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Interval-Valued Time Series Classification Using $D_K$-Distance

Wan Tian, Zhongfeng Qin

TL;DR

This work introduces an imaging-based framework for interval-valued time series classification by leveraging the $D_K$-distance to convert intervals into images via Interval Recurrence Plot (IRP) and Interval Joint Recurrence Plot (IJRP), enabling deep learning-based classification on image data. It provides a theoretical excess-risk bound for multi-classification using offset Rademacher complexity and demonstrates optimal convergence rates under mild conditions. Empirically, the approach consistently outperforms point-valued representations and competing methods across extensive simulations and real weather data, with kernels $K_4$ and $K_5$ that incorporate full interval information yielding the strongest results. The method's effectiveness on both univariate and multivariate interval-valued time series suggests strong practical impact for robust interval-aware analysis, with avenues for future extensions to alternative interval metrics and imaging schemes.

Abstract

In recent years, modeling and analysis of interval-valued time series have garnered increasing attention in econometrics, finance, and statistics. However, these studies have predominantly focused on statistical inference in the forecasting of univariate and multivariate interval-valued time series, overlooking another important aspect: classification. In this paper, we introduce a classification approach that treats intervals as unified entities, applicable to both univariate and multivariate interval-valued time series. Specifically, we first extend the point-valued time series imaging methods to interval-valued scenarios using the $D_K$-distance, enabling the imaging of interval-valued time series. Then, we employ suitable deep learning model for classification on the obtained imaging dataset, aiming to achieve classification for interval-valued time series. In theory, we derived a sharper excess risk bound for deep multiclassifiers based on offset Rademacher complexity. Finally, we validate the superiority of the proposed method through comparisons with various existing point-valued time series classification methods in both simulation studies and real data applications.

Interval-Valued Time Series Classification Using $D_K$-Distance

TL;DR

This work introduces an imaging-based framework for interval-valued time series classification by leveraging the -distance to convert intervals into images via Interval Recurrence Plot (IRP) and Interval Joint Recurrence Plot (IJRP), enabling deep learning-based classification on image data. It provides a theoretical excess-risk bound for multi-classification using offset Rademacher complexity and demonstrates optimal convergence rates under mild conditions. Empirically, the approach consistently outperforms point-valued representations and competing methods across extensive simulations and real weather data, with kernels and that incorporate full interval information yielding the strongest results. The method's effectiveness on both univariate and multivariate interval-valued time series suggests strong practical impact for robust interval-aware analysis, with avenues for future extensions to alternative interval metrics and imaging schemes.

Abstract

In recent years, modeling and analysis of interval-valued time series have garnered increasing attention in econometrics, finance, and statistics. However, these studies have predominantly focused on statistical inference in the forecasting of univariate and multivariate interval-valued time series, overlooking another important aspect: classification. In this paper, we introduce a classification approach that treats intervals as unified entities, applicable to both univariate and multivariate interval-valued time series. Specifically, we first extend the point-valued time series imaging methods to interval-valued scenarios using the -distance, enabling the imaging of interval-valued time series. Then, we employ suitable deep learning model for classification on the obtained imaging dataset, aiming to achieve classification for interval-valued time series. In theory, we derived a sharper excess risk bound for deep multiclassifiers based on offset Rademacher complexity. Finally, we validate the superiority of the proposed method through comparisons with various existing point-valued time series classification methods in both simulation studies and real data applications.

Paper Structure

This paper contains 17 sections, 3 theorems, 43 equations, 2 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. The $D_K$-distance
  4. Interval recurrence plot
  5. Interval joint recurrence plot
  6. Classification based on imaging datasets
  7. Excess risk bound
  8. Simulation Studies
  9. Simulation design
  10. Univariate interval-valued time series classification
  11. Multivariate interval-valued time series classification
  12. Real Data Examples
  13. Univariate interval-valued time series classification
  14. Multivariate interval-valued time series classification
  15. Fine-grained classification
  16. ...and 2 more sections

Key Result

Lemma 3.1

Suppose Conditions 1 and 2 hold, then for any $f_1, f_2 \in {\mathcal{F}}$, we have where $\ell$ is the Lipschitz constant of the auxiliary function $L(\cdot)$.

Figures (2)

  • Figure 1: The technical roadmap of classifying interval-valued time series by transforming them into images.
  • Figure 2: Imaging results for DGP1 based on IRP under different correlation coefficients and kernel functions. Each row represents the images obtained with different correlation coefficients under the same kernel function. Similarly, each column displays the images for the same correlation coefficient using different kernel functions.

Theorems & Definitions (7)

  • Lemma 3.1
  • Theorem 3.1
  • Definition 3.1: Modified empirical covering number
  • Theorem 3.2
  • proof
  • proof
  • proof