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Bounds in Wasserstein Distance for Locally Stationary Functional Time Series

Jan Nino G. Tinio, Mokhtar Z. Alaya, Salim Bouzebda

TL;DR

The paper develops a Nadaraya-Watson (NW) approach to estimate the conditional distribution of locally stationary functional time series (LSFTS) in a semi-metric covariate space, and analyzes its convergence using the Wasserstein distance. By leveraging small-ball probabilities and β-mixing, the authors derive a finite-sample convergence rate of the NW estimator in Wasserstein distance as $E[W_1( hat{π}_t(⋅|x), π_t^*(⋅|x))] = O_P(1/(√T h φ(h)) + h)$ on a central time band $I_h$, and discuss bandwidth selection via leave-one-out cross-validation. Theoretical results are complemented by extensive synthetic and real-data experiments (SST and Nikkei225) that demonstrate decreasing Wasserstein error with larger $T$ and validate the practical performance of the proposed estimator. The work advances distributional regression for nonstationary functional data, offering principled guidance for bandwidth choices and demonstrating applicability to climate and financial time series. These results provide a robust framework for predicting entire conditional distributions in nonstationary functional settings, with potential extensions to missing data and integrated KDE-type approaches.

Abstract

Functional time series (FTS) extend traditional methodologies to accommodate data observed as functions/curves. A significant challenge in FTS consists of accurately capturing the time-dependence structure, especially with the presence of time-varying covariates. When analyzing time series with time-varying statistical properties, locally stationary time series (LSTS) provide a robust framework that allows smooth changes in mean and variance over time. This work investigates Nadaraya-Watson (NW) estimation procedure for the conditional distribution of locally stationary functional time series (LSFTS), where the covariates reside in a semi-metric space endowed with a semi-metric. Under small ball probability and mixing condition, we establish convergence rates of NW estimator for LSFTS with respect to Wasserstein distance. The finite-sample performances of the model and the estimation method are illustrated through extensive numerical experiments both on functional simulated and real data.

Bounds in Wasserstein Distance for Locally Stationary Functional Time Series

TL;DR

The paper develops a Nadaraya-Watson (NW) approach to estimate the conditional distribution of locally stationary functional time series (LSFTS) in a semi-metric covariate space, and analyzes its convergence using the Wasserstein distance. By leveraging small-ball probabilities and β-mixing, the authors derive a finite-sample convergence rate of the NW estimator in Wasserstein distance as on a central time band , and discuss bandwidth selection via leave-one-out cross-validation. Theoretical results are complemented by extensive synthetic and real-data experiments (SST and Nikkei225) that demonstrate decreasing Wasserstein error with larger and validate the practical performance of the proposed estimator. The work advances distributional regression for nonstationary functional data, offering principled guidance for bandwidth choices and demonstrating applicability to climate and financial time series. These results provide a robust framework for predicting entire conditional distributions in nonstationary functional settings, with potential extensions to missing data and integrated KDE-type approaches.

Abstract

Functional time series (FTS) extend traditional methodologies to accommodate data observed as functions/curves. A significant challenge in FTS consists of accurately capturing the time-dependence structure, especially with the presence of time-varying covariates. When analyzing time series with time-varying statistical properties, locally stationary time series (LSTS) provide a robust framework that allows smooth changes in mean and variance over time. This work investigates Nadaraya-Watson (NW) estimation procedure for the conditional distribution of locally stationary functional time series (LSFTS), where the covariates reside in a semi-metric space endowed with a semi-metric. Under small ball probability and mixing condition, we establish convergence rates of NW estimator for LSFTS with respect to Wasserstein distance. The finite-sample performances of the model and the estimation method are illustrated through extensive numerical experiments both on functional simulated and real data.

Paper Structure

This paper contains 42 sections, 8 theorems, 154 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Layout.
  4. Notation.
  5. Background on LSTS and Wasserstein distance
  6. Locally stationary time series
  7. Example.
  8. Wasserstein distance
  9. Small ball probability
  10. Fractional Brownian Motion.
  11. Gaussian process.
  12. Ornstein-Unhlenbeck process.
  13. Mixing condition
  14. Nadaraya-Watson estimation with Wasserstein distance
  15. Remark.
  16. ...and 27 more sections

Key Result

Theorem 1

Suppose Assumptions Assumption: X is lsp - assumption: blocking are satisfied and define $I_h = [C_1h, 1 - C_1h]$. Then

Figures (7)

  • Figure 1: Realizations of Gaussian tvFAR(1) $X_{t,T}(\tau)$ for all $t$ and some $\tau$ for $T=100$ with $J=7$ and $N=100$ discretization points of $\tau\in[0,1]$.
  • Figure 2: Realizations of Gaussian tvFAR(2) $X_{t,T}(\tau)$ for all $t$ and some $\tau$ for $T=100$ with $J=7$ and $N=100$ discretization points of $\tau\in[0,1]$.
  • Figure 3: Time plots of response variables for $T=1000$
  • Figure 4: Wasserstein distances $\pm$ standard deviation at different $u = \frac{t}{T}$ for $T = 500, 1000, 5000, 10000$ using $L=500$ replications and 50 Monte Carlo runs.
  • Figure 5: SST monthly time series from Jan. 1950 - Dec. 2024
  • ...and 2 more figures

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • ...and 1 more