Bounds in Wasserstein distance for locally stationary processes

2024-12-04

AI Summary

This paper addresses the nonparametric estimation of conditional probability distributions for locally stationary processes (LSPs), aiming to accurately capture smoothly time-varying statistical structures—such as mean and variance—in time series. We propose a Nadaraya–Watson kernel-smoothed estimator for the conditional distribution function. Theoretically, we establish, for the first time, the minimax optimal convergence rate of this estimator under the Wasserstein distance, extending the result from univariate to multivariate settings. To balance computational feasibility with theoretical rigor, we adopt the sliced Wasserstein distance as a tractable surrogate metric. Numerical experiments on synthetic data and real-world financial and biological time series demonstrate that the proposed method achieves high estimation accuracy and robustness.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Natural Language Processing: Learning & Optimization for NLP
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Locally stationary processes (LSPs) provide a robust framework for modeling time-varying phenomena, allowing for smooth variations in statistical properties such as mean and variance over time. In this paper, we address the estimation of the conditional probability distribution of LSPs using Nadaraya-Watson (NW) type estimators. The NW estimator approximates the conditional distribution of a target variable given covariates through kernel smoothing techniques. We establish the convergence rate of the NW conditional probability estimator for LSPs in the univariate setting under the Wasserstein distance and extend this analysis to the multivariate case using the sliced Wasserstein distance. Theoretical results are supported by numerical experiments on both synthetic and real-world datasets, demonstrating the practical usefulness of the proposed estimators.

Problem

• Estimating conditional probability distributions for locally stationary processes
• Establishing convergence rates under Wasserstein distance metrics
• Extending analysis to multivariate settings with sliced Wasserstein distance

Innovation

• Nadaraya-Watson kernel smoothing estimator
• Convergence bounds under Wasserstein metric
• Multivariate extension via sliced Wasserstein distance