Second-order Markov

A portion of the database for this polymer is shown in Figure 6. Literature reports that this polymer follows second-order Markov statistics ( 21 ). And, in fact, probabilities that produced simulated spectra comparable to the experimental spectrum could not be obtained with Bernoullian or first-order Markov models. Figure 7 shows the experimental and simulated spectra for these ten pentads using the second-order Markov probabilities Pil/i=0.60, Piv/i=0.35, Pvi/i=0.40, and Pvv/i=0.55 and a linewidth of 14.8 Hz. 

While other programs require modification of the actual code in changing the polymer, spectra, or model, only changes in the user database is required here. Changes in the program since a brief report (22) in 1985 include improvement of the menu structure, added utilities for spectral manipulations, institution of demo spectra and database. Inclusion of Markov statistics, and automation for generation of the coefficients in Equation 1. Current limitations are that only three models (Bernoul llan, and first- and second-order Markov) can be applied, and manual input Is required for the N. A. S. L.. 

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 probabilistic aspect of error propagation in isotactic polypropylene was treated both as a second-order Markov chain (in terms of m and r dyads) (408) and, in terms of a model of enantiomorphic catalyst sites, as asymmetric Ber-... 

Analysis of the poly(methyl methacrylate) sequences obtained by anionic polymerization was undertaken at the tetrad level in terms of two different schemes (10) one, a second-order Markov distribution (with four independent conditional probabilities, Pmmr Pmrr, Pmr Prrr) (44), the other, a two-state mechanism proposed by Coleman and Fox (122). In this latter scheme one supposes that the chain end may exist in two (or more) different states, depending on the different solvation of the ion pair, each state exerting a specific stereochemical control. A dynamic equilibrium exists between the different states so that the growing chain shows the effects of one or the other mechanism in successive segments. The deviation of the experimental data from the distribution calculated using either model is, however, very small, below experimental error, and, therefore, it is not possible to make a choice between the two models on the basis of statistical criteria only. 

For the statistical copolymer the distribution may follow different statistical laws, for example, Bemoullian (zero-order Markov), first- or second-order Markov, depending on the specific reactants and the method of synthesis. This is discussed further in Secs. 6-2 and 6-5. Many statistical copolymers are produced via Bemoullian processes wherein the various groups are randomly distributed along the copolymer chain such copolymers are random copolymers. The terminology used in this book is that recommended by IUPAC [Ring et al., 1985]. However, most literature references use the term random copolymer independent of the type of statistical distribution (which seldom is known). 

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 ... 

A second-order Markov model has also been described to show the effect on stereochemistry of the monomer unit behind the penultimate unit [Bovey, 1972],... 

It can be shown that every theoretical mechanism is automatically nth-order reversible for n = 1, 2, 3, 4 since in any polymer all n(ads) of these orders are automatically equifrequent with their reversals. Also, all first-and second-order Markov models are completely reversible. Third (and higher)-order Markov models are not completely reversible in general. 

Granjeon and Tarroux (1995) studied the compositional constraints in introns and exons by using a three-layer network, a binary sequence representation, and three output units to train for intron, exon, and counter-example separately. They found that an efficient learning required a hidden layer, and demonstrated that neural network can detect introns if the counter-examples are preferentially random sequences, and can detect exons if the counter-examples are defined using the probabilities of the second-order Markov chains computed in junk DNA sequences. 

Coleman and Fox published an alternative mechanism [82], According to these authors, the propagating centres exist in two forms, each of which favours the generation of either the m or r configuration. When both centres are in equilibrium, and when this equilibrium is rapidly established, the chain structure can be described by a modified Bernoulli statistics [83, 84]. The configurations of some polymers agrees better with this model than with first-or even second-order Markov models [84, 85]. 

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-... 

For a second-order Markov model, one would assume that the prediction of response for the fourth month is independent of the response obtained in the first month, given the responses in the second and third months. Such a model could similarly be fitted in S-Plus as... 

The second-order Markov model requires the specification of eight conditional probabilities. This is due to the influence of the last three pseudoasymmetric centers of the growing chain. The details of the second-order Markov model were described by Bovey. For details, the reader is advised to consult the reference. For convenience, the eight conditional probabilities are designated by Greek letters ... 

This software uses statistical probabilities to calculate ) configurational sequences of polymers from NMR data. ) First-order and Second-order Markov propagation models ) are developed from initial giiess values of the dyads ) which are fitted into the statistical equations. If the ) intensities of the triads are known, then the dyads are... 

With Bernoulli mechanisms, the ultimate unit of the growing chain has no influence on the linkage formed by a newly polymerized unit. With first-order Markov mechanisms, the ultimate unit does exert an influence, and in second-order Markov mechanisms, the penultimate, or second last, unit exerts an influence. In third-order Markov mechanisms it is the third last unit that exerts the influence on the linkage of newly joined units. Thus, Bernoulli mechanisms are a special case of Markov mechanisms, and could also be called zero-order Markov mechanisms. Second- and higher-order Markov mechanisms cannot be stated with confidence to occur in polyreactions, and, so, will not be discussed further. In addition, the discussion will be confined to binary mechanisms, that is, polyreactions where the unit possesses only two reaction possibilities. 

Similarly, for second-order Markov mechanisms, the transition probabilities Paa/a,Paa, B,Pba/a,Pba, b,Pab/a,Pab/b,Pbb/a, andpbb/b have to be considered, whereas, for Bernoulli mechanisms, only the transition probabilities ofp a and Pb need be considered. 

The last two diads or last three monomeric units must be considered with second-order Markov statistics. The ratios of rate constants that can be calculated from the diad and triad ratios are given in Table 16-18. The table shows that the ratio of the mole fractions of the two diads is definitely not always a measure of the ratio of the corresponding mean rate constants over all possible individual rates. 

Higher-order Markov models can also be applied [65]. In second-order Markov statistics, the probability of forming m or r depends on the structure of the previous two dyads. There is a total of eight conditional probabilities, of which four are independent. In order to confirm that this model is correct, it is necessary to have accurate pentad probabilities or longer. 

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