Dominance-based Rough Set Approach to decision under uncertainty and time preference

We consider a problem of decision under uncertainty with outcomes distributed over time. We propose a rough set model based on a combination of time dominance and stochastic dominance. For the sake of simplicity we consider the case of traditional additive probability distribution over the set of states of the world, however, we show that the model is rich enough to handle non-additive probability distributions, and even qualitative ordinal distributions. The rough set approach gives a representation of decision maker’s time-dependent preferences under uncertainty in terms of “if…, then…” decision rules induced from rough approximations of sets of exemplary decisions.     

Statistical irreversibility of the Kac reversible circular model

The Kac circular model is a discrete dynamical system which has the property of recurrence and reversibility. Within the framework of this model M. Kac formulated necessary conditions for irreversibility over “short” time intervals to take place and demonstrated Boltzmann’s most important exploration methods and ideas, outlining their advantages and limitations. We study the circular model within the realm of the theory of Gibbs ensembles and offer a new approach to a rigorous proof of the “zeroth” law of thermodynamics based on the analysis of weak convergence of probability distributions.     

A new probabilistic approach to the path criticality in stochastic PERT

The notion of critical path is a key issue in the temporal analysis of project scheduling in deterministic setting. The very essence of the CPM consists in identifying the critical path, i.e., the longest path in a project network, because this path conveys information on how long it should take to complete the project to the project manager. The problem how can a stochastic counterpart of the deterministic critical path be defined is an important question in stochastic PERT. However, in the literature of stochastic PERT this question has so far almost been ignored, and the research into the random nature of a project duration has mainly been concentrated on the completion time in stochastic PERT in which any concrete special path is not specified. In the present paper we attempt to take first steps to fill this gap. We first developed a probabilistic background theory for univariate and bivariate marginal distributions of path durations of stochastic PERT whose joint path durations are modelled by multivariate normal distribution. Then, a new probabilistic approach to the comparison of path durations is introduced, and based on this comparison we define the concept of probabilistically critical path as a stochastic counterpart of the deterministic critical path. Also, an illustrative simple example of PCP and numerical results on the established probability bounds are presented.     

A Bayesian network model for spatial event surveillance

Methods for spatial cluster detection attempt to locate spatial subregions of some larger region where the count of some occurrences is higher than expected. Event surveillance consists of monitoring a region in order to detect emerging patterns that are indicative of some event of interest. In spatial event surveillance, we search for emerging patterns in spatial subregions.A well-known method for spatial cluster detection is Kulldorff’s [M. Kulldorff, A spatial scan statistic, Communications in Statistics: Theory and Methods 26 (6) (1997)] spatial scan statistic, which directly analyzes the counts of occurrences in the subregions. Neill et al. [D.B. Neill, A.W. Moore, G.F. Cooper, A Bayesian spatial scan statistic, Advances in Neural Information Processing Systems (NIPS) 18 (2005)] developed a Bayesian spatial scan statistic called BSS, which also directly analyzes the counts.We developed a new Bayesian-network-based spatial scan statistic, called BNetScan, which models the relationships among the events of interest and the observable events using a Bayesian network. BNetScan is an entity-based Bayesian network that models the underlying state and observable variables for each individual in a population.We compared the performance of BNetScan to Kulldorff’s spatial scan statistic and BSS using simulated outbreaks of influenza and cryptosporidiosis injected into real Emergency Department data from Allegheny County, Pennsylvania. It is an open question whether we can obtain acceptable results using a Bayesian network if the probability distributions in the network do not closely reflect reality, and thus, we examined the robustness of BNetScan relative to the probability distributions used to generate the data in the experiments. Our results indicate that BNetScan outperforms the other methods and its performance is robust relative to the probability distribution that is used to generate the data.     

Estimation of stress-strength reliability based on tail-modelling

In probabilistic design it is common to use parametric statistical models to calculate the probability of failure (P_f) obtained from stress-strength interference theory. In this respect, it is well known that P_f is sensitive to the shape of the upper and lower tails of the stress and strength distributions respectively and also to their interference. If the true shapes of these tails could be discovered, it would be possible to improve the existing techniques. However, in practice there is seldom sufficient real data available to establish these tails with statistical significance. Moreover, often one or both variables are expensive to sample. Thus a separate tail estimation may not always be possible. Motivated to define a theoretical safety factor, the author presents a method here which provides a general solution to the problem by considering the tail estimation for a sample of increased sized. This is obtained by a reformulation of the problem in terms of excesses over a threshold, and by breaking the analysis into two parts; estimation of the number of exceedances and modelling the excesses. The analysis is carried out by non-parametric estimation of the former and parametric modelling of the latter based on data from the critical region.     

Coxian representations of generalized Erlang distributions

This paper studies Coxian representations of generalized Erlang distributions. A nonlinear program is derived for computing the parameters of minimal Coxian representations of generalized Erlang distributions. The nonlinear program is also used to characterize the triangular order and the admissible region of generalized Erlang distributions. It is shown that the admissible region associated with a triangular order may not be convex. For generalized Erlang distributions of ME-order 3, a minimal Coxian representation is found explicitly. In addition, an algorithm is developed for computing a special type of ordered Coxian representations - the bivariate Coxian representation - for generalized Erlang distributions.     

Iterative Approximation of Statistical Distributions and Relation to Information Geometry

The optimal control of stochastic processes through sensor estimation of probability density functions is given a geometric setting via information theory and the information metric. Information theory identifies the exponential distribution as the maximum entropy distribution if only the mean is known and the Γ distribution if also the mean logarithm is known. The surface representing Γ models has a natural Riemannian information metric. The exponential distributions form a one-dimensional subspace of the two-dimensional space of all Γ distributions, so we have an isometric embedding of the random model as a subspace of the Γ models. This geometry provides an appropriate structure on which to represent the dynamics of a process and algorithms to control it. This short paper presents a comparative study on the parameter estimation performance between the geodesic equation and the B-spline function approximations when they are used to optimize the parameters of the Γ family distributions. In this case, the B-spline functions are first used to approximate the Γ probability density function on a fixed length interval; then the coefficients of the approximation are related, through mean and variance calculations, to the two parameters (i.e. μ and β) in Γ distributions. A gradient based parameter tuning method has been used to produce the trajectories for (μ, β) when B-spline functions are used, and desired results have been obtained which are comparable to the trajectories obtained from the geodesic equation. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Note on the characterization of Shannon-Wiener’s measure of information

Summary In [1] the author treated a characterization problem of the ShannonWiener measure of information for continuous probability distributions defined over an abstract measure space (R, S, m), wherem is a σ-finite measure over a σ-field S of subsets ofR, whose rangeM(S) is such thatM(S)=[0, ∞]. This condition on the range of the basic measure, however, can slightly be altered such thatM(S)=[0,1], and this modification is useful for characterization of the Kullback-Leibler mean information. In the present paper, it is shown that the characterization procedure of [1] can be applicable to continuous probability distributions defined on a finite measure space.     

Simulation of random vectors from three-dimensional spherically symmetric stable distributions

A simple approximate algorithm of sampling random vectors from three-dimensional symmetric stable distributions based on analytical approximation of inverse distribution functions is proposed. Explicit expressions for simulating the stable distributions with indices α=0.2(0.1) 1.8 are presented. The errors of the algorithm do not exceed 1–2% in the probability density function. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

Metastability in stochastic dynamics of disordered mean-field models

We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the Wentzell-Freidlin theory in this case, showing that any transition can be decomposed, with probability exponentially close to one, into a deterministic sequence of “admissible transitions”. For these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant factor. The distributions of the rescaled transition times are shown to converge to the exponential distribution. We exemplify our results in the context of the random field Curie-Weiss model. Received: 26 November 1998 / Revised version: 21 March 2000 / Published online: 14 December 2000     

Kronecker representation and decompositional analysis of closed queueing networks with phase-type service distributions and arbitrary buffer sizes

Two approximative fixed-point iterative methods based on decomposition for closed queueing networks with Coxian service distributions and arbitrary buffer sizes are extended to include phase-type service distributions. The irreducible Markov chain associated with each subnetwork in the respective decompositions is represented hierarchically using Kronecker products. The two methods are implemented in a software tool capable of computing the steady-state probability vector of each subnetwork by a multilevel method at each fixed-point iteration and are compared with other methods for accuracy and efficiency. Numerical results indicate that there is a niche filled by the two approximative methods. The authors thank Jean-Michel Fourneau for pointing out Marie’s method and Brouwer’s fixed-point theorem. The first author gratefully acknowledges grant TüBA-GEBİP from the Turkish Academy of Sciences.     

Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra

 We consider stochastic programming problems with probabilistic constraints involving random variables with discrete distributions. They can be reformulated as large scale mixed integer programming problems with knapsack constraints. Using specific properties of stochastic programming problems and bounds on the probability of the union of events we develop new valid inequalities for these mixed integer programming problems. We also develop methods for lifting these inequalities. These procedures are used in a general iterative algorithm for solving probabilistically constrained problems. The results are illustrated with a numerical example. Received: October 8, 2000 / Accepted: August 13, 2002 Published online: September 27, 2002 Key words. stochastic programming – integer programming – valid inequalities     

Bounding Distributions for Stochastic Logic Networks

A stochastic logic network is defined as a connected set of logic and time delay elements. Each of the latter elements has an associated probability distribution describing the nature of that element's delay. When used, for example, in project planning and scheduling, combinations of logic and time delay elements in such networks may represent conditions for the starting of project activities which are themselves represented by time delay elements. It is at present not known how to calculate the probability distributions for the events in such a network. This paper shows how to obtain upper and lower bounds for these probability distributions. The method is not a simulation technique; rather, it is a straightforward computational scheme derived from elementary probability theory. An example is given where the method is applied to a stochastic project scheduling network in which alternative ways exist for carrying out one of the jobs in the network.     

Asymptotics for transient and stationary probabilities for finite and infinite buffer discrete time queues

Consider a discrete time queue with i.i.d. arrivals (see the generalisation below) and a single server with a buffer length m. Let τm be the first time an overflow occurs. We obtain asymptotic rate of growth of moments and distributions of τm as m → ∞. We also show that under general conditions, the overflow epochs converge to a compound Poisson process. Furthermore, we show that the results for the overflow epochs are qualitatively as well as quantitatively different from the excursion process of an infinite buffer queue studied in continuous time in the literature. Asymptotic results for several other characteristics of the loss process are also studied, e.g., exponential decay of the probability of no loss (for a fixed buffer length) in time [0,η], η → ∞, total number of packets lost in [0, η, maximum run of loss states in [0, η]. We also study tails of stationary distributions. All results extend to the multiserver case and most to a Markov modulated arrival process. This revised version was published online in June 2006 with corrections to the Cover Date.     

Large-deviation analysis of Burgers turbulence with white-noise initial data

The statistics of the solution to the inviscid Burgers equation are investigated when the initial velocity is a white-noise function. Explicit representations for the probability distribution functions (PDFs) of the velocity and the distance between two successive shocks are found in terms of the solution to a PDE system. By using these expressions and large-deviation theory, tight bounds are found for the tails of these distributions and for that of the shock-strength distribution. These bounds give the exact rates of decay for the tails. © 1998 John Wiley & Sons, Inc.     

Asymptotics of the solutions to theN-particle Kolmogorov-Feller equations and the asymptotics of the solution to the Boltzmann equation in the region of large deviations

We construct a representation in which the asymptotics of the solution to the Kolmogorov-Feller equation in the Fock space Γ(L ₁(ℝ ⁿ )) is of a form similar to the WKB asymptotic expansion; namely, the Boltzmann equation inL ₁(ℝ ⁿ ) plays the role of the Hamilton equations, the linearized Boltzmann equation extended to Γ(L ₁(ℝ ⁿ )) plays the role of the transport equation, and the Hamilton-Jacobi equation follows from the conservation of the total probability for the solutions of the Boltzmann equation. We also construct the asymptotics of the solution to the Boltzmann equation with small transfer of momentum; this asymptotics is given by the tunnel canonical operator corresponding to the self-consistent characteristic equation. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 694–709, November, 1995. The author is deeply grateful to Prof. A. M. Chebotarev, whose assistance has made the writing of this paper possible. This work was financially supported by the International Science Foundation under grants Nos. MFO000 and MFO300.     

Approximating Probability Distributions Using Small Sample Spaces

We formulate the notion of a "good approximation" to a probability distribution over a finite abelian group ?. The quality of the approximating distribution is characterized by a parameter ɛ which is a bound on the difference between corresponding Fourier coefficients of the two distributions. It is also required that the sample space of the approximating distribution be of size polynomial in and 1/ɛ. Such approximations are useful in reducing or eliminating the use of randomness in certain randomized algorithms. We demonstrate the existence of such good approximations to arbitrary distributions. In the case of n random variables distributed uniformly and independently over the range , we provide an efficient construction of a good approximation. The approximation constructed has the property that any linear combination of the random variables (modulo d) has essentially the same behavior under the approximating distribution as it does under the uniform distribution over . Our analysis is based on Weil's character sum estimates. We apply this result to the construction of a non-binary linear code where the alphabet symbols appear almost uniformly in each non-zero code-word. Received: September 22, 1990/Revised: First revision November 11, 1990; last revision November 10, 1997     

Scheduling projects with stochastic activity duration to maximize expected net present value

Although uncertainty is rife in many project management contexts, little is known about adaptively optimizing project schedules. We formulate the problem of adaptively optimizing the expected present value of a project’s cash flow, and we show that it is practical to perform the optimization. The formulation includes randomness in activity durations, costs, and revenues, so the optimization leads to a recursion with a large state space even if the durations are exponentially distributed. We present an algorithm that partially exercises the “curse of dimensionality” as computational results demonstrate. Most of the paper is restricted to exponentially distributed task durations, but we sketch the adaptation of the algorithm to approximate any probability distribution of task duration.     

Generalized hypergeometric,digamma and trigamma distributions

Summary In this paper we introduce and study new probability distributions named “digamma” and “trigamma” defined on the set of all positive integers. They are obtained as limits of the zero-truncated Type B3 generalized hypergeometric distributions (inverse Pólya-Eggenberger or negative binomial beta distributions), and also by compounding the logarithmic series distributions. The family of digamma distributions has the logarithmic series as a limit and the trigamma as another limit. The trigamma distributions are very close to the zeta (Zipf) distributions. Thus, our new distributions are useful as substitutes of the logarithmic series when the observed frequency data have such a long tail that cannot be fitted by the latter distributions. In the beginning sections we summarize properties of the Type B3 generalized hypergeometric distributions. It is emphasized that the distributions are obtained by compounding a Poisson distribution by “gamma product-ratio” distributions.     

On the existence of ordered couplings of random sets — with applications

Letψ andϕ be two given random closed sets in a locally compact second countable topological spaceS. (They need not be based on the same probability space.) The main result gives necessary and sufficient conditions on the distributions ofψ andϕ, for the existence of two random closed sets and , based on the same probability space and such that their distributions coincide with those ofψ andϕ, resp., and a.s. This coupling result tells us in particular when a probability distribution onS is selectionable w.r.t. (the distribution of) a random closed set. An existence result for realizable thinnings of a simple point process is obtained by specializing it to supports of random measures. The coupling result is extended to random variables in a countably based continuous poset. As examples we mention various kinds of random capacities — in particular random measures — and random compact (saturated) sets. Moreover, the extended result tells us when a probability distribution onS is selectionable w.r.t. the distribution of a random compact (saturated) set.