A Novel Minkowski-Based Fractal Analysis Framework for EEG: Theory and Application in Alzheimer’s Detection

Existing electroencephalography (EEG) complexity methods treat channels as independent univariate series under an isotropic Euclidean metric, which retains the magnitude of inter-channel fluctuations but not the direction of interhemispheric dominance. We reformulate fractal analysis in two-dimensional Minkowski space using the non-positive-definite Lorentzian metric instead of the Euclidean norm, thereby preserving directional information. Each increment vector between paired left–right electrodes is classified by the sign of its Lorentzian self-product as spacelike (left-dominant), timelike (right-dominant) or null, so that complexity and directional asymmetry are quantified jointly. Minkowski Higuchi Fractal Dimension (M-HFD) is a consistent extension: it reduces exactly to the classical Higuchi Fractal Dimension (HFD) under proportional coupling and remains stable under bounded perturbations of that ideal. Synthetic fractional Brownian motion experiments confirm reliable dimension estimation across the full fractal range. On an open-access cohort (36 Alzheimer’s patients, 29 controls), a Mamdani fuzzy inference system under leakage-free nested leave-one-out cross-validation attained AUCs of 0.873 (mixed), 0.883 (female) and 0.880 (male).