Multiscale Dubuc: A New Similarity Measure for Time Series

TL;DR

This study presents a different similarity measure that fuses the notion of Dubuc’s variation from fractal analysis with the Intersection- over-Union (IoU) measure which is widely used in object recognition and proves that it is a metric, possessing desirable properties such as the triangle inequality.

Abstract

Quantifying similarities between time series in a meaningful way remains a challenge in time series analysis, despite many advances in the field. Most real-world solutions still rely on a few popular measures, such as Euclidean Distance (EuD), Longest Common Subsequence (LCSS), and Dynamic Time Warping (DTW). The strengths and weaknesses of these measures have been studied extensively, and incremental improvements have been proposed. In this study, however, we present a different similarity measure that fuses the notion of Dubuc's variation from fractal analysis with the Intersection-over-Union (IoU) measure which is widely used in object recognition (also known as the Jaccard Index). In this proof-of-concept paper, we introduce the Multiscale Dubuc Distance (MDD) measure and prove that it is a metric, possessing desirable properties such as the triangle inequality. We use 95 datasets from the UCR Time Series Classification Archive to compare MDD's performance with EuD, LCSS, and DTW. Our experiments show that MDD's overall success, without any case-specific customization, is comparable to DTW with optimized window sizes per dataset. We also highlight several datasets where MDD's performance improves significantly when its single parameter is customized. This customization serves as a powerful tool for gauging MDD's sensitivity to noise. Lastly, we show that MDD's running time is linear in the length of the time series, which is crucial for real-world applications involving very large datasets.

Figures (3)

• Figure 1: The graphic showing the idea behind the Multisclae Dubuc Similarity ($\mathcal{MDS}$) measure. Two time series $(x_t)_{t=1}^{d}$ and $(y_t)_{t=1}^{d}$ are compared using their identified envelopes (regions with dashed borders) for $\varepsilon = 2$. $\mathcal{MDS}$ quantifies the similarity between the two time series at different granularity levels by changing $\varepsilon$, and computes the intersection ratio (see Eq. \ref{['eq:intersection-ratio']}) at each step. It then aggregates the intersection ratios to return a real value between 0 and 1 as the similarity (or distance) between the two time series. The thick black bars represent the total intersection between the envelopes.
• Figure 2: The bar plot showing the average expected (blue) and actual (orange) accuracy values of the 1-NN classifier on 95 UCR datasets, using TS-MIoU, EuD, DTW, and $\mathcal{MDD}$ distance functions.
• Figure 3: The Texas Sharpshooter plot showing the accuracy gain of $\mathcal{MDD}$ over EuD, DTW, and TS-MIoU.