Generalized Linear Model

Extensions for the classical Linear Regression / Linear Model.

$g\left(E_Y(y\mid x)\right)={\beta }_0+{\beta }_1{x}_1+\dots {\beta }_p{x}_p$

In Logistic Regression we for example assume a Bernoulli Distribution and use the Logit Function as the link function.

The probability function can be from the Exponential Family of Distribution functions. The link function tries to link the Linear Regression output with some parameters from the chosen Distribution (e.g. Mean and Variance).

• Categorical (Bernoulli Distribution)
• Count (Poisson Distribution)
• Duration (Exponential Distribution)

Trained using Maximum Likelihood Estimation.

Example

Logistic Regression is a GLM with Bernoulli Distribution and Logistic Function as link function.

One more example where we chose the Poisson Distribution to not get negative values.

Interpretation of the weights

Example where the Poisson Distribution was used with the log link function. $\ln (E(\mathrm{coffee}\mid \mathrm{str},\mathrm{slp},\mathrm{wrk}))={\beta }_0+{\beta }_{\mathrm{str}}{x}_{\mathrm{str}}+{\beta }_{\mathrm{slp}}{x}_{\mathrm{slp}}+{\beta }_{\mathrm{wrk}}{x}_{\mathrm{wrk}}$ Apply the inverse of the link function to get the effect on the actual expected outcome. $E(\mathrm{coffee}\mid \mathrm{str},\mathrm{slp},\mathrm{wrk})=\exp \left({\beta }_0+{\beta }_{\mathrm{str}}{x}_{\mathrm{str}}+{\beta }_{\mathrm{slp}}{x}_{\mathrm{slp}}+{\beta }_{\mathrm{wrk}}{x}_{\mathrm{wrk}}\right)$