[Paper Review] Elastic Similarity and Distance Measures for Multivariate Time Series

Ahmed Shifaz, Charlotte Pelletier+2 more|arXiv (Cornell University)|Feb 19, 2021

Time Series Analysis and Forecasting28 references4 citations

TL;DR

This paper introduces multivariate extensions of seven elastic similarity and distance measures—DTW, WDTW, DDTW, LCSS, ERP, MSM, and TWE—using independent and dependent strategies inspired by multivariate DTW. It demonstrates that each measure outperforms others on at least one dataset, and that the choice between independent and dependent strategies is fundamentally dataset-dependent, with an ensemble of these measures achieving state-of-the-art accuracy in multivariate time series classification.

ABSTRACT

This paper contributes multivariate versions of seven commonly used elastic similarity and distance measures for time series data analytics. Elastic similarity and distance measures are a class of similarity measures that can compensate for misalignments in the time axis of time series data. We adapt two existing strategies used in a multivariate version of the well-known Dynamic Time Warping (DTW), namely, Independent and Dependent DTW, to these seven measures. While these measures can be applied to various time series analysis tasks, we demonstrate their utility on multivariate time series classification using the nearest neighbor classifier. On 23 well-known datasets, we demonstrate that each of the measures but one achieves the highest accuracy relative to others on at least one dataset, supporting the value of developing a suite of multivariate similarity and distance measures. We also demonstrate that there are datasets for which either the dependent versions of all measures are more accurate than their independent counterparts or vice versa. In addition, we also construct a nearest neighbor-based ensemble of the measures and show that it is competitive to other state-of-the-art single-strategy multivariate time series classifiers.

Motivation & Objective

• To extend seven widely used univariate elastic similarity and distance measures to the multivariate time series setting.
• To evaluate the performance of independent and dependent strategies across multiple multivariate time series classification tasks.
• To determine whether the relative superiority of independent vs. dependent strategies in DTW is coincidental or fundamental to data structure.
• To develop and evaluate an ensemble classifier based on multiple multivariate similarity measures for improved classification accuracy.
• To establish a benchmark for multivariate time series classification using similarity-based methods, paving the way for future hybrid models.

Proposed method

• Adapt the independent strategy by applying each univariate measure separately to each dimension and summing the results.
• Adapt the dependent strategy by treating each time step as a multidimensional point and computing distances using multivariate distance metrics.
• Extend seven univariate measures—DTW, WDTW, DDTW, LCSS, ERP, MSM, and TWE—to multivariate form using both independent and dependent strategies.
• Implement nearest neighbor classification using each multivariate measure and evaluate performance on 23 benchmark datasets.
• Construct a multivariate Elastic Ensemble (MEE) that combines predictions from all multivariate measures using training accuracy for weighting.
• Use leave-one-out cross-validation to train the ensemble and evaluate generalization performance against state-of-the-art classifiers.

Experimental results

Research questions

• RQ1 Do multivariate extensions of commonly used univariate elastic similarity and distance measures improve classification performance on real-world datasets?
• RQ2 Is the performance advantage of independent vs. dependent strategies in multivariate DTW a coincidental artifact or a fundamental property of the data?
• RQ3 Are there datasets where the independent strategy consistently outperforms the dependent strategy across all measures, or vice versa?
• RQ4 Can an ensemble of multivariate similarity and distance measures achieve competitive accuracy with state-of-the-art single-strategy classifiers?
• RQ5 How do the computational costs of the proposed multivariate measures and ensemble compare to existing methods, and what are the scalability implications?

Key findings

• Each of the seven multivariate similarity and distance measures achieved the highest classification accuracy on at least one of the 23 benchmark datasets, demonstrating the value of a diverse toolkit.
• There exist datasets where the independent strategy consistently outperforms the dependent strategy across all measures, and others where the dependent strategy is superior, indicating a fundamental data-dependent relationship.
• The dependent strategy tends to underperform on high-dimensional datasets, suggesting limitations in capturing complex interdependencies across many dimensions.
• The independent strategy performs well when individual dimensions are strong univariate classifiers, indicating potential synergy when dimensions are relatively independent.
• The multivariate Elastic Ensemble (MEE), combining all proposed measures, achieved classification accuracy competitive with state-of-the-art single-strategy classifiers.
• Training time for the MEE ensemble is high, with DTWF_D taking just over one hour per fold, but the goal was accuracy benchmarking rather than scalability, mirroring the univariate Elastic Ensemble's role in advancing the field.