Stochastic complexity of a data set is deened as the shortest possible code length for the data obtainable by using some 񫝐xed set of models. This measure is of great theoretical and practical importance as a tool for tasks such as determining model complexity, or performing predictive inference. Unfortunately , for cases where the data has missing information, computing the stochastic complexit...

We deal with the problem of minimizing the expectation of a real valued random function over the weakly Pareto or Pareto set associated with a Stochastic MultiObjective Optimization Problem (SMOP) whose objectives are expectations of random functions. Assuming that the closed form of these expectations is difficult to obtain, we apply the Sample Average Approximation method (SAA-N, where N is t...

To model combinatorial decision problems involving uncertainty and probability, we introduce stochastic constraint programming. Stochastic constraint programs contain both decision variables (which we can set) and stochastic variables (which follow a probability distribution). They combine together the best features of traditional constraint satisfaction, stochastic integer programming, and sto...

This paper presents a fuzzy stochastic dynamic programming (FSDP) approach to derive steady-state multipurpose reservoir operating policies. The vagueness associated with some operating objectives as well as with the decision-making process is apprehended through fuzzy set theory. Operating objectives are considered as fuzzy sets and their membership functions represent decision maker’s prefere...

In order to discover interesting patterns and dependencies in data, an approach based on rough set theory can be used. In particular, Dominance-based Rough Set Approach (DRSA) has been introduced to deal with the problem of ordinal classification with monotonicity constraints (also referred to as multicriteria classification in decision analysis). However, in real-life problems, in the presence...

1, and thus in any left eigen­ vector π = π̂(1), π̂(2), · · · , π̂(r), π̂(t) of eigenvalue 1, we have π̂(t) = 0. This means that π̂(i) must be a left eigenvector of [Pi] for each i, 1 ≤ i ≤ r. Since π̂(i) can be given an arbitrary scale factor for each i, we get exactly r independent left� eigenvectors, each a probability vector as desired. We take these eigenvectors to be π(i) = 0, · · · , 0, π̂(i),...

In a Euclidean space , the Lebesgue-Stieltjes integral of set-valued stochastic processes d R       , 0, t F F t T    with respect to real valued finite variation process       , 0, t A t T   t is defined directly by employing all integrably bounded selections instead of taking the decomposable closure appearing in some existed references. We shall show that this kind of integr...

Several investigations has recently been made concerning Poisson and composed Poisson stochastic processes. The ordinary Poisson process is conceivable as a sequence of points, distributed at random on the time axis and this idea can be generalized to more than onedimensional spaces. The latter case occurs in making a blood-count, in counting stars, etc. In [8], [4], [6] and [15] conditions are...

Let W [n] be an independent random sequence with constant mean µ W = 0 and variance σ 2 W. Define a new random sequence X[n] as follows: X[0] = 0 X[n] = ρX[n − 1] + W [n] for n ≥ 1. (a) Find the mean value of X[n] for n ≥ 0. (b) Find the autocovariance of X[n], denoted as K XX [m, n]. (c) For what values of ρ does K XX [m, n] tend to G[m − n], for some finite-valued function G, as m and n becom...

In the present paper we are concerned with two topics: the probability distributions of the random variables consisting of a completely additive s t o c hastic set function and the discontinuities of the realizations by supposing the latters to be completely additive number-valued set functions. Some theorems of Chapter III are analogous to those well-known in the theory of stochastic processes...