MATLAB Implementation of KPCA (Kernel Principal Component Analysis) with Code Descriptions

KPCA (Kernel Principal Component Analysis) is a nonlinear dimensionality reduction method that maps original data to a high-dimensional feature space using kernel functions before applying PCA, effectively extracting nonlinear features. In fault diagnosis applications, KPCA is commonly used to monitor system anomalies and quantitatively evaluate them using T2 and SPE statistics.

### Core Implementation Approach for KPCA Kernel Function Selection: Commonly used kernels include Gaussian (RBF) or polynomial kernels, which map input data to high-dimensional spaces. The Gaussian kernel's parameters (such as bandwidth) need to be determined through cross-validation. In MATLAB implementation, the kernel matrix can be computed using vectorized operations for efficiency. Kernel Matrix Centralization: The kernel matrix requires double-centering processing to ensure zero-mean data in the high-dimensional space, which is a critical step in KPCA implementation. This involves subtracting row and column means from the original kernel matrix. Eigenvalue Decomposition: Perform eigenvalue decomposition on the centralized kernel matrix and select the eigenvectors corresponding to the top k largest eigenvalues to form the projection space. The MATLAB 'eig' function or 'svd' can be used for this computation. Statistical Calculations: T2 Statistic: Measures the variation degree of samples in the principal component space, reflecting deviations in main features. Implementation involves calculating the Mahalanobis distance in the feature space. SPE Statistic (Squared Prediction Error): Captures residual information not explained by principal components, used for detecting anomalies in secondary features. This is computed as the reconstruction error in the original space.

### Fault Diagnosis Workflow Training Phase: Train the KPCA model using normal state data and determine control limits for T2 and SPE statistics (e.g., using kernel density estimation). The MATLAB implementation should include threshold calculation based on historical normal data. Testing Phase: Real-time calculation of T2 and SPE values for new samples, with faults detected when values exceed control limits. This requires efficient online computation of kernel similarities. Visualization: Display T2 and SPE trend changes using line plots or scatter plots to intuitively identify fault occurrence points. MATLAB's plotting functions can create monitoring charts with threshold lines.

### Application Extensions Multi-fault Discrimination: Combined with contribution plot analysis to identify specific variables causing statistical limit violations. Implementation requires calculating variable contributions to each statistic. Dynamic KPCA: Suitable for time-varying systems, updating model parameters through sliding windows. This involves recursive kernel matrix updates and incremental eigenvalue decomposition.

MATLAB implementation of KPCA requires attention to numerical stability in kernel matrix computations and memory management for large-scale datasets. Diagnostic plots for T2 and SPE should include threshold lines for quick system state assessment. Code optimization techniques like matrix precomputation and memory-efficient operations are essential for practical applications.