Generalized Linear Model

Extensions for the classical Linear Regression / Linear Model.

g(EY(y∣x))=β0+β1x1+…βpxpg\left(E_Y(y \mid x)\right)=\beta_0+\beta_1 x_1+\ldots \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).

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.

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Interpretation of the weights

Example where the Poisson Distribution was used with the log link function.

ln⁡(E(coffee⁡∣str⁡,slp,wrk))=β0+βstrxstr+βslpxslp+βwrkxwrk\ln (E(\operatorname{coffee} \mid \operatorname{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(coffee⁡∣str⁡,slp,wrk)=exp⁡(β0+βstrxstr+βslpxslp+βwrkxwrk)E(\operatorname{coffee} \mid \operatorname{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)