  Kernel methods are widely used for probability estimation by measuring the
distribution of low-passed vector distances in reconstructed state spaces.
However, the information conveyed by the vector distances that are greater than
the threshold has received little attention. In this paper, we consider the
probabilistic difference of the kernel transformation in reconstructed state
spaces, and derive a novel nonequilibrium descriptor by measuring the
fluctuations of the vector distance with respect to the tolerance. We verify
the effectiveness of the proposed kernel probabilistic difference using three
chaotic series (logistic, Henon, and Lorenz) and a first-order autoregressive
series according to the surrogate theory, and we use the kernel parameter to
analyze real-world heartbeat data. In the heartbeat analysis, the kernel
probabilistic difference, particularly that based on the Kullback--Leibler
divergence, effectively characterizes the physiological complexity loss related
to reduced cardiac dynamics in the elderly and diseased heartbeat data.
Overall, the kernel probabilistic difference provides a novel method for the
quantification of nonequilibria by characterizing the fluctuation theorem in
reconstructed state spaces, and enables reliable detection of cardiac
physiological and pathological information from heart rates.
