Sufficient conditions for the existence of a sample mean of time series under dynamic time warping

• Published in: Annals of mathematics and artificial intelligence Vol. 88; no. 4; pp. 313 - 346
• Main Authors: Jain, Brijnesh, Schultz, David
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.04.2020; Springer; Springer Nature B.V
• Subjects: Algorithms; Artificial Intelligence; Complex Systems; Computer Science; Data mining; Mathematics; Optimization; Time series; Upper bounds; Warping; 68T99; Sample mean; 49J52; Fréchet function; Dynamic time warping; Time series
• ISSN: 1012-2443; 1573-7470
• DOI: 10.1007/s10472-019-09682-2

Time series averaging is an important subroutine for several time series data mining tasks. The most successful approaches formulate the problem of time series averaging as an optimization problem based on the dynamic time warping (DTW) distance. The existence of an optimal solution, called sample mean, is an open problem for more than four decades. Its existence is a necessary prerequisite to formulate exact algorithms, to derive complexity results, and to study statistical consistency. In this article, we propose sufficient conditions for the existence of a sample mean. A key result for deriving the proposed sufficient conditions is the Reduction Theorem that provides an upper bound for the minimum length of a sample mean.