a b s t r a c tComputer simulations are often used to replace physical experiments for exploring the complex relationships between input and output variables. We study the optimal design problem for the prediction of a stationary Ornstein-Uhlenbeck sheet on a monotonic set with respect to the integrated mean square prediction error criterion and the entropy criterion. We show that there is a substantial difference between the shapes of optimal designs for Ornstein-Uhlenbeck processes and sheets. In particular, we show that the optimal prediction based on the integrated mean square prediction error does not necessarily lead to space-filling designs.
The understanding of methane emission and methane absorption plays a central
role both in the atmosphere and on the surface of the Earth. Several important
ecological processes, e.g., ebullition of methane and its natural
microergodicity request better designs for observations in order to decrease
variability in parameter estimation. Thus, a crucial fact, before the
measurements are taken, is to give an optimal design of the sites where
observations should be collected in order to stabilize the variability of
estimators. In this paper we introduce a realistic parametric model of
covariance and provide theoretical and numerical results on optimal designs.
For parameter estimation D-optimality, while for prediction integrated mean
square error and entropy criteria are used. We illustrate applicability of
obtained benchmark designs for increasing/measuring the efficiency of the
engineering designs for estimation of methane rate in various temperature
ranges and under different correlation parameters. We show that in most
situations these benchmark designs have higher efficiency.Comment: 25 pages, 4 figure
Physics, chemistry, biology or finance are just some examples out of the many fields where complex Ornstein-Uhlenbeck (OU) processes have various applications in statistical modelling. They play role e.g. in the description of the motion of a charged test particle in a constant magnetic field or in the study of rotating waves in time-dependent reaction diffusion systems, whereas Kolmogorov used such a process to model the so-called Chandler wobble, the small deviation in the Earth's axis of rotation. A common problem in these applications is deciding how to choose a set of a sample locations in order to predict a random process in an optimal way. We study the optimal design problem for the prediction of a complex OU process on a compact interval with respect to integrated mean square prediction error (IMSPE) and entropy criteria. We derive the exact forms of both criteria, moreover, we show that optimal designs based on entropy criterion are equidistant, whereas the IMSPE based ones may differ from it. Finally, we present some numerical experiments to illustrate selected cases of optimal designs for small number of sampling locations.
The risk of an individual woman having a pregnancy associated with Down's syndrome is estimated given her age, α-fetoprotein, human chorionic gonadotropin, and pregnancy-specific β1-glycoprotein levels. The classical estimation method is based on discriminant analysis under the assumption of lognormality of the marker values, but logistic regression is also applied for data classification. In the present work, we compare the performance of the two methods using a dataset containing the data of almost 89,000 unaffected and 333 affected pregnancies. Assuming lognormality of the marker values, we also calculate the theoretical detection and false positive rates for both the methods.
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