![]() Shannon entropy H s( x) thus appears as the average missing information, that is, the average information required to specify the outcome x when the receiver knows the distribution p i. The source generating the inputs x i ∈ x is characterized by the probability distribution p i. ![]() In this work, the number of bins M of such a histogram was calculated with an optimal estimator proposed by Knuth (2006). (7.80), a histogram is required to infer the probabilities p i of the dataset. The SE is a quantity that increases with the number of possible states: for an unbiased coin, H s( x) = ln (2) ≈ 0.6931 while for an unbiased dice H s( x) = ln (6) ≈ 1.7918. ![]() At the opposite, H s( x) is maximal, equal to ln( M), for a uniform distribution (that is p 1 = p 2 = … = p M). In particular, the minimum H s( x) = 0 is reached for a constant random variable, that is, a variable with a determined outcome, which reflects in a fully localized probability distribution p i = 1 and p j = 0 for i ≠ j. The quantity H s naive( x) is an example of an entropy estimator, in a very similar sense as p i ˆ is an estimator of p i. (7.81) H s x ≈ H ⌢ s naive x = − ∑ i = 1 M p ⌢ i ln p ⌢ i = − ∑ i = 1 M l i N ln l i N 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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