These Lorenz-consistent statistics are compared with statistics based on Shannon’s entropy and Rényi’s second-order entropy to show that the former have better mathematical behavior than the latter. Here, d C 1 refers to the average absolute difference between the relative abundances of dominant and subordinate symbols, with its value being equivalent to the maximum vertical distance from the Lorenz curve to the 45-degree line of equiprobability d C 2 refers to the average absolute difference between all pairs of relative symbol abundances, with its value being equivalent to twice the area between the Lorenz curve and the 45-degree line of equiprobability N is the number of different symbols or maximum expected diversity. Xn ) bits / message, the amount of entropy rate per symbol is given by H ( X ) H. to e ciently compress text, allowing for random sequences of symbols not to. Note: the b in b-ary entropy is the number of different symbols of the ideal.
Novel measures of symbol dominance ( d C 1 and d C 2), symbol diversity ( D C 1 = N (1 − d C 1) and D C 2 = N (1 − d C 2)), and information entropy ( H C 1 = log 2 D C 1 and H C 2 = log 2 D C 2) are derived from Lorenz-consistent statistics that I had previously proposed to quantify dominance and diversity in ecology. Shannon estimated both the upper and lower bounds for entropy and. The original de nition of entropy, due to Clausius, was thermodynamic. Entropy is a concept in thermodynamics (see thermodynamic entropy).
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