Rate Matrix Research Articles

Asymptotical Stability and Stabilization of Continuous-Time Probabilistic Logic Networks

Discrete-time probabilistic logic networks (DT-PLNs), of which probabilistic Boolean networks (PBNs) are a special type, are an important qualitative model for gene regulatory networks (GRNs). Although a DT-PLN can predict the long-term behavior of a GRN, using it to describe the transient kinetics at the microtimescale level remains inconvenient. In this article, we investigate the problems associated with the stability and stabilization of continuous-time probabilistic logic networks (CT-PLNs). First, we demonstrate that the concept of finite-time stability for DT-PLNs cannot be extended to CT-PLNs owing to the nonsingularity of transitional probability matrices. Thus, we introduce the concept of asymptotical stability, which is defined as the convergence in distribution of the network state. Second, by developing the theory of invariant subsets for CT-PLNs, a necessary and sufficient condition for asymptotical stability with respect to a subset is proposed, which is expressed in terms of the transition rate matrix of probability. Third, for a CT-PLN with input nodes, termed a continuous-time probabilistic logic control network, we discuss the subsets that are invariant under piece-wise constant input. Based thereupon, we propose a necessary and sufficient condition under which asymptotically stabilizing sampled-data feedback exists. A method for designing sampled-data feedback is proposed. Last, Monte Carlo simulation algorithms are proposed to efficiently simulate a CT-PLN in the time domain. Examples are provided to demonstrate the methods proposed in this article.

Risk-Based Methodology for Determining Priority Directions for Improving Occupational Safety in the Mining Industry of the Arctic Zone

Over the past 10 years, the mining industry of Russia has seen a greater than three-fold decrease in injury rates, thanks to the successful implementation of innovative labor safety technologies. Despite this, injury levels remain unacceptably high compared to the leading mining countries, which results in increased mining costs. For the mining areas of the Arctic Zone—unlike other regions located in areas with a more favorable climate—the injury rates are influenced not only by the underground labor conditions, but also by the adverse environmental factors. For the Russian Arctic zone, the overall injury risk is proposed to be calculated as the combined impact of occupational and background risk. In this article, we have performed correlation analysis of the overall injury risks in regions of the Arctic zone and regions with favorable climate conditions. Using the Kirov branch of “Apatit”, Joint-stock company (JSC) as an example, we have calculated the risks related specifically to occupational injury rates. We have constructed the relative injury risks and their changes over time and have developed a “basic injury rate matrix” that makes it possible to visualize the results of the comparative analysis of the injury rates on the company’s production sites and to determine priority avenues for improving the occupational safety and lowering the injury rates.

Computing stationary distributions of the D-MAP/D-MSP$$^{(a,b) }/1$$ queueing system

The present study investigates a single-server batch service queueing model with arrival process as discrete-time Markovian arrival process and service process as discrete-time Markovian service process. The server serves the customers in batches followed by the general bulk-service rule. We first determine the random epoch probabilities using the matrix-geometric method, where the rate matrix $${\mathbf{R}}$$ is determined by an efficient approach based on the eigenvalues and the corresponding eigenvectors of the associated characteristic equation. Next we obtain the explicit closed-form expressions for the pre-arrival, intermediate, outside observer’s and post-departure epoch probabilities by developing the relations among them in equilibrium state. Further, we provide an analytically simple approach to carry out the waiting-time distribution in the queue measured in slots as well as the distribution of the size of a service batch of an arriving customer. We also demonstrate a cost function to evaluate the optimum value of the minimum service batch size and the corresponding expected cost of the system. Finally, an adequate variety of numerical experiments are performed for validation purpose of our analytical results and they are conferred in the form of tables and graphs.

On reliability analysis of a load-sharing k-out-of-n: G system with interacting Markov subsystems

Dependency plays a key role in the design stage of multi-component manufacturing systems. To capture stochastic dependencies between subsystems, this paper considers a load-sharing -out-of-: G system which contains non-identical multi-state subsystems. The state evolution of each subsystem follows a continuous-time homogeneous Markov chain. Most existing research on -out-of-: G systems focuses on independent subsystems. However, a subsystem failure usually leads to a higher failure rate for each surviving subsystem within the system. This paper aims to study the interaction among subsystems and analyse the system reliability performance. The transition rate matrix for a given subsystem is assumed to depend on the total number of the remaining surviving subsystems, which can be represented as the sum of a baseline transition rate matrix and a stochastic dependency matrix. This paper proves that the transition rate matrix of the system is the generalised Kronecker sum of transition rate matrices of subsystems, and develops an explicit method to calculate the generalised Kronecker sum. The theory of aggregated stochastic processes is employed to obtain closed-form formulas for reliability indexes. A case study of multi-engine aircraft systems is provided where numerical examples are given to illustrate the developed model and obtained results.

Intra-area factors dominate the spatio-temporal transmission heterogeneity of hand, foot, and mouth disease in China: A modelling study

Hand, foot, and mouth disease (HFMD) has been a serious threat to children's health in China since 2008, and quantifying the transmission heterogeneity of HFMD is crucial for understanding its spatio-temporal dynamic mechanisms and developing precise disease containment strategies. In this study, surveillance data of HFMD cases for patients 15 years of age or younger in east Mainland China between 2009 and 2015 was retrieved from the national surveillance system, and then an age and gender–stratified spatiotemporal Bayesian hierarchical model was built with three types of transmission hazards (intra-area transmission, inter-area transmission, and environmental risk factors). Spatial and temporal heterogeneities of HFMD transmission were quantified by the contribution of the hazard components over time and regions. In addition, the transmission heterogeneity by age–gender subgroups was determined by the transmission probabilities derived from the transmission rate matrix. We found that the intra-area transmission hazard dominated HFMD transmission (56.4% to 96.3%), followed by environmental risk factors (1.1% to 42.1%), whereas the contribution of the inter-area transmission hazard component was relatively small (1.5% to 3.6%). During school holidays, the contribution of the inter-area transmission component caused an instantaneous increase in the overall transmission hazard. In addition, the cumulative hazard contribution across time varied with the geographic region, and three distinct clusters were identified and related with the size of the local population. Moreover, the transmission probability was the highest among the same age–gender group within the intra-area transmission hazard, whereas inter-area and environmental transmission hazards had stronger effects among older children. HFMD transmission varied significantly over time, geographical regions, and age–gender subgroups. Development of targeted and individualized precise policies of HFMD control can benefit from illuminating its spatiotemporal transmission heterogeneity and consideration of the characteristics of population at risk.

Quantitative Evaluation Method and Application of Airflow Stability of Mine Ventilation System

A quantitative analysis method is proposed for airflow stability evaluation of complex ventilation networks. The influences of resistance change on airflow and its variation laws are analyzed by mathematical statistics method based on the simulation of airflow state in ventilation network. It is proposed that the airflow change rate is used as the quantitative evaluation index of airflow stability, and the sensitivity branches and dominant branches of ventilation system are determined by the airflow change rate matrix. A set of airflow direction discriminants of complex network is derived based on the deep analysis of the correlation among resistance, airflow and airflow change rate of branches. The results show that the stability quantitative analysis results and airflow direction discrimination results of this method are consistent with the results of computer simulation of airflow state, and the airflow direction discriminants can be applied to any complex network.

Modulome-Fangjiome Association Study (MoFAS) reveals differential target distribution among four similar fangjis (formulas)

Ethnopharmacological relevancePrecise target distribution is a key issue for further appropriate applications of fangjis (formulas) with similar efficacy and herbal constituents to maximize efficacy and minimize toxicity. Aim of the studyTo develop an algorithm for investigating the differential target distributions and characteristic mechanisms of fangjis. Main methodsIn this study, we proposed a Modulome-Fangjiome Association Study (MoFAS) for comparing fangjis from qi-invigorating and xue-nourishing fangjiome (represented by four fangjis: FEJ, SDT, LYG and QOL). Firstly, the database-driven target network of these 4 fangjis was constructed as qi-xue network and decomposed into modules. Then, the modular map with functional landscape were constructed based on consistency score and enrichment analysis. Finally, we employed a targeting rate (TR) matrix to assess the contribution of this fangjiome to modulome (a set of modules) and compared characteristic effect of fangjis by principal component analysis (PCA). ResultsA qi-xue network constituted by 579 proteins and 23 modules were constructed. In the functional landscape, 3 primary modules were mainly involved in the endocrine system and environmental adaptation. For the target distribution, SDT and QOL were more similar; the FEJ and LYG were located distant from other fangjis according to PCA. The common effects of FEJ, SDT, and QOL focused on stress response and organism development in environmental perturbation, but the FEJ was superior in regulating critical targets, primarily focusing on hormone and neurotransmitter processes. SDT and QOL were concentrated on the majority scale of the qi-xue network, especially for the mitotic cell cycle and development. LYG only targeted lymphocyte costimulation and icosanoid biosynthetic processes. ConclusionIn this study, for the first time, we investigated the difference in the target distribution of qi-invigorating and xue-nourishing fangjiome and provided direct evidence for the characteristic therapeutic effect of these fangjis, which may promote the precise application of fangjis and support the identification of appropriate populations.

Markov models from the square root approximation of the Fokker–Planck equation: calculating the grid-dependent flux

Molecular dynamics (MD) are extremely complex, yet understanding the slow components of their dynamics is essential to understanding their macroscopic properties. To achieve this, one models the MD as a stochastic process and analyses the dominant eigenfunctions of the associated Fokker–Planck operator, or of closely related transfer operators. So far, the calculation of the discretized operators requires extensive MD simulations. The square-root approximation of the Fokker–Planck equation is a method to calculate transition rates as a ratio of the Boltzmann densities of neighboring grid cells times a flux, and can in principle be calculated without a simulation. In a previous work we still used MD simulations to determine the flux. Here, we propose several methods to calculate the exact or approximate flux for various grid types, and thus estimate the rate matrix without a simulation. Using model potentials we test computational efficiency of the methods, and the accuracy with which they reproduce the dominant eigenfunctions and eigenvalues. For these model potentials, rate matrices with up to states can be obtained within seconds on a single high-performance compute server if regular grids are used.

AN IMPROVEMENT OF MARKOVIAN INTEGRATION BY PARTS FORMULA AND APPLICATION TO SENSITIVITY COMPUTATION

This paper establishes a new version of integration by parts formula of Markov chains for sensitivity computation, under much lower restrictions than the existing researches. Our approach is more fundamental and applicable without using Girsanov theorem or Malliavin calculus as did by past papers. Numerically, we apply this formula to compute sensitivity regarding the transition rate matrix and compare with a recent research by an IPA (infinitesimal perturbation analysis) method and other approaches.

LocalDrop: A Hybrid Regularization for Deep Neural Networks.

In neural networks, developing regularization algorithms to settle overfitting is one of the major study areas. We propose a new approach for the regularization of neural networks by the local Rademacher complexity called LocalDrop. A new regularization function for both fully-connected networks (FCNs) and convolutional neural networks (CNNs), including drop rates and weight matrices, has been developed based on the proposed upper bound of the local Rademacher complexity by the strict mathematical deduction. The analyses of dropout in FCNs and DropBlock in CNNs with keep rate matrices in different layers are also included in the complexity analyses. With the new regularization function, we establish a two-stage procedure to obtain the optimal keep rate matrix and weight matrix to realize the whole training model. Extensive experiments have been conducted to demonstrate the effectiveness of LocalDrop in different models by comparing it with several algorithms and the effects of different hyperparameters on the final performances.

The adjustment strategy of ecological land use in the process of urbanization in the suburbs of big cities: A case study of Jizhou District, Tianjin

With the rapid development of rural urbanization, ecological problems have become increasingly serious, so the protection of ecological land is very important. Using vector images from 2010 to 2014 in Jizhou District, Tianjin City, supported by GIS technology, the land use types in Jizhou District, Tianjin City are divided into three categories: cultivated land, construction land and ecological land, and the ecological land is divided into three categories. It is divided into five categories: plantation land, forest land, grassland, water area and water conservancy facility land, and unused land, using methods such as ecological land area change rate and land use transfer matrix model to conduct research and analysis. Studies have shown that the ecological land in Jizhou District of Tianjin has been decreasing year by year in the past five years, and most of the ecological land has been transformed into construction land; the amount of woodland has changed the most drastically, the land for plantation has changed the fastest, while the amount of unused land has shown an overall upward trend.

Fast estimation of multivariate spatiotemporal Hawkes processes and network reconstruction

We present a fast, accurate estimation method for multivariate Hawkes self-exciting point processes widely used in seismology, criminology, finance and other areas. There are two major ingredients. The first is an analytic derivation of exact maximum likelihood estimates of the nonparametric triggering density. We develop this for the multivariate case and add regularization to improve stability and robustness. The second is a moment-based method for the background rate and triggering matrix estimation, which is extended here for the spatiotemporal case. Our method combines them together in an efficient way, and we prove the consistency of this new approach. Extensive numerical experiments, with synthetic data and real-world social network data, show that our method improves the accuracy, scalability and computational efficiency of prevailing estimation approaches. Moreover, it greatly boosts the performance of Hawkes process-based models on social network reconstruction and helps to understand the spatiotemporal triggering dynamics over social media.

Stochastic modeling on M/M/1/N inventory system with queue-dependent service rate and retrial facility

This paper investigates the queue-dependent service rates(QDSR) in the stochastic queueing-inventory system(SQIS). This SQIS consists a single server service channel, $ S $ number of inventories, and a finite queue. An arriving customer gets the service immediately if the server is free and there exists a positive stock in the SQIS. When the server is busy, they have to wait in the finite queue. Suppose they find that the waiting hall is full, either they leave the system or enter into an infinite orbit under the Bernoulli trial. The service rate of any arrival is dependent on the number of customers in the queue at present. The orbital customer can compete for the service only through joining into the waiting hall based on the classical retrial policy. Whenever the number of inventories in SQIS is reached a certain limit $ s $, the replenishment of $ Q( = S-s) $ items is placed. Due to the structure of rate matrix, the stability analysis, minimal non-negative solutions of the quadratic systems are derived through the Neuts matrix-geometric approximation(MGA). Further, the waiting time distribution(WTD) of arrival and necessary system characteristics are derived. Finally, adequate numerical examples are presented to highlight the proposed SQIS.

A study of disproportionately affected populations by race/ethnicity during the SARS-CoV-2 pandemic using multi-population SEIR modeling and ensemble data assimilation

<p style='text-indent:20px;'>The disparity in the impact of COVID-19 on minority populations in the United States has been well established in the available data on deaths, case counts, and adverse outcomes. However, critical metrics used by public health officials and epidemiologists, such as a time dependent viral reproductive number (<inline-formula><tex-math id="M1">\begin{document}$ R_t $\end{document}</tex-math></inline-formula>), can be hard to calculate from this data especially for individual populations. Furthermore, disparities in the availability of testing, record keeping infrastructure, or government funding in disadvantaged populations can produce incomplete data sets. In this work, we apply ensemble data assimilation techniques which optimally combine model and data to produce a more complete data set providing better estimates of the critical metrics used by public health officials and epidemiologists. We employ a multi-population SEIR (Susceptible, Exposed, Infected and Recovered) model with a time dependent reproductive number and age stratified contact rate matrix for each population. We assimilate the daily death data for populations separated by ethnic/racial groupings using a technique called Ensemble Smoothing with Multiple Data Assimilation (ESMDA) to estimate model parameters and produce an <inline-formula><tex-math id="M10000">\begin{document}$R_t(n)$\end{document}</tex-math></inline-formula> for the <inline-formula><tex-math id="M2000">\begin{document}$n^{th}$\end{document}</tex-math></inline-formula> population. We do this with three distinct approaches, (1) using the same contact matrices and prior <inline-formula><tex-math id="M30000">\begin{document}$R_t(n)$\end{document}</tex-math></inline-formula> for each population, (2) assigning contact matrices with increased contact rates for working age and older adults to populations experiencing disparity and (3) as in (2) but with a time-continuous update to <inline-formula><tex-math id="M4">\begin{document}$R_t(n)$\end{document}</tex-math></inline-formula>. We make a study of 9 U.S. states and the District of Columbia providing a complete time series of the pandemic in each and, in some cases, identifying disparities not otherwise evident in the aggregate statistics.</p>

Fault Estimation for Continuous-Time Markovian Jump Systems: An Auxiliary System Approach

This paper addresses the fault estimation problem for continuous-time Markovian jump systems (MJSs). Because the system is stochastic but the estimated fault is deterministic, it is not easy to study the fault estimation problem directly. To overcome this contradiction, an auxiliary system approach is proposed, where a deterministic intermediate variable is constructed by taking expectation on some stochastic variables and the error state and fault are also estimated. Moreover, based on some novel enlarging techniques, more general estimation situations about partially unknown transition rate matrix and semi-Markovian switching are firstly considered. Linear matrix inequality (LMI) conditions are given to guarantee the error states uniformly bounded. Two numerical examples are presented to illustrate the utility and advantage of established methods.

Nonequilibrium Temperature Evolution of Ionization Fronts during the Epoch of Reionization

Abstract The epoch of reionization (EoR) marks the end of the Cosmic Dawn and the beginning of large-scale structure formation in the universe. The impulsive ionization fronts (I-fronts) heat and ionize the gas within the reionization bubbles in the intergalactic medium (IGM). The temperature during this process is a key yet uncertain ingredient in current models. Typically, reionization simulations assume that all baryonic species are in instantaneous thermal equilibrium with each other during the passage of an I-front. Here we present a new model of the temperature evolution for the ionization front by studying nonequilibrium effects. In particular, we include the energy transfer between major baryon species (e −, H i, H ii, He i, and He ii) and investigate their impacts on the post-ionization front temperature T re. For a better step-size control when solving the stiff equations, we implement an implicit method and construct an energy transfer rate matrix. We find that the assumption of equilibration is valid for a nonrelativistic ionization front (speed less than ), but deviations from equilibrium occur for faster fronts. The post-front temperature T re is lower by up to 19.7% (at s−1) or 30.8% (at 1010 cm s−1) relative to the equilibrium case.

An open set of 4×4 embeddable matrices whose principal logarithm is not a Markov generator

ABSTRACT A Markov matrix is embeddable if it can represent a homogeneous continuous-time Markov process. It is well known that if a Markov matrix has real and pairwise-different eigenvalues, then the embeddability can be determined by checking whether its principal logarithm is a rate matrix or not. The same holds for Markov matrices that are close enough to the identity matrix. In this paper we exhibit open sets of Markov matrices that are embeddable and whose principal logarithm is not a rate matrix, thus proving that the principal logarithm test above does not suffice generically.

Incorporating compositional heterogeneity into Lie Markov models for phylogenetic inference

Phylogenetics uses alignments of molecular sequence data to learn about evolutionary trees. Substitutions in sequences are modelled through a continuous-time Markov process, characterised by an instantaneous rate matrix, which standard models assume is time-reversible and stationary. These assumptions are biologically questionable and induce a likelihood function which is invariant to a tree’s root position. This hampers inference because a tree’s biological interpretation depends critically on where it is rooted. Relaxing both assumptions, we introduce a model whose likelihood can distinguish between rooted trees. The model is nonstationary with step changes in the instantaneous rate matrix at each speciation event. Exploiting recent theoretical work, each rate matrix belongs to a nonreversible family of Lie Markov models. These models are closed under matrix multiplication, so our extension offers the conceptually appealing property that a tree and all its subtrees could have arisen from the same family of nonstationary models. We adopt a Bayesian approach, describe an MCMC algorithm for posterior inference and provide software. The biological insight that our model can provide is illustrated through an analysis in which nonreversible but stationary and nonstationary but reversible models cannot identify a plausible root.

Expression of p57KIP2 reduces growth and invasion, and induces syncytialization in a human placental choriocarcinoma cell line, BeWo

IntroductionSyncytiotrophoblasts are the major components of the human placenta involved in fetal maternal exchange and hormone secretion. The syncytiotrophoblasts arise from the fusion of villous cytotrophoblasts. The cell cycle suppressor p57KIP2 is known to be an essential molecule for proper trophoblast differentiation during placental formation. MethodsWe generated p57KIP2-expressing BeWo transfectant cells. Proliferation assay and matrigel invasion assay were used to characterize p57KIP2-expressing BeWo transfectant cells. To reveal the role of p57KIP2 in syncytialization, we proceeded syncytium formation analysis and qRT-PCR for detection of the expression levels Syncytin-1, Syncytin-2 and their receptors. ResultsThe human choriocarcinoma cell line, BeWo has undetectable levels of p57KIP2 expression. Expression of p57KIP2 reduced cell proliferation rate and extracellular matrix invasion activity. p57KIP2 expressing cells displayed multinucleated cells associated with syncytiotrophoblast differentiation. In the syncytialization event, p57KIP2 was found to potentiate forskolin-induced upregulation of Syncytin-2 in a cAMP-independent manner. DiscussionThese results indicate that the expression of p57KIP2 may act on the proliferation/invasion inhibitory factor and enhance the expression of Syncytin-2, which are associated with syncytialization in cytotrophoblasts.

Phylosymmetric Algebras: Mathematical Properties of a New Tool in Phylogenetics

In phylogenetics, it is of interest for rate matrix sets to satisfy closure under matrix multiplication as this makes finding the set of corresponding transition matrices possible without having to compute matrix exponentials. It is also advantageous to have a small number of free parameters as this, in applications, will result in a reduction in computation time. We explore a method of building a rate matrix set from a rooted tree structure by assigning rates to internal tree nodes and states to the leaves, then defining the rate of change between two states as the rate assigned to the most recent common ancestor of those two states. We investigate the properties of these matrix sets from both a linear algebra and a graph theory perspective and show that any rate matrix set generated this way is closed under matrix multiplication. The consequences of setting two rates assigned to internal tree nodes to be equal are then considered. This methodology could be used to develop parameterised models of amino acid substitution which have a small number of parameters but convey biological meaning.