پژوهشهای ریاضی (Jun 2022)
Nonparametric Estimation of Spatial Risk for a Mean Nonstationary Random Field}
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.βinεi-εj2-α-βT si-sj-u2 KGsi-sj-u, where G is the corresponding bandwidth matrix, that obtained from minimizing cross-validation relative squared error of semi-variogram estimate. Algorithm1: Semiparametric Bootstrap Obtain estimates of the error covariance and nonparametric residuals covariance. Generate bootstrap samples with the estimated spatial trend μHs and adding bootstrap errors generated as a spatially correlated set of errors. Compute the kriging prediction Z*s0 at each unsampled location s0 from the bootstrap sample Z*s1,…,Z*sn. Repeat steps 2 and 3 a larger number times B. Therefore, for each un-sampled location s0, B bootstrap replications Z*(1)s0.…. Z*(B)(s0) are obtained. Calculate (1) at position s_0 by calculating the relative frequency of Bootstrap repetition as follows to estimate the unconditional probability of excess of boundary c. rcs0= 1B j=1BIZ*js0≥ c Results and discussion To analysis the practical behavior of the proposed methods a simulation study is conducted under different scenarios. For N=150 samples and n=16×16 were generated on a regular grid in the unit square following model (2), with mean function μs=2.5 + sin( 2π x1) + 4x2 - 0.5 2, and random errors normally distributed with zero mean and isotropic exponential covariogram Ch= 0.04 + 2.01 1- exp-3 ∥ h∥0.5, h∈ R2. For comparing nonparametric spatial methods for estimate unconditional risk, conditional risk, and Indicator kriging, we considered 7 missing observations in certain situations. Empirical spatial risk and its estimates are presented in Table 1. The Indicator kriging is overestimating and estimate spatial risk larger than 1. Generally, an estimated risk with unconditional and conditional methods is near value to empirical value. Table 1. Empirical spatial risk and its estimates Locations (0.13, 0) (0.87, 0.87) (0.80, 0.20) (0.94, 0.27) (0, 0.47) (074, .60) (0.34, 0.60) Methods 0.999 0.300 0.069 0.317 0.504 0.011 0.989 Empirical 0.998 0.351 0.054 0.347 0.494 0.057 0.954 Conditional 1.002 0.230 0.091 0.091 0.652 0.006 0.996 Indicator 1.000 0.388 0.418 0.481 0.602 0.024 0.994 Unconditional The spatial risk mapping for the maximum temperature means of Iran in 364 stations in March 2018 is obtained. By applying Algorithm 1 final trend and semi-variogram estimates are smoother than the pilot version. The conditional and unconditional spatial risk with 150 bootstrap replicates for two values of threshold 25 and 31 on a 75×75 grid are estimated. The unconditional risk estimate is smoother than the conditional risk estimate. Because of this in the unconditional version, biased residual unused directly in the spatial prediction but in the conditional risk estimating, original residuals and simple kriging used. Conclusion The spatial risk estimated with the nonparametric spatial method. For the trend and variability of the random field, modeling applied a local linear nonparametric model. In the simulation study, this method better results than Indicator kriging. Because the flexibility of the nonparametric spatial method could apply for the construction of confidence or prediction intervals and hypothesis testing.
