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CORAL: Concept Drift Representation Learning for Co-evolving Time-series

Kunpeng Xu, Lifei Chen, Shengrui Wang

TL;DR

CORAL addresses concept drift in co-evolving time series by learning a kernel-induced self-representation that yields a time-varying representation matrix $\mathbf{Z}$ with a block-diagonal structure corresponding to distinct concepts. The method maps series into a high-dimensional kernel space via $\boldsymbol{\mathcal{K}}$,optimizes for $k$-block diagonal structure, and uses spectral clustering to identify concepts, track drift across sliding windows, and forecast concept evolution. It integrates with deep learning backbones through Auto-CORAL, combining reconstruction, self-representation, sparsity, and temporal smoothness losses, and it scales with Nyström approximations and low-rank kernel techniques. Empirical results across synthetic and real-world datasets show CORAL effectively identifies concepts, tracks drift, and achieves competitive forecasting performance, while providing interpretable insights into inter-series dynamics. The framework offers a new paradigm for adaptive, interpretable concept drift analysis in multi-series ecosystems with potential applications in finance, healthcare, and climate modeling.

Abstract

In the realm of time series analysis, tackling the phenomenon of concept drift poses a significant challenge. Concept drift -- characterized by the evolving statistical properties of time series data, affects the reliability and accuracy of conventional analysis models. This is particularly evident in co-evolving scenarios where interactions among variables are crucial. This paper presents CORAL, a simple yet effective method that models time series as an evolving ecosystem to learn representations of concept drift. CORAL employs a kernel-induced self-representation learning to generate a representation matrix, encapsulating the inherent dynamics of co-evolving time series. This matrix serves as a key tool for identification and adaptation to concept drift by observing its temporal variations. Furthermore, CORAL effectively identifies prevailing patterns and offers insights into emerging trends through pattern evolution analysis. Our empirical evaluation of CORAL across various datasets demonstrates its effectiveness in handling the complexities of concept drift. This approach introduces a novel perspective in the theoretical domain of co-evolving time series analysis, enhancing adaptability and accuracy in the face of dynamic data environments, and can be easily integrated into most deep learning backbones.

CORAL: Concept Drift Representation Learning for Co-evolving Time-series

TL;DR

CORAL addresses concept drift in co-evolving time series by learning a kernel-induced self-representation that yields a time-varying representation matrix with a block-diagonal structure corresponding to distinct concepts. The method maps series into a high-dimensional kernel space via ,optimizes for -block diagonal structure, and uses spectral clustering to identify concepts, track drift across sliding windows, and forecast concept evolution. It integrates with deep learning backbones through Auto-CORAL, combining reconstruction, self-representation, sparsity, and temporal smoothness losses, and it scales with Nyström approximations and low-rank kernel techniques. Empirical results across synthetic and real-world datasets show CORAL effectively identifies concepts, tracks drift, and achieves competitive forecasting performance, while providing interpretable insights into inter-series dynamics. The framework offers a new paradigm for adaptive, interpretable concept drift analysis in multi-series ecosystems with potential applications in finance, healthcare, and climate modeling.

Abstract

In the realm of time series analysis, tackling the phenomenon of concept drift poses a significant challenge. Concept drift -- characterized by the evolving statistical properties of time series data, affects the reliability and accuracy of conventional analysis models. This is particularly evident in co-evolving scenarios where interactions among variables are crucial. This paper presents CORAL, a simple yet effective method that models time series as an evolving ecosystem to learn representations of concept drift. CORAL employs a kernel-induced self-representation learning to generate a representation matrix, encapsulating the inherent dynamics of co-evolving time series. This matrix serves as a key tool for identification and adaptation to concept drift by observing its temporal variations. Furthermore, CORAL effectively identifies prevailing patterns and offers insights into emerging trends through pattern evolution analysis. Our empirical evaluation of CORAL across various datasets demonstrates its effectiveness in handling the complexities of concept drift. This approach introduces a novel perspective in the theoretical domain of co-evolving time series analysis, enhancing adaptability and accuracy in the face of dynamic data environments, and can be easily integrated into most deep learning backbones.
Paper Structure (42 sections, 6 theorems, 31 equations, 16 figures, 8 tables, 2 algorithms)

This paper contains 42 sections, 6 theorems, 31 equations, 16 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The landscape of Concept Drift
  3. Preliminaries
  4. Self-representation Learning in Time Series
  5. Kernel Trick for Modeling Time Series
  6. CORAL
  7. Kernel-Induced Representation Learning
  8. Adaptation to Concept Drift
  9. Integration into Deep Learning Backbones
  10. Theoretical Analysis
  11. Behavior of the Representation Matrix
  12. Kernel-Induced Representation
  13. Experiments
  14. Data and experimental setup
  15. Q1: Effectiveness
  16. ...and 27 more sections

Key Result

Theorem 4.1

If the multiple time series $\mathbf{S}$ contains $k$ distinct concepts, then $\min \sum_{i=N-k+1}^N \lambda_i(\mathbf{L_{\mathbf{Z}}})$ is equivalent to $\mathbf{Z}$ being $k$-block diagonal.

Figures (16)

  • Figure 1: Modeling power of CORAL for co-evolving time series: CORAL treats time series as an ecosystem - , and automatically identifies, tracks, and predicts dynamic concepts. (a) Original time series; (b) CORAL-generated representation matrix with distinct concepts (C1-C5) in block diagonal form. The red star marks S1, and purple dashed lines trace S1’s concept drift over time; (c) Identified concepts within the series; (d) Concept drift (red dashed lines) and forecasted trends (grey areas).
  • Figure 2: Visualized results on Stock1.
  • Figure 3: Visualization of N-BEATS and CORAL on ETTh1.
  • Figure 4: Online forecasting results on Stock2
  • Figure 5: Partitioning of WS into two sets $E$ and $F$. The optimal window size is determined by the size corresponding to the border.
  • ...and 11 more figures

Theorems & Definitions (10)

  • Theorem 4.1
  • proof
  • Theorem 5.1
  • Theorem 5.2
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof