HyperAI超神经

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 $\text{[math]}$ comes from an exponential family distribution, then the mean ${\mu}$ of the distribution can be explained by the independent $\text{[math]}$ at that point:

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

Among them, ${E{ \left( {y} \right) }}$ is the expected value of $\text{[math]}$ , ${X \beta }$ is the linear estimation formula composed of the unknown to-be-estimated parameters ${\beta }$ and the known variables $\text{[math]}$ , and $\text{[math]}$ is the link function.

In this mode, the variance $\text{[math]}$ of $\text{[math]}$ can be expressed as:

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

where $\text{[math]}$ can be viewed as a function of an exponential random variable, and the unknown parameter ${\beta }$ 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 $\text{[math]}$ from the exponential family.

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

3. The link function $\text{[math]}$ such that ${E{ \left( {y} \right) }\text{ }=\text{ } \mu \text{ }=\text{ }g\mathop{{}}\nolimits^{{-1}}{ \left( {\eta } \right) }}$ .

References

【1】Generalized linear model - Wikipedia