#### Degree Type

Dissertation

#### Date of Award

1986

#### Degree Name

Doctor of Philosophy

#### Department

Statistics

#### Abstract

A spatial experiment is a comparative experiment in which the experimental units are distributed throughout a region in d-dimensional Euclidean space. The classical analysis of variance of such an experiment ignores spatial correlation among the responses. An alternative approach to the analysis of spatial experiments is proposed that does not ignore spatial correlation. In this approach, the outcome of a spatial experiment is regarded as a single realization of a collection of random variables indexed by points in d-dimensional Euclidean space. Such a collection of random variables is called a random field;In conjunction with this approach, we adopt the linear model y = X(beta) + e, where y is an nxl random vector whose elements are observable members of a d-dimensional random field F(,Y) = Y(,s): s (epsilon) (//R)('d) , e is an nxl random vector whose elements are unobservable members of a random field F(,Z) = Z(,s): s (epsilon) (//R)('d) satisying E Z(,s) = 0 for all s (epsilon) (//R)('d), X is an nxp matrix whose elements are functions of s, and (beta) is a pxl vector of unknown parameters. The elements of the covariance matrix V of e are given by evaluating a generally non- linear function C((.),(.);(theta)) of 2d variables, where (theta) is an mxl vector of unknown parameters, at the sites where F(,Y) is observed; this func- tion is called the covariogram of F(,Y). We refer to this model as the random field linear model (RFLM);The estimation of the RFLM parameters, (beta) and (theta) by maximum likelihood approaches is studied extensively. These estimation pro- cedures are generally quite burdensome computationally; however, certain features of the RFLM can, in many cases, be exploited to reduce the amount of computation. The required amount of com- putation is primarily related to the structure of V which, in turn, is greatly affected by the spatial configuration of the sites at which F(,Y) is observed and by properties of the covariogram of F(,Y). Also inves- tigated are conditions under which the estimators of (beta) and (theta) are consistent and asymptotically normal;Estimation of treatment contrasts in a spatial experiment is of major interest. Results from "pseudo-experiments" based on uni- formity trial data suggest that the random field approach is superior to the classical analysis of variance and to several recently proposed "nearest-neighbor" methods.

#### DOI

https://doi.org/10.31274/rtd-180813-12338

#### Publisher

Digital Repository @ Iowa State University, http://lib.dr.iastate.edu/

#### Copyright Owner

Dale Lee Zimmerman

#### Copyright Date

1986

#### Language

en

#### Proquest ID

AAI8703782

#### File Format

application/pdf

#### File Size

242 pages

#### Recommended Citation

Zimmerman, Dale Lee, "A random field approach to spatial experiments " (1986). *Retrospective Theses and Dissertations*. 8501.

https://lib.dr.iastate.edu/rtd/8501