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J. Risk Financial Manag. 2021, 14, 440 5 of 15 Figure 1. Plot of price, volume and daily log-returns. 3. Entropy Profiles Method 3.1. Brief Introduction to Information Theory and Entropy Consider two variables X and Y. Correlation between them may be ill defined when they are discrete, and may be a poor measure of ”relation" when nonlinearity and/or non-Gaussianity is involved. Let R = {a1, a2, ..., aM} be a finite set and p be a proper probability mass function (PDF) on R. The amount of information needed to fully characterize all of the elements of this set consisting of M discrete elements is defined by I(RM) = log2M and is known as Hartley’s formula. Shannon (1948) built on Hartley’s formula in the context of digitization and communications, to develop Shannon’s entropy: M H(p) = − ∑ pilog(pi), (2) i=1 with xlog(x) tending to zero as x tends to zero. This information criterion measures the uncertainty or informational content that is implied by p. The entropy-uncertainty measure H(p) reaches a maximum when p1 = p2 = ... = pM = 1/M (and is equal to Hartley’s formula) and a minimum with a point mass function. It is emphasized here that H(p) is a function of the probability distribution. For example, if η is a random variable with possible distinct realizations x1, x2, ..., xM with probabilities p1, p2, ..., pM, the entropy H(p) does not depend on the values x1, x2, ..., xM of η. If, on the other hand, η is a continuous random variable, then the entropy of a continuous density is H(x) = − p(x)log(p(x))dx, (3) a differential entropy. Renyi (1961) showed that, for a (sufficiently often) repeated experiment, one needs on average the amount H(p) + ε of zero-one symbols (for any positive ε) in order to characterize an outcome of that experiment. Thus, it seems logical to “expect” that the outcome of an experiment contains H(p) information. Similarly, H(p) is a measure of uncertainty about a specific possible outcome before observing it, which is equivalent to the amount of randomness represented by p. It is pro- portional to “variance” in the case of a Normal distribution. Thus, entropy is a far superiorPDF Image | Contrasting Cryptocurrencies with Other Assets
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