Gunter Spöck

Spatial sampling design based on spectral approximations to the random field

During the last 20 years several software packages have become available for spatial statistics. Spatial statistics deals with geo-referenced data and loosely speaking may be subdivided into the areas point processes analysis, areal and lattice data analysis and geostatistics. The topic of this article is geostatistics, the science of continuous stochastic processes that are defined either over some region in 2- or 3-dimensional geographic space or in space-time. Geostatistics is best known under the heading of kriging and covariance function estimation. A lot of free and commercial software packages are nowadays available for these tasks of optimal spatial interpolation and determination of the roughness of spatial random fields. When interpolating a spatial random field by means of kriging the uncertainty and accuracy of the kriging predictions are communicated by means of the so-called kriging variances. The kriging variances are dependent on the number and the density of the available gauged data locations. The denser the grid of available gauged data locations the smaller become the kriging variances and the better become the kriging predictions. Unfortunately, for the task of optimal planning prior to data gathering where to locate the monitoring stations or samples almost no software is freely available up to date. This article reports on a MATLAB and Octave toolbox whose main task is the optimal planning of monitoring networks. Both, addition of sampling locations to available networks and the reduction of monitoring networks are considered in an optimal way by means of borrowing ideas from convex experimental design theory and regression models with random coefficients. Both, design criteria for optimal interpolation with the covariance function assumed to be fixed and certain as well as a criterion where the uncertainty of the covariance function estimation is taken into account are developed and optimal designs are calculated by means of deterministic algorithms that fully make use of the mathematical structure of the considered design criteria.

Spatial Prediction and Optimized Sampling Design for Sodium Concentration in Groundwater

Sodium is an integral part of water, and its excessive amount in drinking water causes high blood pressure and hypertension. In the present paper, spatial distribution of sodium concentration in drinking water is modeled and optimized sampling designs for selecting sampling locations is calculated for three divisions in Punjab, Pakistan. Universal kriging and Bayesian universal kriging are used to predict the sodium concentrations. Spatial simulated annealing is used to generate optimized sampling designs. Different estimation methods (i.e., maximum likelihood, restricted maximum likelihood, ordinary least squares, and weighted least squares) are used to estimate the parameters of the variogram model (i.e, exponential, Gaussian, spherical and cubic). It is concluded that Bayesian universal kriging fits better than universal kriging. It is also observed that the universal kriging predictor provides minimum mean universal kriging variance for both adding and deleting locations during sampling design.

Incorporating covariance estimation uncertainty in spatial sampling design for prediction with trans-Gaussian random fields

Recently, Spock and Pilz [38], demonstrated that the spatial sampling design problem for the Bayesian linear kriging predictor can be transformed to an equivalent experimental design problem for a linear regression model with stochastic regression coefficients and uncorrelated errors. The stochastic regression coefficients derive from the polar spectral approximation of the residual process. Thus, standard optimal convex experimental design theory can be used to calculate optimal spatial sampling designs. The design functionals considered in Spock and Pilz [38] did not take into account the fact that kriging is actually a plug-in predictor which uses the estimated covariance function. The resulting optimal designs were close to space-filling configurations, because the design criterion did not consider the uncertainty of the covariance function. In this paper we also assume that the covariance function is estimated, e.g., by restricted maximum likelihood (REML). We then develop a design criterion that fully takes account of the covariance uncertainty. The resulting designs are less regular and space- filling compared to those ignoring covariance uncertainty. The new designs, however, also require some closely spaced samples in order to improve the estimate of the covariance function. We also relax the assumption of Gaussian observations and assume that the data is transformed to Gaussianity by means of the Box-Cox transformation. The resulting prediction method is known as trans-Gaussian kriging. We apply the Smith and Zhu [37] approach to this kriging method and show that resulting optimal designs also depend on the available data. We illustrate our results with a data set of monthly rainfall measurements from Upper Austria.

A Bayesian Logistic Regression approach in Asthma Persistence Prediction

Background : A number of models based on clinical parameters have been used for the prediction of asthma persistence in children. The number and significance of factors that are used in a proposed model play a cardinal role in prediction accuracy. Different models may lead to different significant variables. In addition, the accuracy of a model in medicine is really important since an accurate prediction of illness persistence may improve prevention and treatment intervention for the children at risk. Methods : Data from 147 asthmatic children were analyzed by a new method for predicting asthma outcome using Principal Component Analysis (PCA) in combination with a Bayesian logistic regression approach implemented by the Markov Chain Monte Carlo (MCMC). The use of PCA is required due to multicollinearity among the explanatory variables. Results : This method using the most appropriate models seems to predict asthma with an accuracy of 84.076% and 86.3673%, a Sensitivity of 84.96% and 87.25% and a Specificity of 83.22% and 85.52%, respectively. Conclusion : Our approach predicts asthma with high accuracy, gives steadier results in terms of positive and negative patients and provides better information about the influence of each factor (demographic, symptoms etc.) in asthma prediction.

Evaluation of Bayesian classifiers in asthma exacerbation prediction after medication discontinuation

ObjectiveThe achievement of the optimal control of the disease is of cardinal importance in asthma treatment. As the control of the disease is sustained the medication should be gradually reduced and then stopped. Nevertheless, the discontinuation of asthma medication may lead to loss of disease control and eventually to an exacerbation of the disease. The goal of this paper is to examine the performance of Bayesian network classifiers in predicting asthma exacerbation based on several patient’s parameters such as objective measurements and medical history data.ResultsIn this study several Bayesian network classifiers are presented and evaluated. It is shown that the proposed semi-naive network classifier with the use of Backward Sequential Elimination and Joining algorithm is able to predict if a patient will have an exacerbation of the disease after his last assessment with 93.84% accuracy and 90.9% sensitivity. In addition, the resulting structure and the conditional probability tables give a clear view of the probabilistic relationships between the used factors. This network may help the clinicians to identify the patients who are at high risk of having an exacerbation after stopping the medication and to confirm which factors are the most important.

Optimal design of experiments for semiconductor lifetime data

Performing experiments is necessary to find influences of different factors on the measured output. In semiconductor industry experiments are mainly performed following predefined specifications and guidelines, given by experts for the device under test (DUT). The statistical method design of experiments (DoE) provides an objective solution to the question: which experiments have to be performed to get the most information concerning main influencing factors and interaction between factors. In practical usage classical DoE often reach their limits, especially when resources for experiments are meagre. A remedy is given by optimal DoEs. They are more flexible and offer the possibility to optimize e.g. the prediction accuracy on a pre-defined area, where performing measurements is difficult. For this purpose the IV-optimality criterion is used in this paper. On the basis of already performed experiments, an exchange algorithm proposed by Spoöck and Pilz [2] was used to select 3 further desired experiments. After their performance they were evaluated and, as expected, an improvement in the mean squared error of prediction (MSEP) was observed.