Markov processes discrete

Markov chains or processes are named after the Russian mathematician A.A.Markov (1852-1922) who introduced the concept of chain dependence and did basic pioneering work on this class of processes [1]. A Markov process is a mathematical probabilistic model that is very useful in the study of complex systems. The essence of the model is that if the initial state of a system is known, i.e. its present state, and the probabilities to move forward to other states are also given, then it is possible to predict the future state of the system ignoring its past history. In other words, past history is immaterial for predicting the future this is the key-element in Markov chains. Distinction is made between Markov processes discrete in time and space, processes discrete in space and continuous in time and processes continuous in space and time. This book is mainly concerned with processes discrete in time and space. 

In the present chapter, Markov processes discrete in time and space, processes discrete in space and continuous in time as well as processes continuous in space and time, have been presented. The major aim of the presentation has been to provide the reader with a concise summary of the above topics which should give the reaaer an overview of the subject. 

The solution of SDE (2.238) was originally defined by Stratonovich [30] as the limit At 0 of a sequence of discrete Markov processes, for which... 

The discrete Markov process used to define a kinetic SDE in Eq. (2.315) or (2.318) can be directly implemented as a numerical algorithm for the integration of a set of SDEs. The resulting simulation algorithm would require the evaluation of neither derivatives of the mobility nor any corrective pseudoforce. It would, however, require an efficient method of calculating the elements of the mobility tensor and derivatives of U and in in the chosen system of generalized coordinates. 

In this section, we have considered four possible ways of formulating and interpreting a set of SDE to describe Brownian motion, and tried to clarify the relationships among them. Because each interpretation may be defined as the At 0 limit of a discrete Markov processes, this discussion of SDEs provides a useful starting point for the discussion of possible simulation algorithms. 

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. 

To solve the equation by simulation, we replace the discrete equation by the following Markov process. A number of particles is distributed in... 

Exercise. A gambler plays heads and tails. Let Yt be the amout of his capital after t throws. Show that Yt is a discrete-time Markov process and find its transition probability. 

This is called the Chapman-Kolmogorov equationIt is an identity, which must be obeyed by the transition probability of any Markov process. The time ordering is essential t2 lies between and t3. Of course, the equation also holds when y is a vector with r components or when y only takes discrete values so that the integral is actually a sum. 

Application. Consider a Markov process with discrete sites and governed by the M-equation... 

Remark. The essential feature of our composite process is that i is an independent process by itself, while the transition probabilities of r are governed by i. This situation can be formulated more generally. Take a Markov process Y(t), discrete or continuous, having an M-equation with kernel... 

These equations and the example shown in Section 4.3.1 can be related if we consider that, when v(t) is a Markov process with discrete values = 1,.n and... 

The monomolecular reaction systems of chemical kinetics are examples of linear coupled systems. Since linear coupled systems are the simplest systems with many degrees of freedom, their importance extends far beyond chemical kinetics. The linear coupled systems in which we are interested may be characterized, in general terms, as arising from stochastic or Markov processes that are continuous in time and discrete in an appropriate space. In addition, the principle of detailed balancing is observed and the total amount of material in the system is conserved. The system is characterized by discrete compartments or states and material passes between these compartments by first order processes. Such linear systems are good models for a large number of processes. 

Each of the optional dynamical models mentioned above involves a homogeneous Markov process Xt = Xt teT in either continuous or discrete time on some state space X. The motion of Xt is given in terms of the stochastic... 

A possible reason for this phenomenon might be that books containing material on this subject have been written in such a way that the simplicity of discrete Markov chains has been shadowed by the tedious mathematical derivations. This caused one to abandon the book, thus loosing a potential tool for handling his problems. In a humorous way, this situation might be demonstrated as follows. Suppose that a Chemical Engineer wishes to study Markov processes and has been suggested several books on this subject. Since the mathematics is rather complex... 

In the preceding chapter Markov chains has been dealt with as processes discrete in space and time. These processes involve countably many states Si, S2,. .. and depend on a discrete time parameter, that is, changes occur at fixed steps n = 0, 1, 2,. ... 

In the following, we derive the Kolmogorov differential equation on the basis of a simple model and report its various versions. In principle, this equation gives the rate at which a certain state is occupied by the system at a certain time. This equation is of a fundamental importance to obtain models discrete in space and continuous in time. The models, later discussed, are the Poisson Process, the Pure Birth Process, the Polya Process, the Simple Death Process and the Birth-and-Death Process. In section 2.1-3 this equation, i.e. Eq.2-30, has been derived for Markov chains discrete in space and time. 

The characteristics of the state space being measured can be used to classify the Markov process. For most purposes, a discrete or finite space is assumed and this implies that there are a finite number of states that will be reached by the process (14). A continuous or infinite process is also possible. Time intervals of observation of a process can be used to classify a Markov process. Processes can be observed at discrete or restricted intervals, or continuously (15). 

Markov chain is the term used to describe a process observed at discrete intervals. However, some investigators prefer to describe Markov chains as a special case of a continuous-time Markov process. That is, the process is only observed at discrete intervals, but in reality it is a continuous-time Markov process (16). Therefore, the Markov process can be used to collectively describe all processes and chains. 

A discrete state stochastic Markov process simulates the movement of the evacuees. Transition from node-to-node is simulated as a random process where the prohahility of transition depends on the dynamically changed states of the destination and origin nodes and on the link between them. Solution of the Markov process provides the expected distribution of the evacuees in the nodes of the area as a function of time. 

Discrete-time Markov processes are a third type of problem we shall discuss. One of the challenges in this case is to compute the correlation time of such a process in the vicinity of a critical point, where the correlation time goes to infinity, a phenomenon called critical slowing down. Computationally, the problem amounts to the evaluation of the second largest eigenvalue of the Markov matrix, or more precisely its difference from unity. The latter goes to zero as the correlation time approaches infinity. 

As a (complete) convolution of regular Markov processes, are stochastic matrices (with dimension y x fcl) and constitute regular discrete Markov processes. Hence, steady state vectors for each location j exist which represent the long-term probabilities of a site s state combinations. ... 

To model the state of the plant, a discrete Markov process is used. To calculate the transition matrix Q of a discrete Markov process, the transition probabilities between both states have to be estimated. All transitions of the recorded inflow data is used. The time series of plant states LOt are calculated by... 

The study of the development of these profiles over time by using a probabilistic graph of transitions between the clusters inferred by k-TSSI (k-Testable Languages in the Strict Sense Inference) algorithm. The objective is to deduce Markov process which has a discrete (finite or countable) state-space. 

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