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Time warping with Hellinger elasticity

Yuly Billig

TL;DR

The Elastic Time Warping algorithm is introduced with a cubic computational complexity to optimize this matching problem for time series with values in an arbitrary metric space, with the stretching penalty given by the Hellinger kernel.

Abstract

We consider a matching problem for time series with values in an arbitrary metric space, with the stretching penalty given by the Hellinger kernel. To optimize this matching, we introduce the Elastic Time Warping algorithm with a cubic computational complexity.

Time warping with Hellinger elasticity

TL;DR

The Elastic Time Warping algorithm is introduced with a cubic computational complexity to optimize this matching problem for time series with values in an arbitrary metric space, with the stretching penalty given by the Hellinger kernel.

Abstract

We consider a matching problem for time series with values in an arbitrary metric space, with the stretching penalty given by the Hellinger kernel. To optimize this matching, we introduce the Elastic Time Warping algorithm with a cubic computational complexity.
Paper Structure (4 sections, 10 theorems, 39 equations)

This paper contains 4 sections, 10 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Hellinger metric on $\text{Diff}([0,1])$.
  3. Distance on the space of functions with a Hellinger penalty for stretching
  4. Elastic Time Warping Algorithm

Key Result

Lemma 1

Let $X$ be a metric space with distance function $\rho: X \times X \rightarrow S \subset \mathbb R_{\geq 0}$. Suppose function $F: \mathbb R_{\geq 0} \rightarrow \mathbb R_{\geq 0}$ satisfies $F(a+b) \leq F(a) + F(b)$ for all $a, b \in S$. Then $F \circ \rho$ is also a metric on $X$.

Theorems & Definitions (15)

  • Lemma 1
  • Proposition 2
  • proof
  • Corollary 3
  • Corollary 4
  • Lemma 5
  • Proposition 6
  • proof
  • Lemma 7
  • proof
  • ...and 5 more