MATLAB Code Implementation of Principal Component Analysis (PCA)

Principal Component Analysis (PCA) is a widely used dimensionality reduction technique that projects original data onto a new set of orthogonal basis vectors through linear transformation. These basis vectors are called principal components. PCA can be applied not only for data dimensionality reduction but also for noise removal and feature extraction.

The fundamental steps for implementing PCA in MATLAB are as follows:

Data Standardization: First, standardize the input data by ensuring each feature has zero mean and unit variance. This preprocessing step eliminates scale differences between features and prevents dimensional inconsistencies from affecting analysis results. In MATLAB, this can be implemented using zscore(X) or manually by subtracting the mean and dividing by the standard deviation.

Compute Covariance Matrix: Calculate the covariance matrix to capture the correlations between different features. This matrix serves as the foundation for subsequent eigenvalue decomposition. The MATLAB implementation typically uses cov(X) after standardization, which computes the covariance matrix of the standardized data matrix.

Eigenvalue Decomposition: Perform eigenvalue decomposition on the covariance matrix to obtain eigenvalues and their corresponding eigenvectors. The magnitude of eigenvalues indicates the importance of respective principal components - larger eigenvalues correspond to eigenvectors that represent the primary directions of data variation. In MATLAB, this can be achieved using the eig() function, which returns eigenvectors and eigenvalues sorted in ascending order.

Principal Component Selection: Construct a projection matrix by selecting the top k eigenvectors corresponding to the largest eigenvalues, where k represents the number of principal components to retain. The selection criterion can be based on explained variance ratio or specific application requirements. The projection matrix W is formed by arranging selected eigenvectors as columns.

Data Transformation: Project the original data onto the selected principal components to obtain the dimensionally reduced dataset. The transformation is performed by matrix multiplication Y = X_standardized * W, where Y represents the transformed data in the new feature space.

In MATLAB, PCA can be conveniently implemented using built-in functions such as pca() or princomp(), which handle the entire process automatically. Alternatively, developers can manually code the matrix operations for better customization and understanding. Whether applied for dimensionality reduction, noise filtering, or feature extraction, PCA effectively extracts essential information from high-dimensional data, thereby enhancing the efficiency of subsequent machine learning tasks or data analysis procedures.