Dale L. Zimmerman

An Experimental Comparison of Ordinary and Universal Kriging and Inverse Distance Weighting

A factorial, computational experiment was conducted to compare the spatial interpolation accuracy of ordinary and universal kriging and two types of inverse squared-distance weighting. The experiment considered, in addition to these four interpolation methods, the effects of four data and sampling characteristics: surface type, sampling pattern, noise level, and strength of small-scale spatial correlation. Interpolation accuracy was measured by the natural logarithm of the mean squared interpolation error. Main effects of all five factors, all two-factor interactions, and several three-factor interactions were highly statistically significant. Among numerous findings, the most striking was that the two kriging methods were substantially superior to the inverse distance weighting methods over all levels of surface type, sampling pattern, noise, and correlation.

A comparison of spatial semivariogram estimators and corresponding ordinary Kriging predictors

Predicting values of a spatially distributed variable, such as the concentration of a mineral throughout an ore body or the level of contamination around a toxic-waste dump, can be accomplished by a regression procedure known as kriging. Kriging and other types of statistical inference for spatially distributed variables are based on models of stochastic processes {Y t: t ∊ D} called random-field models. A commonly used class of random-field models are the intrinsic models, for which the mean is constant, and half of the variance of Yt , – Ys . is a function, called the semivariogram, of the difference t – s. The type of kriging corresponding to an intrinsic model is called ordinary kriging. The semivariogram, which typically is taken to depend on one or more unknown parameters, must be estimated prior to ordinary kriging. Various estimators of the semivariograms parameters have been proposed. For two Gaussian intrinsic random-field models, we compare, by a Monte Carlo simulation study, the performance o...

Parametric modelling of growth curve data: An overview

In the past two decades a parametric multivariate regression modelling approach for analyzing growth curve data has achieved prominence. The approach, which has several advantages over classical analysis-of-variance and general multivariate approaches, consists of postulating, fitting, evaluating, and comparing parametric models for the datas mean structure and covariance structure. This article provides an overview of the approach, using unified terminology and notation. Well-established models and some developed more recently are described, with emphasis given to those models that allow for nonstationarity and for measurement times that differ across subjects and are unequally spaced. Graphical diagnostics that can assist with model postulation and evaluation are discussed, as are more formal methods for fitting and comparing models. Three examples serve to illustrate the methodology and to reveal the relative strengths and weaknesses of the various parametric models.

Mean squared prediction error in the spatial linear model with estimated covariance parameters

The problem considered is that of predicting the value of a linear functional of a random field when the parameter vector θ of the covariance function (or generalized covariance function) is unknown. The customary predictor when θ is unknown, which we call the EBLUP, is obtained by substituting an estimator Ĝj for θ in the expression for the best linear unbiased predictor (BLUP). Similarly, the customary estimator of the mean squared prediction error (MSPE) of the EBLUP is obtained by substituting Ĝj for θ in the expression f for the BLUPs MSPE; we call this the EMSPE. In this article, the appropriateness of the EMSPE as an estimator of the EBLUPs MSPE is examined, and alternative estimators of the EBLUPs MSPE for use when the EMSPE is inappropriate are suggested. Several illustrative examples show that the performance of the EMSPE depends on the strength of spatial correlation; the EMSPE is at its best when the spatial correlation is strong.

Optimal designs for variogram estimation

The variogram plays a central role in the analysis of geostatistical data. A valid variogram model is selected and the parameters of that model are estimated before kriging (spatial prediction) is performed. These inference procedures are generally based upon examination of the empirical variogram, which consists of average squared differences of data taken at sites lagged the same distance apart in the same direction. The ability of the analyst to estimate variogram parameters efficiently is affected significantly by the sampling design, i.e., the spatial configuration of sites where measurements are taken. In this paper, we propose design criteria that, in contrast to some previously proposed criteria oriented towards kriging with a known variogram, emphasize the accurate estimation of the variogram. These criteria are modifications of design criteria that are popular in the context of (nonlinear) regression models. The two main distinguishing features of the present context are that the addition of a single site to the design produces as many new lags as there are existing sites and hence also produces that many new squared differences from which the variograrn is estimated. Secondly, those squared differences are generally correlated, which inhibits the use of many standard design methods that rest upon the assumption of uncorrelated errors. Several approaches to design construction which account for these features are described and illustrated with two examples. We compare their efficiency to simple random sampling and regular and space-filling designs and find considerable improvements. (authors abstract)

Geocoding accuracy and the recovery of relationships between environmental exposures and health

BackgroundThis research develops methods for determining the effect of geocoding quality on relationships between environmental exposures and health. The likelihood of detecting an existing relationship – statistical power – between measures of environmental exposures and health depends not only on the strength of the relationship but also on the level of positional accuracy and completeness of the geocodes from which the measures of environmental exposure are made. This paper summarizes the results of simulation studies conducted to examine the impact of inaccuracies of geocoded addresses generated by three types of geocoding processes: a) addresses located on orthophoto maps, b) addresses matched to TIGER files (U.S Census or their derivative street files); and, c) addresses from E-911 geocodes (developed by local authorities for emergency dispatch purposes).ResultsThe simulated odds of disease using exposures modelled from the highest quality geocodes could be sufficiently recovered using other, more commonly used, geocoding processes such as TIGER and E-911; however, the strength of the odds relationship between disease exposures modelled at geocodes generally declined with decreasing geocoding accuracy.ConclusionAlthough these specific results cannot be generalized to new situations, the methods used to determine the sensitivity of results can be used in new situations. Estimated measures of positional accuracy must be used in the interpretation of results of analyses that investigate relationships between health outcomes and exposures measured at residential locations. Analyses similar to those employed in this paper can be used to validate interpretation of results from empirical analyses that use geocoded locations with estimated measures of positional accuracy.

Another look at anisotropy in geostatistics

A thorough geostatistical data analysis includes a careful study of how the datas second-order variation, as characterized by the semivariogram, depends on the relative orientation of data locations. If the semivariogram depends on only the (Euclidean) distance between locations, then the semivariogram is isotropic; otherwise, it is anisotropic. In this article, I take another look at the modeling of anisotropy in geostatistics. A new, more specific classification of types of anisotropy is proposed. More importantly, some heretofore inadequately understood implications of the dependence of various semivariogram attributes on direction are discussed, and the wisdom of some current practices for modeling the direction-dependence of these attributes is questioned.

Computationally exploitable structure of covariance matrices and generalized convariance matrices in spatial models

Many spatial analyses based on random field models, as varied as kriging, likelihood-based estimation of autocovariance functions, and optimal design of spatial experiments, require the repeated evaluation of a covariance matrix V or a kth-order generalized covariance matrix K and its subsequent inversion. This is generally a formidable computing problem for moderate and large data sets. In this article, however, it is shown that under certain model assumptions, V (or K) possesses one of several types of patterned structure that, if exploited, can significantly reduce the computational burden of the analysis. These patterned structures are characterized, and their implications for matrix evaluation and inversion are considered. The usefulness of the results is illustrated with a soil pH data set

Computationally efficient restricted maximum likelihood estimation of generalized covariance functions

Computational aspects of the estimation of generalized covariance functions by the method of restricted maximum likelihood (REML) are considered in detail. In general, REML estimation is computationally intensive, but significant computational savings are available in important special cases. The approach taken here restricts attention to data whose spatial configuration is a regular lattice, but makes no restrictions on the number of parameters involved in the generalized covariance nor (with the exception of one result) on the nature of the generalized covariance functions dependence on those parameters. Thus, this approach complements the recent work of L. G. Barendregt (1987), who considered computational aspects of REML estimation in the context of arbitrary spatial data configurations, but restricted attention to generalized covariances which are linear functions of only two parameters.

Antedependence Models for longitudinal Data

Introduction Longitudinal data Classical methods of analysis Parametric modeling Antedependence models, in brief A motivating example Overview of the book Four featured data sets Unstructured Antedependence Models Antedependent random variables Antecorrelation and partial antecorrelation Equivalent characterizations Some results on determinants and traces The first-order case Variable-order antedependence Other conditional independence models Structured Antedependence Models Stationary autoregressive models Heterogeneous autoregressive models Integrated autoregressive models Integrated antedependence models Unconstrained linear models Power law models Variable-order SAD models Nonlinear stationary autoregressive models Comparisons with other models Informal Model Identification Identifying mean structure Identifying covariance structure: summary statistics Identifying covariance structure: graphical methods Concluding remarks Likelihood-Based Estimation Normal linear AD(p) model Estimation in the general case Unstructured antedependence: balanced data Unstructured antedependence: unbalanced data Structured antedependence models Concluding remarks Testing Hypotheses on the Covariance Structure Tests on individual parameters Testing for the order of antedependence Testing for structured antedependence Testing for homogeneity across groups Penalized likelihood criteria Concluding remarks Testing Hypotheses on the Mean Structure One-sample case Two-sample case Multivariate regression mean Other situations Penalized likelihood criteria Concluding remarks Case Studies A coherent parametric modeling approach Case study #1: Cattle growth data Case study #2: 100-km race data Case study #3: Speech recognition data Case study #4: Fruit fly mortality data Other studies Discussion Further Topics and Extensions Alternative estimation methods Nonlinear mean structure Discrimination under antedependence Multivariate antedependence models Spatial antedependence models Antedependence models for discrete data Appendix 1: Some Matrix Results Appendix 2: Proofs of Theorems 2.5 and 2.6 References Index