【Advanced】Principal Component Regression (PCR) in MATLAB

# [Advanced] Principal Component Regression (PCR) in MATLAB

## 1. Introduction to Principal Component Regression (PCR)

Principal Component Regression (PCR) is a multivariate statistical method that combines Principal Component Analysis (PCA) with regression analysis to handle high-dimensional datasets. It simplifies the data structure by projecting the original data onto a low-dimensional space of principal components while retaining information relevant to the response variables. PCR is widely applied in various fields, including spectral data analysis, bioinformatics data analysis, and chemometrics data analysis.

## 2. Theoretical Foundation of PCR

### 2.1 Principal Component Analysis (PCA)

Principal Component Analysis (PCA) is a dimensionality reduction technique that aims to project high-dimensional data onto a lower-dimensional space while retaining the maximum variance of the data. It is achieved through the following steps:

- **Covariance Matrix Computation:** Calculate the covariance matrix of the original data matrix, which contains the covariances between all variables.
- **Eigenvalue and Eigenvector Solution:** Perform eigen-decomposition on the covariance matrix to obtain a set of eigenvalues and corresponding eigenvectors.
- **Principal Component Extraction:** The eigenvalues represent the variance explained by each principal component, while the eigenvectors represent the directions of these components in the original data. By selecting the first k eigenvectors with the largest eigenvalues, the first k principal components are obtained.

### 2.2 Regression Analysis

Regression analysis is a statistical modeling technique used to predict the relationship between one or more dependent variables (response variables) and one or more independent variables (explanatory variables). The most common regression model is linear regression, with the equation:

y = b0 + b1x1 + b2x2 + ... + bnxn + ε

Where:

- y is the dependent variable
- x1, x2, ..., xn are independent variables
- b0 is the intercept
- b1, b2, ..., bn are regression coefficients
- ε is the error term

### 2.3 Mathematical Principles of PCR

PCR combines PCA with regression analysis through the following steps:

- **Principal Component Extraction:** Use PCA to extract principal components from the original data.
- **Regression Model Establishment:** Use the principal components as independent variables to establish a regression model to predict the dependent variable.

The mathematical principles of PCR are as follows:

y = b0 + b1PC1 + b2PC2 + ... + bnpCPn + ε

Where:

- y is the dependent variable
- PC1, PC2, ..., PCn are principal components
- b0 is the intercept
- b1, b2, ..., bn are regression coefficients
- ε is the error term

In this way, PCR can reduce high-dimensional data to a low-dimensional space while retaining information relevant to the dependent variable, thereby improving the predictive accuracy of the regression model.

## 3. Implementation of PCR in MATLAB

### 3.1 Data Preprocessing

Before performing PCR analysis, data preprocessing is necessary to ensure data quality and the accuracy of analysis results. Data preprocessing steps include:

- **Handling Missing Values:** The presence of missing values can affect analysis results. For missing values, the following methods can be applied:
- Delete samples or features containing missing values
- Fill missing values using interpolation or mean value methods
- **Handling Outliers:** Outliers can also affect analysis results. For outliers, the following methods can be applied:
- Delete outliers
- Transform outliers (e.g., logarithmic transformation)
- **Standardization or Normalization:** Standardization or norm***mon standardization methods include:
- Mean normalization: Subtract the mean of each feature and divide by its standard deviation
- Min-max normalization: Scale each feature to a range of [0, 1]
- **Feature Selection:** Feature selection can remove irrelevant or redundant features, ***mon feature selection methods include:
- Filter methods: Select features based on statistical information (e.g., variance, correlation) of features
- Wrapper methods: Select features iteratively to optimize model performance
- Embedded methods: Perform feature selection during model training

### 3.2 Principal Compone