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An autocovariance-based learning framework for high-dimensional functional time series

Jinyuan Chang, Cheng Chen, Xinghao Qiao, Qiwei Yao

Abstract

Many scientific and economic applications involve the statistical learning of high-dimensional functional time series, where the number of functional variables is comparable to, or even greater than, the number of serially dependent functional observations. In this paper, we model observed functional time series, which are subject to errors in the sense that each functional datum arises as the sum of two uncorrelated components, one dynamic and one white noise. Motivated from the fact that the autocovariance function of observed functional time series automatically filters out the noise term, we propose a three-step procedure by first performing autocovariance-based dimension reduction, then formulating a novel autocovariance-based block regularized minimum distance estimation framework to produce block sparse estimates, and based on which obtaining the final functional sparse estimates. We investigate theoretical properties of the proposed estimators, and illustrate the proposed estimation procedure via three sparse high-dimensional functional time series models. We demonstrate via both simulated and real datasets that our proposed estimators significantly outperform the competitors.

An autocovariance-based learning framework for high-dimensional functional time series

Abstract

Many scientific and economic applications involve the statistical learning of high-dimensional functional time series, where the number of functional variables is comparable to, or even greater than, the number of serially dependent functional observations. In this paper, we model observed functional time series, which are subject to errors in the sense that each functional datum arises as the sum of two uncorrelated components, one dynamic and one white noise. Motivated from the fact that the autocovariance function of observed functional time series automatically filters out the noise term, we propose a three-step procedure by first performing autocovariance-based dimension reduction, then formulating a novel autocovariance-based block regularized minimum distance estimation framework to produce block sparse estimates, and based on which obtaining the final functional sparse estimates. We investigate theoretical properties of the proposed estimators, and illustrate the proposed estimation procedure via three sparse high-dimensional functional time series models. We demonstrate via both simulated and real datasets that our proposed estimators significantly outperform the competitors.

Paper Structure

This paper contains 28 sections, 15 theorems, 109 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Autocovariance-based dimension reduction
  3. Methodology
  4. Rates in elementwise $\ell_{\infty}$-norm
  5. Block RMD estimation framework
  6. An illustrative example
  7. A general estimation procedure
  8. Theoretical properties
  9. Applications
  10. High-dimensional SFLR
  11. High-dimensional FFLR
  12. High-dimensional VFAR
  13. Empirical studies
  14. Simulation study
  15. Real data analysis
  16. ...and 13 more sections

Key Result

Theorem 1

Let Conditions cond_fsm--cond_eigen hold, and $d$ be a positive integer possibly depending on $(n,p)$. For $n \gtrsim \log p$, there exist some positive constants $c_1$ and $c_2$ independent of $(n,p,d)$ such that holds with probability greater than $1- c_1 p^{-c_2}$, where ${\cal M}_1^W$ is defined in def_sub_fsm.

Figures (1)

  • Figure 1: The boxplots of relative estimation errors for (a) VFAR, (b) SFLR and (c) FFLR.

Theorems & Definitions (21)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Remark 2.1
  • Theorem 3
  • Remark 3.1
  • Proposition 1
  • Theorem 4
  • Remark 4.1
  • Theorem 5
  • ...and 11 more