Amercing: An Intuitive, Elegant and Effective Constraint for Dynamic Time Warping

TL;DR

This paper introduces Amerced Dynamic Time Warping (ADTW), a new DTW variant that penalizes warping with a fixed additive cost, offering a more intuitive and flexible constraint for time series alignment.

Contribution

The paper presents ADTW, a novel DTW variant with additive warping penalties, formally analyzes its properties, and demonstrates its effectiveness in classification tasks.

Findings

ADTW provides an intuitive parameterization for constrained warping.

ADTW outperforms traditional DTW variants in classification benchmarks.

The method is implemented in a C++ demonstration application.

Abstract

Dynamic Time Warping (DTW), and its constrained (CDTW) and weighted (WDTW) variants, are time series distances with a wide range of applications. They minimize the cost of non-linear alignments between series. CDTW and WDTW have been introduced because DTW is too permissive in its alignments. However, CDTW uses a crude step function, allowing unconstrained flexibility within the window, and none beyond it. WDTW's multiplicative weight is relative to the distances between aligned points along a warped path, rather than being a direct function of the amount of warping that is introduced. In this paper, we introduce Amerced Dynamic Time Warping (ADTW), a new, intuitive, DTW variant that penalizes the act of warping by a fixed additive cost. Like CDTW and WDTW, ADTW constrains the amount of warping. However, it avoids both abrupt discontinuities in the amount of warping allowed and the limitations of a multiplicative penalty. We formally introduce ADTW, prove some of its properties, and discuss its parameterization. We show on a simple example how it can be parameterized to achieve an intuitive outcome, and demonstrate its usefulness on a standard time series classification benchmark. We provide a demonstration application in C++.