Elastic bands across the path: A new framework and methods to lower bound DTW

TL;DR

This paper introduces a new class of lower bounds for DTW that are tighter and more effective across various warping window sizes, significantly improving the efficiency of NN-DTW classification.

Contribution

The authors propose a novel set of lower bounds leveraging DTW constraints, maintaining tightness even at large windows, and offering a better speed-tightness trade-off for NN-DTW.

Findings

New lower bounds are tighter than Keogh's across window sizes.

The bounds significantly reduce DTW computations in classification tasks.

Enhanced efficiency in NN-DTW with minimal additional computation.

Abstract

There has been renewed recent interest in developing effective lower bounds for Dynamic Time Warping (DTW) distance between time series. These have many applications in time series indexing, clustering, forecasting, regression and classification. One of the key time series classification algorithms, the nearest neighbor algorithm with DTW distance (NN-DTW) is very expensive to compute, due to the quadratic complexity of DTW. Lower bound search can speed up NN-DTW substantially. An effective and tight lower bound quickly prunes off unpromising nearest neighbor candidates from the search space and minimises the number of the costly DTW computations. The speed up provided by lower bound search becomes increasingly critical as training set size increases. Different lower bounds provide different trade-offs between computation time and tightness. Most existing lower bounds interact with DTW warping window sizes. They are very tight and effective at smaller warping window sizes, but become looser as the warping window increases, thus reducing the pruning effectiveness for NN-DTW. In this work, we present a new class of lower bounds that are tighter than the popular Keogh lower bound, while requiring similar computation time. Our new lower bounds take advantage of the DTW boundary condition, monotonicity and continuity constraints to create a tighter lower bound. Of particular significance, they remain relatively tight even for large windows. A single parameter to these new lower bounds controls the speed-tightness trade-off. We demonstrate that these new lower bounds provide an exceptional balance between computation time and tightness for the NN-DTW time series classification task, resulting in greatly improved efficiency for NN-DTW lower bound search.