LZ76 complexity as a function of sequence length N, for MLZs and for 102 random sequences. The MLZs upper bound is clearly observed, while the simulated random sequences (rnd) are below the MLZs values and mostly above the curve. In the inset, it can be seen that LZ76 complexity for the random sequences can lie also below the curve.
Number of 00 patterns (#) in the MLZs and random (rnd) sequences as a function of sequence length N. While the # for the random sequences exhibit the expected linear behavior with slope 1/4, the behavior for the MLZs departs from a linear law.
Normalized counts for all patterns of length 6 in the MLZs and random (rnd) sequences (string length N = 106). Patterns are ordered by their binary values. In the rnd curve, all patterns have counts near the expected value 2−6(= 0.0156), while for the MLZs, counts vary from slightly below 0.0141 to slightly above 0.0168.
LZ76 complexity as a function of the sequence length N (log-log scale) for the binary expansion of π, , and sequences, together with the MLZs and random (rnd) sequences. The binary expanded irrational numbers cannot be distinguished from the random LZ76 complexity for all lengths considered. All LZ76 complexities are below the MLZs complexity.
c(u) and for the logistic map given by Eq. (10) , and compared to the random (rnd) sequence. Three values for the logistic map parameter were considered: the chaotic regime r = 1.8; the intermittent point r = 1.7499; and the Feigenbaum point (fb) at r = 1.40115518. See text for details.
The relative error of c(u) and estimates ( ) of the true entropy (h) for the r = 1.8 logistic map as function of the sequence length N.
Finite State Automata for the nearest neighbor interaction range. S is the start state, while F and B are recurrent states. represents the probability of emitting a symbol X conditioned on being in state M.
Estimated entropies (bit/symbol) for the logistic map (Eq. (10) ). The second column is the entropy rate value from Ref. 23 . Third and fifth columns are the value of c(u) and , respectively, for the 105 length sequence, each value is averaged over 50 sequences. Fourth and six columns correspond to the entropy rate estimated by fitting the values of c(u) (Eq. (8) ) and (Eq. (9) ), respectively. Values between round brackets are the relative errors with respect to columns two values.
Estimated entropy (bit/symbol) for a -Bernoulli process. See Table I for a description of the column values.
Estimated entropy (bit/symbol) for nearest neighbor Ising model. represents the probability of emitting a symbol X conditioned by being on state M. The rest of the columns follows the same description than Table I .
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