Werner G. Müller

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)

Optimal design of experiments subject to correlated errors

In this paper we consider optimal design of experiments in the case of correlated observations, when no replications are possible. This situation is typical when observing a random process or random field with known covariance structure. We present a theorem which demonstrates that the computation of optimum exact designs corresponds to solving minimization problems in terms of design measures.

Regional convergence in the European Union (1985-1999). A spatial dynamic panel analysis

The invention relates to a process for distillation of crude oils comprising i) passing a hydrocarbon crude oil into a preflash vessel maintained under conditions to separate the crude oil into a preflash liquid and a preflash vapor, ii) passing the pre-flash liquid into a furnace maintained under conditions to heat and partially vaporize the preflash liquid, iii) passing the heated furnace effluent into the lower part of a distillation column maintained under fractionating conditions, iv) passing the preflash vapor into the distillation column in a zone at the bottom of a stripping zone located below the introduction zone of the furnace effluent, and v) passing steam into the distillation column in a zone at the bottom of the stripping zone, such that liquid furnace effluent is contacted with steam and preflash vapor in the stripping zone under conditions sufficient to strip the liquid furnace effluent.

Design measures and approximate information matrices for experiments without replications

The concept of design measures is fundamental in the classical setup of experiments without restrictions on replications of observations (resulting designs are called continuous or approximate). In this paper we extend this concept to experiments, when no such replications are allowed. The resulting interpretation of the design measure is different from the classical case. We introduce a corresponding information matrix, which approximates the Fisher information matrix for exact designs. The basic aim of the paper is to find an optimum exact design with a restricted total number of observations. The usefulness of the developed approach is illustrated by an algorithm that is employed to calculate optimal or improved designs numerically.

Least-squares fitting from the variogram cloud

Several authors have noticed poor finite sample behavior of the most widely used parametric variogram estimator by Cressie (1985). This behavior is most notable in the case that fitting is done on the collection of squared differences of observations from all point pairs, the so-called variogram cloud. This can indeed be made plausible by asymptotical considerations and these motivate a corresponding simple correction. An alternative, even more efficient computer-intensive estimator is also proposed. The benefits of both methods are demonstrated by Monte-Carlo simulation.

Another view on optimal design for estimating the point of extremum in quadratic regression

In this paper we illustrate how certain design problems can be simplified by reparametrization of the response function. This alternative viewpoint provides further insights than the more traditional approaches, like minimax, Bayesian or sequential techniques. It will also improve a practitioner’s understanding of more general situations and their “classical” treatment.

Moving local regression: The weight function

Moving local regression is a nonparametric technique for smoothing, interpolating and forecasting by means of locally fitted regression models. The paper explores the “optimal” structure of the weight function, taking into account the location of supporting points and the suspected behaviour of the remainder term, and surveys results or choice of weight functions in traditional moving local regression approaches.

Discrimination Between Two Binary Data Models. Sequentially Designed Experiments.

In this paper we propose a sequential procedure to design optimum experiments for discriminating between two binary data models. For the problem to be fully specified, not only the mode1link functions should be provided but also their associated linear predictor structures. Further, we suppose that one of the models is true, albeit it is not known which of them. Under these assumptions the procedure consists of making sequential choices of single experimental units to discriminate between the rival models as efficiently as possible. Depending on whether the models are nested or not, alternative methods are proposed. To illustrate the procedure, a simulation study for the classical case of pro bit versus logit model is presented. It enables us to estimate the total sample sizes required to gain a certain power of discrimination and compare them to sample sizes for methods that were previously suggested in the literature.

Residual diagnostics for variogram fitting

The variogram cloud is increasingly employed in geostatistics both as an exploratory tool as well as for variogram fitting. Unfortunately in its pure form the variogram cloud cannot be effectively used to assess the fit of a corresponding parametric variogram model. This is due to the fact that its entries are highly correlated and heteroscedastic (and possibly distorted by anisotropy). We propose fitting of a model by feasible generalized least squares, suggest some diagnostics for assessing the fit and identifying potential outliers. For illustrative purposes we apply them to an example data set of chloride concentrations in the Sudliche Tullnerfeld in Austria. An R-package called vardiag for the interactive exploration of the proposed diagnostics was provided.

Optimal design for local fitting

Abstract This paper provides optimal design strategies for local (polynomial) fitting by the so-called moving local regression, a well-known nonparametric statistical tool. A Kiefer—Wolfowitz type equivalence theorem is formulated. Some examples are employed to illustrate the relations and differences to parametric techniques.