Kernel Principal Component Analysis (KPCA) Method for Multidimensional Quality Assessment

In multidimensional comprehensive analysis, traditional PCA fails to effectively handle nonlinear data patterns. We therefore employed Kernel Principal Component Analysis (KPCA), a kernel-based nonlinear dimensionality reduction technique that maps original data to high-dimensional feature space using kernel functions. Our implementation utilized the Gaussian radial basis function (RBF) kernel, defined as k(x,y) = exp(-γ||x-y||²), which is particularly suitable for continuous variables. The algorithm first computes the kernel matrix from input data, then centers the matrix to ensure feature space mean adjustment, followed by eigenvalue decomposition to extract principal components in the high-dimensional space. In our study assessing 16 fermented bean curd varieties from different regions, KPCA successfully captured nonlinear relationships among quality indicators that conventional PCA would miss. The research revealed significant regional quality variations, with KPCA providing more accurate quality assessments through its ability to model complex data structures. We further analyzed correlations between various quality parameters using kernel-based similarity measures, enabling deeper understanding of product characteristics. The methodology demonstrates that for multidimensional analysis involving nonlinear data, KPCA should be implemented with appropriate kernel selection (like Gaussian RBF) and parameter optimization (through cross-validation) to achieve precise quality evaluations.