Web Server and R Library for the Calculation of Markov Chains Molecular Descriptors

We introduce a new class of Monte Carlo methods, which we call exact estimation algorithms. Such algorithms provide unbiased estimators for equilibrium expectations associated with real-valued functionals defined on a Markov chain. We provide easily implemented algorithms for the class of positive Harris recurrent Markov chains, and for chains that are contracting on average. We further argue that exact estimation in the Markov chain setting provides a significant theoretical relaxation relative to exact simulation methods.

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Exact estimation for Markov chain equilibrium expectations

Journal of Applied Probability ◽

10.1239/jap/1417528487 ◽

2014 ◽

Vol 51 (A) ◽

pp. 377-389 ◽

Cited By ~ 17

Author(s):

Peter W. Glynn ◽

Chang-Han Rhee

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chains ◽

Monte Carlo Methods ◽

Simulation Methods ◽

Exact Simulation ◽

Unbiased Estimators ◽

Estimation Algorithms ◽

New Class

We introduce a new class of Monte Carlo methods, which we call exact estimation algorithms. Such algorithms provide unbiased estimators for equilibrium expectations associated with real-valued functionals defined on a Markov chain. We provide easily implemented algorithms for the class of positive Harris recurrent Markov chains, and for chains that are contracting on average. We further argue that exact estimation in the Markov chain setting provides a significant theoretical relaxation relative to exact simulation methods.

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Exponentially slow transitions on a Markov chain: the frequency of Calcium Sparks

European Journal of Applied Mathematics ◽

10.1017/s0956792505006194 ◽

2005 ◽

Vol 16 (4) ◽

pp. 427-446 ◽

Cited By ~ 34

Author(s):

ROBERT HINCH ◽

S. JON CHAPMAN

Keyword(s):

Markov Chain ◽

Markov Chains ◽

First Passage Time ◽

Passage Time ◽

Projection Methods ◽

Metastable States ◽

Calcium Sparks ◽

Asymptotic Results ◽

Cardiac Muscle Cells ◽

New Class

Calcium sparks in cardiac muscle cells occur when a cluster of Ca2+ channels open and release Ca2+ from an internal store. A simplified model of Ca2+ sparks has been developed to describe the dynamics of a cluster of channels, which is of the form of a continuous time Markov chain with nearest neighbour transitions and slowly varying jump functions. The chain displays metastability, whereby the probability distribution of the state of the system evolves exponentially slowly, with one of the metastable states occurring at the boundary. An asymptotic technique for analysing the Master equation (a differential-difference equation) associated with these Markov chains is developed using the WKB and projection methods. The method is used to re-derive a known result for a standard class of Markov chains displaying metastability, before being applied to the new class of Markov chains associated with the spark model. The mean first passage time between metastable states is calculated and an expression for the frequency of calcium sparks is derived. All asymptotic results are compared with Monte Carlo simulations.

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Markov chains, ${\mathscr R}$-trivial monoids and representation theory

International Journal of Algebra and Computation ◽

10.1142/s0218196715400081 ◽

2015 ◽

Vol 25 (01n02) ◽

pp. 169-231 ◽

Cited By ~ 17

Author(s):

Arvind Ayyer ◽

Anne Schilling ◽

Benjamin Steinberg ◽

Nicolas M. Thiéry

Keyword(s):

Markov Chain ◽

General Theory ◽

Markov Chains ◽

Random Walks ◽

Transition Matrix ◽

Coxeter Groups ◽

Mixing Time ◽

New Class ◽

Free Tree ◽

Sandpile Models

We develop a general theory of Markov chains realizable as random walks on [Formula: see text]-trivial monoids. It provides explicit and simple formulas for the eigenvalues of the transition matrix, for multiplicities of the eigenvalues via Möbius inversion along a lattice, a condition for diagonalizability of the transition matrix and some techniques for bounding the mixing time. In addition, we discuss several examples, such as Toom–Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples previously studied by the authors, such as nonabelian sandpile models and the promotion Markov chain on posets. Many of these examples can be viewed as random walks on quotients of free tree monoids, a new class of monoids whose combinatorics we develop.

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Studies on the EC50 of Natural Monoterpenes as Fungal Inhibitors with Quantitative Structure-Activity Relationships (QSARs)

The Natural Products Journal ◽

10.2174/2210315509666190117150153 ◽

2020 ◽

Vol 10 (1) ◽

pp. 44-60

Author(s):

Mohamed E.I. Badawy ◽

Entsar I. Rabea ◽

Samir A.M. Abdelgaleil

Keyword(s):

Biological Activity ◽

Plant Pathogens ◽

Molecular Descriptors ◽

Microbial Pathogens ◽

Quantitative Structure ◽

Qsar Study ◽

Qsar Analysis ◽

Pharmacophore Modelling ◽

Structure Activity ◽

Wide Range

Background:Monoterpenes are the main constituents of the essential oils obtained from plants. These natural products offered wide spectra of biological activity and extensively tested against microbial pathogens and other agricultural pests.Methods:Antifungal activity of 10 monoterpenes, including two hydrocarbons (camphene and (S)- limonene) and eight oxygenated hydrocarbons ((R)-camphor, (R)-carvone, (S)-fenchone, geraniol, (R)-linalool, (+)-menthol, menthone, and thymol), was determined against fungi of Alternaria alternata, Botrytis cinerea, Botryodiplodia theobromae, Fusarium graminearum, Phoma exigua, Phytophthora infestans, and Sclerotinia sclerotiorum by the mycelia radial growth technique. Subsequently, Quantitative Structure-Activity Relationship (QSAR) analysis using different molecular descriptors with multiple regression analysis based on systematic search and LOOCV technique was performed. Moreover, pharmacophore modelling was carried out using LigandScout software to evaluate the common features essential for the activity and the hypothetical geometries adopted by these ligands in their most active forms.Results:The results showed that the antifungal activities were high, but depended on the chemical structure and the type of microorganism. Thymol showed the highest effect against all fungi tested with respective EC50 in the range of 10-86 mg/L. The QSAR study proved that the molecular descriptors HBA, MR, Pz, tPSA, and Vp were correlated positively with the biological activity in all of the best models with a correlation coefficient (r) ≥ 0.98 and cross-validated values (Q2) ≥ 0.77.Conclusion:The results of this work offer the opportunity to choose monoterpenes with preferential antimicrobial activity against a wide range of plant pathogens.

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Asymptotic behaviour of sample weighted circuits representing recurrent Markov chains

Journal of Applied Probability ◽

10.1017/s0021900200039103 ◽

1990 ◽

Vol 27 (03) ◽

pp. 545-556 ◽

Cited By ~ 2

Author(s):

S. Kalpazidou

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Asymptotic Behaviour ◽

Probabilistic Interpretation ◽

Stationary Markov Chain

The asymptotic behaviour of the sequence (𝒞 n (ω), wc,n (ω)/n), is studied where 𝒞 n (ω) is the class of all cycles c occurring along the trajectory ωof a recurrent strictly stationary Markov chain (ξ n ) until time n and wc,n (ω) is the number of occurrences of the cycle c until time n. The previous sequence of sample weighted classes converges almost surely to a class of directed weighted cycles (𝒞∞, ω c ) which represents uniquely the chain (ξ n ) as a circuit chain, and ω c is given a probabilistic interpretation.

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On the absorption probabilities and mean time for absorption for discrete Markov chains

Monte Carlo Methods and Applications ◽

10.1515/mcma-2021-2084 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Nikolaos Halidias

Keyword(s):

Markov Chain ◽

Random Walk ◽

Markov Chains ◽

Generating Function ◽

Discrete Time ◽

Probability Generating Function ◽

The Mean ◽

Mean Time ◽

Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

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Proportional lumpability and proportional bisimilarity

Acta Informatica ◽

10.1007/s00236-021-00404-y ◽

2021 ◽

Author(s):

Andrea Marin ◽

Carla Piazza ◽

Sabina Rossi

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Upper And Lower Bounds ◽

General Definition ◽

Performance Indices ◽

State Aggregation ◽

Structural Regularity ◽

Original Definition ◽

Aggregation Technique ◽

Definition Of

AbstractIn this paper, we deal with the lumpability approach to cope with the state space explosion problem inherent to the computation of the stationary performance indices of large stochastic models. The lumpability method is based on a state aggregation technique and applies to Markov chains exhibiting some structural regularity. Moreover, it allows one to efficiently compute the exact values of the stationary performance indices when the model is actually lumpable. The notion of quasi-lumpability is based on the idea that a Markov chain can be altered by relatively small perturbations of the transition rates in such a way that the new resulting Markov chain is lumpable. In this case, only upper and lower bounds on the performance indices can be derived. Here, we introduce a novel notion of quasi-lumpability, named proportional lumpability, which extends the original definition of lumpability but, differently from the general definition of quasi-lumpability, it allows one to derive exact stationary performance indices for the original process. We then introduce the notion of proportional bisimilarity for the terms of the performance process algebra PEPA. Proportional bisimilarity induces a proportional lumpability on the underlying continuous-time Markov chains. Finally, we prove some compositionality results and show the applicability of our theory through examples.

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