【Application Inquiry of PCR and PLS】: Application of Principal Component Regression and Partial Least Squares Regression in Linear Regression

# 1. Introduction to PCR and PLS

Principal Component Regression (PCR) and Partial Least Squares Regression (PLS) are common modeling techniques in the field of linear regression. They play a significant role in data processing, feature extraction, and predictive modeling. PCR and PLS help us handle high-dimensional data, mitigate the impact of multicollinearity on modeling results, and enhance the interpretability and predictive accuracy of models. Through the exploration of the principles and applications of PCR and PLS in this article, readers will gain a deeper understanding of the advantages, differences, and practical applications of these two methods, laying a foundation for further learning and application.

# 2. Fundamentals of Linear Regression

Linear regression is a statistical technique used to study the relationship between independent variables (X) and dependent variables (Y). In practical applications, we often need to understand the linear relationship between different variables to make predictions, analyses, and decisions. This chapter will introduce the basic principles of linear regression and model evaluation methods to help readers better understand the core concepts of linear regression.

### 2.1 Principles of Linear Regression
Linear regression describes the relationship between independent variables and dependent variables by fitting a linear equation. The following will delve into the basic principles of linear regression:

#### 2.1.1 Overview of Regression Analysis
Regression analysis is a statistical method used to explore the relationships between variables. In linear regression, we attempt to find the best-fit line that passes as closely as possible through the observed data points to predict the values of the dependent variable.

#### 2.1.2 Ordinary Least Squares
Ordinary least squares is a common fitting method in linear regression, which determines the regression coefficients by minimizing the sum of squared residuals between observed values and fitted values.

# Implementation of Ordinary Least Squares
import numpy as np
from sklearn.linear_model import LinearRegression

# Create a linear regression model
model = LinearRegression()

# Fit the data
model.fit(X, y)

#### 2.1.3 Multiple Linear Regression
Multiple linear regression considers the effects of multiple independent variables on the dependent variable by fitting a multivariate linear equation to describe the relationships between variables.

### 2.2 Evaluation of Linear Regression Models
Evaluating the goodness of fit of linear regression models is crucial for the reliability of the results. The following will introduce several commonly used model evaluation methods:

#### ***
***mon goodness-of-fit indicators include R-squared and Adjusted R-squared.

# Calculate R-squared
r_squared = model.score(X, y)

#### 2.2.2 Significance Testing of Regression Coefficients
In linear regression, we need to perform significance testing on regression coefficients to determine whether independent variables have a significant effect on the dependent variable.

| Independent Variable | Regression Coefficient | P-value |
|---------------------|-----------------------|---------|
| X1 | 0.752 | 0.001 |
| X2 | 1.234 | 0.002 |

#### 2.2.3 Residual Analysis
Residual analysis helps us evaluate the predictive ability of the model, test whether the fit meets statistical assumptions, and identify outliers or anomalous points.

# Residual analysis
residuals = y - model.predict(X)

In this chapter, we delved into the principles and model evaluation methods of linear regression, laying the foundation for subsequent chapters on Principal Component Regression and Partial Least Squares Regression.

# 3. Principles and Applications of Principal Component Regression (PCR)

Principal Component Regression (PCR) is a regression analysis method based on Principal Component Analysis (PCA), often used to deal with multicollinearity and high-dimensional datasets. In this chapter, we will delve into the principles of PCR and its specific applications in practice.

### 3.1 Overview of Principal Component Analysis (PCA)
Principal Component Analysis is a dimensionality reduction technique that can transform high-dimensional data into lower-dimensional data while preserving the main information in the data. In PCR, the application of PCA is to solve the problem of multicollinearity among independent variables.

#### 3.1.1 Eigenvalues and Eigenvectors
In PCA, the eigenvalues and eigenvectors of the data covariance matrix are key. Eigenvectors describe the main directions of the data, while eigenvalues indicate the importance of the data in these directions.

# Calculate the covariance matrix
cov_matrix = np.cov(data.T)
# Calculate eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eig(cov_matrix)

#### ***
***mon methods include retaining a specific proportion of the variance of the principal components or determining the number of components based on the size of the eigenvalues.

# Select the number of principal components
explained_variance_ratio = eigenvalues / np.sum(eigenvalues)
cumulative_variance_ratio = np.cumsum(explained_variance_ratio)

#### 3.1.3 The Idea of Principal Component Regression
The idea of principal component regression is to use the data after dimensionality reduction by PCA for linear regression analysis, thereby solving problems caused by multicollinearity and high-dimensional data.

### 3.2 Construction of PCR Models
The construction of PCR models includes determining the number of principal components, methods for fitting the model, and the selection of model evaluation indicators. The following will explore each in turn.

#### 3.2.1 Determination of the Number of Principal Components
Determining the appropriate number of principal componen