{'fraction_non_alphanumeric': 0.022357723577235773, 'fraction_numerical': 0.0, 'mean_word_length': 5.437908496732026, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Delay independence of mutual-information rate of two symbolic sequences.

Introduced by Shannon as a "rate of actual transmission," mutual information rate (MIR) is an extension of mutual information to a pair of dynamical processes. We show a delay-independence theorem, according to which MIR is not sensitive to a time shift between the two processes. Numerical studies of several benchmark situations confirm that this theoretical asymptotic property remains valid for realistic finite sequences. Estimations based on block entropies and a causal state machine algorithm perform better than an estimation based on a Lempel-Ziv compression algorithm provided that block length and maximum history length, respectively, can be chosen larger than the delay. MIR is thus a relevant index for measuring nonlinear correlations between two experimental or simulated sequences when the transmission delay (in input-output devices) or dephasing (in coupled systems) is variable or unknown.