Fractal Eigen decomposition (FED): A generalized Covariance-Based framework for 1D signal decomposition with application to schizophrenia EEG classification

We propose Fractal Eigen Decomposition (FED), a new spectral approach that incorporates local fractal complexity directly into the metric structure of a generalized eigenvalue decomposition. FED reorders signal-adaptive Hankel covariance modes according to fractal complexity via a generalized eigenvalue problem Cv = λF − 1v. When F = I, FED reduces exactly to classical SSA, guaranteeing backward compatibility. The method’s theoretical foundation rests on five formally proved propositions: positive semi-definiteness, real and non-negative eigenvalues, F − 1-orthogonality, the SSA special case, and spectral monotonicity. An ablation analysis across ten F-transform variants and eight power settings showed that fractal-mode selectivity increases steadily with p, reaching a practical plateau around p = 4–6, while reconstruction error rises from 10.9% at p = 0 to 30.2% at p = 2; p = 2 was retained as a practical default offering substantial selectivity gains at a moderate reconstruction cost. FED was further validated on a real-world schizophrenia EEG dataset. Under a stratified 5 × 5 repeated cross-validation protocol (40 LASSO-selected features per branch, matched for feature count), temporal-morphological and spectral power features extracted directly from each channel achieved 73.1% accuracy (AUC = 0.826) with an SVM classifier, whereas the same features recomputed over four FED modes and the residual raised accuracy to 87.6% (AUC = 0.942; Wilcoxon signed-rank p < 0.001 across 25 paired folds). This advantage was found to be robust to classifier kernel choice under a separate, fold-wise-normalized protocol (linear, RBF, and polynomial kernels compared explicitly) and, by construction, could not be attributed to a larger feature count. .