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Inferring the Most Similar Variable-length Subsequences between Multidimensional Time Series

Thanadej Rattanakornphan, Piyanon Charoenpoonpanich, Chainarong Amornbunchornvej

TL;DR

The paper tackles inferring the most similar subsequences between two multidimensional time series when subsequences may differ in length. It introduces an exact generalized DTW-based framework (VLSubsequenceInferFunction) that leverages lower/upper bounds to prune unlikely pairs and then computes exact DTW on a small candidate set to guarantee optimality. The approach achieves significant runtime improvements over brute-force baselines in simulations (up to about 4x faster) and real-world data (up to 20x faster), and demonstrations on baboon movement and stock-price datasets show its practical value for detecting coordinated patterns and dependencies. The framework is general to any multidimensional time series and is backed by theoretical guarantees, with publicly available code and data for reproducibility.

Abstract

Finding the most similar subsequences between two multidimensional time series has many applications: e.g. capturing dependency in stock market or discovering coordinated movement of baboons. Considering one pattern occurring in one time series, we might be wondering whether the same pattern occurs in another time series with some distortion that might have a different length. Nevertheless, to the best of our knowledge, there is no efficient framework that deals with this problem yet. In this work, we propose an algorithm that provides the exact solution of finding the most similar multidimensional subsequences between time series where there is a difference in length both between time series and between subsequences. The algorithm is built based on theoretical guarantee of correctness and efficiency. The result in simulation datasets illustrated that our approach not just only provided correct solution, but it also utilized running time only quarter of time compared against the baseline approaches. In real-world datasets, it extracted the most similar subsequences even faster (up to 20 times faster against baseline methods) and provided insights regarding the situation in stock market and following relations of multidimensional time series of baboon movement. Our approach can be used for any time series. The code and datasets of this work are provided for the public use.

Inferring the Most Similar Variable-length Subsequences between Multidimensional Time Series

TL;DR

The paper tackles inferring the most similar subsequences between two multidimensional time series when subsequences may differ in length. It introduces an exact generalized DTW-based framework (VLSubsequenceInferFunction) that leverages lower/upper bounds to prune unlikely pairs and then computes exact DTW on a small candidate set to guarantee optimality. The approach achieves significant runtime improvements over brute-force baselines in simulations (up to about 4x faster) and real-world data (up to 20x faster), and demonstrations on baboon movement and stock-price datasets show its practical value for detecting coordinated patterns and dependencies. The framework is general to any multidimensional time series and is backed by theoretical guarantees, with publicly available code and data for reproducibility.

Abstract

Finding the most similar subsequences between two multidimensional time series has many applications: e.g. capturing dependency in stock market or discovering coordinated movement of baboons. Considering one pattern occurring in one time series, we might be wondering whether the same pattern occurs in another time series with some distortion that might have a different length. Nevertheless, to the best of our knowledge, there is no efficient framework that deals with this problem yet. In this work, we propose an algorithm that provides the exact solution of finding the most similar multidimensional subsequences between time series where there is a difference in length both between time series and between subsequences. The algorithm is built based on theoretical guarantee of correctness and efficiency. The result in simulation datasets illustrated that our approach not just only provided correct solution, but it also utilized running time only quarter of time compared against the baseline approaches. In real-world datasets, it extracted the most similar subsequences even faster (up to 20 times faster against baseline methods) and provided insights regarding the situation in stock market and following relations of multidimensional time series of baboon movement. Our approach can be used for any time series. The code and datasets of this work are provided for the public use.
Paper Structure (16 sections, 9 theorems, 1 equation, 8 figures)

This paper contains 16 sections, 9 theorems, 1 equation, 8 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Problem Statement
  4. Methods
  5. Time Complexity
  6. Experiments
  7. Experimental setup
  8. Time series simulation
  9. Real world dataset
  10. Baboon leader/follower time series
  11. Stock prices of companies in similar sector vs those in unrelated sector
  12. Results
  13. Simulation results
  14. Case study: leadership of coordinated movement of baboons
  15. Case study: Stock prices of companies in similar sector vs those in unrelated sector
  16. ...and 1 more sections

Key Result

Proposition 3.1

Given subsequence $U_{i,\omega_U}$, $W_{j,\omega_W}$ where $\omega_U \geq \omega_W$, and a distance matrix $M_{U, W} = M$, the following inequality holds: $DTW_L(U_{i,\omega_U}, W_{j,\omega_W}) \leq DTW(U_{i,\omega_U}, W_{j,\omega_W})$

Figures (8)

  • Figure 1: A high-level overview of the proposed framework. Given a pair of time series and time window $\omega_1,\omega_2$, the framework infers the most similar subsequences of length $\omega_1,\omega_2$ respectively.
  • Figure 2: An example of simulated time series that contains the most similar subsequences with different lengths: 60 for X and 80 for Y (orange). The proposed algorithm correctly identified intervals that contain most similar subsequence in both time series (yellow highlight).
  • Figure 3: A comparison of running time of methods vs. a) the length of time series to find the most similar subsequences and b) the window size with the fix time series length as 2000.
  • Figure 4: A comparison of noise level vs. a) the running time and b) F1 score of inferring correct subsequences. The time series length is 2000 time steps.
  • Figure 5: Heatmaps of lead difference during a) the first 100 seconds and b) the last 500 seconds. The y-axis is leader and x-axis is follower. Given top-1000 most similar subsequences, how much leader leads follower = # of leading subsequence pairs - # of following subsequence pairs (defined in Subsection \ref{['subsec:caseBaboon']}).
  • ...and 3 more figures

Theorems & Definitions (18)

  • Definition 1: Time series and its subsequence
  • Definition 2: Warping path
  • Definition 3: Distance between subsequences of time series
  • Definition 4: Dynamic time warping distance between subsequences of time series
  • Definition 5: Distance matrix of time series
  • Definition 6: Lower bound of DTW
  • Definition 7: Upper bound of DTW
  • Definition 8: Domain of interest
  • Definition 9: Dynamic time warping matrix
  • Proposition 3.1
  • ...and 8 more