The origin of the form of logistic formula from the generalized linear regression

The normal linear regression is in the form of: (The reason for this is that linearity is the linear meaning)

The linear regression model has several characteristics:

1.

2.x,y usually takes a continuous value

The distribution of 3.y is normal or close to normal.

The generalized linear model is extended as follows:

The 1.,h is a well known function for strictly monotone full smoothing. (Inverse function of h) is called the contact function.;

2.x,y can go to continuous or discrete values, discrete values are more common.

The distribution of 3.Y is generalized to exponential distribution, the normal state is its special case. The density form of y:

B (•), C (•) For known functions, for natural parameters, for additional parameters or scatter parameters.

At this point, it can be proved that B adds a little to represent the first derivative of B, and two points represents its second derivative.

(Y1,y2,y3,y4 ...) The Union distribution function (likelihood function) is:

Among them, because

So when the inverse function is exactly equal to H (h= b), the likelihood function has the simplest form:

Below we discuss the two classification (0-1,logic) issues:

For Y=f (x), the value of Y is only 0 1 problem,

Remember, the density expression of y is, to be written in exponential form, deduced, can be another (corresponding,),

So the density expression () has an exponential form:

。 Equivalent.

So

Are the simplest form we want.

At this point, this is the well-known logistic model.

In addition, the theorem can be verified,

, the mean value

Variance

Note: Most of the content is from the Zhang San Guo Teacher courseware.

The origin of the form of logistic formula from the generalized linear regression