The original multiscale entropy (MSE) method [1,2] quantifies the complexity of the temporal changes in one specific feature of a time series: the local mean values of the fluctuations. The method comprises two steps: 1) coarsegraining of the original time series, and 2) quantification of the degree of irregularity of the coarsegrained (CG) time series using an entropy measure such as sample entropy (SampEn) [3].
The generalized multiscale entropy (GMSE) method [4] quantifies the complexity of the dynamics of a set of features of the time series related to local sample moments. The method differs from the original MSE in the way that the CG time series are computed. In the original method, the mean value is used to derive a set of representations of the original signal at different levels of resolution. This choice implies that information encoded in features related to higher moments is discarded. The coarsegraining procedure in the generalized algorithm extracts statistical features such as the variance (standard deviation [SD] or mean absolute deviation [MAD]), skewness, kurtosis, etc, over a range of time scales. This tutorial focuses primarily on the quantification of the information encoded in fluctuations in standard deviation.
We use a subscript after MSE to designate the type of coarsegraining employed. Specifically, MSE_{μ}, MSE_{σ} and MSE_{σ2} refer to MSE computed for mean, SD and variance CG time series, respectively.
For a dynamical property of interest, such as mean or standard deviation, MSE algorithms comprise two sequential procedures:
As noted above, in the original MSE method (MSE_{μ}) the property of interest is the local mean value. The CG time series capture fluctuations in local mean value for preselected time scales. In the original application, such CG time series were obtained by dividing the original time series into nonoverlapping segments of equal length and calculating the mean value of each of them. However, other approaches for extracting the same “type” of information (local mean) can also be considered, including low pass filtering the original time series using Fourier analysis, among others (e.g., the empirical mode decomposition) methods.
The GMSE method expands the original MSE framework to other properties of a signal. Here, we address the quantification of information encoded in the fluctuations of the “volatility” of the signal.
Figure 1 shows the interbeat interval (RR) time series from a healthy subject, simulated 1/f noise and their SD CG time series for scales 5 and 20. The fluctuation patterns of the physiologic CG time series appear more unpredictable, “less uniform” and more “bursty,” than those of simulated 1/f noise.

Figure 2 shows MSE_{σ} (top panels) and MSE_{σ2} (bottom panels) analyses of physiologic and simulated longrange correlated (1/f) noise time series. The physiologic time series are the RR intervals (left panels) from healthy young to middleaged (≤ 50 years) and healthy older (> 50 years) subjects and patients with chronic (congestive) heart failure (CHF). The time series are available on PhysioNet: i) 26 healthy young subjects and 46 healthy older subjects (nsrdb, nsr2db) ii) 32 patients with CHF class III and IV (chfdb, chf2db).

Entropy over the preselected range of scales was higher for 1/f than white noise, both for SD and variance CG time series. With respect to the RR interval time series, entropy values were on average higher for the group of healthy young subjects than for the group of healthy older subjects. In addition, the entropy values for the group of CHF patients were, on average, the lowest. The results were qualitatively the same for SD and variance CG time series.
These findings are consistent with those derived from traditional (mean CG) MSE analyses. They indicate that: 1) 1/f noise processes are more complex than uncorrelated random ones; 2) the complexity of heart rate dynamics degrades with aging and heart disease.
Madalena Costa (mcosta3@bidmc.harvard.edu)