Abstract
Non-linear analysis found many applications in biomedicine. Approximate entropy (ApEn) is a popular index of complexity often applied to biomedical data, as it provides quite stable indications when processing short and noisy epochs. However, ApEn strongly depends on parameters, which were chosen in the literature in wide ranges. This paper points out that ApEn depends on sampling rate of continuous time signals, embedding dimension, tolerance (under which a match is identified), epoch duration and low frequency trends. Moreover, contradicting results can be obtained changing parameters. This was found both in simulations and in experimental EEG. These limitations of ApEn suggest the introduction of an alternative index, here called modified ApEn, which is based on the following principles: oversampling is compensated, self-recurrences are ignored, a fixed percentage of recurrences is selected and low frequency trends are removed. The modified index allows to get more stable measurements of the complexity of the tested simulated data and EEG. The final conclusions are that, in order to get a reliable estimation of complexity using ApEn, parameters should be chosen within specific ranges, data must be sampled close to the Nyquist limit and low frequency trends should be removed. Following these indications, different studies could be more easily compared, interpreted and replicated. Moreover, the modified ApEn can be an interesting alternative, which extends the range of parameters for which stable indications can be achieved.
Highlights
Some biomedical data have non-linear properties and complex behavior, e.g., heart rate variability (Lake and Moorman, 2011; Liu et al, 2011), electroencephalogram (EEG; Mesin and Costa, 2014; Li et al, 2016), electromyogram (EMG; Mesin et al, 2016), pupillogram (Mesin et al, 2013; Onorati et al, 2016), center of mass during quiet standing (Liang et al, 2017)
Signal complexity is usually estimated assuming that the data were generated by a dynamical system in stationary conditions which can possibly develop irregular behavior due to its non-linearity (Kantz and Schreiber, 1997)
approximate entropy (ApEn) and the modified index were tested on simulations (Figures 1–5) and experiments (Figure 6)
Summary
Some biomedical data have non-linear properties and complex behavior, e.g., heart rate variability (Lake and Moorman, 2011; Liu et al, 2011), electroencephalogram (EEG; Mesin and Costa, 2014; Li et al, 2016), electromyogram (EMG; Mesin et al, 2016), pupillogram (Mesin et al, 2013; Onorati et al, 2016), center of mass during quiet standing (Liang et al, 2017) The estimation of their level of regularity can be useful in many applications (Seely and Christou, 2000; Gao et al, 2015). The approximate entropy (ApEn) was introduced (Pincus, 1991) as an index which could provide a rough, but quite stable indication of complexity even from short and noisy epochs
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