The time complexity analysis of a class of gene expression programming

The most complex models with the high quality of the results obtained are, as a rule, more expensive in terms of developer qualifications and computational resources. During process design, a detailed description of the object is often not required, and the accuracy of the results obtained should not be higher than the accuracy of the measuring instruments used. Thus, the optimal combination of simplicity and quality of the mathematical description of technological processes is an urgent task of mathematical modeling. To identify a cell model developed based on the theory of Markov chains, data obtained by solving discrete models of the Boltzmann equation are used. A method to identify cell models of a fluidized bed reactor has been developed using data obtained based on solving discrete models of the Boltzmann equation. The adequacy of the identified model of a fluidized bed reactor has been verified. An approach to develop computational support for a cell model based on the theory of Markov chains is presented. The analysis of the results obtained has shown an adequate description of the processes in fluidized bed reactors in terms of cell models. The models are developed based on the theory of Markov chains and identified based on the results obtained within the framework of discrete models of the Boltzmann equation. The proposed method to identify and verify cell models provides the possibility to obtain simultaneously acceptable indicators of model simplicity and the accuracy of calculation of the design and operating parameters of fluidized bed reactors.

The Perron-Frobenius Projection in the Theory of Graphs, Digraphs, Designs and Stochastic Processes

The goal of this paper is to develop a general method to establish conditional ergodicity of infinite-dimensional Markov chains. Given a Markov chain in a product space, we aim to understand the ergodic properties of its conditional distributions given one of the components. Such questions play a fundamental role in the ergodic theory of nonlinear filters. In the setting of Harris chains, conditional ergodicity has been established under general nondegeneracy assumptions. Unfortunately, Markov chains in infinite-dimensional state spaces are rarely amenable to the classical theory of Harris chains due to the singularity of their transition probabilities, while topological and functional methods that have been developed in the ergodic theory of infinite-dimensional Markov chains are not well suited to the investigation of conditional distributions. We must therefore develop new measure-theoretic tools in the ergodic theory of Markov chains that enable the investigation of conditional ergodicity for infinite dimensional or weak-* ergodic processes. To this end, we first develop local counterparts of zero-two laws that arise in the theory of Harris chains. These results give rise to ergodic theorems for Markov chains that admit asymptotic couplings or that are locally mixing in the sense of H. F\"{o}llmer, and to a non-Markovian ergodic theorem for stationary absolutely regular sequences. We proceed to show that local ergodicity is inherited by conditioning on a nondegenerate observation process. This is used to prove stability and unique ergodicity of the nonlinear filter. Finally, we show that our abstract results can be applied to infinite-dimensional Markov processes that arise in several settings, including dissipative stochastic partial differential equations, stochastic spin systems and stochastic differential delay equations.

Abstract. We develop the theory of cyclic Markov chains and apply it to the El Niño-Southern Oscillation (ENSO) predictability problem. At the core of Markov chain modelling is a partition of the state space such that the transition rates between different state space cells can be computed and used most efficiently. We apply a partition technique, which divides the state space into multidimensional cells containing an equal number of data points. This partition leads to mathematical properties of the transition matrices which can be exploited further such as to establish connections with the dynamical theory of unstable periodic orbits. We introduce the concept of most and least predictable states. The data basis of our analysis consists of a multicentury-long data set obtained from an intermediate coupled atmosphere-ocean model of the tropical Pacific. This cyclostationary Markov chain approach captures the spring barrier in ENSO predictability and gives insight also into the dependence of ENSO predictability on the climatic state.

The quality of engineers is an important issue in industrialised countries. This quality is commonly assured by the accreditation of engineering programs and the certification, licensure, and registration of graduates. A high-quality engineering program should be concerned with the quality of education of its graduates. This paper presents an analytical method for understanding the cumulative effects of competence acquisition in courses on the quality of engineering programs, thus allowing quality enhancement from a global perspective of the program. The proposed method was developed from a model established through the theory of Markov chains, considering the cumulative effects of acquisition of competencies over the semesters on the quality of engineering graduates. In the proposed model, the quality of engineering graduates is represented by discrete levels, numbered from 1 to 5, representing the worst to the best quality, respectively. This paper does not answer the question of how to achieve the desired quality or what learning methods could be used; however, the method can predict the quality of engineering education in a program and provide information to allow the faculty an opportunity to adjust their teaching during the program. The model explains how the heterogeneity of learning affects the average quality of engineering graduates. Understanding this process is critical to proactive quality management in engineering undergraduate programs.

One of the central considerations in the theory of Markov chains is their convergence to an equilibrium. Coefficients of ergodicity provide an efficient method for such an analysis. Besides giving sufficient and sometimes necessary conditions for convergence, they additionally measure its rate. In this paper we explore coefficients of ergodicity for the case of imprecise Markov chains. The latter provide a convenient way of modelling dynamical systems where parameters are not determined precisely. In such cases a tool for measuring the rate of convergence is even more important than in the case of precisely determined Markov chains, since most of the existing methods of estimating the limit distributions are iterative. We define a new coefficient of ergodicity that provides necessary and sufficient conditions for convergence of the most commonly used class of imprecise Markov chains. This so-called weak coefficient of ergodicity is defined through an endowment of the structure of a metric space to the class of imprecise probabilities. Therefore we first make a detailed analysis of the metric properties of imprecise probabilities.

We propose an inter-disciplinary approach to particle swarm optimization (PSO) by establishing a molecular dynamics (MD) formulation of the algorithm, leading to a physical theory for the swarm environment. The physical theory provides new insights on the operational mechanism of the PSO method. In particular, a thermodynamic analysis, which is based on the MD formulation, is introduced to provide deeper understanding of the convergence behavior of the basic classical PSO algorithm. The thermodynamic theory is used to propose a new acceleration technique for the PSO. This technique is applied to the problem of synthesis of linear array antennas and very good improvement in the convergence performance is observed. A macroscopic study of the PSO is conducted by formulating a diffusion model for the swarm environment. The Einstein’s diffusion equation is solved for the corresponding probability density function (pdf) of the particles trajectory. The diffusion model for the classical PSO is used, in conjunction with Schrodinger’s equation for the quantum PSO, to propose a generalized version of the PSO algorithm based on the theory of Markov chains. This unifies the two versions of the PSO, classical and quantum, by eliminating the velocity and introducing position-only update equations based on the probability law of the method.

Research on query cost estimation for local database systems in a multidatabase system (MDBS) has attracted many researchers in the database area recently. Obtaining good query cost estimates is crucial for performing effective global query optimization in an MDBS. However, most techniques suggested so far, including database calibrating and query sampling, consider only a static multidatabase environment. Recently, we proposed a qualitative approach to developing cost models for a dynamic multidatabase environment. It has been shown that this approach is promising in estimating the cost of a query run in any given contention state for a dynamic environment. However, a large (cost) query in practice may experience multiple contention states during its execution, which cannot be directly handled by the qualitative approach. In this paper, we propose two new techniques, i.e., fractional analysis and probabilistic approach, to solve the problem. The fractional analysis technique, which is suitable for a system environment that changes contention states gradually and smoothly, estimates a query cost by analyzing its fractions. The probabilistic approach, which is suitable for a system environment that changes contention states rapidly and randomly, estimates a query cost based on the theory of Markov chains. Cost estimation formulas for both techniques are derived, and their properties are studied. Our experimental results demonstrate that the suggested techniques are quite promising in estimating costs for large queries in a dynamic multidatabase environment.

A major challenge of the heterogeneous wireless networks is how to jointly utilize the resources of different radio access technologies (RATs) in an efficient manner while simultaneously achieving the desired multi-service quality of service (QoS) and minimizing the service cost. To meet the challenge, this paper proposes an adaptive common radio resource management (CRRM) scheme in the context of wireless wide area network (WWAN) and wireless local area network (WLAN). The key parameters of the scheme are service type, user mobility and location information, load information, and service cost. The effectiveness of the proposed CRRM scheme is assessed analytically using the theory of Markov chains. Numerical results show that the proposed scheme minimizes the rate of unnecessary vertical handoffs (VHO), thereby providing stable communication without degrading call blocking performance and increasing service cost.

We have suggested that sine-wave based acuity gratings (such as Gabor functions) should be replaced by a new set of test functions - functions which act as the eigenfunctions of an appropriate operator (Stewart & Pinkham, in press). The Hermite functions emerge as an important special case, and have recently been applied to problems in image analysis (Martens, 1990). The Hermite functions are generated as the eigenfunctions of a differential operator which produces a simple and appealing mathematical model of space-variant acuity. The operator’s mathematical properties are in better agreement with the spatial inhomogeneity of the visual field than is the Fourier kernel. Surprisingly, these functions are their own Fourier transformations, and so are easily related to the Fourier transforms which are used in image analysis. If we are correct, human factors work on visual acuity can be placed on a coherent mathematical foundation, and new approaches to evaluating the quality of visual images will emerge. Just as important, a space-variant model of human visual acuity includes as a special case, the space-invariant models of image analysis used in engineering studies of image processing. The Hermite decomposition of the stimulus is directly related to the eigenfunctions of our space-variant acuity operator. The theory of Markov chains can be tied directly to the components of such a Hermite decomposition. The Hermite breakdown of two-dimensional and one-dimensional fractional Brownian motion produces components having corresponding spectra. Consequently, the information inherent in two- dimensional fractional Brownian motion is preserved under the Hermite decomposition, and finds a natural expression as a Markov process. Markov models are widely used for automatic speech recognition. Similar methods can be used to capture the inherent structure of visual images. Some natural scenes can be represented by fractals. Many images can be treated as

Nonlinear GARCH Models for Highly Persistent Volatility

SUMMARY First a new explanation in probability terms is given for the occurrence of geometric or conditional geometric distributions in some queueing problems. The methods used allow some simple but useful generalizations to be obtained. In §§2 and 3, these results are used to find the equilibrium distribution for a queueing system for which customers can arrive late, and for which the service time has a negative exponential distribution. The effects of different assumptions about the distribution of lateness on the queue distribution are investigated. In §4, a generalization of a result in the theory of Markov chains given in §1 is derived.

In this work we develop a bootstrap method based on the theory of Markov chains. The method moves from the two competing objectives that a researcher pursues when performing a bootstrap procedure: (i) to preserve the structural similarity – in statistical sense – between the original and the bootstrapped sample; (ii) to assure a diversification of the latter with respect to the former. The original sample is assumed to be driven by a Markov chain. The approach we follow is to implement an optimization problem to estimate the memory of a Markov chain (i.e. its order) and to identify its relevant states. The basic ingredients of the model are the transition probabilities, whose distance is measured through a suitably defined functional. We apply the method to the series of electricity prices in Spain. A comparison with the Variable Length Markov Chain bootstrap, which is a well established bootstrap method, shows the superiority of our proposal in reproducing the dependence among data.KeywordsMarkov ChainInformation LossElectricity PriceBootstrap ProcedureTransition Probability MatrixThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A delegation process is proposed for setting the weights when a weighted sum of individual utility functions is used to make group decisions. Conditions under which a unique set of weights exist and methods for calculating these weights are obtained using the theory of Markov chains. It is shown that the Pareto optimal set at each step of delegation is a subset of the Pareto optimal set at the previous step.

The article deals with the method for the assessment of the fiber optic communication lines (FOCL) reliability taking into account the effect of the optical fiber tension, the temperature influence and the built-in diagnostic equipment errors of the first kind. The reliability is assessed in terms of the availability factor using the theory of Markov chains and probabilistic mathematical modeling. To obtain a mathematical model, the following steps are performed: the FOCL state is defined and validated; the state graph and system transitions are described; the system transition of states that occur at a certain point is specified; the real and the observed time of system presence in the considered states are identified. According to the permissible value of the availability factor, it is possible to determine the limiting frequency of FOCL maintenance.