Generalized Linear ModelsIt is a flexible linear regression model that allows the dependent variable to have a distribution form other than the normal distribution.

definition

The generalized linear model is an extension of simple least squares regression. Assuming that each data observation $latex {Y}$latexY comes from an exponential family distribution, then the mean $latex {\mu}$latexμ of the distribution can be explained by the independent $latex {X}$latexX at that point:

$latex {E{ \left( {y} \right) }\text{ }=\text{ } \mu \text{ }=\text{ }g\mathop{{}}\nolimits^{{-1}}{ \left( {X \beta } \right) }}$latexE(y) = μ = g−1(Xβ)

Among them, $latex {E{ \left( {y} \right) }}$latexE(y) is the expected value of $latex {y}$latexy, $latex {X \beta }$latexXβ is the linear estimation formula composed of the unknown to-be-estimated parameters $latex {\beta }$latexβ and the known variables $latex {X}$latexX, and $latex {g}$latexg is the link function.

In this mode, the variance $latex {y}$latexy of $latex {V}$latexV can be expressed as:

$latex {Var{ \left( {y} \right) }\text{ }=\text{ }V{ \left( { \mu } \right) }\text{ }=\text{ }V{ \left( {g\mathop{{}}\nolimits^{{-1}}{ \left( {X \beta } \right) }} \right) }}$latexVar(y) = V(μ) = V(g−1(Xβ))

where $latex {V}$latexV can be viewed as a function of an exponential random variable, and the unknown parameter $latex {\beta }$latexβ is usually estimated using the maximum likelihood estimator, the almost maximum likelihood estimator, or the Bayesian method.

Model composition

The generalized linear model consists of the following main parts:

1. Distribution function $latex {f}$latexf from the exponential family.

2. Linear predictor $latex { \eta \text{ }=\text{ }X \beta }$latexη = Xβ .

3. The link function $latex {g}$latexg such that $latex {E{ \left( {y} \right) }\text{ }=\text{ } \mu \text{ }=\text{ }g\mathop{{}}\nolimits^{{-1}}{ \left( {\eta } \right) }}$latexE(y) = μ = g−1(η) .

References

【1】Generalized linear model - Wikipedia