Time Series Decomposition using the Fréchet Distance

2026-07-05

AI Summary

This work proposes the Fréchet decomposition problem, which seeks to approximate a collection of univariate time series as Fréchet combinations of a small set of basis curves while minimizing the total Fréchet distance. Analogous to principal component analysis, this is the first effort to incorporate the Fréchet distance into time series decomposition, and it introduces two distinct variants of the problem. For the single-basis setting, the authors design a (1+ε)-approximation algorithm; for an arbitrary number k of basis curves, they present an exact solution method based on projection distance, assuming a given candidate set of basis curves. By delivering both an efficient approximation scheme and a general exact algorithm, this study establishes a novel paradigm for modeling nonlinear structures in time series data.

Abstract

In this paper, we introduce a new data analysis problem that aims to decompose a set of univariate time series into a small set of $k$ base curves of length at most $l$ such that the sum of Fréchet distances of the time series to a ``Fréchet combination'' of the base curves is minimized. Here, a Fréchet combination allows to combine individually scaled base curves using a $k$-dimensional traversal. We call the problem of finding a set of optimal base curves the Fréchet decomposition problem and we consider two variants: (a) the base curves can be arbitrary curves of bounded length and (b) the curves come from a given finite set of candidate curves. We think of the Fréchet decomposition problem as a Fréchet variant of principal component analysis. For the case of a single base curve we develop a $(1+\varepsilon)$-approximation algorithm for the Fréchet decomposition problem. Additionally we give an exact algorithm for the projection distance problem that asks to compute the distance of one given time series to a given set of $k$ base curves. This allows us to design an exact algorithm for the Fréchet decomposition problem for general $k$ when curves come from a fixed candidate set.

Problem

• Time Series Decomposition
• Fréchet Distance
• Base Curves
• Fréchet Combination
• Univariate Time Series

Innovation

• Fréchet distance
• time series decomposition
• approximation algorithm
• projection distance
• principal component analysis