Measuring Congruence on High Dimensional Time Series

TL;DR

This paper introduces a new distance measure called congruence distance for high-dimensional time series, proves its computational hardness, and offers approximation algorithms with theoretical bounds, motivated by motion gesture recognition.

Contribution

It generalizes classical congruence to multi-dimensional time series and provides the first algorithms with bounds for approximating this NP-hard measure.

Findings

Congruence distance is NP-hard to compute.

Two approximation algorithms with theoretical bounds are proposed.

Application to motion gesture recognition demonstrates relevance.

Abstract

A time series is a sequence of data items; typical examples are videos, stock ticker data, or streams of temperature measurements. Quite some research has been devoted to comparing and indexing simple time series, i.e., time series where the data items are real numbers or integers. However, for many application scenarios, the data items of a time series are not simple, but high-dimensional data points. Motivated by an application scenario dealing with motion gesture recognition, we develop a distance measure (which we call congruence distance) that serves as a model for the approximate congruency of two multi-dimensional time series. This distance measure generalizes the classical notion of congruence from point sets to multi-dimensional time series. We show that, given two input time series $S$ and $T$, computing the congruence distance of $S$ and $T$ is NP-hard. Afterwards, we present two algorithms that compute an approximation of the congruence distance. We provide theoretical bounds that relate these approximations with the exact congruence distance.