【Lasso Regression Principle Analysis】: The Principle and Practical Application of Lasso Regression

# 1. Introduction to Ridge Regression
Ridge Regression is a classic linear regression algorithm designed to address the issue of poor performance of ordinary least squares in the presence of multicollinearity. By incorporating an L2 regularization term, Ridge Regression effectively controls the complexity of the model, avoiding overfitting. In practical applications, Ridge Regression is often used to handle high-dimensional data and scenarios where features are strongly correlated, demonstrating excellent stability and generalization capabilities.

【Three Key Techniques for Content Creation】:
- Valuable: Ridge Regression is one of the important models in machine learning. Understanding the overview of Ridge Regression can help readers grasp its role and value in data modeling.
- Practical: Through this chapter, readers can gain a basic understanding of the fundamental principles of Ridge Regression and its applications in solving real-world problems, laying a foundation for more in-depth learning and practice.

# 2. Linear Regression Basics
Linear Regression, being one of the simplest and most commonly used algorithms in machine learning, is a staple for beginners. In this chapter, we will delve into the basics of linear regression, including the method of least squares, residual analysis, and the concepts of overfitting and underfitting.

## 2.1 Principles of Linear Regression
### 2.1.1 Method of Least Squares
The method of least squares is a commonly used parameter estimation method in linear regression, which finds the best-fitting line or hyperplane by minimizing the sum of squared residuals between actual observed values and model predictions. Specifically, for a set of observed data \((x_1, y_1), (x_2, y_2), ..., (x_n, y_n)\), the expression of a linear regression model is \(y = β_0 + β_1x + ε\), where \(β_0\) and \(β_1\) are the intercept and slope, respectively, and \(\hat{y_i} = β_0 + β_1x_i\). The values of \(β_0\) and \(β_1\) are solved by minimizing the sum of squared residuals \(\sum_{i=1}^{n}(y_i - \hat{y_i})^2\).

import numpy as np
from sklearn.linear_model import LinearRegression

# Create a linear regression model
model = LinearRegression()
# Fit the model
model.fit(X, y)
# Get the intercept and slope
intercept = model.intercept_
coefficients = model.coef_

### 2.1.2 Residual Analysis
A residual is the difference between an observed value and a model's fitted value. Residual analysis is an important means of evaluating the goodness of model fit. By observing the distribution of residuals, one can determine if the model has systematic errors or outliers, allowing for adjustments to the model or removal of outliers to improve the fit.

# Calculate residuals
residuals = y - model.predict(X)
# Plot a scatter graph of residuals
plt.scatter(y, residuals)
plt.axhline(y=0, color='r', linestyle='-')
plt.xlabel('Actual values')
plt.ylabel('Residuals')
plt.title('Residual Plot')
plt.show()

### 2.1.3 Overfitting and Underfitting
In linear regression, both overfitting and underfitting are common problems. Overfitting occurs when a model is overly tailored to the training data, leading to poor generalization; while underfitting indicates that the model does not fit the data well, resulting in low prediction accuracy. To address this issue, it is necessary to use an appropriate model complexity and training set size, and to perform cross-validation.

# Using linear regression models with different complexities
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import make_pipeline

# Create polynomial features
degree = 10
model = make_pipeline(PolynomialFeatures(degree), LinearRegression())
model.fit(X, y)

In this section, we have delved into the principles of linear regression, including the method of least squares, residual analysis, and the problems of overfitting and underfitting. With an understanding of the basics of linear regression algorithms, we can better apply them to solve real-world problems.

# 3. Principles of Ridge Regression
Ridge Regression is a widely used regularized linear regression method in statistical modeling and machine learning. This chapter will delve into the principles of Ridge Regression, including the basic concepts, loss function forms, and specific problems that Ridge Regression aims to solve.

### 3.1 Introduction to Ridge Regression
Before introducing Ridge Regression, let's briefly explain what regularization is. Regularization is a penalty term introduced during model training to prevent overfitting, constraining the complexity of the model to improve generalization.

#### 3.1.1 Penalty Term
The penalty term in Ridge Regression is the L2 norm, which constrains the size of the model parameters to prevent overfitting due to excessively large parameters. Its mathematical expression is as follows:

\text{Cost}_{\text{Ridge}} = \text{Cost}_{\text{OLS}} + \lambda \sum_{i=1}^{n} \beta_{i}^2

Where \(\text{Cost}_{\text{OLS}}\) represents the loss function of ordinary least squares, \(\lambda\) is a hyperparameter controlling the strength of the penalty term, and \(\beta_{i}\) are the coefficients of the model.

#### 3.1.2 Ridge Regression Loss Function
The loss function of Ridge Regression combines the loss function of ordinary least squares with the penalty term. It is formulated as:

\text{Loss}_{\text{Ridge}} = \sum_{i=1}^{n} (y_{i} - \hat{y_{i}})^2 + \lambda \sum_{i=1}^{n} \beta_{i}^2

In Ridge Regression, aside from minimizing the sum of squared residuals between predictions and actual values, the L2 norm of the parameters is also minimized to achieve the purpose of constraining the parameters.

#### 3.1.3 Problems Solved by Ridge Regression
Ridge Regression