![]() ![]() RNA secondary structure was first formally described by Smith and Waterman. The secondary structure schematic of the RNA also has a context-free nature, allowing it to be described by such grammars. Grenander, Kuich, Hutchins, Soule and others have investigated important properties of CFGs and SCFGs, such as convergence of average length of derivations, capacity, or the Shannon’s entropy ( Shannon entropy) as defined in, using generating-functions techniques. Stochastic CFGs (SCFGs) assign probabilities to each rule, which in turn assigns a likelihood value to each word by multiplying the probabilities of the rules used to derive that word. A grammar is said to be unambiguous if there is a one-to-one correspondence between the words and their descriptive derivations. The set of words generated by a grammar is referred to as the language of that grammar. CFGs have a recursive nature and are composed of sets of rules that can drive strings of alphabets, also referred to as words. Since then, they have been used broadly in a variety of computer science applications. Manzourolajdad, in Emerging Trends in Applications and Infrastructures for Computational Biology, Bioinformatics, and Systems Biology, 2016 9.1 IntroductionĬhomsky initially formalized context-free grammars (CFGs) as an attempt to model languages. Rényi entropies have applications to random search problems, questionnaire theory, optimal coding (the greatest lower bounds of the arithmetic or exponential mean codeword lengths are the Shannon and the Rényi entropies, respectively), even to differential geometry. The Rényi entropies α H n (α ≠ 1) need not be (s) subadditive. The weighted quasiariihrnetic mean (3) is (a) additive and(c) small for small probabilities if and only if it is either the Shannon's entropy (1) or a Ré nyi entropy (2) of positive ( but ≠ 1 )order. The Rényi entropies of positive order (including the Shannon entropy as of order 1) have the following characterization (, see also ). Where f is continuous and strictly monotonia Notice that f − 1 gives the weighted arithmetic, geometric, exponential, harmonic, and power means when f ( x ) = x, log x, e x, 1 / x, x c, respectively. Hence, what are the Shannon entropies of the state variables at these filter parameters? What features will such phase portraits exhibit? To address these issues, we have the following observation. However (as discussed in Chapter 9), the quasi periodic behaviors are exhibited for some exceptional filter parameters. When both the eigenvalues of the second order digital filters associated with two's complement arithmetic are outside the unit circle, random-like chaotic patterns are typically exhibited all over the phase plane and the Shannon entropies of the state variables are independent of the initial conditions and the filter parameters. So it is difficult to determine the type of trajectories by the Shannon entropies when the eigenvalues of the system matrix are complex and inside or on the unit circle. (a) Set of initial conditions for different types of trajectories when b = −1 and a = 0.5 (b) Shannon entropies of symbolic sequences for different initial conditions when b = −1 and a = 0.5.Īs can be seen from Figure 10.4, the Shannon entropies of the symbolic sequences for the type II trajectory may be higher than that for the type III trajectory, even though the symbolic sequences of the type II trajectory are periodic and have limit cycle behaviors – those of the type III trajectory are aperiodic and have chaotic behavior. ![]()
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