pyhctsa.operations.entropy.bubble_entropy¶
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pyhctsa.operations.entropy.bubble_entropy(y, m=
10, tau=1)¶ Bubble entropy of a time series.
Manis et al.’s bubble entropy [1], an ordinal entropy that depends very little on its embedding dimension. The series is cut into overlapping runs of
mvalues spacedtausamples apart. For each run, the number of swaps a bubble sort needs to put it in order (equivalently, the number of pairs in which an earlier value exceeds a later one, from 0 tom(m-1)/2) is counted. The Renyi entropy of order 2 of the distribution of this swap count,H_m = -log(sum_k p_k**2), is computed for runs ofmand ofm + 1values. The bubble entropy is the increase in entropy on going frommtom + 1values,H_(m+1) - H_m, divided bylog((m+1)/(m-1))to normalize for the dimension. Low values indicate series whose runs have predictable orderings. For white noise the long-series value is about 0.64 form = 5and 0.69 form = 10(rising slowly toward 0.75 asmgrows). Port of hctsa’sEN_BubbleEn.A swap is counted only when an earlier value is strictly greater than a later one, so tied values are never swapped (as in a standard bubble sort). The estimate is a small difference between two entropies, so it is noisy when the series is short relative to the number of possible swap counts: for series of about 1000 samples, embedding dimensions much above 10 give poorly reproducible values.
With a delay set by the timescale of the series, the runs of
mvalues span(m-1)*tausamples, so for a slowly decorrelating series (a large delay) there are few runs and the value is noisy. hctsa uses'mi', which is rarely NaN: it falls back on the first automutual-information minimum when the autocorrelation function never decays to 1/e. The'ac1e'delay is NaN when the autocorrelation function never decays to 1/e (as for many random walks) or the runs are too few, and the first zero crossing ('ac') is less stable still.References