On the construction of component importance measures for semi-Markov systems

In this paper we consider semi-Markov reliability models of systems with discrete state space in a setup general enough to cover systems with maintenance and repair. The systems are assumed to consist of several components which can either be up or down in each state. In this framework we propose two different types of component importance measures which are based on transition rates and interval availability, respectively. For these importance measures we study both the time-dependent and the steady state situation, and express them in terms of quantities easily calculated from the building blocks of the semi-Markov process.     

A modified algorithm for statistical simulation of random-structure systems with distributed transitions

An algorithm for statistical simulation of random-structure systems with distributed transitions is constructed. The algorithm is based on numerical methods for solving stochastic differential equations, and uses a modified maximum cross-section method when the transition intensity depends on the vector of state.     

A modified EM algorithm for mixture models based on Bregman divergence

The EM algorithm is a sophisticated method for estimating statistical models with hidden variables based on the Kullback–Leibler divergence. A natural extension of the Kullback–Leibler divergence is given by a class of Bregman divergences, which in general enjoy robustness to contamination data in statistical inference. In this paper, a modification of the EM algorithm based on the Bregman divergence is proposed for estimating finite mixture models. The proposed algorithm is geometrically interpreted as a sequence of projections induced from the Bregman divergence. Since a rigorous algorithm includes a nonlinear optimization procedure, two simplification methods for reducing computational difficulty are also discussed from a geometrical viewpoint. Numerical experiments on a toy problem are carried out to confirm appropriateness of the simplifications.     

浅析几类双输入排队系统模型

从矩阵几何解的角度出发,分析了七种类型的Geom/Geom/(Geom/Geom)/H 双输入排队系统.对这几个模型进行了描述,使用拟生灭过程探讨了各类模型的矩阵结构,并给出了其状态转移概率矩阵.     

Suboptimal security solution of a linear-quadratic pursuit-evasion game with mixed noises

The discrete-time, linear-quadratic pursuit-evasion game is considered in this paper. The difference system contains the state-dependent, control-dependent, and additive noises. In addition, each player has additive noises in his own measurement vector, and he is restricted to implement a linear estimator based on his measurements. Each player's philosophy is not to run a risk but to adopt a security control policy. Because of the linear estimation restriction, the security solution is suboptimal, which is found to be the closed-loop controls which are linear in the best estimate of the state. Equations for calculating the state estimation are derived. An algorithm for off-line computing the filter gain is also given.     

An EM Algorithm for Batch Markovian Arrival Processes and its Comparison to a Simpler Estimation Procedure

Although the concept of Batch Markovian Arrival Processes (BMAPs) has gained widespread use in stochastic modelling of communication systems and other application areas, there are few statistical methods of parameter estimation proposed yet. However, in order to practically use BMAPs for modelling, statistical model fitting from empirical time series is an essential task. The present paper contains a specification of the classical EM algorithm for MAPs and BMAPs as well as a performance comparison to the computationally simpler estimation procedure recently proposed by Breuer and Gilbert. Furthermore, it is shown how to adapt the latter to become an estimator for hidden Markov models.     

Error-Correcting Codes and Phase Transitions

The theory of error-correcting codes is concerned with constructing codes that optimize simultaneously transmission rate and relative minimum distance. These conflicting requirements determine an asymptotic bound, which is a continuous curve in the space of parameters. The main goal of this paper is to relate the asymptotic bound to phase diagrams of quantum statistical mechanical systems. We first identify the code parameters with Hausdorff and von Neumann dimensions, by considering fractals consisting of infinite sequences of code words. We then construct operator algebras associated to individual codes. These are Toeplitz algebras with a time evolution for which the KMS state at critical temperature gives the Hausdorff measure on the corresponding fractal. We extend this construction to algebras associated to limit points of codes, with non-uniform multi-fractal measures, and to tensor products over varying parameters.     

Decision diagrams for the approximate analysis of Markov models

Decision diagrams of various types can be used to encode the exact state space and transition rate matrix of large Markov models. However, the exact solution of such models still requires to store at least one real vector with one entry per reachable state, a formidable limitation to the practical use of these encodings. Thus, we discuss automatic techniques for the approximate computation of performance measures when the Markov model can be compactly encoded but not exactly solved. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A simulation-based multivariate Bayesian control chart for real time condition-based maintenance of complex systems

When complex systems are monitored, multi-observations from several sensors or sources may be available. These observations can be fused through Bayesian theory to give a posterior probabilistic estimate of the underlying state which is often not directly observable. This forms the basis of a Bayesian control chart where the estimated posterior probability of the state can be compared with a preset threshold level to assess whether a full inspection is needed or not. Maintenance can then be carried out if indicated as necessary by the inspection. This paper considers the design of such multivariate Bayesian control chart where both the transition between states and the relationship between observed information and the state are not Markovian. Since analytical or numerical solutions are difficult for the case considered in this paper, Monte Carlo simulation is used to obtain the optimal control chart parameters, which are the monitoring interval and the upper control limit. A two-stage failure process characterised by the delay time concept is used to describe the underlying state transition process and Bayesian theory is used to compute the posterior probability of the underlying state, which is embedded in the simulation algorithm. Extensive examples are shown to demonstrate the modelling idea.     

Modifications of the standard vector Monte Carlo estimate for characteristics analysis of scattered polarized radiation

There are two versions of weighted vector algorithms for the statistical modeling of polarized radiative transfer: a “standard” one, which is convenient for parametric analysis of results, and an “adaptive” one, which ensures finite variances of estimates. The application of the adaptive algorithm is complicated by the necessity of modeling the previously unknown transition density. An optimal version of the elimination algorithm used in this case is presented in this paper. A new combined algorithm with a finite variance and an algorithm with a mixed transition density are constructed. The comparative efficiency of the latter is numerically studied as applied to radiative transfer with a molecular scattering matrix.     

The Stochastic Theory of Fluvial Landsurfaces

A stochastic theory of fluvial landsurfaces is developed for transport-limited erosion, using well-established models for the water and sediment fluxes. The mathematical models and analysis are developed showing that some aspects of landsurface evolution can be described by Markovian stochastic processes. The landsurfaces are described by nondeterministic stochastic processes, characterized by a statistical quantity, the variogram, that exhibits characteristic scalings. Thus the landsurfaces are shown to be self-organized critical (SOC) systems, possessing both an initial transient state and a stationary state, characterized by respectively temporal and spatial scalings. The mathematical theory of SOC systems is developed and used to identify three stochastic processes that shape the surface. The SOC theory of landsurfaces reproduces established numerical results and measurements from digital elevation models (DEMs).     

Gibbs ensembles,equidistribution of the energy of sympathetic oscillators and statistical models of thermostat

The paper develops an approach to the proof of the “zeroth” law of thermodynamics. The approach is based on the analysis of weak limits of solutions to the Liouville equation as time grows infinitely. A class of linear oscillating systems is indicated for which the average energy becomes eventually uniformly distributed among the degrees of freedom for any initial probability density functions. An example of such systems are sympathetic pendulums. Conditions are found for nonlinear Hamiltonian systems with finite number of degrees of freedom to converge in a weak sense to the state where the mean energies of the interacting subsystems are the same. Some issues related to statistical models of the thermostat are discussed.      

On modeling left-truncated loss data using mixtures of distributions

A new statistical methodology is developed for fitting left-truncated loss data by using the G-component finite mixture model with any combination of Gamma, Lognormal, and Weibull distributions. The EM algorithm, along with the emEM initialization strategy, is employed for model fitting. We propose a new grid map which considers the model selection criterion (AIC or BIC) and risk measures at the same time, by using the entire space of models under consideration. A simulation study validates our proposed approach. The application of the proposed methodology and use of new grid maps are illustrated through analyzing a real data set that includes left-truncated insurance losses.     

A Markov chain Monte Carlo Expectation Maximization Algorithm for Statistical Analysis of DNA Sequence Evolution with Neighbor-Dependent Substitution Rates

The evolution of DNA sequences can be described by discrete state continuous time Markov processes on a phylogenetic tree. We consider neighbor-dependent evolutionary models where the instantaneous rate of substitution at a site depends on the states of the neighboring sites. Neighbor-dependent substitution models are analytically intractable and must be analyzed using either approximate or simulation-based methods. We describe statistical inference of neighbor-dependent models using a Markov chain Monte Carlo expectation maximization (MCMC-EM) algorithm. In the MCMC-EM algorithm, the high-dimensional integrals required in the EM algorithm are estimated using MCMC sampling. The MCMC sampler requires simulation of sample paths from a continuous time Markov process, conditional on the beginning and ending states and the paths of the neighboring sites. An exact path sampling algorithm is developed for this purpose.     

A Finite Field Framework for Modeling,Analysis and Control of Finite State Automata

In this paper, we address the modeling, analysis and control of finite state automata, which represent a standard class of discrete event systems. As opposed to graph theoretical methods, we consider an algebraic framework that resides on the finite field <formula form="inline">${\Op F}_2$</formula> which is defined on a set of two elements with the operations addition and multiplication, both carried out modulo 2. The key characteristic of the model is its functional completeness in the sense that it is capable of describing most of the finite state automata in use, including non-deterministic and partially defined automata. Starting from a graphical representation of an automaton and applying techniques from Boolean algebra, we derive the transition relation of our finite field model. For cases in which the transition relation is linear, we develop means for treating the main issues in the analysis of the cyclic behavior of automata. This involves the computation of the elementary divisor polynomials of the system dynamics, and the periods of these polynomials, which are shown to completely determine the cyclic structure of the state space of the underlying linear system. Dealing with non-autonomous linear systems with inputs, we use the notion of feedback in order to specify a desired cyclic behavior of the automaton in the closed loop. The computation of an appropriate state feedback is achieved by introducing an image domain and adopting the well-established polynomial matrix method to linear discrete systems over the finite field <formula form="inline">${\Op F}_2$</formula>. Examples illustrate the main steps of our method.     

An extensive-form analysis for comparing fuzzy information systems by means of the worth and quietness of information

Statistical decision problems deal with the choice among several actions, whose consequences depend upon the state of nature, and this choice is usually made on the basis of the outcome of an experiment selected from a collection of experiments whose probability measures depend upon the state. The selection criteria among the experiments are generally based on some measures of the information contained in each experiment about that true state.In this paper we suggest a new selection criterion among the experiments associated with a statistical decision problem, when the experimental outcomes cannot be exactly perceived by the decision maker, but rather the available information from each experimental outcome can be regarded as an element in a fuzzy information system (as defined by Tanaka, Okuda and Asai).In a previous paper we have proposed a method for comparing fuzzy information systems on the basis of the maximization of the ‘oworth of information of a fuzzy park information system’ (Tanaka et al). When the application of such a method leads to indifference between two fuzzy information systems, this indifference can be avoided by applying the criterion in this paper.This selection criterion is an extensive-form analysis which is based on two measures of the information contained in a fuzzy information system about the true state: the ‘worth of information of a fuzzy information system’ and the ‘expected quietness of information of a fuzzy information system’.The suitability of the criterion stated above is corroborated by studying its main properties and contrasting this procedure with other ones making use of different measures of information.     

Some sufficient conditions for the non-negativity preservation property in the discrete heat conduction model

In the course of the numerical approximation of mathematical models there is often a need to solve a system of linear equations with a tridiagonal or a block-tridiagonal matrices. Usually it is efficient to solve these systems using a special algorithm (tridiagonal matrix algorithm or TDMA) which takes advantage of the structure. The main result of this work is to formulate a sufficient condition for the numerical method to preserve the non-negativity for the special algorithm for structured meshes. We show that a different condition can be obtained for such cases where there is no way to fulfill this condition. Moreover, as an example, the numerical solution of the two-dimensional heat conduction equation on a rectangular domain is investigated by applying Dirichlet boundary condition and Neumann boundary condition on different parts of the boundary of the domain. For space discretization, we apply the linear finite element method, and for time discretization, the well-known Θ-method. The theoretical results of the paper are verified by several numerical experiments.     

Mixed simulation-state estimation of water distribution systems based on a least squares loop flows state estimator

This paper presents combined simulation and state estimation algorithm for water distribution systems based on the loop corrective flows and the variation of nodal demands as independent variables and it optimizes the Least Squares (LS) criterion. The combination of the two algorithms for simulation and state estimation is based on the delimitation of regions in the water network that are state estimated while for the remaining parts of the water network the simulation task is realized. The sizes of the respective delimitations can be based either on the hydraulic or topological distances from the real pressure measurements, flow measurements or measured nodal consumptions. The delimitations are realized through modifications of the inverse of the upper form tree incidence matrix which is used in order to construct the respective state estimated or simulated water network areas: the simulated nodes and pipes have the corresponding incidence columns zeroed in the inverse of the upper form tree incidence matrix while the state estimated nodes and pipes keep the values of their incidence described in the corresponding columns of the inverse of the upper form tree incidence matrix. The combined novel algorithm can be also applied to regions of water distribution systems which contain low pipe flows so that to avoid any convergence problems in the numerical algorithm. It results an efficient and effective novel mixed simulation-state estimation which is implemented on realistic water distribution systems.     

Error Estimates for the Approximation of a Class of Optimal Control Systems Governed by Linear PDEs

This paper deals with a class of optimal control problems in which the system is governed by a linear partial differential equation and the control is distributed and with constraints. The problem is posed in the framework of the theory of optimal control of systems. A numerical method is proposed to approximate the optimal control. In this method, the state space as well as the convex set of admissible controls are discretized. An abstract error estimate for the optimal control problem is obtained that depends on both the approximation of the state equation and the space of controls. This theoretical result is illustrated by some numerical examples from the literature.