We study statistical properties of the Jensen-Shannon divergence D, which quantifies the difference between probability distributions, and which has been widely applied to analyses of symbolic sequences. We present three interpretations of D in the framework of statistical physics, information theory, and mathematical statistics, and obtain approximations of the mean, the variance, and the probability distribution of D in random, uncorrelated sequences. We present a segmentation method based on D that is able to segment a nonstationary symbolic sequence into stationary subsequences, and apply this method to DNA sequences, which are known to be nonstationary on a wide range of different length scales.

We introduce a segmentation algorithm to probe the temporal organization of heterogeneities in human heartbeat interval time series. We find that the lengths of segments with different local mean heart rates follow a power-law distribution and show that this scale-invariant structure is not a simple consequence of the long-range correlations present in the data. The differences in mean heart rates between consecutive segments display a common functional form, but with different parameters for healthy individuals and for heart-failure patients. These findings suggest that there is relevant physiological information hidden in the heterogeneities of the heartbeat time series.

We investigate how various linear and nonlinear transformations affect the scaling properties of a signal, using the detrended fluctuation analysis (DFA). Specifically, we study the effect of three types of transforms: linear, nonlinear polynomial and logarithmic filters. We compare the scaling properties of signals before and after the transform. We find that linear filters do not change the correlation properties, while the effect of nonlinear polynomial and logarithmic filters strongly depends on (a) the strength of correlations in the original signal, (b) the power of the polynomial filter and (c) the offset in the logarithmic filter. We further investigate the correlation properties of three analytic functions: exponential, logarithmic, and power-law. While these three functions have in general different correlation properties, we find that there is a broad range of variable values, common for all three functions, where they exhibit identical scaling behavior. We further note that the scaling behavior of a class of other functions can be reduced to these three typical cases. We systematically test the performance of the DFA method in accurately estimating long-range power-law correlations in the output signals for different parameter values in the three types of filters, and the three analytic functions we consider.

A segmentation algorithm based on the Jensen-Shannon entropic divergence is used to decompose long-range correlated DNA sequences into statistically significant, compositionally homogeneous patches. By adequately setting the significance level for segmenting the sequence, the underlying power-law distribution of patch lengths can be revealed. Some of the identified DNA domains were uncorrelated, but most of them continued to display long-range correlations even after several steps of recursive segmentation, thus indicating a complex multi-length-scaled structure for the sequence. On the other hand, by separately shuffling each segment, or by randomly rearranging the order in which the different segments occur in the sequence, shuffled sequences preserving the original statistical distribution of patch lengths were generated. Both types of random sequences displayed the same correlation scaling exponents as the original DNA sequence, thus demonstrating that neither the internal structure of patches nor the order in which these are arranged in the sequence is critical; therefore, long-range correlations in nucleotide sequences seem to rely only on the power-law distribution of patch lengths. © 1996 The American Physical Society.