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Nonparametric regression of spatio-temporal data using infinite-dimensional covariates

Subhrajyoty Roy, Soudeep Deb, Sayar Karmakar, Rishideep Roy

Abstract

In spatio-temporal analysis, we often record data at specific time intervals but with varying spatial locations between these timepoints. We propose a conditional model to analyze such spatio-temporal data that accommodates the dependencies alongside second-order stationary explanatory variables, which may be infinite-dimensional and accommodate spatio-temporal covariates. Because of the absence of a mixing-type dependence condition in this case, which is typically required by the existing studies, we consider a weaker polynomially decaying moment contraction (PMC) condition on the covariates. In this paper, we obtain nonparametric point estimates of the mean and covariate functions of such a regression model, which we then show to be statistically consistent. We also obtain a simultaneous confidence interval of the mean function using the central limit theorem for the proposed estimator. Such simultaneous inference tools can be used to test for certain specifications of the mean function. Some simulation studies and two real-data analyses have been illustrated to corroborate the findings.

Nonparametric regression of spatio-temporal data using infinite-dimensional covariates

Abstract

In spatio-temporal analysis, we often record data at specific time intervals but with varying spatial locations between these timepoints. We propose a conditional model to analyze such spatio-temporal data that accommodates the dependencies alongside second-order stationary explanatory variables, which may be infinite-dimensional and accommodate spatio-temporal covariates. Because of the absence of a mixing-type dependence condition in this case, which is typically required by the existing studies, we consider a weaker polynomially decaying moment contraction (PMC) condition on the covariates. In this paper, we obtain nonparametric point estimates of the mean and covariate functions of such a regression model, which we then show to be statistically consistent. We also obtain a simultaneous confidence interval of the mean function using the central limit theorem for the proposed estimator. Such simultaneous inference tools can be used to test for certain specifications of the mean function. Some simulation studies and two real-data analyses have been illustrated to corroborate the findings.

Paper Structure

This paper contains 29 sections, 16 theorems, 116 equations, 10 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Existing literature and our contribution
  3. Notations
  4. Nonparametric estimation
  5. Mathematical Framework
  6. Estimation methodology
  7. Choice of the Kernel and the Bandwidth
  8. Asymptotic properties
  9. Consistency of the estimator of mean
  10. Asymptotic distribution of the estimate of mean
  11. Simultaneous Confidence Interval Estimation
  12. Real-life example: Analyzing air pollution in Delhi
  13. Application of the proposed method
  14. Sensitivity analysis
  15. Conclusion
  16. ...and 14 more sections

Key Result

Proposition 1

Let $f : \chi \rightarrow \mathbb{R}$ be a function such that it is measurable and Lipschitz (under the scaled norm given in eqn:scaled-l2-space) with a constant $C$, i.e., $\left\lvert f(\boldsymbol{x}) - f(\boldsymbol{y}) \right\rvert \leqslant C\Vert{\boldsymbol{D}^{-1}(\boldsymbol{x} - \boldsymb $\blacktriangleleft$$\blacktriangleleft$

Figures (10)

  • Figure 1: A heatplot indicating the missingness of the $\mathrm{PM}2.5$ observations across locations (columns) and timepoints (rows).
  • Figure 2: Position and expected goals (xG) of shots taken against Chelsea during the league 2014-15 by the players of Arsenal, Liverpool, and Manchester City.
  • Figure 3: The locations of the measurement stations across India and their status (Left), and in the union territory of Delhi region (Right)
  • Figure 4: Hourly measurements of PM2.5 in different measurement stations in Delhi for the year 2020; some representative locations are highlighted by colour (y-axis is in log-scale).
  • Figure 5: Estimated mean levels of PM$2.5$-concentration across the entire Delhi region for three different months and during two specific times of the day (Morning 10 AM and evening 8 PM). The black plus points indicate the locations of the measurement stations.
  • ...and 5 more figures

Theorems & Definitions (21)

  • Proposition 1
  • Corollary 1
  • Proposition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 2
  • Theorem 4
  • Corollary 3
  • Theorem 5
  • ...and 11 more