Higher-order Markov chains

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). 

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. 

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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