Random Function Model Research Articles

A selective review of recent characterizations of stochastic choice models using distribution and functional equation techniques

Abstract There are two major classes of stochastic choice models that have received extensive study: random function models assume that the choice probabilities can be represented by some deterministic function of the values of an appropriate random vector over the potential choice options; constant function models assume that the choice probabilities can be represented by some deterministic function of appropriate vectors of real-valued scale values over the potential choice options. This paper summarizes recent characterizations of broad classes of such stochastic choice models, illustrating the use of distribution and functional equation theory techniques in this domain, and states various open problems associated with such characterizations. The emphasis is on random and constant function models related to generalized extreme value models, with some comment on recent work on generalized normal models.

Prediction of tracer plume migration in disordered porous media by the method of conditional probabilities

The purpose of this study is to develop a method for predicting the spatial moments and concentration of a tracer plume conditional to locally measured data. We set this study in a probabilistic frame due to the inherent inability to characterize deterministically the flow domain. Our starting point is the Dagan (1984) conceptual model which relates the concentration to the translation of tagged elements of solute mass. Then, using Darcy's law and the flow equations, we describe that translation using a random function model for the velocity. Comparison with numerical simulations shows this model to be favorable for variances of the log conductivity as large as 2.56 (Dykstra‐Parsons coefficient of 0.8). It is shown how this random function model is correlated with the hydraulic head and conductivity fields, and furthermore, how it can be made conditional to measured data. It is concluded that conditioning improves predictions of the spatial moments of the tracer plume by making a more comprehensive use of available data and has a potential for a large range of applications.

Detecting a Smooth Signal: Optimality of Cusum Based Procedures

Cox et al. (1988) show that under a random smooth function model with signal covariance matrix V, a locally most powerful test for the presence of a nonpolynomial signal is based on the statistic yTVy. We show that if a well-known difference-based roughness measure is used to derive a natural choice for V then this statistic is the sum of squares of a residual cusum sequence. Thus we confirm theoretically the long-held impression that the residual cusum sequence is powerful as a tool for the detection of smooth signal.

Detecting a smooth signal: Optimality of cusum based procedures

SUMMARY Cox et al. (1988) show that under a random smooth function model with signal covariance matrix V, a locally most powerful test for the presence of a nonpolynomial signal is based on the statistic yTVy. We show that if a well-known difference-based roughness measure is used to derive a natural choice for V then this statistic is the sum of squares of a residual cusum sequence. Thus we confirm theoretically the long-held impression that the residual cusum sequence is powerful as a tool for the detection of smooth signal.

Use of Geostatistics For Accurate Mapping of Earthquake Ground Motion

The geostatistical method of kriging is applied for the analysis, estimation and mapping of earthquake ground motion. No modification to this method was required for this new application. Along with kriging, the concept of the semi-variogram is introduced and shown to be important for estimation and mapping. In geostatistical analyses of earthquake ground motion the ground motion is modelled by a random function. This random function model is shown to be accurate for providing estimates of earthquake ground motion at unsampled locations. Kriging is applied in this paper for the mapping of peak horizontal acceleration recorded during the 1971 San Fernando, and the 1983 Coalinga, California earthquakes. These results are compared to maps predicted using attenuation expressions and differences are analysed. Results demonstrate the utility of kriging for the analysis of earthquake ground motion.

Application of universal kriging for estimation of earthquake ground motion: Statistical significance of results

Universal kriging is compared with ordinary kriging for estimation of earthquake ground motion. Ordinary kriging is based on a stationary random function model; universal kriging is based on a nonstationary random function model representing first-order drift. Accuracy of universal kriging is compared with that for ordinary kriging; cross-validation is used as the basis for comparison. Hypothesis testing on these results shows that accuracy obtained using universal kriging is not significantly different from accuracy obtained using ordinary kriging. Tests based on normal distribution assumptions are applied to errors measured in the cross-validation procedure;t andF tests reveal no evidence to suggest universal and ordinary kriging are different for estimation of earthquake ground motion. Nonparametric hypothesis tests applied to these errors and jackknife statistics yield the same conclusion: universal and ordinary kriging are not significantly different for this application as determined by a cross-validation procedure. These results are based on application to four independent data sets (four different seismic events).

Bootstrapped models for intrinsic random functions

Use of intrinsic random function stochastic models as a basis for estimation in geostatistical work requires the identification of the generalized covariance function of the underlying process. The fact that this function has to be estimated from data introduces an additional source of error into predictions based on the model. This paper develops the sample reuse procedure called the “bootstrap” in the context of intrinsic random functions to obtain realistic estimates of these errors. Simulation results support the conclusion that bootstrap distributions of functionals of the process, as well as their “kriging variance,” provide a reasonable picture of variability introduced by imperfect estimation of the generalized covariance function.