Markov chain second-order

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. 

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 ... 

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-... 

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 ... 

Based on the estimated density functions, the mean and the 95%-quantile are superimposed by the dashed lines in Figure 4-13a and Figure 4.13b whereby the latter coincides with the re-order level ensuring a 95% a-service level. For both sites the density functions of total consumption are bimodal and skewed to the right. The first mode is at zero consumption and is inherited from the Markov chain of the production models. It corresponds to situations when a pipeline inspection coincides with a cracker. shut-down. The second mode is inherited from the Weibull distribution determining the pipeline inspection time which also causes the skewness. 

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 ... 

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. 

Second step Probability computation In order to compute probabilities, we consider that we have a system that receives segments ofbits. According to whether the segment received is error free or not, the system transitions from state to state. Thus, we can build a Markov chain. 

Other statistical models. The second-order Markov model is sometimes also applied to copolymers. Here, the probability of addition of a given monomer depends not only on the identity of the chain end monomer, but also on the nature of the preceding or penultimate monomer unit. As there are then four possible types of chain end to consider (namely, -AA, -AB, -BA, and -BB), there are eight addition probabilities which describe addition of the A and B monomers (e.g. Pg g represents the probability of B adding to a -BA chain end). As with the first-order Markov case, only half of these are independent because (Paaa + aab) = ( aba + abb) = Equations for... 

In the case of the unsubstituted C -symmetric metallocenes 8 and 11, copolymerization proceeds under control of the last inserted monomer unit (chain-end control), that is, it can be described by a second order Markov model. Ethylene is inserted with these zirconocenes three times faster than norbornene. No norbornene block sequences longer than two (NN units) are formed, in agreement with parameters calculated for 8 (rsE = 2.40, r E = 4.34, rEN = 0.03, and rm = 0.00). This result easily explains the maximum observed Xn = 0.66. 

The f22 values are in general lower than those obtained for propene or other a-olefms, in agreement with the low homopolymerization activity of norbomene. The f22 value for catalyst 1-5 is much greater than ri2 this shows the tendency of this catalyst to insert a third norbomene after the second one. It was clear that the next-to-last E or norbomene monomer unit exerts an influence on the reactivity of the propagating Mt-E or Mt-N species, which depends upon the catalyst stmc-ture. The second-order Markov model must be used to describe E-N copolymerizations promoted by metallocenes I-l, 1-2, and IV-1. A third-order or a more complex model may be required to fit the experimental data obtained with catalyst 1-6, where more sterically hindered indene substitutions are dominant. At higher norbomene concentrations, copolymers with all catalysts may need more complex models. These results allowed the conclusion that E-N copolymerization is dominated by the bulkiness of the norbomene monomer and of the copolymer chain. 

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