Markov chain analysis

Johnson GE, Hedgebeth JB, Skalski JR et al (2004) A Markov chain analysis of fish movements to determine entrainment zones. Fish Res 69 349-358... 

Mtihlenbein (1992) studied the optimal mutation rate for genetic algorithms. By using a Markov chain analysis, he found that the optimal mutation rate pm is... 

Doveton, J. H. (1971) An application of Markov chain analysis to the Ayrshire Cole Measures succession. Scottish J. Geology 1, 11-27. 

Doveton, J. H. (1994) Theory and applications of vertical variability measures from Markov Chain analysis. In Yams, X Chambers, R. (Eds.) Stochastic modeUing and geostatistics., AAPG Comput. Appl. Geol. 3, 55-63. 

Information flow is related to the mean first passage time (MFPT) concept used in Markov chain analysis. MFPT represents the mean time it takes for a signal released at a node to reach another node in the Markov chain. 

In the next subsection, I describe how the basic elements of Bayesian analysis are formulated mathematically. I also describe the methods for deriving posterior distributions from the model, either in terms of conjugate prior likelihood forms or in terms of simulation using Markov chain Monte Carlo (MCMC) methods. The utility of Bayesian methods has expanded greatly in recent years because of the development of MCMC methods and fast computers. I also describe the basics of hierarchical and mixture models. 

Rigorous kinetic analysis has shown [41] that the products of binary copolymerization, formed under the conditions of constant concentrations of monomers, may be described by the extended Markov chain with four states Sa, if to label monomeric units conventionally coloring them in red and black. Unit Ma is presumed to be black when the corresponding monomer Ma adds to the radical as the first monomer of the complex. In other cases, when monomer Ma adds individually or as the second monomer of the complex, the unit Ma is assumed to be red. As a result the state of a monomeric unit is characterized by two attributes, one of which is its type (a=l,2) while the second one is its color (r,b). For example, we shall speak about the unit being in the state Sx provided it is of the first type and red-colored, i.e. Mrx. The other states Sa are determined in a similar manner ... 

I3G. Winkler, Image Analysis, Random Fields and Markov Chain Monte Carlo Methods, Springer-Verlag, New York, 2003. 

Berg, B.A. Markov Chain Monte Carlo Simulations and Their Statistical Analysis, World Scientific,... 

In the three dimensional chain on a cubical lattice, N/3 elements are considered to be directed along the three directions x, y and z. The analysis of Krigbaum and Kaneko shows that the partition function A for the three dimensional chain can be written as a product AXAVAX, in which each of the terms are given by the solution of a one-dimensional Markov chain of Nj3 steps ... 

Modeling Electrochemical Phenomena via Markov Chains and Processes gives an introduction to Markov Theory, then discusses applications to electrochemistry, including modeling electrode surface processes, electrolyzers, the repair of failed cells, analysis of switching-circuit operations, and other electrochemical systems... 

Gamer C, Mclnnes LA, Service SK, et al. Linkage analysis of a complex pedigree with severe bipolar disorder, using a Markov chain Monte Carlo method. Am J Hum Genet 2001 68(4) 1061-1064. 

Covington TR, Gentry PR, Van Landingham CB, Andersen ME, Kester JE, Clewell HJ. 2007. The use of Markov chain Monte Carlo uncertainty analysis to support a public health goal for perchloroethylene. Regul Toxicol Pharmacol 47 1-18. 

The consistent kinetic analysis of the copolymerization with the simultaneous occurrence of the reactions (2.1) and (2.5) leads to the conclusion that the probabilities of the sequences of the monomer units M, and M2 in the macromolecules can not be described by a Markov chain of any finite order. Consequently, in this very case we deal with non-Markovian copolymers, the general theory for which is not yet available [6]. However, a comprehensive statistical description of the products of the complex-radical copolymerization within the framework of the Seiner-Litt model via the consideration of the certain auxiliary Markov chain was carried out [49, 59, 60]. 

The general formulae (5.1), (5.3), and (5.7) are still valid under the transition to the more complicated models described in Sect. 2. In the case of the penultimate model it concerns also the dynamic Eqs. (5.2) into which now one should substitute the dependence j (i) obtained after the solution of the problem of the calculation of the stationary vector tE(x) of the Markov chain corresponding to this model. Substituting the function X1(x1) obtained via the above procedure (see Sect. 3.1) into Eq. (5.2) for the binary copolymerization we can find its explicit solution expressed through the elementary functions. However, this solution is rather cumbersome and has no practical importance. It is not needed even for the classification of the dynamic behavior of the systems, which can be carried out only via analysis of Eq. (5.5) by determining the number n = 0,1,2, 3 of the inner azeotropes in the 2-simplex [14], The complete set of phase portraits of the binary... 

The term Markov chain frequently appears in this chapter. This term is named after the Russian mathematician Andre Markov (1856-1922). The Markov theory is widely applied in many fields, including the analysis of stock-markets, traffic flows, queuing theories (e.g. modelling a telephone customer service hotline), reliability theories (e.g. modelling the time for a component to wear out) and many other systems involving random processes. 

Graph theory Describes topology of networks and subnetworks, based on quantification of the number of nodes (signaling components) and links between them. Dynamic properties of networks through Boolean analysis. Network analysis based on probabilities (Markov chain and Bayesian) to identify paths and relationships between different nodes in the network. (75-81)... 

B. A. Berg (2003) Multicanonical simulations step by step. Comp. Phys. Comm. 153, pp. 397-406 ibid. (2004) Markov Chain Monte Carlo Simulations and their Statistical Analysis. World Scientific P. 380... 

R. Elber and M. Karpins (1987) Multiple conformational states of proteins A molecular dynamics analysis of Myoglobin. Science 235, pp. 318-321 P. Deuflhard, W. Huisinga, A. Fischer, and C. Schiitte (2000) Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains. Lin. Alg. Appl. 315, pp. 39-59... 

More recently Brochot et al. [89] reported an extension of the isobolographic approach to interaction studies for convulsant interaction among pelloxacin, norfloxacin, and theophylline in rats. Their contribution is unique in that they started out by explaining pharmacodynamic interactions for two drugs, but then extended the approach to derive an isobol for three drug interaction. In addition they included Bayesian analysis and developed a population model with Markov chain Monte Carlo methods. 

A statistical analysis of the sequence distribution can be performed in terms of direct and inverted units (D and I), i.e. of units written with carbon Cl at the left or at the right, respectively. Dyads DD and II, which differ in the sense of observation, correspond to head-to-tail, ID to head-to-head and DI to tail-to-tail junctions respectively. In the same way triads of D or I units are related to longer sequences. Remember that DDD and III, IDD and IID, DDI and DII, IDI and DID cannot be distinguished from each other. An interpretation according to a first-order Markov chain requires the use of two conditional probabilities, p... 

O. Kempthorne and L. Folks, Markov chains, in Probability, Statistics, and Data Analysis. Iowa University Press, Ames, 1971, pp. 188-197. 

Baum, L.E., et ah, A Maximization Technique Occurring in the Statistical Analysis of Prohabilistic Functions of Markov Chains, Ann. Math. Statist., 41, 164, 1970. 

Baum, L. E., Pertie, T., Soules, G. Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann Math Stat 41(1), 164—171. 

Thus, we take advantage of the accuracy, robustness and efficiency of the direct problem solution, to tackle the associated inverse heat transfer problem analysis [26, 27] towards the simultaneous estimation of momentum and thermal accommodation coefficients in micro-channel flows with velocity slip and temperature jump. A Bayesian inference approach is adopted in the solution of the identification problem, based on the Monte Carlo Markov Chain method (MCMC) and the Metropolis-Hastings algorithm [28-30]. Only simulated temperature measurements at the external faces of the channel walls, obtained for instance via infrared thermography [30], are used in the inverse analysis in order to demonstrate the capabilities of the proposed approach. A sensitivity analysis allows for the inspection of the identification problem behavior when the external wall Biot number is also included among the parameters to be estimated. 

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