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Block Toeplitz Sparse Precision Matrix Estimation for Large-Scale Interval-Valued Time Series Forecasting

Wan Tian, Zhongfeng Qin

TL;DR

This work tackles large-scale interval-valued time series forecasting by introducing a feature extraction framework that jointly automates segmentation, clustering, and representation learning. Core ideas include estimating a collection of block Toeplitz sparse precision matrices to capture cluster-specific conditional dependencies, and transforming subsequences into image data via Joint Recurrence Plot for transfer-learning-based feature extraction using WS-DAN. The approach is optimized through a majorization-minimization scheme that alternates between assigning subsequences to clusters (via Viterbi) and estimating the precision matrices (via ADMM with SCAD), with proven convergence. Empirical results on stock data demonstrate that the learned representations substantially improve forecasting accuracy over traditional methods and many deep-learning baselines. The methodology offers a scalable pathway to leverage invariant structure in ITS for enhanced predictive performance in finance and related domains.

Abstract

Modeling and forecasting interval-valued time series (ITS) have attracted considerable attention due to their growing presence in various contexts. To the best of our knowledge, there have been no efforts to model large-scale ITS. In this paper, we propose a feature extraction procedure for large-scale ITS, which involves key steps such as auto-segmentation and clustering, and feature transfer learning. This procedure can be seamlessly integrated with any suitable prediction models for forecasting purposes. Specifically, we transform the automatic segmentation and clustering of ITS into the estimation of Toeplitz sparse precision matrices and assignment set. The majorization-minimization algorithm is employed to convert this highly non-convex optimization problem into two subproblems. We derive efficient dynamic programming and alternating direction method to solve these two subproblems alternately and establish their convergence properties. By employing the Joint Recurrence Plot (JRP) to image subsequence and assigning a class label to each cluster, an image dataset is constructed. Then, an appropriate neural network is chosen to train on this image dataset and used to extract features for the next step of forecasting. Real data applications demonstrate that the proposed method can effectively obtain invariant representations of the raw data and enhance forecasting performance.

Block Toeplitz Sparse Precision Matrix Estimation for Large-Scale Interval-Valued Time Series Forecasting

TL;DR

This work tackles large-scale interval-valued time series forecasting by introducing a feature extraction framework that jointly automates segmentation, clustering, and representation learning. Core ideas include estimating a collection of block Toeplitz sparse precision matrices to capture cluster-specific conditional dependencies, and transforming subsequences into image data via Joint Recurrence Plot for transfer-learning-based feature extraction using WS-DAN. The approach is optimized through a majorization-minimization scheme that alternates between assigning subsequences to clusters (via Viterbi) and estimating the precision matrices (via ADMM with SCAD), with proven convergence. Empirical results on stock data demonstrate that the learned representations substantially improve forecasting accuracy over traditional methods and many deep-learning baselines. The methodology offers a scalable pathway to leverage invariant structure in ITS for enhanced predictive performance in finance and related domains.

Abstract

Modeling and forecasting interval-valued time series (ITS) have attracted considerable attention due to their growing presence in various contexts. To the best of our knowledge, there have been no efforts to model large-scale ITS. In this paper, we propose a feature extraction procedure for large-scale ITS, which involves key steps such as auto-segmentation and clustering, and feature transfer learning. This procedure can be seamlessly integrated with any suitable prediction models for forecasting purposes. Specifically, we transform the automatic segmentation and clustering of ITS into the estimation of Toeplitz sparse precision matrices and assignment set. The majorization-minimization algorithm is employed to convert this highly non-convex optimization problem into two subproblems. We derive efficient dynamic programming and alternating direction method to solve these two subproblems alternately and establish their convergence properties. By employing the Joint Recurrence Plot (JRP) to image subsequence and assigning a class label to each cluster, an image dataset is constructed. Then, an appropriate neural network is chosen to train on this image dataset and used to extract features for the next step of forecasting. Real data applications demonstrate that the proposed method can effectively obtain invariant representations of the raw data and enhance forecasting performance.

Paper Structure

This paper contains 15 sections, 2 theorems, 86 equations, 2 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. The feature extraction procedure
  3. Auto-segmentation and clustering of ITS
  4. Multivariate time series imaging
  5. The construction of feature extraction network
  6. The optimization aspect
  7. Estimation of assignment set
  8. Estimation of the precision matrices
  9. Hyperparameters selection
  10. Convergence of the algorithm
  11. Empirical analysis
  12. Conclusions
  13. Proofs for Results
  14. The computation procedure of equation (\ref{['partialdevwork']})
  15. Table of selected stocks

Key Result

Lemma 4.1

Assume that ($\widehat{{\bm{\Theta}}}^{+}, \widehat{{\bm{\Gamma}}}^{+}$) is an optimal solution of (Mstep), and $\widehat{{\bm{\Lambda}}}^{+}$ is the corresponding optimal dual variable associate with the equality constrain ${\bm{\Theta}} ={\bm{\Gamma}}$. Then the sequence $\left\{\left({\bm{\Theta} where ${\bm{U}}^+ = \left(\widehat{{\bm{\Theta}}}^{+}, \widehat{{\bm{\Gamma}}}^{+}\right)^\top$ and

Figures (2)

  • Figure 1: The technical roadmap of the proposed method includes auto-segmentation and clustering, multivariate time series imaging, feature extraction network training, feature extraction, and forecasting.
  • Figure 2: From top to bottom, the figures show the loss on the training set and the root mean squared error on the test set during the training process of the deep learning method ResCNN for window widths of 10, 15, 20, 25, 30, and 40, respectively.

Theorems & Definitions (4)

  • Lemma 4.1
  • Theorem 4.1
  • proof
  • proof