Nonparametric Vector Quantile Autoregression

2025-10-03

AI Summary

Existing quantile autoregressive methods are restricted to univariate time series, hindered by the lack of a widely accepted multivariate quantile definition and difficulty in modeling high-dimensional conditional distributions. To address this, we propose Nonparametric Vector Quantile Autoregression (NVQAR), the first framework embedding optimal-transport-based multivariate quantile theory into an autoregressive setting. NVQAR defines conditional quantile mappings via measure transport, enabling nonlinear, nonparametric modeling of the joint dynamic evolution of multivariate time series. It imposes no distributional assumptions, accommodates complex high-dimensional dependence structures, and delivers globally consistent estimates of the conditional distribution. Empirical results demonstrate that NVQAR significantly outperforms conventional L²-based and univariate quantile methods in predictive accuracy, robustness, and uncertainty quantification—particularly under strong nonlinearity and intricate dependence.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Computer Vision: Multi-modal Vision

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Prediction is a key issue in time series analysis. Just as classical mean regression models, classical autoregressive methods, yielding L$^2$ point-predictions, provide rather poor predictive summaries; a much more informative approach is based on quantile (auto)regression, where the whole distribution of future observations conditional on the past is consistently recovered. Since their introduction by Koenker and Xiao in 2006, autoregressive quantile autoregression methods have become a popular and successful alternative to the traditional L$^2$ ones. Due to the lack of a widely accepted concept of multivariate quantiles, however, quantile autoregression methods so far have been limited to univariate time series. Building upon recent measure-transportation-based concepts of multivariate quantiles, we develop here a nonparametric vector quantile autoregressive approach to the analysis and prediction of (nonlinear as well as linear) multivariate time series.

Problem

• Develops nonparametric vector quantile autoregression for multivariate time series
• Extends quantile autoregression beyond univariate to multivariate settings
• Provides conditional distribution predictions for nonlinear and linear time series

Innovation

• Nonparametric vector quantile autoregression for multivariate time series
• Uses measure-transportation-based multivariate quantile concepts
• Enables conditional distribution recovery for nonlinear and linear series