Multivariate Model for Soil Parameters Based on Johnson Distributions

Improved bounds and simulation procedures on the value of the multivariate normal probability distribution function value are given in the paper. The author's variance reduction technique was based on the Bonferroni bounds involving the first two binomial moments only. The new variance reduction technique is adapted to the most refined new bounds developed in the last decade for the estimation the probability of union respectively intersection of events. Numerical test results prove the efficiency of the simulation procedures described in the paper.

An adaptive probability state variable (PSV) parameter search algorithm Possessing long-term memory has been formulated to cope with systems that must avoid high performance-penalty operating regions. The information gained from all previous experiments is efficiently encoded in multivariate probability distribution functions (pdf's). This long-term memory capability enables the PSV algorithm effectively to avoid future experiments in high penalty regions. The systems considered are resource-limited, and catastrophic failure may occur if parameter values lying in high penalty regions are implemented. Those cases in which the high penalty regions are not known in advance were investigated. The PSV algorithm has the capability of adaptively learning the location and hypervolume of these regions as the search proceeds. The algorithm is explicitly guided in its internal strategies as a function of the remaining system resources and the updated probability distribution functions. Clustering analysis is used both in the discovery of new operating regions and for updating the pdf's, As a by-product of this research, clustering was also investigated as a presearch scheme. It is shown that this procedure has great promise as a means of assessing the complexity of an optimization problem. Experimental results are presented to demonstrate the utility of the adaptive PSV search algorithm.

Copulas have become an important analytic tool for characterizing multivariate distributions and dependence. One is often interested in simulating data from copula estimates. The process can be analytically and computationally complex and usually involves steps that are unique to a given parametric copula. We describe an alternative approach that uses \probability{proportional{to{size" (PPS) random sampling with weights formed from the copula likelihood function. The method is flexible and can be applied to parametric and nonparametric marginal density estimates. The precision of the simulation can be calibrated by adjusting the density of the multidimensional grid used in the simulation process. The approach is fully transparent to any copula function with continuous random variables. An example evaluates a number of goodness-of-fit criteria and provides strong support for the validity and practicality of the method.

On multivariate extensions of Conditional-Tail-Expectation

Site investigation data are Multivariate, Uncertain and unique, Sparse and Incomplete (MUSIC) which poses significant challenges when it comes to the probabilistic site-characterisation. In the state of practice statistical uncertainties in geotechnical parameters are considered by univariate distributions, that neglect the dependencies among them. This paper proposes a probabilistic calibration of the HS-small model using multivariate distributions, which account for the correlations among the parameters, in an incomplete database of stiff clay till properties from Copenhagen. Two multivariate distribution functions were constructed, one based on eight parameters from triaxial and soil classification tests, and the other on seven parameters from oedometer and soil classification tests on soil samples from 26 sites in Copenhagen. The two distribution functions were combined by merging the generated samples that encountered the minimum difference in the common variables. The proposed procedure was compared with a Gibbs sampler approach for constructing site-specific multivariate distribution functions from incomplete and sparse databases. The scatter on the correlated behaviour among geotechnical parameters was significantly reduced using the proposed approach. Generated samples were able to reproduce the stress–strain behaviour observed from the triaxial tests and follow similar trend as existing transformation models on Copenhagen clay till.

A multivariate conditional joint probability distribution of a set of K normalized structure factors has been developed using a novel approach. The covariance matrix of the distribution is calculated for all the space groups in terms of linear combinations of specific unitary structure factors. It is shown that if any number of off-diagonal elements are set arbitrarily to zero, an approximation is obtained to the covariance matrix which corresponds to a particular set of a priori conditions. The importance of this result in practical phase-determining methods is pointed out. Group theory is used to obtain results valid for all space groups. The multivariate distribution is used to calculate more general versions of the Cochran and Woolfson sign probability and the Karle and Hauptman tangent formulae.

In mathematical epidemiology, a well-known formula describes the impact of heterogeneity on the basic reproductive number, R0, for situations in which transmission is separable and for which there is one source of variation in susceptibility and one source of variation in infectiousness. This formula is written in terms of the magnitudes of the heterogeneities, as quantified by their coefficients of variation, and the correlation between them. A natural question to ask is whether analogous results apply when there are multiple sources of variation in susceptibility and/or infectiousness. In this paper we demonstrate that with three or more coupled heterogeneities, R0 under separable transmission depends on details of the distribution of the heterogeneities in a way that is not seen in the well-known simpler situation. We provide explicit formulae for the cases of multivariate normal and multivariate log-normal distributions, showing that R0 can again be expressed in terms of the magnitudes of the heterogeneities and the pairwise correlations between them. The formulae, however, differ between the two multivariate distributions, demonstrating that no formula of this type applies generally when there are three or more coupled heterogeneities. We see that the results of the formulae are approximately equal when heterogeneities are relatively small and show that an earlier result in the literature (Koella, 1991) should be viewed in this light. We provide numerical illustrations of our results and discuss a setting in which coupled heterogeneities are likely to have a major impact on the value of R0. We also describe a rather surprising result: in a system with three heterogeneities, R0 can exhibit non-monotonic behavior with increasing levels of heterogeneity, in marked contrast to the familiar two heterogeneity setting in which R0 either increases or decreases with increasing heterogeneity.

Many modern day collaborative systems/system of systems rely heavily on the sharing of information in order to improve performance, manage resources, and maximize overall capability. These types of systems are characterized by many decentralized nodes that can either all be identical or partitioned into a finite set of specialized types. When information is shared within any system of systems, the overall performance hinges on its ability to correctly associate the information received. The primary hypothesis evaluated by any system of systems after the receipt of new information is to determine whether this information belongs to a previously observed entity, or not. When this hypothesis is false, the new information belongs to a new entity, which includes both real and false entities. To evaluate this hypothesis, and to determine the optimal assignments to make at each time step, a data association discriminator or scoring function that performs like a distance function between two probability distributions with common support is defined. This paper defines the properties desired of a data association discriminator, highlights the measures of information that satisfy these properties, and develops the corresponding gating and scoring equations for use during the data association process. The most commonly employed gating and scoring function in the data association literature is the square of the Mahalanobis distance and the log-likelihood score function, which are only defined between two multivariate Gaussian distributions. One of the objectives of this paper is to demonstrate the superior characteristics of the data association discriminators presented herein when used in determining the optimal assignments as compared with the log-likelihood score function. Toward this end, and due to the prevalence of the multivariate Gaussian distribution function in the general entity tracking and information fusion literature, the closed-form equations for the data association discriminators based on the statistical divergences, along with the other measures of information, namely, entropy and mutual information, for multivariate Gaussian distributions will be presented. The architecture upon which a system of systems is developed and designed plays a fundamental role when selecting an appropriate association discriminator. This paper discusses various measures of information commonly used between and within the components of these types of systems and compares and contrasts their behavior in a common framework with a focus on the data association problem. Commonly used measures of information, namely, differential entropy, mutual information, and the statistical divergences, along with the log-likelihood score function are all examined analytically in a common mathematical framework, the advantages and disadvantages of each are discussed, new results are derived and presented, and several numerical examples based on synthetic data are presented that illuminate their behaviors and characteristics. Lastly, it is demonstrated that the Kullback–Leibler discriminator/divergence is not the best choice for use as a data association discriminator.

Continuous thermodynamics provides a simple way for the thermodynamic treatment of polydisperse systems. Such systems consist of a very large number of similar species whose composition is described not by the mole fractions of the individual components but by continuous distribution functions. For copolymers, multivariate distribution functions have to be used for describing the dependence of thermodynamic behavior on molar mass, chemical composition, sequence length, branching, etc. In this paper, we review continuous thermodynamics as applied to copolymer systems. Special attention is focused on liquid-liquid equilibria and thermodynamic stability. Equilibria in solutions of random copolymers, blends of random copolymers with homo- or copolymers, and also the high pressure phase equilibrium in the mixture of copoly(ethene vinylacetate) with its monomers are also discussed. A special examination of polydispersity effects in solutions with block copolymers is made. Thus, the paper reviews in a comprehensive way how to build up continuous thermodynamics with multivariate distribution functions and how to derive relations necessary for solving special problems. Some short remarks on possible future prospects will round up the paper.

In this study, the probability distribution of migrant population is examined on the basis that inter-regional migration is a combined process of, firstly, the household decision making, and, secondly, the probability distribution of sizes of migrating households. Though in the area of migration studies, the multinomial distribution which is used in the entropy maximizing model is well known, there are many other multivariate discrete probability distributions developed and utilized in mathematical ecology, financial engineering and other fields (Mosimann (1962, 1963), Shibuya & Shimizu (2002), Wang (2000)). Significance of these multivariate discrete probability distributions in the area of the inter-regional migration studies is explored.Further, on the assumption that the probability distribution of migrant population is a compound distribution of those of migrating households and migrating household sizes, migrating household models based on these multivariate discrete distributions are expanded for examining the characteristics of the distributions of migrating population. Expected values, variances and covariances of the migrant population are derived from the characteristic functions of the compound distributions, and it is found that the variance can be expressed as a quadratic function (without a constant) of the expected value, and that the first-order parameter is closely related to the average size of migrating households.JEL classification: C16, R23

One specific problem statistical offices and research institutes are faced with when releasing microdata is the preservation of confidentiality. Traditional methods to avoid disclosure often destroy the structure of the data, and information loss is potentially high. In this paper an alternative technique of creating scientific–use files is discussed, which reproduces the characteristics of the original data quite well. It is based on Fienberg (1997, 1994) who estimates and resamples from the empirical multivariate cumulative distribution function of the data in order to get synthetic data. The procedure creates data sets – the resample – which have the same characteristics as the original survey data. The paper includes some applications of this method with (a) simulated data and (b) innovation survey data, the Mannheim Innovation Panel (MIP), and a comparison between resampling and a common method of disclosure control (disturbance with multiplicative error) with regard to confidentiality on the one hand and the appropriateness of the disturbed data for different kinds of analyses on the other. The results show that univariate distributions can be better reproduced by unweighted resampling. Parameter estimates can be reproduced quite well if the resampling procedure implements the correlation structure of the original data as a scale or if the data is multiplicatively perturbed and a correction term is used. On average, anonymization of data with multiplicatively perturbed values protects better against re–identification than the various resampling methods used.

An important feature of Bayesian statistics is the opportunity to do sequential inference: the posterior distribution obtained after seeing a dataset can be used as prior for a second inference. However, when Monte Carlo sampling methods are used for inference, we only have a set of samples from the posterior distribution. To do sequential inference, we then either have to evaluate the second posterior at only these locations and reweight the samples accordingly, or we can estimate a functional description of the posterior probability distribution from the samples and use that as prior for the second inference. Here, we investigated to what extent we can obtain an accurate joint posterior from two datasets if the inference is done sequentially rather than jointly, under the condition that each inference step is done using Monte Carlo sampling. To test this, we evaluated the accuracy of kernel density estimates, Gaussian mixtures, mixtures of factor analyzers, vine copulas and Gaussian processes in approximating posterior distributions, and then tested whether these approximations can be used in sequential inference. In low dimensionality, Gaussian processes are more accurate, whereas in higher dimensionality Gaussian mixtures, mixtures of factor analyzers or vine copulas perform better. In our test cases of sequential inference, using posterior approximations gives more accurate results than direct sample reweighting, but joint inference is still preferable over sequential inference whenever possible. Since the performance is case-specific, we provide an R package mvdens with a unified interface for the density approximation methods.

We present a new algorithm for time series prediction using temporal fuzzy clustering. The algorithm is based on the framework of temporal clustering that was applied successfully to analyze, segment and recognize patterns of nonstationary signals in applications such as speech recognition and biomedical signal analysis. We combine fuzzy clustering in the observation space and cluster validation in the time axis in order to generate a prediction according to the online estimation of a time varying multivariate mixture distribution function that underlies the series elements. The resulting temporal behavior of the membership matrices can also be used to extract a prediction on the future probability distribution function (PDF) of the time series. The algorithm is more feasible than common methods such as hidden Markov models (HMM) in predicting nonstationary signals with a slow drift in their PDF and is also more efficient from a computation standpoint.

In radio wave propagation and wireless communication fields, it is known that rainfall attenuation (or rainfall intensity) in an area with small reception power or time ratio under a fading environment can be well approximated by a Gamma distribution (Nakagami‐m distribution). To investigate the selection diversity method, a joint probability function that takes correlation into consideration for multiple random variables must be used to quantitatively evaluate its results. In this paper, the authors derive an exact representation according to an infinite series for a general multivariate (M‐dimensional) Gamma distribution function with correlation. In addition, besides investigating the numerical behavior of the exact representation, they obtain an extremely effective approximate representation for practical use. That approximation, which is a revision of the probability distribution with no correlation according to the determinant of a matrix having the square roots of the correlation coefficients as its elements, not only is easy to handle, but also provides an excellent physical interpretation. © 2004 Wiley Periodicals, Inc. Electron Comm Jpn Pt 1, 87(8): 1–12, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecja.10224

Expansions of multivariate Pickands densities and testing the tail dependence