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Estimation of High-dimensional Nonlinear Vector Autoregressive Models

Yuefeng Han, Likai Chen, Wei Biao Wu

TL;DR

The paper addresses high-dimensional time series with nonlinear dynamics by proposing a sparse additive nonlinear VAR framework built from basis expansions, enabling flexible modeling beyond linear VARs. It develops sharp Bernstein-type tail inequalities for nonlinear VAR processes with non-Gaussian innovations, providing the probabilistic foundation for subsequent high-dimensional analysis. The authors establish convergence rates and model-selection consistency for an $\ell_1$-regularized estimator based on a truncated basis expansion, with explicit bias-variance trade-offs tied to sparsity, smoothness ($\beta$), and truncation level ($L$). A scalable block-coordinate-descent algorithm is proposed, and the framework is validated through simulations and a real gene-regulatory-network time series, demonstrating improved performance over linear VAR in capturing nonlinear interactions. Overall, the work extends VAR theory to nonlinear, non-Gaussian, high-dimensional settings and offers practical tools for inference in complex time-series systems such as genomics and neuroscience.

Abstract

High-dimensional vector autoregressive (VAR) models have numerous applications in fields such as econometrics, biology, climatology, among others. While prior research has mainly focused on linear VAR models, these approaches can be restrictive in practice. To address this, we introduce a high-dimensional non-parametric sparse additive model, providing a more flexible framework. Our method employs basis expansions to construct high-dimensional nonlinear VAR models. We derive convergence rates and model selection consistency for least squared estimators, considering dependence measures of the processes, error moment conditions, sparsity, and basis expansions. Our theory significantly extends prior linear VAR models by incorporating both non-Gaussianity and non-linearity. As a key contribution, we derive sharp Bernstein-type inequalities for tail probabilities in both non-sub-Gaussian linear and nonlinear VAR processes, which match the classical Bernstein inequality for independent random variables. Additionally, we present numerical experiments that support our theoretical findings and demonstrate the advantages of the nonlinear VAR model for a gene expression time series dataset.

Estimation of High-dimensional Nonlinear Vector Autoregressive Models

TL;DR

The paper addresses high-dimensional time series with nonlinear dynamics by proposing a sparse additive nonlinear VAR framework built from basis expansions, enabling flexible modeling beyond linear VARs. It develops sharp Bernstein-type tail inequalities for nonlinear VAR processes with non-Gaussian innovations, providing the probabilistic foundation for subsequent high-dimensional analysis. The authors establish convergence rates and model-selection consistency for an -regularized estimator based on a truncated basis expansion, with explicit bias-variance trade-offs tied to sparsity, smoothness (), and truncation level (). A scalable block-coordinate-descent algorithm is proposed, and the framework is validated through simulations and a real gene-regulatory-network time series, demonstrating improved performance over linear VAR in capturing nonlinear interactions. Overall, the work extends VAR theory to nonlinear, non-Gaussian, high-dimensional settings and offers practical tools for inference in complex time-series systems such as genomics and neuroscience.

Abstract

High-dimensional vector autoregressive (VAR) models have numerous applications in fields such as econometrics, biology, climatology, among others. While prior research has mainly focused on linear VAR models, these approaches can be restrictive in practice. To address this, we introduce a high-dimensional non-parametric sparse additive model, providing a more flexible framework. Our method employs basis expansions to construct high-dimensional nonlinear VAR models. We derive convergence rates and model selection consistency for least squared estimators, considering dependence measures of the processes, error moment conditions, sparsity, and basis expansions. Our theory significantly extends prior linear VAR models by incorporating both non-Gaussianity and non-linearity. As a key contribution, we derive sharp Bernstein-type inequalities for tail probabilities in both non-sub-Gaussian linear and nonlinear VAR processes, which match the classical Bernstein inequality for independent random variables. Additionally, we present numerical experiments that support our theoretical findings and demonstrate the advantages of the nonlinear VAR model for a gene expression time series dataset.

Paper Structure

This paper contains 11 sections, 13 theorems, 179 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Bernstein Inequalities for Nonlinear VAR Processes
  3. Proof Sketch.
  4. Sparse additive nonlinear VAR models
  5. The model
  6. Asymptotic properties
  7. Simulation Studies
  8. Real Data Analysis
  9. Proofs
  10. Proofs of Theorems in Section \ref{['section:bern']}
  11. Proofs of Theorems in Section \ref{['section:var']}

Key Result

Theorem 1

Consider the VAR process defined in eq:autoregmodel, where the function $h$ satisfies Assumption asmp:LipconstH. Let $g$ be any function satisfying Assumption asmp:functiong with $\tau=\|G\|_1$. Then:

Figures (5)

  • Figure 1: Three different network matrix patterns used in the simulation studies. Here gray points represent the zero entries and black points represent nonzero entries.
  • Figure 2: The bacterial SOS DNA repair system
  • Figure 3: The target SOS DNA repair network
  • Figure 4: Reconstruction of SOS DNA repair network by nonlinear VAR model
  • Figure 5: Reconstruction of SOS DNA repair network by linear VAR model

Theorems & Definitions (34)

  • Remark 1: Existence of stationary distribution
  • Remark 2
  • Theorem 1
  • Remark 3
  • Corollary 1
  • Remark 4
  • Proposition 1
  • Remark 5
  • Proposition 2
  • Theorem 2
  • ...and 24 more