MATLAB Implementation of PCA with Sample Data

Principal Component Analysis (PCA) is a widely used dimensionality reduction technique that maps high-dimensional data to a lower-dimensional space while preserving the main characteristics of the data. MATLAB implementation of PCA with data typically involves calculating variance contribution rates, determining the number of principal components, and plotting T-squared and SPE charts to evaluate model performance. ### PCA Implementation Steps Data Preprocessing: First standardize the data to ensure each feature has zero mean and unit variance, preventing different measurement scales from affecting PCA results. In MATLAB, this can be implemented using the zscore function or manual standardization: data_std = (data - mean(data)) ./ std(data). Covariance Matrix Calculation: Compute the covariance matrix from the standardized data, which reflects the correlations between different features. MATLAB's cov function can be used: cov_matrix = cov(data_std). Eigenvalue Decomposition: Perform eigenvalue decomposition on the covariance matrix to obtain eigenvalues and corresponding eigenvectors. The magnitude of eigenvalues determines the importance of principal components. MATLAB implementation: [eigenvectors, eigenvalues] = eig(cov_matrix). Determining Principal Components: Typically use cumulative variance contribution rate to select the number of principal components, such as setting a threshold (e.g., 95%) so that the selected principal components explain most of the data variance. Code implementation involves sorting eigenvalues in descending order and calculating cumulative contribution: cum_var = cumsum(eigenvalues)/sum(eigenvalues). Principal Component Scores: Project the original data onto the selected principal components to obtain reduced-dimension data. MATLAB code: scores = data_std * eigenvectors(:,1:k) where k is the number of selected principal components. T-squared Statistic: Measures the abnormality of samples in the principal component space, used for outlier detection. Calculation: T2 = sum((scores ./ std(scores)).^2, 2). SPE Chart (Squared Prediction Error): Measures the reconstruction error of samples in the PCA model, reflecting how well the model fits the data. Implementation: SPE = sum((data_std - scores * eigenvectors(:,1:k)').^2, 2). ### Chart Analysis Variance Contribution Plot: Helps users select appropriate number of principal components, typically choosing the first few components with high cumulative contribution rates. MATLAB implementation can use pareto or plot functions to visualize variance distribution. T-squared Control Chart: If a sample's T-squared statistic exceeds control limits, it may indicate abnormal behavior in the PCA model. Control limits can be calculated using chi-square distribution: control_limit = chi2inv(0.95, k). SPE Chart: If a sample's SPE value is too high, it indicates the model fails to adequately explain the sample's characteristics, suggesting potential abnormalities or noise. Control limits for SPE can be derived using statistical methods based on residual distribution. Through these steps, one can effectively extract the main features of data and perform model diagnostics using T-squared and SPE charts. The complete MATLAB implementation would involve combining these components with appropriate visualization functions like plot, scatter, and control limit calculations for comprehensive PCA analysis.