Frequency analysis Markov models

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.

If we will consider arbitrary random process, then for this process the conditional probability density W xn,tn x, t, ... x i,f i) depends on x1 X2,..., x . This leads to definite temporal connexity of the process, to existence of strong aftereffect, and, finally, to more precise reflection of peculiarities of real smooth processes. However, mathematical analysis of such processes becomes significantly sophisticated, up to complete impossibility of their deep and detailed analysis. Because of this reason, some tradeoff models of random processes are of interest, which are simple in analysis and at the same time correctly and satisfactory describe real processes. Such processes, having wide dissemination and recognition, are Markov processes. Markov process is a mathematical idealization. It utilizes the assumption that noise affecting the system is white (i.e., has constant spectrum for all frequencies). Real processes may be substituted by a Markov process when the spectrum of real noise is much wider than all characteristic frequencies of the system. 

In the recent literature, one can find models of inspection frequency optimization with the use e.g. simulation processes (see e.g. (Alfares 1999, Vaghefi and Sarhangian 2009)), probability analysis (see e.g. (Baohe 2002)), Bayes or quasi-Bayes approach (see e.g. (Durango-Cohen and Madanat 2008, Jones et al. 2010)), or Markov processes (see e.g. (Kenzin and Frostig 2009)). Moreover, the analysed problems include e.g. the periodic inspection optimization for systems with failure interaction (see e.g. (Golmakani and Moakedi 2012, Zeque-ira and Berenguer 2005)), or the introduction of imperfect inspection (see e.g. (Berrade et al. 2013, Zhao et al. 2007)). 

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