The target of this paper is to introduce the Optimal Experimental Design Problem (OEDP) for the Envirometrics. In principle an OEDP is a non-linear one, in the sense that the underlying model describing the phenomenon is a non-linear model, Ford et al. (1989). When there is a "rhythm" an optimal design problem can be introduced for Fourier models through lattices, Bates et al (1998), while a geometrical approach for D a c-optimality can be adopted as in Kitsos et al (1988). For an introduction to Fourier analysis in the Atmospheric Sciences see Wilks (1995), when the problem is to fit the model.
This paper extensively discusses the non-linear exponential model and the optimal design approach for situations adopting this model. As examples consider the following:

1. The amount of dust deposit E on the collecting plates as a function of the amount of outcoming dust-emission, the amount of incoming dust, migration velocity, total collecting area, total gas flow.

2. The change in atmospheric pressure P.

3. The radiation flux I, as a function of the wavelength.

C.P. KITSOS
Department of Human Resources and Development
University of Athens
Athens, Greece

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 talk, 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 (non-linear) 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 variogram 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.

WERNER G. MUELLER
Department of Statistics
University of Economics
Augasse 2-6
Vienna,  A-1090, Austria
werner.mueller@wu-wien.ac.at
 
 

In general an optimum design depends on unknown parameters and can therefore not be used in practice. It is also the case that designs based on guessed values of the parameter values can be very inefficient, even if the guessed values are close to the true values. An alternative approach is to define a range of parameter values along with a probability measure reflecting the experimenters belief and construct a design that is "optimal on the average". The Laplace design is defined as the optimal design when a uniform distribution is used over the specified range of parameter values.

In this paper we consider Laplace designs when the experiment has a finite number of treatments and there is a binary response. In particular, we give an algorithm for computation of Laplace designs and, in a few examples, compare the efficiencies of the Laplace design and the optimum design based on the true parameter values.

HANS NYQUIST
Department of Statistics
University of Umea
S-901 87 Umea, Sweden
hans.nyquist@stat.umu.se