Adaptive law-based feature representation for time series classification

Time series classification (TSC) underpins applications across finance, healthcare, and environmental monitoring, yet real-world series often contain noise, local misalignment, and multiscale patterns. We introduce adaptive law-based transformation (ALT), a multiscale generalization of the earlier linear law-based transformation (LLT). ALT scans each series with variable-length, shifted windows, constructs symmetric delay embeddings, and extracts eigenvectors associated with the eigenvalue of minimal magnitude ("shapelet laws") that capture locally stable patterns. These laws are assembled into class-specific dictionaries, and new series are projected onto them to yield compact, transparent features that enhance linear separability while remaining compatible with standard classifiers. On the BasicMotions dataset with synthetic Gaussian noise, ALT sustained test accuracy roughly 15-20 percentage points (pp) above raw inputs and 5-10 pp above LLT at moderate noise levels. Across ten datasets from the UCR Time Series Classification Archive-spanning motion, biomedical, spectroscopy, and industrial domains-ALT improved median test accuracy by + 7.6 pp with k-nearest neighbors (KNN) and + 4.8 pp with support vector machines (SVMs), with particularly large gains on long, noisy industrial series (FordA/B: + 23.1-25.3 pp). In addition, ALT often reduced SVM training time (median reductions of 340.6 s on FordB and 717.5 s on FordA) while maintaining or improving accuracy. ALT thus offers a lightweight and transparent alternative to heavyweight TSC pipelines: it requires only a small hyperparameter set, produces stable and discriminative features, and delivers competitive or superior accuracy under challenging conditions.