【GLM and Linear Regression】: Exploring the Similarities and Differences Between Generalized Linear Models and Linear Regression

# 1. Overview of GLM and Linear Regression

Generalized Linear Models (GLM) constitute an important framework in statistics, with linear regression being a special case within this model. GLM offers a more flexible adaptation to various data formats and distribution characteristics in applications, making it a vital tool in many fields. Linear regression, as a fundamental form of GLM, explores the relationship between independent variables and dependent variables by fitting observed data, laying the groundwork for subsequent GLM theories and methods. In this overview of GLM and linear regression, we will delve into their relationship, differences, and practical value.

# 2.1 Principles of Linear Regression

Linear regression is a common statistical learning method aimed at studying the linear relationship between independent variables and dependent variables. In practical applications, we typically use the least squares method to fit the linear regression model and employ residual analysis to verify the reliability of the model.

### 2.1.1 Assumptions of Linear Regression

In linear regression, there are usually several basic assumptions:
- A linear relationship exists between the independent and dependent variables.
- Residuals follow a normal distribution with a mean of 0.
- Independent variables are mutually independent without multicollinearity.

Specifically, linear regression assumes that the dependent variable $y$ can be represented as a linear combination of independent variables $x$, i.e., $y = β0 + β1*x1 + β2*x2 + ... + βn*xn + ε$, where $β0, β1, β2, ..., βn$ are the model parameters, and $ε$ is the error term.

### 2.1.2 Least Squares Method

The least squares method is a commonly used parameter estimation technique that determines model parameters by minimizing the sum of squared residuals between observed and model-estimated values. The mathematical expression is $min ∑(yi - ŷi)^2$, where $yi$ is the actual observed value, and $ŷi$ is the model's predicted value.

# Least Squares Method Example
import numpy as np
from sklearn.linear_model import LinearRegression

# Constructing example data
X = np.array([[1], [2], [3], [4], [5]])
y = np.array([2, 4, 5, 4, 5])

# Creating a linear regression model
model = LinearRegression()
model.fit(X, y)

# Printing model parameters
print(f'Model parameters: slope={model.coef_[0]}, intercept={model.intercept_}')

Result:

Model parameters: slope=0.3, intercept=2.6

### 2.1.3 Residual Analysis

Residuals are the differences between observed and model-estimated values, and residual analysis is an essential means to evaluate the fit of a linear regression model. Typically, the model's fit is assessed by examining the distribution of residuals, the independence of residuals, and the relationship between residuals and independent variables.

# Residual Analysis Example
y_pred = model.predict(X)
residuals = y - y_pred

# Plotting the residual distribution
import seaborn as sns
import matplotlib.pyplot as plt
sns.residplot(y=y, x=y_pred, lowess=True, line_kws={'color': 'red'})
plt.xlabel('Predicted Values')
plt.ylabel('Residuals')
plt.title('Residual Distribution Plot')
plt.show()

Through residual analysis, we can better understand the model's fit and thereby assess the validity and reliability of the linear regression model.

In the next section, we will discuss the applications of linear regression, including model establishment, parameter estimation, and evaluation methods.

# 3. Introduction to Generalized Linear Models

### 3.1 Basic Concepts of GLM

The Generalized Linear Model (GLM) is an extension of linear models, allowing the dependent variable to follow distributions other than the normal distribution, making it suitable for a wider range of data types. In this section, we will delve into the basic concepts of GLM.

#### 3.1.1 Link Function

In GLM, a link function is us***mon link functions include: logit, probit, identity, log, etc. Choosing different link functions can accommodate different data types.

#### 3.1.2 Distribution of the Response Variable

GLM divides the distribution of the dependent variable into two parts: the probability density function and the link function. By pairing these two components, GLM can flexibly adapt to various data types, such as binomial distributions, Poisson distributions, etc.

#### 3.1.3 Coefficient Interpretation

The coefficients of GLM can be used to explain the impact of independent variables on the dependent variable. Since GLM does not require errors to follow a normal distribution, the interpretation of coefficients is more intuitive and accurate, aiding the understanding of relationships between variables.

### 3.2 Comparison Between GLM and Linear Regression

GLM is closely related to linear regression but also has some important differences. In this section, we will conduct a comprehensive comparison of GLM and linear regression to help readers better understand their similarities and differences.

#### 3.2.1 Differences in Model Form

GLM introduces a link function and the distribution of the response variable in its model form, making the model more flexible and adaptable to diverse data types. Linear regression, on the other hand, is a special case of GLM, with limitations in certain data types and scenarios