Adaptive Classification of Interval-Valued Time Series

TL;DR

This work tackles the classification of interval-valued time series, where intervals are converted into image representations to leverage powerful CNN-based classifiers. By representing each interval with a convex combination of its upper and lower bounds and learning the combination coefficients via ADMM, the method produces Recurrence Plot (RP) or Joint Recurrence Plot (JRP) images suitable for discriminative deep learning. The authors establish a margin-based multiclass generalization bound tying generalization performance to the network structure and complexity, and they validate the approach through extensive simulations and real-data applications, showing superior performance to representative-point baselines, especially in challenging scenarios with high class overlap. The framework applies to both univariate and multivariate interval-valued time series and offers a flexible, theory-grounded avenue for interval-aware time-series classification with potential extensions to other imaging schemes.

Abstract

In recent years, the modeling and analysis of interval-valued time series have garnered significant attention in the fields of econometrics and statistics. However, the existing literature primarily focuses on regression tasks while neglecting classification aspects. In this paper, we propose an adaptive approach for interval-valued time series classification. Specifically, we represent interval-valued time series using convex combinations of upper and lower bounds of intervals and transform these representations into images based on point-valued time series imaging methods. We utilize a fine-grained image classification neural network to classify these images, to achieve the goal of classifying the original interval-valued time series. This proposed method is applicable to both univariate and multivariate interval-valued time series. On the optimization front, we treat the convex combination coefficients as learnable parameters similar to the parameters of the neural network and provide an efficient estimation method based on the alternating direction method of multipliers (ADMM). On the theoretical front, under specific conditions, we establish a margin-based multiclass generalization bound for generic CNNs composed of basic blocks involving convolution, pooling, and fully connected layers. Through simulation studies and real data applications, we validate the effectiveness of the proposed method and compare its performance against a wide range of point-valued time series classification methods.

Figures (3)

• Figure 1: The graph of $(1 + \tanh(\nu x))/2$ for various values of $\nu$.
• Figure 2: Distributions of optimal convex combination coefficients obtained under different degrees of smoothing for the three DGPs. Each row represents a specific DGP, and each column represents a level of smoothing.
• Figure 3: Images obtained based on RP corresponding to DGP1 at various correlation coefficients and levels of approximation. Each row represents the images obtained at different levels of approximation for the same correlation coefficient (class), while each column indicates the images for different classes at the same level of approximation.

Theorems & Definitions (13)

• Lemma 4.1
• Lemma 4.2
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Lemma A1
• proof
• proof
• proof
• Lemma A2