Data-driven Exponential Framing for Pulsive Temporal Patterns without Repetition or Singularity

TL;DR

This work tackles the challenge of extracting pulsive temporal patterns from small datasets without repetition or singularity, common in manufacturing contexts. It introduces Data-driven Exponential Framing (DEF), a method that uses time-delay embedding to fit a linear dynamical model on delay coordinates, yielding exponential decay bases whose time constants capture pattern durations. By selecting an optimal delay order via AIC and extracting pattern-specific amplitudes and decay rates, DEF identifies multiple, distinct pattern lengths and provides compact, interpretable descriptors for downstream analysis, validation, and control. Comparative experiments on toy and turret-punch data show DEF outperforms Hankel-based baselines such as SSA and Delay-DMD in isolating non-repeating pulsive structures, highlighting its practical relevance for monitoring and diagnosing pulse-driven dynamics in manufacturing.

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 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.

Figures (10)

• Figure 1: Trajectory of $x(t)$ and $v(t)$ in Eq. (\ref{['eq:toy_model']}) driven by the PWC input $u(t)$.
• Figure 2: Preliminary results for the toy model data: (a) Degree $d$ vs. $\mathrm{AIC}(d)$, where the orange circle denotes the optimal value $d^*$, and (b) Eigenvalues of $\tilde{\sf A}_d$ with $d = 1200$.
• Figure 3: Main results for the toy model data: (a) The target subsequences ${\hbox{\boldmath$x$}}_{1200}[6200]$ and ${\hbox{\boldmath$x$}}_{1200}[11200]$ highlighted by the bold orange and green lines, respectively. (b,c) The magnitude $\tilde{A}_{i,d}[n]$ at each time constant $\tilde{T}_i$ for (b) $n = 6200$ and (c) $n = 11200$.
• Figure 4: Schematic diagram of turret punch blanking a metal sheet.
• Figure 5: Target current data measured in the servo motor that drives the press, where the representative locations are denoted by L0, L1, …, L8.