Markov transition model

Hierarchical protein strucmre superposition STRUCTAL-based program Markov transition model of evolution... 

Structure Comparison Using the Markov Transition Model of Evolution. 

A Markov process model describes several discrete health states in which a person can exist at time t, as well as the health states into which the person may move at time t +1. A person can reside in just one health state at any given time. The progression from time t to time t +1 is known as a cycle. All clinically important events are modeled as transitions in which a person moves from one health state to another. The probabilities associated with each change between health states are known as transition probabilities. Each transition probability is a function of the health state and the treatment. 

There are many advantages, detailed in Chapter 1, of using the discrete Markov-chain model in Chemical Engineering. Probably, the most important advantage is that physical models can be presented in a unified description via state vector and a one-step transition probability matrix. Consequently, a process is demonstrated solely by the probability of a system to occupy a state or not to occupy it. William Shakespeare profoundly stated this in the following way " to be (in a state) or not to be (in a state), that is the question". 

Transition networks have also been constructed for biomolecules directly from MD data, - where they are sometimes referred to as Markov state models. In order to gather sufficient statistics, the transitions between states must be observable on an MD timescale. Alternative methods exist, not based on MD, where connections between conformations are inferred based on distance criteria. The edge weights of the corresponding graphs are then based on the energies of the two geometries that are assumed to be connected, rather than calculated barriers or rate constants. 

In a KMC method, it is typically assumed that various possible state-to-state transitions from a given state are well modelled by the Arrenhius law and then molecular dynamics is used to calculate the prefactor A and energy difference AE in order to understand the timescales and relative probabilities of different rare events. A Markov state model can be developed to help understand the global dynamics and simplify the model as a whole. For references on many interesting approaches to this important topic, the reader is referred to [36,42,137,149,391]. Andersen Thermostat. Of particular interest is the simple and useful Andersen thermostat [11]. This method works by selecting atoms at random and randomly perturbing their momenta in a way consistent with prescribed thermodynamic conditions. It has been rigorously proven to sample the canonical distribution [114],... 

The General Markov Reward Model considers the continuous time Markov chain with a set of states and transition intensity matrix... 

AU of them require simplifying assumptions about time to failure behavior of the system components. Moreover, Markov method analyses the system by identifying all the different states in which the system can reside and is able to produce accurate system reUabUity measures by assigning rates of transition between these states. However, the Markov method has its own drawbacks in its appUcation for a relatively large system to establish the state transition model is an intractable task. 

The Markov chain model can be used to analyze chemotaxis experiments. If there is a preferred direction for cell movement, its steady-state probability wUl be significantly higher than the steady-state probabUities of the other directions. The Markov chain approach can also provide a more detaUed description of ceU migration since it accounts for stops in ceU movement and uses more than one descriptor (e.g., transition state probabilities and average waiting times) to define persistence [148]. 

As we mentioned in section 2.1 the period of 20 years is divided into sup-periods according to the environment state "ES . The transition between different states is controlled by an external covariate which follow a Markov Chain model. The Markov Chain is constructed by three states (Nl, N2 and N3) that represent respectively the Calm, Normal and Agitated ES. Considering the nine indexes influenced by this factor, the transition matrix between these states and the average duration in a state are defined by the following matrix and vector ... 

The state-transition model can be analyzed using a number of approaches as a Markov chains, using semi-Markov processes or using Monte Carlo simulation (Fishman 1996). The applicability of each method depends on the assumptions that can be made regarding faults occurrence and a repair time. In case of the Markov approach, it is necessary to assume that both the faults and renewals occur with constant intensities (i.e. exponential distribution). Also the large number of states makes Markov or semi-Markov method more difficult to use. Presented in the previous section reliability model includes random values with exponential, truncated normal and discrete distributions as well as some periodic relations (staff working time), so it is hard to be solved by analytical methods. 

Markov modeling is a technique for calculating system reliability as exponential transitions between various states of operability, much like atomic transitions. In addition to the use of constant transition rates, the model depends only on the initial and final states (no memory). 

Emersed electrode, 12 Energy scales and electrode potentials, 7 Energy transitions via polaronic and bipolaronic levels, 362 Engineering models, for fluorine generation cells, 539 Esin and Markov plots, 259-260 Experimental data comparison thereof, 149 on potential of zero charge, 56... 

In the framework of this ultimate model [33] there are m2 constants of the rate of the chain propagation kap describing the addition of monomer to the radical Ra whose reactivity is controlled solely by the type a of its terminal unit. Elementary reactions of chain termination due to chemical interaction of radicals Ra and R is characterized by m2 kinetic parameters k f . The stochastic process describing macromolecules, formed at any moment in time t, is a Markov chain with transition matrix whose elements are expressed through the concentrations Ra and Ma of radicals and monomers at this particular moment in the following way [1,34] ... 

This is the simplest of the models where violation of the Flory principle is permitted. The assumption behind this model stipulates that the reactivity of a polymer radical is predetermined by the type of bothjts ultimate and penultimate units [23]. Here, the pairs of terminal units MaM act, along with monomers M, as kinetically independent elements, so that there are m3 constants of the rate of elementary reactions of chain propagation ka ]r The stochastic process of conventional movement along macromolecules formed at fixed x will be Markovian, provided that monomeric units are differentiated by the type of preceding unit. In this case the number of transient states Sa of the extended Markov chain is m2 in accordance with the number of pairs of monomeric units. No special problems presents writing down the elements of the matrix of the transitions Q of such a chain [ 1,10,34,39] and deriving by means of the mathematical apparatus of the Markov chains the expressions for the instantaneous statistical characteristics of copolymers. By way of illustration this matrix will be presented for the case of binary copolymerization ... 

An exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

Besides the fugacity models, the environmental science literature reports the use of models based on Markov chain principle to evaluate the environmental fate of chemicals in multimedia environment. Markov chain is a random process, and its theory lies in using transition matrix to describe the transition of a substance among different states [39,40]. If the substance has all together n different kinds of states,... 

The exact computation of P W) in this simple one-dipole model is already a very arduous task that, to my knowledge, has not yet been exactly solved. We can, however, consider a limiting case and try to elucidate the properties of the work (heat) distribution. Here we consider the limit of large ramping speed r, where the dipole executes just one transition from the down to the up orientation. A few of these paths are depicted in Fig. 13b. This is also called a first-order Markov process because it only includes transitions that occur in just one direction (from down to up). In this reduced and oversimplified description, a path is fully specified by the value of the field H at which the dipole reverses orientation. The work along one of these paths is given by... 

The Markov model uses the clinical data to calculate the probability of transitioning from a severe disease state ( OFF time >25% of the day), to a less severe disease state ("OFF time <25% of the day) for entacapone therapy. This enables a calculation of the total amount of time a cohort of patients will... 

For many physical applications, modeled by a homogeneous Markov process in time and space, the rate of transition is time independent and depends only on the difference of the starting and arriving states. Therefore, one can see that the master equation is given by... 

Although a theoretical approach has been desecrated as to how one can apply the generalized coupled master equations to deal with ultrafast radiationless transitions taking place in molecular systems, there are several problems and limitations to the approach. For example, the number of the vibrational modes is limited to less than six for numerical calculations. This is simply just because of the limitation of the computational resources. If the efficient parallelization can be realized to the generalized coupled master equations, the limitation of the number of the modes can be relaxed. In the present approach, the Markov approximation to the interaction between the molecule and the heat bath mode has been employed. If the time scale of the ultrashort measurements becomes close to the characteristic time of the correlation time of the heat bath mode, the Markov approximation cannot be applicable. In this case, the so-called non-Markov treatment should be used. This, in turn, leads to a more computationally demanding task. Thus, it is desirable to develop a new theoretical approach that allows a more efficient algorithm for the computation of the non-Markov kernels. Another problem is related to the modeling of the interaction between the molecule and the heat bath mode. In our model, the heat bath mode is treated as... 

Lithium clusters have been a popular model for the calculation of metal properties because of their low atomic number. Lasarov and Markov (49) used a Hiickel procedure to determine the properties of a 48-atom Li crystal. They found a transition to metal properties with the binding energy per atom approaching 1.8 eV at 30 atoms. The ionization potential approached the bulk value since some electrons occupy antibonding molecular orbitals, as observed for Ag clusters. The calculated properties of the largest cluster were not those of a bulk metal. 

Consider now a multicompartment structure aiming not only to describe the observed data but also to provide a rough mechanistic description of how the data were generated. This mechanistic system of compartments is envisaged with the drug flowing between the compartments. The stochastic elements describing these flows are the transition probabilities as previously defined. In addition, with each compartment in this mechanistic structure, one can associate a retention-time distribution (a). The so-obtained multicompartment model is referred to as the semi-Markov formulation. The semi-Markov model has two properties, namely that ... 

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