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A Locally Optimal Algorithm for Estimating a Generating Partition from an Observed Time Series and Its Application to

Najah F Ghalyan1, David J Miller2, Asok Ray3

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This study introduces a new algorithm for generating partitions in dynamical systems, improving time series symbolization for better analysis and anomaly detection. The method guarantees local optimality, outperforming existing techniques in chaotic map symbolization and material science applications.

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Area of Science:

  • Dynamical Systems and Time Series Analysis
  • Information Theory
  • Applied Mathematics

Background:

  • Symbolization of discrete-time dynamical systems is crucial for characterizing system dynamics and enabling applications like time series classification and anomaly detection.
  • Existing methods for estimating generating partitions, like the Hirata et al. objective function, suffer from non-convergent algorithms and heuristic nearest neighbor assignments.
  • The Hirata et al. objective function, while effective, is often misinterpreted as a state-space partitioning problem, rather than a clustering problem in an infinite-dimensional space.

Purpose of the Study:

  • To develop a novel, locally optimal algorithm for estimating generating partitions by minimizing the Hirata et al. objective function.
  • To address the non-convergence issues of previous algorithms and provide guaranteed discrepancy descent for joint, locally optimal symbolization.
  • To demonstrate the effectiveness of the new approach in improving time series symbolization and its application in anomaly detection.

Main Methods:

  • Development of an iterative nearest-neighbor symbol assignment algorithm with guaranteed discrepancy descent.
  • Framing generating partition estimation as a clustering problem in the space of the entire time series, rather than state-space partitioning.
  • Application of the algorithm to symbolizing chaotic maps and time-series anomaly detection in a polycrystalline alloy material.

Main Results:

  • The proposed algorithm achieves joint, locally optimal symbolization of the entire time series.
  • Demonstrated improvement over popular methods for symbolizing chaotic maps using several performance measures.
  • Successful application of the approach to time-series anomaly detection, including in a real-world material science context.

Conclusions:

  • The novel algorithm provides a robust and convergent method for generating partition estimation, advancing the field of time series analysis.
  • The approach offers a new perspective on lossy source coding and clustering in infinite-dimensional spaces.
  • The method shows significant potential for enhancing the accuracy and reliability of time series classification and anomaly detection in diverse applications.