semi-Markovian Random Walk Research Articles

Moment-based approximation for variance of semi-Markovian random walk with gamma distributed interference of chance

Moment-based approximation for variance of semi-Markovian random walk with gamma distributed interference of chance

On the Boundary Functional of a Semi-Markov Process

In this paper, we consider the semi-Markov random walk process with negative drift, positive jumps. An integral equation for the Laplace transform of the conditional distribution of the boundary functional is obtained. In this work, we define the residence time of the system by generalized exponential distributions with different parameters via fractional order integral equation. The purpose of this paper is to reduce an integral equation for the Laplace transform of the conditional distribution of a boundary functional of the semi-Markov random walk processes to fractional order differential equation with constant coefficients.

Limit Theorem for a Semi-Markovian Random Walk with General Interference of Chance

A semi-Markovian random-walk process with general interference of chance was constructed and investigated. The key point of this study is the assumption that the discrete interference of chance has a general form. Under some conditions, it is proved that the process is ergodic, and the exact forms of the ergodic distribution and characteristic function of the process are obtained. By using basic identity for random walks, the characteristic function of the process is expressed by the characteristic function of a boundary functional. Then, two-term asymptotic expansion for the characteristic function of the standardized process is found. Using this asymptotic expansion, a weak convergence theorem for the ergodic distribution of the standardized process is proved, and the limiting form for the ergodic distribution is obtained. The obtained limit distribution coincides with the limit distribution of the residual waiting time of the renewal process generated by a sequence of random variables expressing the discrete interference of chance.

Distribution of the Lower Boundary Functional of the Step Process of Semi-Markov Random Walk with Delaying Barrier

In this paper, a step process of semi-Markovian random walk with delaying barrier on the zero-level is constructed and the Laplace transformation of the distribution of first crossing time of this process into the delaying barrier is obtained. Also, the expectation and standard diversion of a boundary functional of the process are given.

Three-term asymptotic expansion: A semi-Markovian random walk with a generalized beta distributed interference of chance

ABSTRACTA semi-Markovian random walk process (X(t)) with a generalized beta distribution of chance is considered. The asymptotic expansions for the first four moments of the ergodic distribution of the process are obtained as E(ζn) → ∞ when the random variable ζn has a generalized beta distribution with parameters (s, S, α, β); α, β > 1, 0 ⩽ s < S < ∞. Finally, the accuracy of the asymptotic expansions is examined by using the Monte Carlo simulation method.

Determination of the mean and mean-square deviations of the system level

The stepwise process of a semi-Markovian random walk with two delaying screens is considered. A mathematical model of the considered process is constructed, and a double Laplace transform for conditional and unconditional distributions of the process is obtained, as well as a Laplace transform for the ergodic distribution of the process and its first and second moments, which depend on two parameters, specifically, the input parameter and output parameters. These parameters may be used to control the state of the process.

Laplace-Stieltjes transform of the distribution of the first moment of crossing the level a(a &gt; 0) by a semi-Markovian random walk with positive drift and negative jumps

Using a sequence of independent equally distributed two-dimensional random variables, the process of a semi-Markovian random walk with positive drift and negative jumps is constructed. The Laplace-Stieltjes transform of the distribution of the first moment of crossing the level a(a > 0) is found.

On the Moments of a Semi-Markovian Random Walk with Gaussian Distribution of Summands

In this article, a semi-Markovian random walk with delay and a discrete interference of chance (X(t)) is considered. It is assumed that the random variables ζ n , n = 1, 2,…, which describe the discrete interference of chance form an ergodic Markov chain with ergodic distribution which is a gamma distribution with parameters (α, λ). Under this assumption, the asymptotic expansions for the first four moments of the ergodic distribution of the process X(t) are derived, as λ → 0. Moreover, by using the Riemann zeta-function, the coefficients of these asymptotic expansions are expressed by means of numerical characteristics of the summands, when the process considered is a semi-Markovian Gaussian random walk with small drift β.

SMETANA: Accurate and Scalable Algorithm for Probabilistic Alignment of Large-Scale Biological Networks

In this paper we introduce an efficient algorithm for alignment of multiple large-scale biological networks. In this scheme, we first compute a probabilistic similarity measure between nodes that belong to different networks using a semi-Markov random walk model. The estimated probabilities are further enhanced by incorporating the local and the cross-species network similarity information through the use of two different types of probabilistic consistency transformations. The transformed alignment probabilities are used to predict the alignment of multiple networks based on a greedy approach. We demonstrate that the proposed algorithm, called SMETANA, outperforms many state-of-the-art network alignment techniques, in terms of computational efficiency, alignment accuracy, and scalability. Our experiments show that SMETANA can easily align tens of genome-scale networks with thousands of nodes on a personal computer without any difficulty. The source code of SMETANA is available upon request. The source code of SMETANA can be downloaded from http://www.ece.tamu.edu/~bjyoon/SMETANA/.

Limit distribution for a semi-Markovian random walk with Weibull distributed interference of chance

In this paper, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered. In this study, it is assumed that the sequence of random variables {ζn}, n =1 , 2, ... , which describes the discrete interference of chance, forms an ergodic Markov chain with the Weibull stationary distribution. Under this assumption, the ergodic theorem for the process X(t) is discussed. Then the weak convergence theorem is proved for the ergodic distribution of the process X(t) and the limit form of the ergodic distribution is derived. MSC: 60G50; 60K15; 60F99

Laplace transform of the distribution of the semi-Markov walk process with a positive drift, negative jumps, and a delay screen at zero

In this paper, we investigate the process of semi-Markov random walk with a positive drift, negative jumps, and a delay screen at zero. Explicit forms are found of the Laplace transform in time, the Laplace-Stieltjes transform by the state (phase) of the conditional and unconditional distributions of the process states, and the explicit form of the Laplace-Stieltjes transform of the ergodic distribution of the process.

RESQUE: Network reduction using semi-Markov random walk scores for efficient querying of biological networks

Recent technological advances in measuring molecular interactions have resulted in an increasing number of large-scale biological networks. Translation of these enormous network data into meaningful biological insights requires efficient computational techniques that can unearth the biological information that is encoded in the networks. One such example is network querying, which aims to identify similar subnetwork regions in a large target network that are similar to a given query network. Network querying tools can be used to identify novel biological pathways that are homologous to known pathways, thereby enabling knowledge transfer across different organisms. In this article, we introduce an efficient algorithm for querying large-scale biological networks, called RESQUE. The proposed algorithm adopts a semi-Markov random walk (SMRW) model to probabilistically estimate the correspondence scores between nodes that belong to different networks. The target network is iteratively reduced based on the estimated correspondence scores, which are also iteratively re-estimated to improve accuracy until the best matching subnetwork emerges. We demonstrate that the proposed network querying scheme is computationally efficient, can handle any network query with an arbitrary topology and yields accurate querying results. The source code of RESQUE is freely available at http://www.ece.tamu.edu/~bjyoon/RESQUE/

Enhancing the accuracy of HMM-based conserved pathway prediction using global correspondence scores

BackgroundComparative network analysis aims to identify common subnetworks in biological networks. It can facilitate the prediction of conserved functional modules across different species and provide deep insights into their underlying regulatory mechanisms. Recently, it has been shown that hidden Markov models (HMMs) can provide a flexible and computationally efficient framework for modeling and comparing biological networks.ResultsIn this work, we show that using global correspondence scores between molecules can improve the accuracy of the HMM-based network alignment results. The global correspondence scores are computed by performing a semi-Markov random walk on the networks to be compared. The resulting score naturally integrates the sequence similarity between molecules and the topological similarity between their molecular interactions, thereby providing a more effective measure for estimating the functional similarity between molecules. By incorporating the global correspondence scores, instead of relying on sequence similarity or functional annotation scores used by previous approaches, our HMM-based network alignment method can identify conserved subnetworks that are functionally more coherent.ConclusionsPerformance analysis based on synthetic and microbial networks demonstrates that the proposed network alignment strategy significantly improves the robustness and specificity of the predicted alignment results, in terms of conserved functional similarity measured based on KEGG ortholog (KO) groups. These results clearly show that the HMM-based network alignment framework using global correspondence scores can effectively find conserved biological pathways and has the potential to be used for automatic functional annotation of biomolecules.

Semi-Markov failure rates processes

Abstract Usually, a reliability function is defined by a failure rate which is a real function taking the non-negative real values. In this paper the failure rate is assumed to be a stochastic process with non-negative and right continuous trajectories. The reliability function is defined as an expectation of a function of that random process. Particularly, the failure rate defined by the semi-Markov processes is considered here. The theorems dealing with the renewal equations for the conditional reliability functions with a semi-Markov process as a failure rate are presented in this paper. A system of that kind of equations for the discrete state space semi-Markov process is applied for calculating the reliability function for the 3-states semi-Markov random walk. Using the introduced system of renewal equations for the countable state space, the reliability function for the Furry–Yule failure rate process is obtained.

A Novel Low-Complexity HMM Similarity Measure

In this letter, we propose a novel similarity measure for comparing Hidden Markov models (HMMs) and an efficient scheme for its computation. In the proposed approach, we probabilistically evaluate the correspondence, or goodness of match, between every pair of states in the respective HMMs, based on the concept of semi-Markov random walk. We show that this correspondence score reflects the contribution of a given state pair to the overall similarity between the two HMMs. For similar HMMs, each state in one HMM is expected to have only a few matching states in the other HMM, resulting in a sparse state correspondence score matrix. This allows us to measure the similarity between HMMs by evaluating the sparsity of the state correspondence matrix. Estimation of the proposed similarity score does not require time-consuming Monte-Carlo simulations, hence it can be computed much more efficiently compared to the Kullback–Leibler divergence (KLD) thas has been widely used. We demonstrate the effectiveness of the proposed measure through several examples.

Three-term asymptotic expansions for the moments of the random walk with triangular distributed interference of chance

In this study, a semi-Markovian random walk with a discrete interference of chance ( X( t)) is considered and under some weak assumptions the ergodicity of this process is discussed. The exact formulas for the first four moments of ergodic distribution of the process X( t) are obtained when the random variable ζ 1, which is describing a discrete interference of chance, has a triangular distribution in the interval [ s, S] with center ( S + s)/2. Based on these results, the asymptotic expansions with three-term are obtained for the first four moments of the ergodic distribution of X( t), as a ≡ ( S − s)/2 → ∞. Furthermore, the asymptotic expansions for the variance, skewness and kurtosis of the ergodic distribution of the process X( t) are established. Finally, by using Monte Carlo experiments it is shown that the given approximating formulas provide high accuracy even for small values of parameter a.

Asymptotic Expansions for the Moments of the Semi-Markovian Random Walk with Gamma Distributed Interference of Chance

In this study, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered and under some weak assumptions the ergodicity of this process is discussed. The exact formulas for the first four moments of the ergodic distribution of the process X(t) are obtained when the random variable ζ1, which describes a discrete interference of chance, has a gamma distribution with parameters (α, λ), α > 1, λ > 0. Based on these results, the asymptotic expansions are obtained for the first four moments of the ergodic distribution of the process X(t), as λ → 0. Furthermore, the asymptotic expansions for the skewness and kurtosis of the ergodic distribution of the process X(t) are established. Finally, it is discussed that the alternative estimations for the stationary characteristics of this process can be offered by using obtained asymptotic expansions.

On the Inventory Model with Two Delaying Barriers

In this paper the process of semi-Markovian random walk with negative drift under angle α (0° &lt; α &lt; 90°), and positive jumps with probability ρ (0 &lt; ρ &lt; 1) having two delaying screens at level zero and a (a &gt; 0) is constructed. The exact expressions for Laplace transforms of the distributions of the first moments in order to reach to these screens by the process and, in particularly, the expectations and the variances of indicated distributions are obtained.

Asymptotic expansions for the moments of a semi-Markovian random walk with exponential distributed interference of chance

In this paper, a semi-Markovian random walk process ( X ( t ) ) with a discrete interference of chance is constructed and the ergodicity of this process is discussed. Some exact formulas for the first- and second-order moments of the ergodic distribution of the process X ( t ) are obtained, when the random variable ζ 1 has an exponential distribution with the parameter λ > 0 . Here ζ 1 expresses the quantity of a discrete interference of chance. Based on these results, the third-order asymptotic expansions for mathematical expectation and variance of the ergodic distribution of the process X ( t ) are derived, when λ → 0 .

Path-probability density functions for semi-Markovian random walks

In random walks, the path representation of the Green's function is an infinite sum over the length of path probability density functions (PDFs). Recently, a closed-form expression for the Green's function of an arbitrarily inhomogeneous semi-Markovian random walk in a one-dimensional (1D) chain of L states was obtained by utilizing path-PDFs calculations. Here we derive and solve, in Laplace space, the recursion relation for the n order path PDF for the same system. The recursion relation relates the n order path PDF to L/2 (round towards zero for an odd L) shorter path PDFs and has n independent coefficients that obey a universal formula. The z transform of the recursion relation straightforwardly gives the generating function for path PDFs, from which we recover the Green's function of the random walk, but, moreover, derive an explicit expression for any path PDF of the random walk. These expressions give the most detailed description of arbitrarily inhomogeneous semi-Markovian random walks in 1D.