Multiscale entropy extends sample entropy to multiple time scales or signal resolutions to provide an additional perspective when the time scale of relevance is unknown.
Multiscale entropy (MSE) provides insights into the complexity of fluctuations over a range of time scales and is an extension of standard sample entropy measures described here. Like any entropy measure, the goal is to make an assessment of the complexity of a time series. One of the main reasons to use a multi-scale approach is when the time scale of relevance in the time series is not known. For example, if looking at speech it would be of relevance to consider the time scales of words rather than individual sounds, but if you did not have any idea that the audio signal represented speech, or perhaps even any idea of the concept of speech, you would not know what time scale would be most informative. It would therefore be more informative to look across a range of time scales. In the EEG, the underlying code is unknown and therefore the time scale of relevance is unknown.
See related post What Does the EEG Signal Measure?
Computing Multiscale Entropy
The basic principle of multiscale entropy involves coarse graining or down sampling the timeseries – essentially looking at the time series at increasingly coarser time resolutions. The operation is as follows:
Let’s say the time series contains the points x1, x2, x3, …..xN sampled every millisecond (so that the original time scale T is 1 ms). Coarse graining the data basically means averaging different numbers of consecutive points to create different scales or resolutions of the signal.
1. At scale 1, the coarse-grained time series is the original time series at hand.
2. At scale 2, the coarse-grained time series is formed by averaging two consecutive time points as shown in (A) in the figure below. That is define y1 = (x1 + x2)/2 ; y2 = (x3 + x4)/2 and so on.
3 . At scale 3, the coarse-grained time series is formed as the average of three consecutive time points as shown in (B) in the figure below. That is define y1 = (x1 + x2 + x3)/3; y2 = (x4 + x5 + x6)/3 and so on.
You can repeat the procedure as many times as relevant for the time series of study.
Figure adapted from [1]
Mathematically this is given as
where τ is the time scale.
Subsequently, sample entropy is computed for each of the scales or resolutions and plotted vs the scale. The area under this curve, which is essential the sum of sample entropy values over the range of scales, is used as the multiscale entropy measure. A time-series that has a lot of fluctuations will generate higher values of entropy and thus can be regarded as signal with higher complexity. Similarly, signals with high degree of regularity will have lower values of entropy.
Application of MSE in EEG analysis
Since MSE computes entropy of a signal at different scales, it is an interesting tool to understand how complexity of biological signals like EEG changes at different time scales. The curve of entropy vs time scale may yield a peak which indicates a time scale at which there is maximal entropy and may therefore be of greater relevance. Indeed Escudero et al. [2] showed that MSE could find significant differences between Alzheimer’s patients and control subjects even on large time scales at 10 electrodes, and that EEG activity is less complex in Alzheimer’s patients compared to controls. Although their study used only 11 controls and Alzheimer’s patients.
Figure adapted from [3]
In another study by Partk et al., [3] MSE analysis was done on Normal subjects, Alzheimer’s patients and people with mild cognitive impairment (MCI). The results again showed a loss of complexity for Alzheimer’s Disease (AD) subjects indicated by lower entropy values as seen in the figure above. Both Normal and MCI subjects had local maxima in the entropy value at scales 6 and 7 and then the entropy values decrease. Their analysis also showed that at scale factor of 1, sample entropy of Mild Cognitive Impairment, Alzheimer’s and normal subjects are statistically indistinguishable and thus entropy measures like sample or approximate entropy would have failed in this context to discriminate healthy subjects from pathological ones
In another study Catarino et al. [4] performed MSE analysis on healthy subjects and subjects with autistic spectrum disorder (ASD) performing social and non-social task (visual stimuli comprising of faces and chairs). Their results showed significant decrease in EEG complexity in Autism group compared to controls in occipital and parietal regions (shown below, with p-values).
Figure adapted from [4]
Thus, various disease states are associated with decreases in MSE measures albeit in a nonspecific manner.
Parameters in MSE
MSE simply extends the Sample Entropy measure to different time scales. Therefore the underlying parameters m (the length of the segments compared) and r (the distance measure between two segments) are the same. Discussed in an earlier blogpost, these are critical to the performance of MSE. There are no standard guidelines on how to choose these parameters. With respect to m, the length of the data, it has been shown that application of MSE requires sufficient amount of data at each time scale. Costa et al. [5] showed that the mean values of sample entropy (over 30 simulations) diverges as the number of data points decrease for white and 1/f noise. Particularly in case of 1/f, due to non-stationarity, the divergence is faster compared to white noise. To see the effect of the parameters m, r and data length on sample entropy refer to our earlier blogpost and another excellent read on these issues article by Costa et al. [5].
With respect to r what needs to be kept in mind is that in order to avoid a significant contribution of noise in the estimation of sample entropy, r must be higher than most of the signal noise. Another criteria for choosing r is based on signal dynamics.
See more details in related post The Impact of Parameter Choices on EEG Entropy Measures.
However, the most significant aspect is the manner in which r is computed and whether the distance measure chosen (usually Euclidian distance) is really a relevant to the signal at hand. For instance, in the case of audio signal of speech, Euclidian distance may not represent the most accurate measure of distance between two words. It is entirely possible that this is the case in EEG as well.
References
[1] Busa, Michael A., and Richard EA van Emmerik. “Multiscale entropy: A tool for understanding the complexity of postural control.” Journal of Sport and Health Science 5.1 (2016): 44-51.
[2] Escudero, J., et al. “Analysis of electroencephalograms in Alzheimer’s disease patients with multiscale entropy.” Physiological measurement 27.11 (2006): 1091.
[3] Park, Jeong-Hyeon, et al. “Multiscale entropy analysis of EEG from patients under different pathological conditions.” Fractals15.04 (2007): 399-404.
[4] Catarino, Ana, et al. “Atypical EEG complexity in autism spectrum conditions: a multiscale entropy analysis.” Clinical neurophysiology 122.12 (2011): 2375-2383.
[5] Costa, Madalena, Ary L. Goldberger, and C-K. Peng. “Multiscale entropy analysis of biological signals.” Physical review E 71.2 (2005): 021906.
I have a quesiton:”Sample entropy is computed for each of the scales or resolutions and plotted vs the scale. The area under this curve, which is essential the sum of sample entropy values over the range of scales, is used as the multiscale entropy measure.”
In 2005, Costa stated that: if for the majority of the scales the entropy values are higher for one time series than for another, the former is considered more complex than the latter, but not propose how to quantify the complexity in his paper.
How did you come to this conclusion?
Thank you so much for the crystal clear explanation of multiscale entropy algorithm.