Non-stationary Probability Distribution Research Articles

A Global Chance-Constraint for Stochastic Inventory Systems Under Service Level Constraints

We consider a class of production/inventory control problems that has a single product and a single stocking location, for which a stochastic demand with a known non-stationary probability distribution is given. Under the widely-known replenishment cycle policy the problem of computing policy parameters under service level constraints has been modeled using various techniques. Tarim and Kingsman introduced a modeling strategy that constitutes the state-of-the-art approach for solving this problem. In this paper we identify two sources of approximation in Tarim and Kingsman's model and we propose an exact stochastic constraint programming approach. We build our approach on a novel concept, global chance-constraints, which we introduce in this paper. Solutions provided by our exact approach are employed to analyze the accuracy of the model developed by Tarim and Kingsman.

Partial distortion entropy maximization for online data clustering

Competitive learning neural networks are regarded as a powerful tool for online data clustering to represent a non-stationary probability distribution with a fixed number of weight vectors. One difficulty in practical applications of competitive learning neural networks to online data clustering is that most of them require heuristically-predetermined threshold parameters for balancing a trade-off between convergence accuracy, i.e. error minimization performance, and speed of adaptation to the changes in source statistics. Although adaptation acceleration is achievable by relocating a "useless" node so that it becomes useful, excessive relocation often disturbs error minimization. Hence, both of the adaptation speed and the error minimization performance sensitively depend on threshold parameters to determine whether a node should be relocated or not. In general, it is difficult to know adequate threshold parameters a priori. This paper proposes a novel criterion for decision making of node relocation without heuristically predetermined thresholds. According to the proposed criterion, a node is relocated only if the relocation task improves partial distortion entropy, which is an online optimality metric reliable from the viewpoint of error minimization. Hence, node relocation is carried out without disturbing error minimization. As a result, both quick adaptation and error minimization are simultaneously accomplished without any carefully predefined parameters. Experimental results clarify the validity of the proposed criterion. Competitive learning with the criterion is clearly superior to other representative algorithms in terms of both quick adaptation and error minimization performance.

Stochastic Measure of Fatigue Crack Damage for Health Monitoring of Ductile Alloy Structures

This paper models a stochastic measure of fatigue crack damage in ductile alloys that are commonly encountered in structures and machinery components of complex mechanical systems such as land, air, ocean, and space vehicles. The constitutive equations of the damage measure are built upon the physics of fracture mechanics and are substantiated by Karhunen-Loe` ve decomposition of fatigue test data where statistical orthogonality of the estimated measure and the resulting estimation error is demonstrated in a Hilbert space setting. The non-stationary probability distribution (PDF) function of the damage estimate is generated in a closed form without numerically solving stochastic differential equations in the Wiener integral or It integral setting. The model of crack damage measure allows real-time execution of decision algorithms for health monitoring, risk assessment, and life prediction of mechanical structures on inexpensive platforms such as a Pentium processor. The stochastic model of fatigue crack damage measure is in good agreement with experimental data sets for 2024-T3 and 7075-T6 aluminum alloys.

Optimal encoding of non-stationary sources

The usual assumption for proofs of the optimality of lossless encoding is a stationary ergodic source. Dynamic sources with non-stationary probability distributions occur in many practical situations where the data source is formed from a composition of distinct sources, for example, a document with multiple authors, a multimedia document, or the composition of distinct packets sent over a communication channel. There is a vast literature of adaptive methods used to tailor the compression to dynamic sources. However, little is known about optimal or near optimal methods for lossless compression of strings generated by sources that are not stationary ergodic. Here, we do not assume the source is stationary. Instead, we assume that the source produces an infinite sequence of concatenated finite strings s1…sn, where: (i) Each finite string si is generated by a sampling of a (possibly distinct) stationary ergodic source Si, and (ii) the length of each of the si is lower bounded by a function L(n) such that L(n)/log(n) grows unboundedly with the length n of all the text within s1…si. Thus each input string is a sequence of substrings generated by possibly distinct and unknown stationary ergodic sources. The optimal expected length of a compressed coding of a finite prefix s1…sk is ∑i=1kniHi, where ni is the length of si and Hi is the entropy of Si. We give a window-based LZ77-type method for compression that we prove gives an encoding with asymptotically optimal expected length. We give another LZ77-type method for compression where the expected time for encoding and decoding is nearly linear (approaching arbitrarily close to linear O(n) for large n). We also prove that this later method gives an encoding with asymptotically optimal expected length. In addition, give a dictionary-based LZ78-type method for compression, which takes linear time with small constant factors. This final algorithm also gives an encoding with asymptotically optimal expected length, assuming the Si are stationary ergodic sources that satisfy certain mixing conditions and L(n)⩾ne for some e>0.

Relaxation of probability characteristics of dynamical systems, described by Ferhulst stochastic equation with «pink» noise

Probability characteristics relaxation of а solution of Ferhulst stochastic equation is considered. It was supposed, that random influence on the system is Gaussian Markov process with arbitrary values of correlation time. Two models of nonstationary probability distributions with exact stationary athymptotic are proposed on the basis of numerical solutions of relaxation equations for moments: in the case of delta—correlated random force and quasistationary force. Relaxation of main probability characteristics is examined in dependence оп nonlinearity, initial conditions, noise intensity and kind of noise spectrum. Evolution of the model probability distributions has been constructed.

Analytic-Numerical Construction of Nonstationary Probability Distributions for one Class of Nonlinear Stochastic Systems

We propose a method for constructing nonstationary model probability distributions for nonlinear dynamic systems related to the Verhulst stochastic equation. The proposed procedure is based on the numerical solution of relaxation differential equations for the mean and the variance. The set of the moment equations is closed and the probability density is constructed on the basis of rigorous analytical relations for the stationary probability characteristics. As a result, these distributions have correct stationary asymptotics. We show the possibility of numerical control of the accuracy of the proposed procedure. We consider the examples of relaxation of the probability characteristics of the amplitude of a self-oscillator and a parametric oscillator with a noise pump. The evolution of the amplitude probability distribution is found.

Stochastic Modeling of Fatigue Crack Damage for Risk Analysis and Remaining Life Prediction

This paper presents a stochastic model of fatigue crack damage in metallic materials that are commonly encountered in structures and machinery components of complex mechanical systems (e.g., aircraft, spacecraft, and power plants). The constitutive equation of the damage model is based on the physics of fracture mechanics and is validated by Karhunen-Loe`ve analysis of test data. The (nonstationary) probability distribution function (PDF) of fatigue crack damage is generated in a closed form without numerically solving stochastic differential equations in the Wiener integral or Itoˆ integral setting. The crack damage model thus allows real-time execution of decision algorithms for risk assessment and life prediction on inexpensive platforms such as a Pentium processor. The model predictions are in close agreement with experimental data of fatigue crack growth statistics for 2024-T3 and 7075-T6 aluminum alloys.

Statistical dynamics of sine-square map

In this paper we study (i) the variation of local Lyapunov exponent, (ii) characterization of chaotic attractor at bifurcations using the variance σ n ( q) of fluctuations of coarse-grained local expansion rates of nearby orbits and (iii) characterization of weak and strong chaos in a sine-square map which describes the dynamics of the liquid crystal hybrid optical bistable device. The standard deviation of local Lyapunov exponent λ( X, L) calculated after every L time steps and Allan variance are found to approach zero in the limit L → ∞ as L − α . For all chaotic attractors of the map the σ n ( q) versus q plot exhibits a peak at q = q α . We show that additional peaks, however, occur only for the attractors just before and just after the bifurcations. We investigate the characteristics of the probability distributions of a k-step difference quantity ΔX k = X i+ k − X i . We show that a nonstationary probability distribution occurs for weak chaos and a stationary distribution occurs for strong chaos.

Maximum entropy principle and non-stationary distributions of stochastic systems

The principle of maximum entropy (in its classical form), successfully applied in many fields (e.g. statistics, reliability, estimation), has recently been extended to analyze the systems governed by stochastic differential equations and especially to determining the stationary probability distribution of the solution process. In this paper we develop the maximum entropy approach to characterize non-stationary probability distributions of the solutions of stochastic systems. The variational problem for the entropy functional includes time-dependent constraints in the form of differential equations for moments. The general scheme of the method is given along with the effective treatment of a number of first and second order stochastic systems. The maximum entropy probability distributions are compared with the exact solutions and with the simulation results.

Scheduling as a multi-dimensional placement problem

This paper describes research in developing an algorithm for a constraint-satisfaction problem (CSP) which solves certain classes of scheduling problems by formulating them as multi-dimensional placement problems. The test domain is check-in counter scheduling at an international airport. In this airport the counters are managed and allocated centrally by the local aviation authority. This scheduling problem is complicated by the fact that the resource requirement is stochastic—passenger arrival is a non-stationary probability distribution; the resources are not uniform—the physical location and available free space for queuing are different for different counters; and the resources to be allocated have other associated resources with their own sets of constraints—counters have associated baggage belts that have different capacity constraints. The approach presented combines simulation techniques with constraint-based reasoning to handle the stochastic nature of the problem. Scheduling requirements are modelled as multi-dimensional objects. The CSP algorithm presented here treats scheduling as a process of kneading and moulding these multi-dimensional objects into unallocated free space until a place is found for each of these objects. The constraints of the problem define how much the shape can be moulded.

Pattern retrieval and learning in nets of asynchronous binary threshold elements

A training algorithm for network of asynchronous learning-threshold elements is presented and analyzed. The algorithm is based on the Hebbian hypothesis, and it allows adaptation of the learning-network parameters to changing pattern environments. In particular, the network's properties can be quantified in environments where pattern occurrence is random, with nonequal, nonstationary probability distributions. The state reassessment probabilities of neurons during information retrieval can also be nonstationary and not equal for all neurons. The trained network is a content-addressable memory. The authors evaluate its stabilization properties with respect to a given set of patterns, using the theory of Markov processes. The results are applicable for the determination of efficient coding for information that has to be stored, and for prediction of actual pattern-retrieval capabilities of the trained network. The authors include the popular sum-of-outer-products assignment as an analyzable specific case of their training procedure, and allow the steady-state analysis of a large group of sigmoidal learning curves. >

Equivalent Stochastic Systems

Equivalent stochastic systems are defined as randomly excited dynamical systems whose response vectors in the state space share the same probability distribution. In this paper, the random excitations are restricted to Gaussian white noises; thus, the system responses are Markov vectors, and their probability densities are governed by the associated Fokker-Planck equations. When the associated Fokker-Planck equations are identical, the equivalent stochastic systems must share both the stationary probability distribution and the transient nonstationary probability distribution under identical initial conditions. Such systems are said to be stochastically equivalent in the strict (or strong) sense. A wider class, referred to as the class of equivalent stochastic systems in the wide (or weak) sense, also includes those sharing only the stationary probability distribution but having different Fokker-Planck equations. Given a stochastic system with a known probability distribution, procedures are developed to identify and construct equivalent stochastic systems, both in the strict and in the wide sense.