Markov property

See, for instance [20] for a detailed discussion of the Markov property and of Markov processes. 

Markovnikov rule, 20 774 Markov property, 26 1022 Markush chemical structures, indexing and searching, 18 242. See also WPI entries... 

This chapter defines and describes the subclass of stochastic processes that have the Markov property. Such processes are by far the most important in physics and chemistry. The rest of this book will deal almost exclusively with Markov processes. 

This example exhibits several features that are of general validity. First it is clear that the Markov property holds only approximately. If the previous displacement Xk — Xk-x happened to be a large one, then the chances are slightly in favor of a large velocity at the time when Xk is observed. This velocity will survive for a short time of the order of the autocorrelation time of the velocity, and thereby favor a large value of Xk + 1 — Xk. Thus the fact that the autocorrelation time of the velocity is not strictly zero gives rise to some correlation between two successive displacements. This effect is small, provided that the time between two observations is much longer than the autocorrelation time of the velocity. 

Exercise. The Chapman-Kolmogorov equation (2.1) expresses the fact that a process starting at with value yx reaches y3 at t3 via any one of the possible values y2 at the intermediate time t2. Where does the Markov property enter into this argument ... 

A finite Markov chain is one whose range consists of a finite number N of states. They have been extensively studied, because they are the simplest Markov processes that still exhibit most of the relevant features. The first probability distribution Pi(y, t) is an iV-component vector pn(t) (n = 1,2,...,JV). The transition probability Tz(y2 yi) is an N x N matrix. The Markov property (3.3) leads to the matrix equation... 

The Markov property states that the conditional probability of X tn) for given values of A f,), X(t2),.. ., X(tn 1) depends only on the most recent value X tn l). 

To find the (absolute) probability that the particle has n radicals of sizes t/i, t/2, yn at time t -f- r, we use the following relation based on the Markov property ... 

This design can be represented by a Hidden Markov Model (HMM). A HMM abstractly consists of two related stochastic processes a hidden process j, that fulfills the Markov property and an observed process Of that depends on the state of the hidden process jt at time t. A HMM is fully specified by the initial distribution tt, the rate matrix R of the hidden Markov process j, as well as by the law that governs the observable Of depending on the respective hidden state jt. 

Example 2.31. In examples 2.29 and 2.30 the Markov property clearly held, i.e., the new step solely depends on the previous step. Thus, the forecast of the weather could only be regarded as an approximation since the knowledge of the weather of the last two days, for example, might lead us to different predictions than knowing the weather only on the previous day. One way of improving this approximation is to take as states the weather of two successive days. 

Chapter 26), the probability of a subject taking a dose or doses (one or more) at any given dosing time is a function of whatever doses were taken at the immediate past dosing time preceding the one in question. This, of course, is independent of all other previous dosing events—a Markov property. 

A series of probable transitions between states can be described with Markov modeling. The natural course of a disease, for example, can be viewed for an individual subject as a sequence of certain states of health (12). A Markovian stochastic process is memoryless. To predict what the future state will be, knowledge of the current state is sufficient and is independent of where the process has been in the past. This is termed the strong Markov property (13). 

With first-order Markov chains, considering all t, the conditional distribution of yt+ given (yo, yi, y2,..., yi) is identical to the distribution of y,+i given only y,. That is, we only need to consider the current state in order to predict the state at the next time point. The predictability of the next state is not influenced by any states prior to the current state—the Markov property. 

The joint distribution for a first-order Markov chain depends only on the one-step transition probabilities and on the marginal distribution for the initial state of the process. This is because of the Markov property. A first-order Markov chain can be fit to a sample of realizations from the chain by fitting the log-linear (or a nonlinear mixed effects) model to [To, Li, , YtiYt] for T realizations because association is only present between pairs of adjacent, or consecutive, states. This model states that the odds ratios describing the association between To and Yt are the same at any combination of states at the time points 2,..., T, for instance. 

As shown in Figure 6.23, Crouse et al. [43] proposed a model, where the underlying backbone becomes a tree structure, and the Markov property... 

Hidden Markov models (HMMs) are statistical models based on the Markov property [34]. A stochastic process has the Markov property if the conditional probability... 

HMMs are statistical models for systems that behave like a Markov chain, which a discrete-time stochastic process formed on the basis of the Markov property. 

A Markov chain is a stochastic process Po, Pi,. .. such that the distribution of Pi, given all previous values Po,. .., Pm, depends only on Pm. That is, it interprets the fact that for a process satisfying the Markov property of Eq. (15), given the present, the past is irrelevant to predict its position in a future instant [29] ... 

In essence, the DPS approach reduces the problem of global kinetics to a discrete space of stationary points. Phenomenological rate constants can then be extracted under the assumption of Markovian dynamics within this space, which requires that the system has time to equilibrate between transitions and lose any memory of how it reached the current minimum. The Markovian assumption is therefore an essential part of the framework. However, we can regroup the stationary points into states whose members are separated by low barriers so that the Markov property is likely to be better obeyed between the groups (Section 14.2.3). 

Suppose now that we have a stochastic process X(t), t a 0, observed for till times, not just integer times. We ttike the state space to be the finite set 1, 2,. .., as before. The Markov property in this situation is just as described in Subsection 3.1 for the discrete-time case knowing the state of X at a time t, its evolution after time t is independent of its history before time t. Briefly, the future is independent of the past, given the present. 

By the Markov property of X, the process X(0), X(h), X(2h),. . . that results when we observe X at intervals of length (i is a discrete-time Markov chain with transition matrix P h). If we think of h as a small time interval, then equation (26) shows that the transition probabilities of this discretetime chain are given approximately by... 

When the chain does leave state i, it chooses its next state j i according to the probabilities exponentially distributed Because the Markov property of X implies that T must have the memoryless property (13), and this in turn implies that T has the exponential distribution. 

In this subsection we treat several queues of the Ml Ml si K type. These queues have Poisson arrivals, exponential service times, s servers, and capacity K. For these queues, the number-in-system L(f), t 0, is a continuous-time Markov chain, in fact, a birth-and-death process (Subsection 3.6). The Markov property arises from the exponentitility of service and interarrival times—see the discussion following (33). The queueing discipline is taken to be FIFO in every case. The results presented here follow fairly directly from (38). 

Market power, in supply cheiin management, 2127-2128 Market research, 269 Market turbulence, 311-314 Markov chains, 2150-2156 in continuous time, 2154-2156 and Markov property, 2150-2151 queueing model based on, 2153-2154, 2158-2159... 

The Markov property greatly simplifies the task of finding the joint probability distribution of the values of the stochastic process at two times. 

The Langevin dynamics equations have the Markov property in the sense that the state of the system at any future time has a dependency on the current state of the system, but not on the prior history. 

To use these equations for recognition, we need to connect state sequences with what we wish eventually to find, that is, word sequences. We do this by using the lexicon, so that, if the word hello has a lexicon pronunciation /h eh 1 ou/, then a model for the whole word is created by simply concatenating the individual HMMs for the phones /h/, /eh/, /y and /ou/. Since the phone model is made of states, a sequence of concatenated phone models simply generates a new word model with more states there is no qualitative difference between the two. We can then also join words by concatenation the result of this is a sentence model, which again is simply made from a sequence of states. Hence the Markov properties of the states and the language model (explained below) provide a nice way of moving from states to sentences. 

When solving the Markov-time dependent equations, states representing identical capacities are merged into one state to avoid an explosion in the number of states. This will, however, compromise the Markov properties. 

The RAM tool comprises several approximation and simplifications. An important issue to treat is the procedure to merge states with equal capacity into one state where the Markov property does not hold. The use of phase-type distributions is considered to improve the accuracy of the current modeling, see e.g. Neuts (1981). 

---