  The association of equal heartbeat intervals with cardiac conditions and the
effect of the equality on permutation-based time irreversibility are
investigated in this paper. We measure the distributions of equal heartbeat
intervals under three conditions, namely congestive heart failure (CHF),
healthy young and elderly, whose time irreversibility of heartbeats is detected
by measuring the probabilistic difference between permutations instead of raw
vectors. We demonstrate that heartbeats contain high rates of equal states,
particularly the CHF with around 20% equalities, and the distributions of equal
values discriminate the heartbeats at very short data length. The CHF have more
equal values than the healthy young (p<1.47*10^{-15}) and elderly
(p<2.48*10^{-11}), and the healthy young have less equalities than the elderly
(p<3.16*10^{-4}). Time irreversibility considering equal values is promising to
extract nonlinear behaviors of heartbeats, confirming the decreased nonlinear
complexity of the diseased and aging heart rates, while that involving no
equality leads to erroneous nonlinearity detection. In our contribution, we
highlight the pathological or physiological information contained by the
distribution of equal heartbeat intervals that might contribute to develop
relevant biomarkers in the area of heart analysis, and demonstrate the
effectiveness of equality-based time irreversibility in the nonlinearity
detection of heartbeats.
