On the use of sample entropy to analyze human postural sway data

Abstract

We analyze the irregularity of human postural sway data during quiet standing using the sample entropy (SampEn) algorithm. By considering recent methodological developments, we show that the SampEn parameter is able to characterize the irregularity of the center of pressure fluctuations through the analysis of the velocity variable. We present a practical method to select the input parameters of the SampEn algorithm. We show that the computed SampEn successfully discriminates two sensory conditions (eyes-open and eyes-closed) in a group of healthy young adults. We also perform surrogate data tests to investigate the nature of the underlying dynamics of our experimental data. Finally, the results of the proposed approach are compared to those obtained with the multiscale entropy algorithm.

Introduction

Within the large set of human movement behavioral data, the displacements of the center of pressure (COP) are certainly among the most studied. During quiet standing, the displacements of the COP display highly irregular and non-stationary [1] fluctuations. The complexity (in the common sense) of such data have naturally motivated studies devoted to the analysis of their temporal dynamics and to characterize the effects of a broad set of functionally relevant factors (i.e. visual perception, cognitive task, aging, and disease) on postural stability. However, the clinical utility of dynamical measures is still an open question.

In this field, some studies focused on the analysis of the correlation and fractal properties of COP signal [2], [3], [4], [5], [6] generally leading to a fractional Brownian motion (fBm) statistical model [7]. Within this context, it has been reported [8], [9] that COP time series show long-range correlations with single scaling exponent over a broad range of time windows. Other approaches belong to the nonlinear deterministic paradigm. Using different analysis methods, many studies based on this paradigm investigated the deterministic features of COP dynamics [10], [11], [12], [13], [14] and their potentially chaotic behavior [15], [16], [17], [18].

Another direction for the analysis of COP dynamics, is based on the quantification of the complexity of COP time series in terms of their irregularity. Newell [19] used the approximate entropy to characterize the dynamical structure of human postural sway. The aim was to investigate the effect of motor development and aging on the complexity of COP trajectories during quiet standing. The results were consistent with the hypothesis of a reduction of the degrees of freedom of the system with aging. Sabatini [8] combined different algorithms for the estimation of entropy to extract a complexity index and investigated the effect of visual perception. In more recent papers, approximate entropy was used to investigate the effect of cerebral concussion [20], [21] and cognitive task [22] on postural dynamics. Roerdink et al. [23] published a study on the dynamical structures of COP trajectories in patients recovering from stroke. To the best of our knowledge, this study was the first to apply the sample entropy (SampEn) algorithm to postural sway recordings.

In the framework of time series analysis, entropy is defined as a quantity measuring the rate of generation of information (see [24], Chapter 11.4, for a detailed overview on the subject). Since the seminal work of Kolmogorov [25] and Sinai [26] on the quantification of the predictability of complex dynamical systems, several algorithms were proposed to improve and adapt the estimation of the entropy of a system from a time series. Grassberger and Procaccia [27] were the first to propose an algorithm to estimate the Kolmogorov–Sinai (K–S) entropy from time series generated by chaotic systems. Using a modification introduced by Takens [28], Eckmann and Ruelle [29] proposed a new formula to estimate the K–S entropy. In order to extend their formula to time series generated by stochastic processes, Pincus [30] introduced a family of statistics, namely the approximate entropy (ApEn). One advantage of the ApEn analysis lies in the possibility to apply it both to deterministic and stochastic systems. More recently, Richman and Moorman [31] modified the ApEn method and proposed the sample entropy analysis to study time series of lengths ranging from 100 to 20,000 points. The SampEn basically quantifies the irregularity of a sequence of numbers and hence it naturally provides a complexity index. Basically, the SampEn increases with the irregularity of the time series. It has been estimated for physiological data such as heart rate variability [31], [32], [33], neural respiratory signals [34] and EMG recordings [35]. In [34], the authors compared ApEn and SampEn analyses. They addressed issues related to the choice of the input parameters and showed that the SampEn approach produced more consistent results. They also showed that SampEn is less sensitive to the length of data. In addition, they investigated the sampling rate effect and found that both ApEn and SampEn are sensitive to over-sampling. More recently, Govindan et al. [36] addressed specifically the sampling issue and modified the original method by taking into account a time delay in the definition of the template vectors. They clearly showed that the choice of the time delay (or lag time) has an important effect on the SampEn value and that its use improved the characterization of the system complexity.

In [23], the authors applied the SampEn method without considering the long-range correlations of the COP fluctuations. It has been shown recently [36] that such correlations should be taken into account for the estimation of SampEn. The presence of temporal correlations was not considered in two more recent studies using the SampEn algorithm [37], [38]. In [39], Costa et al. performed a complexity analysis of COP data based on an index derived from the multiscale entropy (MSE) algorithm. In this work, the non-stationarity of the position data was specifically addressed. The MSE analysis [40] is basically performed by computing the SampEn at different scales by coarse-graining the time series.

In the present paper, we use the SampEn algorithm [31] by taking into account the recent methodological advances [36] which appear particularly relevant for the analysis of COP data. An adapted practical solution for choosing the parameters of the SampEn method is derived from a statistical criterion proposed by Lake et al. [32]. We explore the role of visual feedback on the estimated entropy of our data. The visual perception manipulation we applied is classical in the investigation of postural control mechanisms [4], [19], [11], [8], [23], [37], [38]. Surrogate data tests are also performed to investigate the intrinsic nature of the underlying dynamical process generating COP data. In order to compare our approach to a directly related available method, we also analyse our data using the MSE complexity index [39].