This paper introduces the Tangent–Secant Similarity Index (TSI), a novel geometric method for measuring structural similarity between time series. TSI jointly analyzes directional and behavioral coherence through local tangents and meso-scale secant structures, maintaining O(n) computational complexity. A key theoretical property is that the Big Secant formulation achieves 4× lower noise variance than the Standard Secant (Var(m″) = σ²/(8k²) versus Var(m′) = σ²/(2k²)), which is proven analytically and confirmed experimentally. TSI can detect regime shifts, structural divergence, and critical transitions at early stages without explicit filtering. We validate TSI across three domains: (1) synthetic signals with controlled frequency transitions, demonstrating early-warning via Event Horizon; (2) EEG inter-session analysis on N = 15 subjects from the SEED-IV dataset, showing that TSI produces significantly more consistent inter-session estimates than Pearson correlation (σTSI = 0.063 vs. σPearson = 0.363, Levene p = 0.0008); and (3) thermodynamic phase-transition detection, where all three single-signal metrics (GSCI, SGM, MIA) jointly identify the water boiling point without manual thresholds. Mathematically, TSI unifies the central difference quotient from numerical analysis, the turning function from computational geometry, and Laplace kernel functions from machine learning into a coherent multiscale framework. Applications span biomedical signal analysis, financial system monitoring, and real-time anomaly detection. Index Terms — Time series analysis, geometric signal processing, structural similarity, regime detection, secant methods, central difference quotient, scale-space theory, Laplace kernel.
Berk Yücetin (Wed,) studied this question.