Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-Stationary Covariance Functions for Spatial Data on Linear Networks

Alfredo Alegría

Abstract

We introduce a novel class of non-stationary covariance functions for random fields on linear networks that allows both the variance and the correlation range of the random field to vary spatially. The proposed covariance functions are useful to model random fields with a spatial dependence that is locally isotropic with respect to the resistance metric, a distance that reflects the topology of the network. The framework admits explicit stochastic representations of the associated random fields and can be naturally extended to matrix-valued covariance functions for vector-valued random fields. We assess the statistical and computational performance of a weighted local likelihood estimator for the proposed models using synthetic data generated on the street network of the University of Chicago neighborhood.

Non-Stationary Covariance Functions for Spatial Data on Linear Networks

Abstract

We introduce a novel class of non-stationary covariance functions for random fields on linear networks that allows both the variance and the correlation range of the random field to vary spatially. The proposed covariance functions are useful to model random fields with a spatial dependence that is locally isotropic with respect to the resistance metric, a distance that reflects the topology of the network. The framework admits explicit stochastic representations of the associated random fields and can be naturally extended to matrix-valued covariance functions for vector-valued random fields. We assess the statistical and computational performance of a weighted local likelihood estimator for the proposed models using synthetic data generated on the street network of the University of Chicago neighborhood.
Paper Structure (16 sections, 3 theorems, 40 equations, 5 figures, 1 table)

This paper contains 16 sections, 3 theorems, 40 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Resistance metric
  3. Non-stationary covariance functions
  4. Main result
  5. Examples
  6. Illustration
  7. Extensions of the framework
  8. Stochastic representation
  9. Vector-valued GRFs
  10. Weighted local likelihood
  11. Description of the method
  12. Simulation
  13. Discussion
  14. Proof of Theorem \ref{['teo1']}
  15. Proof of Proposition \ref{['prop:stochastic']}
  16. ...and 1 more sections

Key Result

Theorem 3.1

Let $a: \mathcal{L} \rightarrow (0,\infty)$ and $b: \mathcal{L} \rightarrow (0,\infty)$, and define $\alpha(s,t)= [a(s) + a(t)]/2$. Let $\text{d}_R(s,t)$ be the resistance metric between $s$ and $t$, and let $F$ denote a positive finite measure on $(0,\infty)$. Then, the function is positive semidefinite on $\mathcal{L}\times \mathcal{L}$.

Figures (5)

  • Figure 3.1: Functions $a$ (left) and $c$ (right), defined in Equations (\ref{['eq:aa']}) and (\ref{['eq:bb']}), respectively, on the spiders linear network. The circle indicates the point $s^{*}\in\mathcal{L}$ involved in the definitions of $a$ and $c$.
  • Figure 3.2: Mapping $\text{d}_R \mapsto \sqrt{a_0}/\sqrt{a_0 + \text{d}_R }$ for selected values of $a_0$, illustrating how the local isotropic correlation structure of the non-stationary covariance function (\ref{['cauchy']}) depends on $a_0$.
  • Figure 3.3: (Left) Simulated GRF on the spiders linear network, with the non-stationary covariance function given in Equation (\ref{['cauchy']}). (Right) Corresponding realization of a LGCP, with line thickness proportional to the magnitude of the log Gaussian field.
  • Figure 5.1: (Left) The function $a$ used in our simulations on the Chicago street network. Six randomly selected locations where $a$ will be estimated are highlighted. (Right) Decay of the weights as a function of the resistance metric for the compactly supported ($\tau=120$) and Gaussian ($\eta=120$) weighting schemes.
  • Figure 5.2: Centered boxplots of the estimates of $a$, at the six selected sites, across 100 independent realizations of a GRF with covariance function (\ref{['cauchy']}). Results are shown for the compactly supported and Gaussian weighting schemes.

Theorems & Definitions (6)

  • Theorem 3.1
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Proposition 4.1
  • Theorem 4.1