GLM (Generalized linear model) and LR (logistic regression) detailed

GLM Generalized linear model

George Box said: "All models is wrong, some is useful" 1. Starting with the Linear Model

As a foundation of GLM, this section review the classic Linear Regression, and expounds some basic terms.
The basic formula for our linear regression is essentially to look at x x and then predict Y y with a simple linear function h (x) H (x):
Y=h (x) =wtx y=h (x) =w^tx

1.1 dependent variable y y

This is our forecast target, also called response variable. There is an easily confusing point, in fact Y y can express three meanings (modeled by distribution, observed by sampling, predicted by expectation): distribution; When response variable is discussed abstractly, We are actually concerned about the given data and parameters, y| X,w y|\ x,w the distribution of obedience. Linear Regression y y obeys the Gaussian distribution, the specific value is the real number, but here we are concerned about the distribution. observed outcome; Our label, sometimes denoted by T-T, is the true observed result, just a value. Expected outcome; Y=e[y|x]=h (x) y=\mathbf{e}[y|x]=h (x) represents the prediction of the model; Note y y actually obeys a distribution, but the predicted result is the mean value of the entire distribution μ\mu, just a value. 1.2 Independent variable x x

This is our feature, which can contain many dimensions, a feature also known as a predictor. 1.3 hypothesis H (x) H (x)

The assumption of the linear model is very simple, i.e. h (x) =wtx h (x) = W^tx inner product of the weight vector and feature vector, known as linear predictor. This is the linear model, and GLM is also a generalization of the base.
In depth, the various dimension features (Predictor) XJ x_j through the coefficients WJ w_j linear Plus, this process integrates the information, while the different weight (coefficient) reflect the different contribution of the relevant characteristics. 2. Extended to generalized Linear Model 2.1 Motive & Definition

The linear model has a very strong limitation, that is, response variable y y must obey the Gaussian distribution; The main limitation is that the scale of the fitted target y y is a real number (−∞,+∞) (-\infty,+\infty). In particular, there are two problems: the range of values for Y Y and some common problems do not match. For example, count (number of visitors is constant) and binary (a two classification problem) the variance of Y Y is constant constant. Some problems may depend on the mean value of Y Y, for example, I predict the more generous the target value (the less accurate predictions)

So then we use the generalized Linear Model to overcome these two problems.
Define GLM as a sentence (from wiki):
In statistics, the generalized linear Model (GLM) was a flexible generalization of ordinary linear regression that allows F Or response variables that has the error distribution models other than a normal distribution.
in detail, we can decompose GLM into Random Component, System Component, and Link Function three parts . 2.2 Random Component

An exponential family model for the response

Here refers to the response variable must obey a exponential family distribution exponential family distribution, namely Y|x,w