The mathematical path-data analysis advanced-Generalized linear model

in theStatisticson,Generalized linear Model (Generalized linear Model) is a widely usedlinear regressionmode. This model assumes that the distribution function of the random variables measured by the experimenter and the systemic effects (i.e., non-random effects) in the experiment can bechain-knot function(link Function) to establish a function to explain its relevance.

The generalized linear model (generalized linear model, GLM) is an extension of the simple least squares regression (OLS), in the generalized linear pattern, assuming that the observed value of each data comes from an exponential family distribution. The average of the distribution can be explained by an X independent of that point:

The expected value is the linear estimator consisting of the unknown parameter and the known variable, then the link function.

In this mode, the variance can be expressed as:

General assumptions can be considered as a function of an exponential family of random variables .

Unknown parameters are usually estimated in the most approximate likelihood , almost as much as estimated, or in the Bayesian method .

chain-knot function [ edit ]

The link function explains the relationship between the linear predictor and the distribution expectation . The choice of the link function depends on the case. Usually as long as the value of the link function can contain the values of the distribution of the conditions.

When using the distribution of the regular parameter θ , the link function must conform to the condition that X^{TY is the sufficient statistic of β . This is true when θ is equal to the value of the link function of the linear predictor. The following is a list of the canonical chain functions and their inverse functions (sometimes called mean functions) for the distribution of exponential families:}

the canonical chain-knot function

Distribution	name	chain-knot function	mean value function
Normal state	Identical
Index	Countdown
Gamma
Inverse Gauss	Two-time Countdown
Poisson	Natural logarithm
Two-item	Logit
Polynomial

Generalized linear regression is suitable for the following 2 scenarios:

1. The conditional average of the dependent variable is a nonlinear function of the regression parameter
2, the dependent variable is non-normal distribution of data

The mathematical path-data analysis advanced-Generalized linear model