Constrained Sparse Dynamic Time Warping

Dynamic time warping (DTW) has been applied to a wide range of machine learning problems involving the comparison of time series. An important feature of such time series is that they can sometimes be sparse in the sense that the data takes zero value at many epochs. This corresponds for example to quiet periods in speech or to a lack of physical activity. However, employing conventional DTW for such sparse time series runs a full search ignoring the zero data. So a fast dynamic time warping algorithm that is exactly equivalent to DTW was developed for the unconstrained case where there is no global constraint on the permissible warping path. It was called sparse dynamic time warping (SDTW). In this paper we focus on the development and analysis of a fast dynamic time warping algorithm for the constrained case where there is a global constraint on the permissible warping path, specifically limit the width along the diagonal of the permissible path domain. We call this constrained sparse dynamic time warping (CSDTW). A careful formulation and analysis are performed to determine exactly how CSDTW should treat the zero data. It is shown that CSDTW reduces the computational complexity relative to constrained DTW by about three times the sparsity ratio, which is defined as the arithmetic mean of the fraction of non-zero's in the two time series. Numerical experiments confirm the speed advantage of CSDTW relative to constrained DTW for sparse time series with sparsity ratio up to 0.2-0.3. This study provides a benchmark and also background to potentially understand how to exploit such sparsity when the underlying time series is approximated to reduce complexity.