Hierarchical entropy analysis for biological signals

We develop a hierarchical entropy (HE) method to quantify the complexity of a time series based on hierarchical decomposition and entropy analysis. The proposed method is applied to the Gaussian white noise and the 1/f1/f noise. We prove that the difference frequency components of the Gaussian white noise with the same scale factor have the same value of entropies, and the values decline as the scale factor increases. We also apply the HE method to the 1/f1/f noise, and prove mathematically that a lower frequency component of a 1/f1/f noise is also a 1/f1/f noise and verify numerically that a higher frequency component of a 1/f1/f random vector is approximately equal to a Gaussian random vector. The theoretical results are confirmed by numerical results. Moreover, we show that the HE method is an efficient method to analyze heartbeat signals by applying it to the cardiac interbeat interval time series of healthy young and elderly subjects, congestive heart failure (CHF) subjects and atrial fibrillation (AF) subjects.