Markov distributions, first-order

If, instead, the dyad probability depends on the nature (m or r) of the preceding dyad, the distribution follows a first-order Markov process, with two independent statistical parameters and Pr , (the probability that after a m dyad a r dyad follows and vice versa, respectively). The corresponding equations are listed in Table 4, column 3. They correspond to those of a nonideal copolymerization and are reduced to the previous case when p + p = 1 ... 

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 exact computation of P W) in this simple one-dipole model is already a very arduous task that, to my knowledge, has not yet been exactly solved. We can, however, consider a limiting case and try to elucidate the properties of the work (heat) distribution. Here we consider the limit of large ramping speed r, where the dipole executes just one transition from the down to the up orientation. A few of these paths are depicted in Fig. 13b. This is also called a first-order Markov process because it only includes transitions that occur in just one direction (from down to up). In this reduced and oversimplified description, a path is fully specified by the value of the field H at which the dipole reverses orientation. The work along one of these paths is given by... 

From the analysis of 13C NMR spectra of polypropylenes, Doi96) found that the sequence distribution of inverted propylene units follows first-order Markov statistics. Table 4 lists the two reactivity ratios rQ and rt, for the polymerization of propylene with the soluble catalysts composed of VC14 and alkylaluminums at — 78 °C ... 

To analyse bond breakage under steady loading, we take advantage of the enormous gap in time scale between the ultrafast Brownian diffusion (r 10 — 10 s) and the time frame of laboratory experiments ( 10 s to min). This means that the slowly increasing force in laboratory experiments is essentially stationary on the scale of the ultrafast kinetics. Thus, dissociation rate merely becomes a function of the instantaneous force and the distribution of rupture times can be described in the limit of large statistics by a first-order (Markov) process with time-dependent rate constants. As force rises above the thermal force scale, i.e. rj-t> k T/x, the forward transition... 

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... 

Girard et al. (34) proposed a hierarchical Markov model for patient compliance with oral medications that was conditioned on a set of individual-specific nominal daily dose times. The individual random effects for the model were assumed to be multivariate normally distributed. Assuming first-order Markov hypothesis (see... 

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. 

The probabilities of the regiosequence pentads for commercial PVF and urea PVF are shown in Table III. For the former sample it is apparent simply by inspection that the regiosequence distribution is not Bernoullian, since Pobs(C5) and Pobs (D5) are different (2). The distributions conform to first-order Markov statistics, characterized by two reactivity ratios r0 and r 5 where r0 = k /lq, and rj — ku/k10 (kjj is the rate constant for monomer addition to terminal radical i which generates the new terminal radical j). The present pentad data is insufficient to check the validity of this model, but it is unlikely that there is any deviation, as the same model has been tested and found adequate to describe the regiosequence distribution in PVF2 (2). 

Table IV shows the reactivity ratios rG and r, derived from the probabilities in Table III in accord with a first-order Markov model (2), where it is assumed that the more likely propagating terminal radical structure is 1 (—CHF-) and not 0 (—CH2). This assumption is consistent with gas phase reactions of VF with mono-, di-, and trifluoromethyl radicals, which add more frequently to the CH2 carbon than to the CHF carbon (20). The reactivity ratio product is unity if Bernoullian statistics apply, and we see this is not the case for either PVF sample, although the urea PVF is more nearly Bernoullian in its regiosequence distribution. Polymerization of VF in urea at low temperature also reduces the frequency of head-to-head and tail-to-tail addition, which can be derived from the reactivity ratios according to %defect — 100(1 + ro)/(2 + r0 + r,). Our analysis of the fluorine-19 NMR spectrum shows that commercial PVF has 10.7% of these defects, which compares very well with the value of 10.6% obtained from carbon-13 NMR (13). Therefore the values of 26 to 32% reported by Wilson and Santee (21) are in error.

Whereas the stereosequence distribution in isoregic and aregic PVF is nearly ideal random (Bernoullian with p(m) — 0.5), the latter has a first-order Markov regiosequence distribution. Accordingly the monomer sequence isomerism in PVF cannot be described by a single parameter such as the % defect, and requires two reactivity ratios for complete specification. 

Expressions for the mole fractions of longer sequences can be built up in a similar fashion. Expressions for first-order Markov dyad and triad distributions are given in Table 2.3, and can be compared with the analogous expressions for stereochemical sequence distributions in chapter 1. 

As with the Bernoullian model, comparison between an observed and calculated sequence distribution is required to check for conformity to first-order Markov statistics. Obviously, with only two independent observations, a dyad distribution is insufficient for determining the two independent probabilities of the model. In contrast, a triad distribution provides five independent observations, so this can be used to check conformity to first-order Markov statistics. Trial values of the monomer addition probabilities can be obtained by taking appropriate combinations of the expressions shown in Table 2.3. For example, is given by... 

Several examples of NMR studies of copolymers that exhibit Bernoullian sequence distributions but arise from non-Bernoullian mechanisms have been reported. Komoroski and Schockcor [11], for example, have characterised a range of commercial vinyl chloride (VC)/vinylidene chloride (VDC) copolymers using carbon-13 NMR spectroscopy. Although these polymers were prepared to high conversion, the monomer feed was continuously adjusted to maintain a constant comonomer composition. Full triad sequence distributions were determined for each sample. These were then compared with distributions calculated using Bernoullian and first-order Markov statistics the better match was observed with the former. Independent studies on the variation of copolymer composition with feed composition have indicated that the VDC/VC system exhibits terminal model behaviour, with reactivity ratios = 3.2 and = 0.3 [12]. As the product of these reactivity ratios is close to unity, sequence distributions that are approximately Bernoullian are expected. 

Copolymers with first-order Markov sequence distributions... 

The vast majority of copolymers described in the literature conform to the terminal model for copolymerisation and therefore exhibit sequence distributions which will in principle conform to first-order Markov statistics. Of... 

An (A, (p) dynamic system is deterministic if knowing the state of the system at one time means that the system is uniquely specified for all r 6 T. In many cases, the state of a system can be assigned to a set of values with a certain probability distribution, therefore the future behaviour of the system can be determined stochastically. Discrete time, discrete state-space (first order) Markov processes (i.e. Markov chains) are defined by the formula... 

K. R. Sharma, First Order Markov Model Representation of Chain Seqnence Distribution of Alphamethylstyrene Acrylonitrile Prepared by Reversible Free Radical Polymerization, 229th ACS National Meeting, San Diego, CA, March 2005. 

First-order Markov model to represent chain seqnence distribution of SAN... 

The dyad probabilities for copolymer were calculated as a function of reactivity ratios and monomer composition. A first-order Markov model was developed to predict the chain sequence distribution of SAN and AMS-AN copolymers. The six triad concentrations for SAN copolymer were calculated. Nin dyad and 27 triads for random terpolymers were calculated and tabulated in Tables 11.3, 11.5-11.7. 

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