[Basic] Principal Component Analysis (PCA) in MATLAB

# Introduction to Principal Component Analysis (PCA) in MATLAB

Principal Component Analysis (PCA) is a widely used dimensionality reduction technique in the fields of data analysis and machine learning. It projects high-dimensional data onto a lower-dimensional space through a linear transformation while retaining as much of the original data's information as possible. The main goal of PCA is to find a set of orthogonal bases that maximize the variance of the projected data.

Advantages of PCA include:

***Dimensionality Reduction:** PCA can reduce high-dimensional data to a more manageable and visualizable lower-dimensional space.
***Feature Extraction:** PCA can extract the most important features from the original data, thereby simplifying the modeling and analysis process.
***Interpretability:** The base vectors of PCA can explain the variations in the original data, providing an in-depth understanding of the data structure.

# 2. Theoretical Foundation of PCA

### 2.1 Mathematical Principles of PCA

Principal Component Analysis (PCA) is a linear transformation technique aimed at projecting high-dimensional data onto a lower-dimensional space while maximizing the variance of the projected data. The mathematical principles of PCA are based on the following steps:

1. **Centering the Data:** Subtract the mean of each feature from the data set so that the data is distributed around the origin.
2. **Computing the Covariance Matrix:** The covariance matrix represents the covariance between different features in the data set. It is a symmetric matrix, and the diagonal elements represent the variance of each feature.
3. **Eigenvalue Decomposition:** Perform eigenvalue decomposition on the covariance matrix to obtain eigenvalues and eigenvectors. The eigenvalues represent the variance of each eigenvector in the covariance matrix, and the eigenvectors represent the directions of these eigenvectors.
4. **Selecting Principal Components:** Select principal components based on the size of the eigenvalues. Typically, the first k eigenvectors with the largest eigenvalues are chosen as the principal components.

### 2.2 Covariance Matrix and Eigenvalue Decomposition in PCA

The covariance matrix C is an n×n matrix, where n is the number of features in the data set. The (i, j) element of the covariance matrix represents the covariance between feature i and feature j.

import numpy as np

# Sample data
data = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])

# Compute the covariance matrix
cov_matrix = np.cov(data.T)

# Output the covariance matrix
print(cov_matrix)

Eigenvalue decomposition breaks down the covariance matrix into eigenvalues and eigenvectors:

# Compute eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eig(cov_matrix)

# Output eigenvalues and eigenvectors
print("Eigenvalues:", eigenvalues)
print("Eigenvectors:", eigenvectors)

The eigenvalues represent the variance of each eigenvector in the covariance matrix, while the eigenvectors represent the directions of these eigenvectors.

### Logical Analysis

* Centering the data can eliminate scale differences between features in the data set, making them comparable.
* The covariance matrix represents the correlation between different features in the data set. The diagonal elements represent the variance of each feature, while the off-diagonal elements represent the covariance between features.
* Eigenvalue decomposition breaks down the covariance matrix into eigenvalues and eigenvectors. The eigenvalues represent the variance of each eigenvector, and the eigenvectors represent the directions of these eigenvectors.
* Principal components are eigenvectors with the largest eigenvalues, representing the directions of the largest variance in the data set.

# 3.1 PCA Data Preprocessing

Data preprocessing is crucial before applying PCA. The purpose of data preprocessing is to eliminate noise and outliers from the data and make the data distribution more closely resemble a normal distribution, thereby enhancing the effectiveness of PCA dimensionality reduction.

**3.1.1 Missing Value Handling**

Missing values are a common issue in data preprocessing. Methods for handling missing values include:

- **Removing Missing Values:** If the number of missing values is small, samples or features containing missing values can be directly deleted.
- **Imputing Missing Values:** If there are many missing values, ***mon imputation methods include:
- Mean imputation: Fill in missing values with the mean of the feature.
- Median imputation: Fill in missing values with the median of the feature.
- K-nearest neighbor imputation: Estimate missing values based on the feature values of neighboring samples with missing values.

**3.1.2 Outlier Handling**

Outliers are values in the data that significantly deviate from other samples. The presence of outliers can affect the results of PCA dimensionality reduction. Methods for handling outliers include:

- **Removing Outliers:** If the number of outliers is small, samples containing outliers can be directly deleted.
- **Transforming Outliers:** Outliers can be transformed into the normal range using methods such as log transformation, square root transformation, etc.
- **Truncating Outliers:** Outliers are truncated to a reasonable range.

**3.1.3 Data Standardization**

Data standardization refers to scaling each feature