses that violate the linear model assumptions through two mechanisms: a link function and a variance function. The link function transforms the target range to potentially -infinity to +infinity so that the simple form of linear models can be maintained. The variance function expresses the variance as a function of the predicted response, thereby accommodating responses with non-constant variances (such as the binary responses).

Oracle Machine Learning for SQL includes two of the most popular members of the GLM family of models with their most popular link and variance functions:

• Linear regression with the identity link and variance function equal to the constant 1 (constant variance over the range of response values).

• Logistic regression

In other words, the methods of linear regression assume that the target value ranges from minus infinity to infinity and that the target variance is constant over the range. The logistic regression target is either 0 or 1. A logistic regression model estimate is a probability. The job of the link function in logistic regression is to transform the target value into the required range, minus infinity to infinity.

GLM Function	Default Link Function	Other Supported Link Functions
Linear regression (gaussian)	identity	none
Logistic regression (binomial)	logit	probit, cloglog, cauchit, and binomial variance