Markov chain first-order

MCMC methods are essentially Monte Carlo numerical integration that is wrapped around a purpose built Markov chain. Both Markov chains and Monte Carlo integration may exist without reference to the other. A Markov chain is any chain where the current state of the chain is conditional on the immediate past state only—this is a so-called first-order Markov chain higher order chains are also possible. The chain refers to a sequence of realizations from a stochastic process. The nature of the Markov process is illustrated in the description of the MH algorithm (see Section 5.1.3.1). 

We have tacitly assumed that the rate constants depend only on the last unit of the chain. In such a situation, the copolymerization is called a Markov copolymerization of first order. The special case (i), r r- = 1, is a Markov copolymerization of order zero. If reactivity also depends on the penultimate unit of the chain, the polymerization is a Markov copolymerization of second order. 

In order to obtain the expression for the components of the vector of instantaneous copolymer composition it is necessary, according to general algorithm, to firstly determine the stationary vector ji of the extended Markov chain with the matrix of transitions (13) which describes the stochastic process of conventional movement along macromolecules with labeled units and then to erase the labels. In this particular case such a procedure reduces to the summation ... 

Styrene-SQ., Copolymers. I would now like to discuss two systems which illustrate the power of C-13 nmr in structural studies. The first is the styrene-SO system. As already indicated, this is of the type in which the chain composition varies with monomer feed ratio and also with temperature at a constant feed ratio (and probably with pressure as well.) The deviation of the system from simple, first-order Markov statistics, —i.e. the Lewis-Mayo copolymerization equation—, was first noted by Barb in 1952 ( ) who proposed that the mechanism involved conplex formation between the monomers. This proposal was reiterated about a decade later by Matsuda and his coworkers. Such charge transfer com-... 

More complex schemes have been proposed, such as second-order Markov chains with four independent parameters (corresponding to a copolymerization with penultimate effect, that is, an effect of the penultimate member of the growing chain), the nonsymmetric Bernoulli or Markov chains, or even non-Maikov models a few of these will be examined in a later section. Verification of these models calls for the knowledge of the distribution of sequences that become longer, the more complex the proposed mechanism. Considering only Bernoulli and Markov processes it may be said that at the dyad level all models fit the experimental data and hence none can be verified at the triad level the Bernoulli process can be verified or rejected, while all Markov processes fit at the tetrad level the validity of a first-order Markov chain can be confirmed, at the pentad level that of a second-order Maikov chain, and so on (10). 

The copolymer described by Eq. 6-1, referred to as a statistical copolymer, has a distribution of the two monomer units along the copolymer chain that follows some statistical law, for example, Bemoullian (zero-order Markov) or first- or second-order Markov. Copolymers formed via Bemoullian processes have the two monomer units distributed randomly and are referred to as random copolymers. The reader is cautioned that the distinction between the terms statistical and random, recommended by IUPAC [IUPAC, 1991, in press], has often not been followed in the literature. Most references use the term random copolymer independent of the type of statistical process involved in synthesizing the copolymer. There are three other types of copolymer structures—alternating, block, and graft. The alternating copolymer contains the two monomer units in equimolar amounts in a regular alternating distribution ... 

Having established that a particular polymerization follows Bemoullian or first-order Markov or catalyst site control behavior tells us about the mechanism by which polymer stereochemistry is determined. The Bemoullian model describes those polymerizations in which the chain end determines stereochemistry, due to interactions between either the last two units in the chain or the last unit in the chain and the entering monomer. This corresponds to the generally accepted mechanism for polymerizations proceeding in a nonco-ordinated manner to give mostly atactic polymer—ionic polymerizations in polar solvents and free-radical polymerizations. Highly isoselective and syndioselective polymerizations follow the catalyst site control model as expected. Some syndioselective polymerizations follow Markov behavior, which is indicative of a more complex form of chain end control. 

Within the framework of the above models the problem of the calculation of the sequence distribution is solved in a quite simple way [51-53, 6]. In order to find the probability of any sequence Uk consisting of k units, it should be expressed through the sequence of Markov chain states, the probability of which is calculated usually by means of the routine procedure as a product of the few factors. The first factor 7i corresponds to the initial state Sh and each of the following factors, Vy, corresponds to the transition from the state Sj to Sj at the conditional movement along the sequence of Markov chain states. For instance, in this manner one can calculate the probability of the sequence U3 = S3[1M2M2 in the both cases of terminal model ... 

A key facet of copolymerization is the possible disparity of reactivities of the monomers. Traditional procedure is to assume, at least as an approximation, that the reactivity of a growing propagating center depends only on the identity of its reactive end unit (i.e., the last monomer added), not on the composition and length of the rest of its chain [124-126] (first-order Markov or terminal model see also... 

The simple copolymer model is a first-order Markov chain in which the probability of reaction of a given monomer and a macroradical depends only on the terminal unit in the radical. This involves consideration of four propagation rate constants in binary copolymerizations, Eqs. (7-2)-(7-4). The mechanism can be extended by including a penultimate unit effect in the macroradical. This involves eight rate constants. A third-order case includes antepenultimate units and 16 rate coefficients. A true test of this model is not provided by fitting experimental and predicted copolymer compositions, since a match must be obtained sooner or later if the number of data points is not saturated by the adjustable reactivity ratios. 

The r.h.s. of flg. 3.31 presents liquid-gas coexistence curves, of which curve I relates to the conditions of fig. 3.31a. Curve II, arises from somewhat improved lattice statistics. For curve I the chain is fully flexible, implying that each bond can bend back to coincide with the previous one. In statistical parlance it is said that the chain has no self-avoidance and obeys first-order Markov statistics. In curve II a second-order Markov approximation was used ) in which three consecutive bonds in the chain are forbidden to overlap and an energy difference of 1/kT is assigned to local sets of three that have a bend conformation. The figure demonstrates the extent of this variation T is reduced as a result of the loss of conform-... 

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... 

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. 

A hrst-order Markov chain (MC) would assume an association between the responses at the hrst and second months. The prediction of the response for the third month would be independent of the response in the first month, given the response in the second month. Given the response in the third month, the prediction for the fourth month would be independent of the response in the second month. Thus, once the response for a previous month is known, other months are not needed to predict the response for a current month. Data of the type described above could be fitted in S-Plus as follows using the loglm function in the S-Plus library MASS ... 

A (first-order) Markov process is defined as a finite-state probability model in which only the current state and the probability of each possible state change is known. Thus, the probability of making a transition to each state of the process, and thus the trajectory of states in the future, depends only upon the current state. A Markov process can be used to model random but dependent events. Given the observations from a sequence of events (Markov chain), one can determine the probability of one element of the chain (state) being followed by another, thus constructing a stochastic model of the system being observed. For instance, a first-order Markov chain can be defined as... 

Syndiotactic propagation of propylene is know to be catalyzed by homogeneous vanadium catalyst (1 ). In the polypropylene samples prepared with the homogeneous catalysts, the relative population of iso-, hetero- and syndiotactic triads is in accordance with that predicted from the first order Markov model (25, 26). There is no chiral structure around the homogeneous vanadium species. The stereochemistry of the entering monomer is controlled by the chirality of the growing chain end, in contrast with the isotactic propagation. 

From the Ah-NMR analysis of poly(3,8-d2-styrene oxide) obtained using ZnEt2/H20 as initiator Figure 2), the formation of the partly crystalline fraction can be described by first order Markov statistics, while that for the amorphous fraction follows Bemoulllan statistics. Different chain propagation mechanisms are, therefore, responsible for the formation of the two different polymer fractions obtained from this particular catalyst. Consequently, the existence of two different active centers, responsible for the two polymerization mechanisms and for formation of fractions I and II, are clearly indicated. 

To predict the course of a copolymerization we need to be able to express the composition of a copolymer in terms of the concentrations of the monomers in the reaction mixture and the relative reactivities of these monomers. In order to develop a simple model, it is necessary to assume that the chemical reactivity of a propagating chain (which may be free-radical in a radical chain copolymerization and carbocation or carboanion in an ionic chain copolymerization) is dependent only on the identity of the monomer unit at the growing end and independent of the chain composition preceding the last monomer unit [2-5]. This is referred to as the first-order Markov or terminal model of copolymerization. 

When the manner of addition is affected by the growing chain end, the configurations of the added units will not obey Bernoulli statistics. In the simplest case, first-order Markov statistics will operate. Addition will be characterized by two parameters because the probability of r diad generation by monomer addition to an m end unit, will not be identical with monomer addition to an r end unit, P . The probabilities of m or r diad generation by addition to m or r chain ends will be bound by the relations P = (1 — Pmr) rm 0 rr) According to first-order Markov... 

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