Multiscale entropy (MSE) analysis [1,2] is a new method of measuring the complexity of finite length time series. This computational tool can be applied both to physical and physiologic data sets, and can be used with a variety of measures of entropy. We have developed and applied MSE for the analysis of physiologic time series, for which we prefer to estimate entropy using the sample entropy (SampEn) measure [3]. SampEn is a refinement of the approximate entropy family of statistics introduced by Pincus [4]. Both have been widely used for the analysis of physiologic data sets [5,6].

Traditional entropy measures quantify only the regularity (predictability) of time series on a single scale. There is no straightforward correspondence, however, between regularity and complexity. Neither completely predictable (e.g., periodic) signals, which have minimum entropy, nor completely unpredictable (e.g., uncorrelated random) signals, which have maximum entropy, are truly complex, since they can be described very compactly. There is no consensus definition of complexity. Intuitively, complexity is associated with ``meaningful structural richness'' [7] incorporating correlations over multiple spatio-temporal scales.

For example, we and others have observed that traditional single-scale entropy estimates tend to yield lower entropy in time series of physiologic data such as inter-beat (RR) interval series than in surrogate series formed by shuffling the original physiologic data. This happens because the shuffled data are more irregular and less predictable than the original series, which typically contain correlations at many time scales. The process of generating surrogate data destroys the correlations and degrades the information content of the original signal; if one supposes that greater entropy is characteristic of greater complexity, such results are profoundly misleading. The MSE method, in contrast, shows that the original time series are more complex than the surrogate ones, by revealing the dependence of entropy measures on scale [8,9,10,11,12].

The MSE method incorporates two procedures:

- A ``coarse-graining'' process is applied to the time series. For a
given time series, multiple coarse-grained time series are constructed
by averaging the data points within non-overlapping windows of
increasing length, (see Figure 1). Each
element of the coarse-grained time series, , is
calculated according to the equation:

where represents the scale factor and . The length of each coarse-grained time series is . For scale 1, the coarse-grained time series is simply the original time series. - SampEn is calculated for each coarse-grained time series, and then plotted as a function of the scale factor. SampEn is a ``regularity statistic.'' It ``looks for patterns'' in a time series and quantifies its degree of predictability or regularity (see Figure 2).

2005-06-24