Nonparametric Estimation of Spatial Risk for a Mean Nonstationary Random Field}

Authors
Mohammad Moghadam
Mohsen Mohammadzadeh
Tarbiat Modares University
Abstract
Introduction

Estimating the spatial hazard, or in other words, the probability of exceeding a certain boundary is one of the important issues in environmental studies that are used to control the level of pollution and prevent damage from natural disasters. Risk zoning provides useful information to decision-makers; For example, in areas where spatial hazards are high, zoning is used to design preventive policies to avoid adverse effects on the environment or harm to humans.

Generally, the common spatial risk estimating methods are for stationary random fields. In addition, a parametric form is usually considered for the distribution and variogram of the random field. Whereas in practice, sometimes these assumptions are not realistic. For an example of these methods, we can point to the Indicator kriging, Disjunctive kriging, Geostatistical Markov Chain, and simple kriging. In practice utilize the parametric spatial models caused unreliable results. In this paper, we use a nonparametric spatial model to estimate the unconditional probability or spatial risk:

rcs0=PZs0⩾c. (1)

Because the conditional distribution at points close to the observations has less variability than the unconditional probability, nonparametric spatial methods will be used to estimate the unconditional probability.

Material and methods

Let Z=Zs1,…,ZsnT be an observation vector from the random field {Zs;s∈D⊆Rd} which is decomposed as follows

Zs=μs+εs, (2)

where μ(s) is the trend and ε(s) is the error term, that is a second-order stationary random field with zero mean and covariogram Ch=Covεs,εs+h. The local linear model for the trend is given by

μHs= e1TSsTWsSs-1 SsTWsZ≡ ϕTsZ,

where e1 is a vector with 1 in the first entry and all other entries 0, Ss is a matrix with ith row equal to (1, (si-s)T), Ws = diag {KHs1 – s,…,KH(sn-s)}, KHu=H-1K(H-1u), K is a triple multiplicative multivariate kernel function and H is a nonsingular symmetric d×d bandwidth matrix. In this model, the bandwidth matrix obtained from a bias corrected and estimated generalized cross-validation (CGCV).

From nonparametric residuals ε(s) = Z(s) -μ(s) a local linear estimate of the variogram 2 γ(⋅)is obtained as the solution of the following least-squares problem




minα.βi
Keywords
Bandwidth
Bias corrected of variogram
Bootstrap
Local linear estimator
Spatial risk
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Mathematical Research
Volume 8, Issue 2 - Serial Number 21
Spring 2022
Pages 242-265