Data-Driven Exponential Framing for Pulsive Temporal Patterns Without Repetition or Singularity

Abstract Extracting pulsive temporal patterns from a small dataset without their repetition or singularity shows significant importance in manufacturing applications but does not sufficiently attract scientific attention. We propose to quantify how long temporal patterns appear without relying on their repetition or singularity, enabling us to extract such temporal patterns from a small dataset. Inspired by the celebrated time-delay embedding and data-driven Hankel matrix analysis, we introduce a linear dynamical system model on the time-delay coordinates behind the data to derive the discrete-time bases, each of which has a distinct exponential decay constant. The derived bases are fitted onto subsequences that are extracted with a sliding window in order to quantify how long patterns are dominant in the set of subsequences. We call the quantification method data-driven exponential framing (DEF). A toy model-based experiment shows that DEF can identify multiple patterns with distinct lengths. DEF is also applied to electric current measurement on a punching machine, showing its possibility to extract multiple patterns from real-world oscillatory data.