Time irreversibility, a fundamental property of complex system, can be characterized by the directional asymmetry of probabilistic distributions, which is quantified by equalities-involved permutation relative entropy in our nonlinear dynamics analysis of heart rates . Equal values are not rare in discrete heartbeats because of the limits of resolution of signals collection, and more importantly equal states contain underlying important cardiac regulation information which is neglected by the original permutative method and some measurements of time irreversibility . The relative entropy of modified permutation associated with equal states  shows advantages to these time irreversibility parameters and has great improvements to that of the original permutation. The temporal irreversibility researches on cardiac activities suggest the highest probabilistic asymmetry of the healthy young heart rates and highlight the facts that heart diseases and aging broke asymmetry, or reduce the nonlinear dynamical complexity , of human heartbeats .
Symbolic relative entropy, an efficient nonlinear complexity parameter measuring probabilistic divergences of symbolic sequences, is proposed in our nonlinear dynamics analysis of heart rates considering equal states. Equalities are not rare in discrete heartbeats because of the limits of resolution of signals collection, and more importantly equal states contain underlying important cardiac regulation information which is neglected by some chaotic deterministic parameters and temporal asymmetric measurements . The relative entropy of symbolization associated with equal states has satisfied nonlinear dynamics complexity detections in heartbeats and shows advantages to some nonlinear dynamics parameters without considering equalities. Researches on cardiac activities suggest the highest probabilistic divergence of the healthy young heart rates and highlight the facts that heart diseases and aging reduce the nonlinear dynamical complexity of heart rates .
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Time irreversibility, a fundamental property of complex system, can be characterized by the directional asymmetry of probabilistic distributions, which is quantified by equalities-involved permutation relative entropy
Symbolic relative entropy, an efficient nonlinear complexity parameter measuring probabilistic divergences of symbolic sequences, is proposed
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. Equal values
considering equal states. Equalities
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the original permutative method and some measurements of time irreversibility
some chaotic deterministic parameters and temporal asymmetric measurements
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modified permutation
symbolization
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has satisfied nonlinear dynamics complexity detections in heartbeats and
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these time irreversibility parameters and has great improvements to that of the original permutation. The temporal irreversibility researches
some nonlinear dynamics parameters without considering equalities. Researches
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asymmetry
divergence
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broke asymmetry, or
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, of human heartbeats
of heart rates