A General Metric-Space Formulation of the Time Warp Edit Distance (TWED)

TL;DR

This paper extends the Time Warp Edit Distance (TWED) to arbitrary metric spaces, providing a theoretical foundation for applying elastic distances beyond traditional time series to diverse data types.

Contribution

It introduces a generalized TWED (GTWED) that is a true metric in arbitrary metric spaces, broadening the applicability of elastic distance measures.

Findings

GTWED is a true metric under mild conditions.

Classical TWED is a special case of GTWED.

Enables elastic distances on sequences over arbitrary domains.

Abstract

This short technical note presents a formal generalization of the Time Warp Edit Distance (TWED) proposed by Marteau (2009) to arbitrary metric spaces. By viewing both the observation and temporal domains as metric spaces $(X, d)$ and $(T, \Delta)$, we define a Generalized TWED (GTWED) that remains a true metric under mild assumptions. We provide self-contained proofs of its metric properties and show that the classical TWED is recovered as a special case when $X = \mathbb{R}^d$, $T \subset \mathbb{R}$, and $g(x) = x$. This note focuses on the theoretical structure of GTWED and its implications for extending elastic distances beyond time series, which enables the use of TWED-like metrics on sequences over arbitrary domains such as symbolic data, manifolds, or embeddings.