Quasi-Monte Carlo Methods in Designs of Spatial Sampling Points
The key ingredient of the quasi-Monte Carlo method is to generate a set of points with a low discrepancy. One of the basic components in statistical inference for continuous spatial processes is to design a set of sampling points properly so as to minimiz
- PDF / 1,259,023 Bytes
- 12 Pages / 439.37 x 666.14 pts Page_size
- 40 Downloads / 243 Views
1
Introduction
The quasi-Mon t e Carlo method is a mechani sm to generate a set of points wit h a low discrepan cy. It has been used to pr ovide reliable and efficient numerical computations of multi-dimensional int egrat ions, as t he resul ts in [1] indicate. It also pr ovides solut ions to a variety of pr oblems in probability and statist ics, such as generations of representative points of random vecto rs and design of compute r experiments , as introduced in [2] and [3]. Here, we discuss its application t o pr oblems of sampling designs in est imating an int egral of a spatial pr ocess and in regression est imation with correlated errors. These pr oblems ar ise in statist ical inference for spatial processes . It is genera lly straightforward to formulat e t he best linear unbiased est imator when statistical pr operti es of spat ial processes are eit her completely known or partially known up to a regression form , since t his pr oblem is linear in nature. There is also a relatively easy solution to findin g sa mpling points with good asympt ot ic properties in one dimension (see [4] and [5]) . However, designing sampling points of good qualities in two or high er dim ensions is not an easy task, as witnessed by th e work in [6]-[9] and the results surveyed in [4]. In th e setup of averaged case of det erministi c int egration, th e relevan ce of th e quasi-Monte Carlo method in est imat ion of an integral by a sample mean est imato r is established in [10]. It is our int enti on to enforce t his finding in t he conte xt of int egral est imat ion of spatial processes and regression est imation with corr elat ed err ors . * This resear ch work is supp orted by t he 1999 Facult y Resear ch Grant of Southwest
Missour i State Uni versity.
K.‒T. Fang et al. (eds.)., Monte Carlo and Quasi-Monte Carlo Methods 2000 © Springer-Verlag Berlin Heidelberg 2002
476
The problems of integral estimation and regression estimation with correlated errors are introduced first. For the self-containedness of the paper, the concept of the mean squared discrepancy of the quasi-Monte Carlo method is then considered. Its relevance to spatial sampling designs in integral estimation and regression estimation is discussed for spatial processes with no quadratic mean derivatives. For integral estimation, discussions are done in cases where the spatial process has zero mean, regression mean and unknown mean, respectively. Several equalities and inequalities concerning mean squared errors of integral estimation and regression estimation are established.
2
Estimation of Integrals of Spatial Processes
Consider a spatial process X(t) on the d-dimensional unit cube C = [O,l]d with zero mean EX(t) = O. The values of X(t) are correlated according to a covariance Cov(X(t), X(s)) = R(t, s). Throughout the paper, observations of X at a number of N sampling points TN = {ti ,N }~l' ti,N E C are denoted by XlrN = (X(t1,N), . . . ,X(tN,N)). The variance-covariance matrix of XTN ' denoted by RTN = (COV(X(ti ,N),X(tj,N)))NxN = (R(ti ,N' tj,N))NXN
Data Loading...
