Computational and Mathematical Methods in Medicine

Volume 2016 (2016), Article ID 1737953, 9 pages

http://dx.doi.org/10.1155/2016/1737953

## Symmetry Analysis of Gait between Left and Right Limb Using Cross-Fuzzy Entropy

^{1}School of Electrical Engineering and Automation, Anhui University, Hefei 230601, China^{2}Information Technology Research Centre, Nanjing Sport Institute, Nanjing 210014, China

Received 16 October 2015; Accepted 24 January 2016

Academic Editor: Didier Delignières

Copyright © 2016 Yi Xia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The purpose of this paper is the investigation of gait symmetry problem by using cross-fuzzy entropy (C-FuzzyEn), which is a recently proposed cross entropy that has many merits as compared to the frequently used cross sample entropy (C-SampleEn). First, we used several simulation signals to test its performance regarding the relative consistency and dependence on data length. Second, the gait time series of the left and right stride interval were used to calculate the C-FuzzyEn values for gait symmetry analysis. Besides the statistical analysis, we also realized a support vector machine (SVM) classifier to perform the classification of normal and abnormal gaits. The gait dataset consists of 15 patients with Parkinson’s disease (PD) and 16 control (CO) subjects. The results show that the C-FuzzyEn values of the PD patients’ gait are significantly higher than that of the CO subjects with a value of less than , and the best classification performance evaluated by a leave-one-out (LOO) cross-validation method is an accuracy of 96.77%. Such encouraging results imply that the C-FuzzyEn-based gait symmetry measure appears as a suitable tool for analyzing abnormal gaits.

#### 1. Introduction

Human gait is a complex process. The locomotor system incorporates input from the cerebellum, the motor cortex, and the basal ganglia, as well as feedback from visual, vestibular, and proprioceptive sensors [1]. Under healthy conditions, this multilevel control system produces a periodic and complementary movement of legs, which can be further subdivided into eight sequential subphases [2]. Following this delicate control strategy, a considerable degree of symmetry or similarity exists on the moving cadence and the stride length between left and right limb [3]. However, factors such as aging [4], peripheral neuropathy [5], and neurodegenerative disorders [6] could undermine such control mechanism in normal gait and lead to disturbance of gait phases and inconsistent stride length and disrupt rhythm. As a result, increased stride-to-stride variability and asymmetry often happened in abnormal gait [7, 8]. Hence, gait analysis was an important component during the clinical diagnosis or therapy assessment for those gait-related diseases [9].

During the past decades, with the rapid development of sensor technology and the emergence of corresponding signal processing methods, a lot of research efforts have been devoted to providing a quantitative and long-term gait evaluation methodology [2, 8, 10, 11]. Our main interest in this study is the quantitative assessment of gait symmetry, which has been addressed in several different studies [8, 12–15]. Among these studies, one frequently used clinical measure of symmetry [12, 15, 16] is , where and are the values of feature that is extracted from a time series for right and left limb, respectively. The degree of symmetry can also be quantified by the Pearson correlation coefficient between two time series, and an example of such studies was presented by Su et al. [13]. In addition to the above two methods, gait symmetry was also investigated by other methods. Sant’Anna and Wickström [14] proposed a symbol-based gait symmetry measure. Liao et al. [8] introduced multiresolution entropy analysis into the evaluation of gait symmetry.

Though most methods have achieved a certain success, relatively few studies have tried to apply cross entropy to the analysis of gait symmetry. Cross entropy is a kind of complexity measure that is generalized from entropy. By definition, the cross entropy measures the synchrony of two time series as the studies in [17–19] have done, but it can be also used for measuring the degree of dissimilarity of two concurrent, nonstationary biological signals [20–22]. In this study, the application of cross entropy to the evaluation of gait symmetry is inspired by the following observation: a symmetric gait must be similar, and a certain degree of dissimilarity must exist in an asymmetric gait. Therefore, by way of measuring gait similarity, the gait symmetry can also be measured.

From the literature, the first cross entropy, that is, cross-approximate entropy (C-ApEn), was proposed by Pincus and Singer [23]. However, it has two obvious limitations. First, because there are no self-matches, C-ApEn is not always defined. Second, there is “direction dependence” of C-ApEn analysis due to its template-wise approach. Then, Richman and Moorman [24] proposed cross sample entropy (C-SampleEn) between two time series. C-SampleEn is not direction dependent since it does not use a template-wise approach when estimating conditional probabilities. However, due to the same reason as C-ApEn that it does not count self-match, the definition of C-SampleEn is not always guaranteed. Recently, Xie et al. [25] proposed a new measure called cross-fuzzy entropy (C-FuzzyEn), which was derived from fuzzy entropy [26] that was based on the concept of fuzzy sets. Contrary to the common practice in C-ApEn and C-SampleEn, the similarity between any two embedded vectors is not measured by a Heaviside function, but with an exponential function. Thus, the discontinuity caused by the hard boundary of the Heaviside function is eliminated. As a result, C-FuzzyEn is always defined, and the choice of the parameters is assigned with more freedom.

The main purpose of this paper was to investigate the utility of the cross-fuzzy entropy for measuring gait symmetry using gait rhythm signals. The gaits of the patients with Parkinson’s disease (PD) and the healthy control (CO) subjects were analyzed in this study. Parkinson’s disease is a typical neurodegenerative disorder related to the central nervous system, and one of the main symptoms at its early stage is gait disorders, such as reduced stride length [27], freezing of gait [28], and increased gait variability [29]. The gait symmetry problem of PD patients was also investigated in previous studies [30, 31], and it has been reported that the degree of asymmetry of the PD patients’ gaits was larger than that of the normal gaits. In the present study, we reinvestigated the quantification of gait symmetry for PD patients and CO subjects. We hypothesized that the degree of symmetry can also be measured by using the novel gait symmetry measure that was based on C-FuzzyEn. With the encouraging results obtained in the experiments, we hope this study can provide a useful method for evaluating the abnormal gait of PD patients, especially at its early stage.

#### 2. Methodologies

##### 2.1. Gait Dataset

The gait dataset used in this study was contributed by Hausdorff et al. [32]. It includes gait data from fifteen PD subjects aged 44–80 years (age mean ± standard deviation, SD: years; 10 males and 5 females) and sixteen healthy CO subjects aged 20–74 years (age mean ± standard deviation, SD: years; 2 males and 14 females). According to the experimental protocol, the subjects were asked to walk at their normal pace along a straight hallway that was 77 m in length for 300 s. The gait signals were measured with ultrathin force-sensitive switches placed inside each subject’s shoes. Seven different gait rhythm signals for left or right limb were calculated with the algorithm proposed in [33]. In the present study, we have interest only in left and right stride interval (time from initial contact of one foot to the immediate subsequent contact) time series. To remove the outliers data points caused by the turnaround at the end of the hallway, a preprocessing method [34] was also applied.

##### 2.2. Definition of Cross-Fuzzy Entropy

Since cross-fuzzy entropy is developed on the basis of cross sample entropy, we first introduce the cross sample entropy and then point out how cross-fuzzy entropy differs from cross sample entropy. By this way of establishment, it is believed that the comparison of these two cross entropies can be more impressive.

Given two one-dimensional discrete time series with equal length, and , the definition of cross sample entropy is given as follows [24]:(1)Form the vectors:(2)The distance between two such vectors is defined as(3)Define , where is a Heaviside function and is given as where is a threshold value. Thus, can be deemed as the number of within of divided by , and the distance threshold controls the similarity of two vectors.(4)Define , and similarly is defined on the vectors and . Then, cross sample entropy is given as

Notably, there are two modifications in the cross-fuzzy entropy (C-FuzzyEn) proposed by Xie et al. [25] to the above C-SampleEn. First, the Heaviside function that measures the similarity of two vectors is replaced by an exponential function: where controls the width of the exponential function and determines the gradient of its boundary. Thus, C-FuzzyEn can be written as to include its different parameters. We set to be 2 in this study according to the suggestion given in [26]. By this fuzzy function, the discontinuity caused by the hard boundary of the Heaviside function can be eliminated. Second, to highlight the effect of vector’s shape instead of the absolute values in the fuzzy-based similarity measurement, the baseline is subtracted from the vector; that is,where the baseline is the average of all , . Similar preprocessing is performed on , , and .

##### 2.3. Performance Tests of C-FuzzyEn

The performance of C-FuzzyEn was tested on some typical simulation signals, such as i.i.d. uniform random numbers and the processes. The process [35] is a composite of stochastic and deterministic components. For fixed and a sine wave signal with points, the signal is formed by substituting the randomly chosen points of the sine signal with the i.i.d. random numbers. In all the following experiments, to highlight the statistical stability of C-FuzzyEn, the time series pair was generated 200 times randomly for a fixed parameter set, and then the mean and the standard deviation (SD) of the C-FuzzyEn values were computed. As a comparison, C-SampleEn was also calculated with the same testing process.

###### 2.3.1. Effect of Parameter r and

It can be found from the definition that the most relevant parameters of C-FuzzyEn are parameter and data length . The influence of these two parameters on the calculation of C-FuzzyEn was evaluated by two experiments. We first evaluated the effect of parameter . The testing signal pairs were an i.i.d. uniform random time series and a time series. The embedded dimension was set to be 2, and the data length was first set to be 100 and then 50. As for parameter , it varied from a predefined value set of , where meant that the value varied from to in steps of . The second experiment was designed to evaluate the influence of the data length. Two i.i.d. random uniform numbers of different lengths were examined. Data length ranged from 50 to 500 in steps of 50, parameter was set to be 0.3, and the embedded dimension was set to be 2 and 3 successively.

###### 2.3.2. Relative Consistency Analysis

Let denote a parameter of the given cross entropy algorithm. Given two pairs of time series, and , then the relative consistency is defined in the following way: if there is , inducing the cross entropy of to be lower than that of , that is, , then, for all , should be true. We tested the relative consistency of C-FuzzyEn by using the following two pairs of time series: the [MIX(0.2), MIX(0.3)] pair and the [MIX(0.3), MIX(0.4)] pair. Since the MIX(0.4) time series should be more disordered than MIX(0.2) time series, it was hypothesized that the cross entropy of the [MIX(0.2), MIX(0.3)] pair should have a lower value than that of the [MIX(0.3), MIX(0.4)] pair for all the parameters. The data length was set to be 100, 50 in sequential order, while in both cases parameter was changed from the set of .

##### 2.4. Gait Symmetry Analysis Using C-FuzzyEn

To apply C-FuzzyEn on the gait stride interval signals for the analysis of gait symmetry, there were three parameters to be set up. The first parameter was the embedded dimension , that is, the length of the comparing vectors. Since the data series used in this study were very short (with a length from 169 to 269), we used in order to calculate the frequency of and -component vectors with sufficient statistical accuracy. Then, the value of parameter was chosen from a predefined value set, that is, , based on the gait data precision and the results of some preliminary experiments. During the above setting process of parameter , data length was fixed to be 150 since it was close to the maximum length of some gait sequences used in this study. By parameter selection, we hope to find out a parameter set that can provide the largest separation between the C-FuzzyEn values for the normal gait and that for the PD patients’ gait.

##### 2.5. Statistical Analysis and Classification of Gait Patterns

SPSS software (Version 17.0, SPSS Inc., Chicago, IL, USA) was adopted for all statistical analyses. The continuous and categorical variables between the groups were compared using Mann-Whitney test. A was considered statistically significant.

To implement the classification of Parkinson gait and normal gait with the proposed gait symmetry feature, the popular support vector machine (SVM) classifier was utilized in this study. The SVM classifier introduced by Vapnik [36] is the first implementation of structural risk minimization (SRM), which is a theory that enforced the selection of the optimal learning model from a subset of models. To set up a linear separating hyperplane for nonlinear problems, SVM employed kernel methods to map data to a higher dimensional feature space. In the present study, the popular radial basis function (RBF) kernel was adopted.

The classification performance was evaluated by the leave-one-out (LOO) cross-validation method. Moreover, the classification results were measured by sensitivity, specificity, and accuracy. The area under the receiver operator curve (ROC), as a well-established index of diagnostic accuracy, was also calculated by using the software ROCKIT provided by the University of Chicago, Chicago, IL, USA [37].

#### 3. Results

##### 3.1. Results of Performance Tests for C-FuzzyEn

###### 3.1.1. Effect of Parameter and

Figure 1 illustrates the calculation results of and with parameter changing from the value set of . In either Figure 1(a) () or Figure 1(b) (), it can be found that C-SampleEn gives no value when falls below a certain value, and the smaller the value of is, the larger the minimum value of is needed to ensure the definition of C-SampleEn. In contrast, such problems do not bother the calculation of C-FuzzyEn, and the choice of parameter and is much more free.