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Classification of neurodegenerative diseases using gait dynamics via deterministic learning
Information Sciences 317 (2015) 246–258
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Information Sciences journal homepage: www.elsevier.com/locate/ins
Classification of neurodegenerative diseases using gait dynamics via deterministic learning Wei Zeng a,b,⇑, Cong Wang c a
School of Mechanical & Electrical Engineering, Longyan University, Longyan 364012, China School of Mechanical & Automotive Engineering, South China University of Technology, Guangzhou 510640, China c School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China b
a r t i c l e
i n f o
Article history: Received 15 December 2014 Received in revised form 21 April 2015 Accepted 24 April 2015 Available online 2 May 2015 Keywords: Gait analysis Deterministic learning Neurodegenerative diseases Gait dynamics Movement disorders
a b s t r a c t Neurodegenerative diseases (NDDs), such as Parkinson’s disease (PD), Huntington’s disease (HD) and amyotrophic lateral sclerosis (ALS), create serious gait abnormalities. They lead to altered gait rhythm and gait dynamics which can be reflected by a time series of stride-to-stride measures of footfall contact times. The temporal fluctuations in gait dynamics provide us with a non-invasive technique to evaluate the effects of neurological impairments on gait and its variations with diseases. In this paper, we present a new method using gait dynamics to classify (diagnose) NDDs via deterministic learning theory. The classification approach consists of two phases: a training phase and a classification phase. In the training phase, gait features representing gait dynamics are derived from the time series of swing intervals and stance intervals of the left and right feet. Gait dynamics underlying gait patterns of healthy controls and NDDs subjects are locally accurately approximated by radial basis function (RBF) neural networks. The obtained knowledge of approximated gait dynamics is stored in constant RBF networks. Gait patterns of healthy controls and NDDs subjects constitute a training set. In the classification phase, a bank of dynamical estimators is constructed for all the training gait patterns. Prior knowledge of gait dynamics represented by the constant RBF networks is embedded in the estimators. By comparing the set of estimators with a test NDDs gait pattern to be classified, a set of test errors are generated. The average L1 norms of the errors are taken as the classification measure between the dynamics of the training gait patterns and the dynamics of the test NDDs gait pattern according to the smallest error principle. Finally, experiments are carried out to demonstrate that the proposed method can effectively separate the gait patterns between the groups of healthy controls and neurodegenerative patients. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Neurodegenerative diseases (NDDs), including Parkinson’s disease (PD), Huntington’s disease (HD), and amyotrophic lateral sclerosis (ALS), produce changes in altered neuromuscular control. Since flexion and extension motions of two lower limbs are regulated by the central nervous system, the gait of a patient with a neurodegenerative disorder would become abnormal due to deterioration of motor neurons. PD and HD are typical disorders of the basal ganglia and are associated with
⇑ Corresponding author at: School of Mechanical & Electrical Engineering, Longyan University, Longyan 364012, China. Tel./fax: +86 597 2799753. E-mail address: [email protected] (W. Zeng). http://dx.doi.org/10.1016/j.ins.2015.04.047 0020-0255/Ó 2015 Elsevier Inc. All rights reserved.
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characteristic changes in gait rhythm [4]. ALS is a disorder primarily affecting the motoneurons of the cerebral cortex, brain stem, and spinal cord [2]. Because the neurons of the basal ganglia likely play an important role in regulating muscular motor control such as balance and sequencing of movements, it is reasonable to expect that the stride-to-stride dynamics, as well as the gait cycle duration, are affected by these NDDs. Gait information has been widely used for the movement studies in healthy controls and also in subjects with different types of diseases. Analysis of temporal gait parameters is very useful for a better understanding of the mechanisms of movement disorders, and also has the high potential in presenting automatic non-invasive method based on gait dynamics for the classification of NDDs [2]. Gait dynamics are regulated by a complex nervous system. It is believed that the regulatory feedback loops of a physiologic system need to operate across multiple spatial and temporal scales to be able to adapt to an ever changing environment [30]. Thus, the time series of stride, stance or swing intervals are likely to exhibit fluctuations across multiple spatial and temporal scales [15]. In recent related studies, computer-aided tools have been utilized to measure the gait interval parameters in healthy adults, and also to describe the distinct characteristics of the gait in NDDs [8,9,16,19,22,20,24,27–29,32,3 7,40,43,31]. Aziz and Arif [2] converted the stride time series into a kind of symbol sequence, then applied a threshold dependent symbolic entropy method in the analysis of gait complexity. They observed that the normalized corrected Shannon entropy of the symbolic stride sequences is much lower in ALS at different short thresholds. Scafetta et al. [29] used the supercentral pattern generator (SCPG) model to simulate human gait dynamics, and also discussed the stochastic and fractal properties of the gait in PD, HD and ALS. Wu and Shi [41] proposed a statistical analysis method for the classification of gait cadence in subjects with ALS and healthy controls. In their approach, the probability density functions of gait cadence were estimated using Parzen-window method and then the Kullback–Leibler divergence was derived. With this method it was able to classify the stride patterns of the ALS and the control subjects with the accuracy rate of 82.8%. In their another similar study [39], it was hypothesized that the swing interval turns count (SWITC) of the ALS patients might be different from that of healthy subjects. The ALS gait patterns, characterized by the SWITC parameter and the known ASI feature, could be distinguished from the gait patterns of healthy subjects through the linear and nonlinear classifiers. Using the SWITC parameter and the average stride interval could reach to the classification rate of 89.66% for the gaits in subjects with ALS and gaits in healthy control subjects. Daliri [7] presented an approach for the diagnosis of NDDs based on gait dynamics. The proposed method used information from a time series of stride intervals, swing intervals, stance intervals and double support intervals of stride-to-stride measures. The support vector machines using different kernels were examined for the diagnosis. Based on the findings in [3], a reasonable approach to investigate the effects of neurological impairments on patients’ ability to mediate the locomotion of two lower limbs was proposed. It compared the left and right stance-interval series in terms of their regularities at multi-resolution levels. Liao et al. [23] investigated gait asymmetry in NDDs using the multi-resolution entropy analysis of stance time fluctuations. Their results showed that gait symmetry is significantly disturbed in subjects with PD, HD, and ALS, and the degree of disturbance is more prominent in the subjects with ALS. A number of scientists showed that NDDs patients present altered fractal dynamics of gait characterized by reduced stride-interval correlations, that is, the walking of these individuals becomes more random [30,18,21]. The randomness increases with the severity of the neurodegenerative impairment. Our primary interest in this paper is to investigate modeling of gait dynamics using the time series of the left and right feet. Specifically, we would like to know whether the three types of neurological disease (PD, HD and ALS) would impair the patient’s ability to regulate the locomotion of two feet and whether the difference of gait dynamics is related to the particular type of disease that the patient has. Nevertheless, research on the temporal variability of gait dynamics, such as step to step, is limited. It has been postulated that the ability to maintain a steady gait (i.e. low stride-to-stride variability of gait cycle timing and its sub-phases) would be diminished in NDDs [17]. Irregular timing of steps in NDDs suggests a disturbance of rhythmic locomotor activity generation and gait dynamics [11]. The classification (diagnosis) of NDDs based on the difference of gait dynamics between NDDs and healthy controls is what we want to attempt. In this paper, we present a new method using gait dynamics for the classification of NDDs via deterministic learning theory. The time series of swing and stance intervals of the left and right feet are used to model gait dynamics of NDDs and healthy control subjects according to the SCPG model. Gait dynamics underlying gait patterns of healthy controls and NDDs subjects are locally accurately approximated by radial basis function (RBF) neural networks. The obtained knowledge of approximated gait dynamics is stored in constant RBF networks. The gait patterns of healthy controls and NDDs subjects constitute a training set. In the classification phase, a bank of dynamical estimators is constructed for all the training gait patterns. Prior knowledge of gait dynamics represented by the constant RBF networks is embedded in the estimators. By comparing the set of estimators with a test NDDs gait pattern to be classified, a set of test errors are generated. The average L1 norms of the errors are taken as the classification measure between the dynamics of the training gait patterns and the dynamics of the test NDDs gait pattern to be classified according to the smallest error principle. The proposed method can effectively separate the gait patterns between the groups of healthy controls and neurodegenerative patients. Compared with other recently reported results in [41,7,10], our method achieves superior classification performance. The rest of the paper is organized as follows. Section 2 introduces preliminary knowledge about deterministic learning theory and problem formulation. Section 3 describes the proposed method. This includes the data description, feature extraction and selection, learning and classification mechanisms. Section 4 presents experimental results. Section 5 gives some discussions and conclusions.
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2. Preliminaries and problem formulation 2.1. Deterministic learning theory In deterministic learning theory, identification of system dynamics of general nonlinear systems is achieved according to the following elements: (i) employment of localized RBF networks; (ii) satisfaction of a partial persistence of excitation (PE) condition; (iii) exponential stability of the adaptive system along the periodic or recurrent orbit; (iv) locally accurate neural network (NN) approximation of the unknown system dynamics [34,35]. P The RBF networks can be described by f nn ðZÞ ¼ Ni¼1 wi si ðZÞ ¼ W T SðZÞ, where Z 2 XZ Rp is the input vector, W ¼ ½w1 ; . . . ; wN T 2 RN is the weight vector, N is the NN node number, and SðZÞ ¼ ½s1 ðkZ l1 kÞ; . . . ; sN ðkZ lN kÞT , with si ðÞ being a radial basis function, and li ði ¼ 1; . . . ; NÞ being distinct points in state space. The Gaussian function T
si ðkZ li kÞ ¼ exp½ðZligÞ2 ðZli Þ is one of the most commonly used radial basis functions, where i
li ¼ ½li1 ; li2 ; . . . ; liN T is the
center of the receptive field and gi is the width of the receptive field. The Gaussian function belongs to the class of localized radial basis functions in the sense that si ðkZ li kÞ ! 0 as kZk ! 1. It has been shown in [26] that for any continuous function f ðZÞ : XZ ! R where XZ Rp is a compact set, and for the NN approximator, where the node number N is sufficiently large, there exists an ideal constant weight vector W , such that for each > 0; f ðZÞ ¼ W T SðZÞ þ ðZÞ; 8Z 2 XZ , where jðZÞ < j is the approximation error. Moreover, for any bounded trajectory Z f ðtÞ within the compact set XZ ; f ðZÞ can be approximated by using a limited number of neurons located in a local region T Nf along the trajectory: f ðZÞ ¼ W T f Sf ðZÞ þ f , where Sf ðZÞ ¼ ½sj1 ðZÞ; . . . ; sjf ðZÞ 2 R , with N f < N; jsji j > iðji ¼ j1 ; . . . ; jf Þ; i > 0 is a
small positive constant, W f ¼ ½wj1 ; . . . ; wjf T , and
f
is the approximation error, with jjf j jjj being small.
Based on previous results on the PE property of RBF networks [13], it is shown in [36] that for a localized RBF network W T SðZÞ whose centers are placed on a regular lattice, almost any recurrent trajectory ZðtÞ can lead to the satisfaction of the PE condition of the regressor subvector Sf ðZÞ. 2.2. Problem formulation Consider a general nonlinear human gait dynamical system in the following form:
x_ ¼ Fðx; pÞ;
xðt0 Þ ¼ x0
ð1Þ
where x ¼ ½x1 ; . . . ; xn T 2 Rn is the state of the system representing the gait features, p is a constant vector of system parameters (different p will in general generate different gait patterns of healthy controls and NDDs subjects). T
Fðx; pÞ ¼ ½f 1 ðx; pÞ; . . . ; f n ðx; pÞ is a smooth but unknown nonlinear vector field. Assumption 1. The gait system state x remains uniformly bounded. Moreover, the system trajectory starting from x0 , denoted as uf ðx0 Þ, is in either a periodic or periodic-like (recurrent) motion. For the human gait system, Murray et al. [25] suggested that human gait is a form of periodic or quasi-periodic motion. Thus, Assumption 1 is reasonable. Our objective is to choose suitable time series of gait features satisfying Assumption 1 and design a dynamic RBF network to identify and approximate the unknown vector Fðx; pÞ for healthy controls and NDDs subjects. The approximation result can be used to represent gait dynamics which will be stored and used for the classification of NDDs. 3. Method In this section, we propose a method for the classification of NDDs using the information obtained from gait dynamics. Firstly, several gait features are extracted from the time series of the left and right feet as the feature vectors, which represent the variations of gait dynamics. Then, the RBF networks are used to approximate gait dynamics through the supercentral pattern generator (SCPG) model. The difference of gait dynamics will be applied to distinguish the healthy controls from the group of subjects with NDDs. The outline of the proposed method has been shown in Fig. 1. 3.1. Data description Data sets for analysis and classification are taken from gait time series in neurodegenerative database contributed by [18], which is available online in the PhysioNet (http://www.physionet.org/physiobank/database/gaitndd). The database contains 64 recordings of gait from 16 healthy controls, 13 subjects with ALS, 20 subjects with HD and 15 subjects with PD. Healthy control group consists of subjects with age range 20–74, ALS group consists of subjects with age range 36–70, HD group consists of subjects with age range 29–71 and PD group consists of subjects with age range 44–80. Subjects were instructed to walk at their normal pace along a 77 m long hallway for 5 min. Force sensitive switches were placed in the subjects’ shoes, output of these switches provided a force applied on the floor [19]. A 12-bit on-board analog-to-digital converter sampled the output
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Fig. 1. Overview of the proposed method for the (classification) diagnosis of NDDs using gait dynamics.
of foot switches at 300 Hz. From the recorded force applied to the ground during walking, the time series of the left (right) stride time, left (right) stance time and left (right) swing time were derived. To minimize startup effects the first 20 s recorded data were excluded and median filter was used to remove data points (outliers) that were far away from the median value [17]. These outliers were mainly due to the turns at the end of the hallway.
3.2. Feature extraction and selection For each time series, we extract four different features to create our feature vectors for the classification. These features including the left and right swing intervals, the left and right stance intervals are employed to create a feature vector of 4 dimensions for each subject in the database.
Fig. 2. An example of sequence of stride times from different groups of subjects including a subject with PD, a subject with HD, a subject with ALS disease and a healthy control (CO) subject.
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The raw data are obtained using force-sensitive resistors, with the output roughly proportional to the force under the foot. Stride-to-stride measures of footfall contact times are derived from these signals and have been shown in Fig. 2. Some time series data are derived from the signals in the left and right feet. These include left and right swing intervals, left and right stance intervals. Samples of these time series for subjects with PD, HD, ALS and a healthy control subject have been shown in Figs. 3–6. Here, in Table 1, we give the values of measures of the fluctuation magnitude of swing, stance and stride intervals in ALS, PD, HD and healthy control subjects. It is observed from the data sets that: (1) the average swing, stance and stride intervals of patients with ALS are significantly longer than those of healthy controls or of patients with PD or HD; (2) the values of two measures of the fluctuation magnitude of swing, stance and stride intervals in ALS, i.e., the standard deviation (SD) and coefficient of variation (CV = standard deviation/mean 100) of the detrended interval time series, are larger than those of healthy controls, PD and HD patients, respectively; (3) the fluctuation of the stride interval can be reflected by the swing and stance intervals since stride interval equals to swing interval plus stance interval; (4) the fluctuation of the time series between the left and right legs of healthy controls is smaller than other NDDs patients, which shows the good gait symmetry of healthy controls. In other words, gait symmetry is significantly disturbed in subjects with ALS, PD and HD. It is found that gait symmetry in subjects with PD and HD is disturbed less than in subjects with ALS. It is seen from Table 1 that the time series of four groups of subjects are obviously different, which means gait dynamics of the four groups of subjects represented by the time series are different. That is why we attempt the classification of NDDs based on gait dynamics. Hence the left and right swing intervals, the left and right stance intervals are selected as gait features for the classification of NDDs. 3.3. Training and learning mechanism based on selected features In this subsection, we present a scheme for modeling and identification of gait dynamics of healthy controls and NDDs subjects using the above mentioned gait features. Human locomotion is known to be a voluntary process. However, it is also regulated through a network of neurons called a central pattern generator (CPG), which is capable of producing a syncopated output [5]. The early nonlinear dynamical models of CPGs for gait assumed that a single nonlinear oscillator be used for each limb participating in the locomotion process [6,38]. More recent dynamical models, using the property of synchronization of nonlinear dynamical systems, allowed for neurons within an assembly to become enslaved to a single rhythmic muscular activity. Thus, rather than having a separate nonlinear oscillator for each limb, it is possible to have a single CPG to determine how human walk. More recently, West and Scafetta [37] developed a supercentral pattern generator (SCPG) model that reproduced both fractal and multifractal properties of gait dynamics. It has been tested its effectiveness on the dataset available on the web site (http://www.physionet.org/). The potential utility of the model is that it is capable of reproducing some aspects of walking dynamics. Stride time interval sequences of healthy controls and patients with NDDs are characterized by different fractal and multifractal exponents. These exponents can be used as a measure of the degree of maturation or degeneration of the neural network that regulates human movement. In the SCPG model, a forced Van der Pol oscillator is used to mimic human gait (including the healthy controls and NDDs subjects) which is defined by the following equation:
€x þ lðx2 P2 Þx_ þ ð2pf j Þ2 x ¼ A sinð2pf 0 tÞ
ð2Þ
where parameter P controls the amplitude of the oscillations, l controls the degree of nonlinearity of the oscillator, f j is the inner virtual frequency of the oscillator during the jth cycle which is related to the intensity of the jth neural fired impulse, A and f 0 are, respectively, the strength and frequency of the external driver. The frequency of the oscillator would be f ¼ f j if A ¼ 0. The complex gait system can be modeled by assuming that the amplitude of the impulses of the correlated firing neural centers regulates only the unperturbed inner frequency of the nonlinear forced Van der Pol oscillator that mimics the gait cycle. The stride interval is assumed to coincide with the actual period of the Van der Pol oscillator. In this way the gait frequency may differ slightly from the potential frequency induced by the neural firing activity. In fact, the chaotic behavior of nonlinear oscillators, such as the Van der Pol oscillator, allows a more complex behavior that may be controlled also by a constraint that forces the oscillator to follow a particular fixed frequency [37]. A Van der Pol oscillator is herein adopted because it is a prototypical nonlinear oscillator capable of producing stable oscillations (known as limit cycles) which describe the quasiperiodic gait dynamics very well. The observed stride interval is assumed to coincide with the actual period of each cycle of the Van der Pol oscillator, a period that depends on the unperturbed inner frequency of the oscillator, the amplitude, and the frequency of the forcing function. Since the frequency of stepping increases in proportion to the amplitude of the electric stimulation, we can assume that the time series of the intensity of the impulses fired by the neural centers is associated with a time series of virtual frequencies f j . This same mechanism may describe the fractal transition for patients with NDDs as well [30]. Let
x1 ¼ x x2 ¼ x_
ð3Þ
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0.7
right swing interval (s)
left swing interval (s)
0.7 0.6 0.5 0.4 0.3 0.2
0
50
100
0.6 0.5 0.4 0.3 0.2
150
0
50
Time (s) right stance interval (s)
left stance interval (s)
150
2
2
1.5
1
0.5
100
Time (s)
0
50
100
150
1.5
1
0.5
0
50
Time (s)
100
150
Time (s)
Fig. 3. A sample of time series derived from the signals in the left and right feet for a subject with PD.
1
right swing interval (s)
left swing interval (s)
1 0.8 0.6 0.4 0.2
0
50
100
0.8 0.6 0.4 0.2
150
0
50
Time (s)
100
150
Time (s) right stance interval (s)
left stance interval (s)
1.5 1
0.5
0
0
50
100
Time (s)
150
1
0.5
0
0
50
100
150
Time (s)
Fig. 4. A sample of time series derived from the signals in the left and right feet for a subject with HD.
Then, Eq. (2) can be converted into the following form:
(
x_ 1 ¼ x2 2 x_ 2 ¼ lðx21 P2 Þx2 ð2pf j Þ x1 þ A sinð2pf 0 tÞ
ð4Þ
It is seen from [14] that x can be expressed as the function of f i . Then, the gait system dynamics can be represented by the 2
function Fðx1 ; x2 Þ ¼ lðx21 P2 Þx2 ð2pf j Þ x1 þ A sinð2pf 0 tÞ ¼ Fðf i Þ. In order to more accurately describe the human walking, gait dynamics can be modeled as the following form:
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1
right swing interval (s)
left swing interval (s)
1 0.8 0.6 0.4 0.2
0
50
100
0.8 0.6 0.4 0.2
150
0
50
Time (s)
150
1.6
right stance interval (s)
left stance interval (s)
2
1.5
1
0.5
100
Time (s)
0
50
100
150
1.4 1.2 1 0.8
0
50
Time (s)
100
150
Time (s)
Fig. 5. A sample of time series derived from the signals in the left and right feet for a subject with ALS.
0.5
right swing interval (s)
left swing interval (s)
0.5 0.4 0.3 0.2 0.1
0
50
100
0.4 0.3 0.2 0.1
150
0
50
Time (s) right stance interval (s)
left stance interval (s)
0.9 0.8 0.7 0.6 0.5
0
50
100
Time (s)
100
150
Time (s)
150
0.9 0.8 0.7 0.6 0.5
0
50
100
150
Time (s)
Fig. 6. A sample of time series derived from the signals in the left and right feet for a healthy control (CO) subject.
x_ ¼ Fðx; pÞ þ v ðx; pÞ
ð5Þ
where x ¼ ½x1 ; . . . ; xn T 2 Rn are the states of system (5) which represent the time series features (including the left and right swing intervals, the left and right stance intervals) of human gait, p is a constant vector of system parameters. T
Fðx; pÞ ¼ ½f 1 ðx; pÞ; . . . ; f n ðx; pÞ is a smooth but unknown nonlinear vector representing the gait system dynamics, v ðx; pÞ is the modeling uncertainty. The system trajectory starting from initial condition x0 , is denoted as uf ðx0 Þ. Since the modeling uncertainty v ðx; pÞ and the gait system dynamics Fðx; pÞ cannot be decoupled from each other, we consider the two terms together as an undivided term, and define /ðx; pÞ :¼ Fðx; pÞ þ v ðx; pÞ as the general gait system dynamics.
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Table 1 Mean, SD and CV values of the swing, stance and stride intervals for 16 healthy control (CO) subjects, 13 patients with ALS, 15 patients with PD and 20 patients with HD. Gait rhythm (sec)
Healthy control (CO) mean ± SD (CV)
ALS mean ± SD (CV)
Left swing interval Right swing interval Left stance interval Right stance interval Left stride interval Right stride interval
0.3969 ± 0.0400 0.3907 ± 0.0415 0.7007 ± 0.0603 0.7063 ± 0.0589 1.0976 ± 0.0924 1.0970 ± 0.0923
0.4515 ± 0.0733 0.4491 ± 0.0827 1.0187 ± 0.3101 1.0198 ± 0.3020 1.4687 ± 0.3619 1.4704 ± 0.3701
(10.08) (10.62) (8.61) (8.34) (8.42) (8.41)
PD mean ± SD (CV) (16.23) (18.41) (30.44) (29.61) (24.64) (25.17)
0.3744 ± 0.0386 0.3690 ± 0.0555 0.7677 ± 0.0989 0.7681 ± 0.0962 1.1421 ± 0.1145 1.1371 ± 0.1099
HD mean ± SD (CV) (10.31) (15.04) (12.88) (12.52) (10.03) (9.66)
0.4009 ± 0.0670 0.3827 ± 0.0611 0.7596 ± 0.1194 0.7789 ± 0.1352 1.1605 ± 0.1679 1.1616 ± 0.1725
(16.71) (15.97) (15.72) (17.36) (14.47) (14.85)
The objective of the training or learning phase is to identify or approximate the general gait system dynamics /ðx; pÞ to a desired accuracy via deterministic learning. Based on deterministic learning theory, the following dynamical RBF networks are employed to identify the gait system dynamics /ðx; pÞ ¼ ½/1 ðx; pÞ; . . . ; /n ðx; pÞT :
^ T SðxÞ ^x_ ¼ Að^x xÞ þ W
ð6Þ
where ^ x ¼ ½^ x1 ; . . . ; ^ xn is the state vector of the dynamical RBF networks, A ¼ diag½a1 ; . . . ; an is a diagonal matrix, with ai > 0 ^ T SðxÞ ¼ ½W ^ T S1 ðxÞ; . . . ; W ^ T Sn ðxÞT are used to approximate the unknown being design constants, localized RBF networks W 1
n
/ðx; pÞ. The employment of RBF NN is due to its associated properties, including the function approximation ability, the spatially localized structure and a property concerning the PE condition [34,13,36]. The NN weight updating law is given by:
_ ^_ i ¼ W ^i f W xi ri Ci W i ¼ Ci SðxÞ~
ð7Þ
fi ¼ W ^ i W ; W is the ideal constant weight vector such that / ðx; pÞ ¼ where ~ xi ¼ ^ xi xi ; W i i i
T W i SðxÞ
þ i ðxÞ; i ðxÞ < is the
T i
NN approximation error, Ci ¼ C > 0, and ri > 0 is a small value. With Eqs. (5) and (6), the derivative of the state estimation error ~ xi satisfies
^ T SðxÞ / ðx; pÞ ¼ ai ~xi þ W f T SðxÞ i ~x_ i ¼ ai ~xi þ W i i i
ð8Þ
By using the local approximation property of RBF networks, the overall system consisting of dynamical model (8) and the NN weight updating law (7) can be summarized into the following form in the region Xf
"
x~_ i _ f fi W
#
" ¼
ai
Sfi ðxÞT
Cfi Sfi ðxÞ
0
#"
~xi f fi W
#
" þ
fi
# ð9Þ
^ fi ri Cfi W
and
_ f ^_ ¼ W ^ W xi ri Cfi W fi ¼ Cfi Sfi ðxÞ~ fi fi
ð10Þ
f T S ðxÞ. The subscripts ðÞ and ðÞ are used to stand for terms related to the regions close to and far away where fi ¼ i W f f fi f from the trajectory uf ðx0 Þ. The region close to the trajectory is defined as Xf :¼ ZjdistðZ; uf Þ 6 di , where Z ¼ x; di > 0 is a constant satisfying sðdi Þ > i; sðÞ is the RBF used in the network, i is a small positive constant. The related subvectors are given as: Sf ðxÞ ¼ ½sj1 ðxÞ; . . . ; sjf ðxÞT 2 RNf , with the neurons centered in the local region Xf , and W f ¼ ½wj1 ; . . . ; wjf T 2 RNf is the corref T S ðxÞj is small, so fi ¼ Oði Þ. sponding weight subvector, with N f < N. For localized RBF networks, j W fi f
^ fi . For the human gait sysThe nominal part of system (9) is referred to as system (9) without the terms fi and ri Cfi W tem, Murray et al. [25] suggested that human gait is a form of periodic or quasi-periodic motion. In SubSection 3.2, we have shown that the selected gait features are quasi-periodic signals generated from the time series. Hence, the NN input x ¼[left swing interval, right swing interval, left stance interval, right stance interval]T is quasi-periodic. According to Theorem 1 in [33], the regression subvector Sfi ðxÞ satisfies PE condition almost always. This will lead to expof fi Þ ¼ 0 of the nominal part of system (9) [12]. Based on the analysis results given in [33], the NN xi ; W nential stability of ð~ f fi converges to small neighborhoods of zero, with the sizes of the neighborhoods being determined weight estimate error W ^ T SðxÞ can approximate the by fi and kri Cfi W k, both of which are small values. This means that the entire RBF network W fi
i
unknown /i ðx; pÞ along the trajectory uf , and
^ T SðxÞ þ i1 /i ðx; pÞ ¼ W i where
i1 ¼ Oðfi Þ.
ð11Þ
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By the convergence result, we can obtain a constant vector of neural weights according to
^ i ðtÞ W i ¼ meant2½ta ;tb W
ð12Þ
where tb > ta > 0 represent a time segment after the transient process. Therefore, we conclude that accurate identification of the function /i ðx; pÞ is obtained along the trajectory uf ðx0 Þ by using W Ti Si ðxÞ, i.e.,
/i ðx; pÞ ¼ W Ti SðxÞ þ i2 where
ð13Þ
i2 ¼ Oði1 Þ and subsequently i2 ¼ Oð Þ.
3.4. Classification mechanism In this subsection, we present a scheme for the classification of NDDs using the learned gait dynamics. Consider a training dataset containing dynamical gait patterns ukf of healthy controls and NDDs subjects, k ¼ 1; . . . ; M, with the kth gait training pattern ukf generated from
x_ ¼ F k ðx; pk Þ þ v k ðx; pk Þ;
xðt0 Þ ¼ xf0
where F k ðx; pk Þ denotes the gait system dynamics, vector.
ð14Þ
v k ðx; pk Þ denotes the modeling uncertainty, pk
is the system parameter
As shown in SubSection 3.3, the general gait system dynamics /k ðx; pk Þ :¼ F k ðx; pk Þ þ v k ðx; pk Þ can be accurately identified T
and stored in constant RBF networks W k SðxÞ. By utilizing the learned knowledge obtained in the training phase, a bank of M estimators is first constructed for the trained gait systems as follows: T
v_ k ¼ Bðv k xÞ þ W k SðxÞ
ð15Þ kn T
k ¼ ½v k1 ; . . . ; v is the state of the estimator, B ¼ diag½b1 ; . . . ; bn is a where k ¼ 1; . . . ; M is used to stand for the kth estimator, v diagonal matrix which is kept the same for all estimators, x is the state of an input test pattern generated from Eq. (5). In the classification phase, by comparing the test gait pattern (standing for a certain NDDs subject) generated from human gait system (5) with the set of M estimators (15), we obtain the following test error systems: T
v~_ ki ¼ bi v~ ki þ W ki Si ðxÞ /i ðx; pÞ; i ¼ 1; . . . ; n; k ¼ 1; . . . ; M where
ð16Þ
v ¼ v xi is the state estimation (or synchronization) error. We compute the average L1 norm of the error v ~ ki
ki
~ ki ðtÞk1 ¼ kv
~ ki ðtÞ
1 Tc
Z
t
tTc
~ ki ðsÞjds; jv
t P Tc
ð17Þ
where Tc is the cycle of human gait. The fundamental idea of the NDDs classification is that if a test gait pattern generated from a certain NDDs subject to be T
classified is similar to the trained gait pattern sðs 2 f1; . . . ; kgÞ, the constant RBF network W si Si ðxÞ embedded in the matched estimator s will quickly recall the learned knowledge by providing accurate approximation to gait dynamics. Thus, the cor~ si ðtÞk1 will become the smallest among all the errors kv ~ ki ðtÞk1 . Based on the smallest error principle, the responding error kv appearing NDDs pattern can be classified. We have the following classification scheme. Neurodegenerative diseases classification scheme: If there exists some finite time t s ; s 2 f1; . . . ; kg and some ~ si ðtÞk1 < kv ~ ki ðtÞk1 for all t > t s , then the appearing NDDs subject can be classified. i 2 f1; . . . ; ng such that kv 4. Experimental results Experiments are implemented using matlab software and tested on an Intel Corei5 3.5 GHz computer with 4 GB RAM. The computation time is related to the number of the neurons and the size of the feature data employed. We assign feature vector sequences for all the 64 healthy controls and NDDs subjects. The time series for each subject has been divided into two subseries, one for training and the other for test. That is, each subject contains one sequence for training and one sequence for test, respectively. Based on the method described in SubSection 3.2, we extract all the 64 subjects’ gait features through time series, which means the input of the RBF networks x ¼[left swing interval, right swing interval, left stance interval, right stance interval]T. In order to eliminate the data difference between different gait features, all the gait feature data are normalized to ½1; 1. Here we evaluate our proposed method for the classification of NDDs using gait dynamics. Several experiments are conducted to test the ability of the proposed method. First we evaluate each group of NDDs subjects against the healthy control subjects. Then we evaluate all groups of NDDs against the healthy control subjects. For the evaluation, three measurements are used including the Sensitivity, the Specificity and the Accuracy. These measurements are defined as follows [1,42]:
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Sensitivity ¼
TP TP þ FN
ð18Þ
Specificity ¼
TN TN þ FP
ð19Þ
Accuracy ¼
TP þ TN TP þ TN þ FN þ FP
ð20Þ
where TP is the number of true positives, FN is the number of false negatives, TN is the number of true negatives and FP is the number of false positives. The classification results of the NDDs in the all-training-all-test and leave-one-out training–testing styles are presented in the following tables. 4.1. Classification of ALS One group of NDDs considered in our study is the subjects with ALS. The data are available from 13 subjects with ALS disease (numbered ‘als1’, ‘als2’, . . ., ‘als13’) and from 16 healthy control subjects (numbered ‘control1’, ‘control2’, . . ., ‘control16’). The data are divided into the training and test subsets using the strategy explained above, the proposed approach is applied in the selection of the data and the Sensitivity, the Specificity and the Accuracy are measured for the test sets. Figs. 7 and 8 show an example of the training and classification of the patient with ALS numbered ‘als1’ in the 29-person ^ T Si ðxÞ is constructed in a regular lattice, with nodes N ¼ 83521, the centers dataset. In the training phase, the RBF network W i
li evenly spaced on ½1; 1 ½1; 1 ½1; 1 ½1; 1, and the widths g ¼ 0:15. The weights of the RBF networks are updated ^ i ð0Þ ¼ 0. The design parameters for (6) and (7) are ai ¼ 0:5; according to Eq. (7). The initial weights W
C ¼ diagf1:5; 1:5; 1:5; 1:5g; ri ¼ 10; ði ¼ 1; . . . ; 4Þ. The convergence of neural weights is shown in Fig. 7, which demonstrates partial parameter convergence, that is, only the weight estimates of some neurons whose centers close to the orbit are activated and updated. These weights converge to their optimal values W i . Based on the deterministic learning theory, gait T
dynamics /ki ðx; pk Þ can be locally accurately approximated by W ki Si ðxÞ; ðk ¼ 1; . . . ; 29Þ along the quasi-periodic system trajectories, then these constant weights are stored for each training pattern. T
In the classification phase, by using the constant networks W ki Si ðxÞ; ðk ¼ 1; . . . ; 29Þ, 29 RBF network estimators are constructed based on (15). The parameters in (15) and (17) are bi ¼ 500ði ¼ 1; . . . ; 4Þ; T c ¼ 1:1s. Consider classification of the test patient ‘als1’ by 29 training patterns. The average L1 norms of the synchronization errors, that is, k~ xki ðtÞk1 ðk ¼ 1; . . . ; 29Þ are shown in Fig. 8. It is obvious that after certain time, the average L1 norm generated by the training pattern ‘als1’ becomes smaller than the others. Then, the test pattern can be effectively classified. The classification results have been shown in Table 2. 4.2. Classification of PD We also consider the data collected from the subjects with Parkinson’s disorder. In this case we have the data from 15 subjects with PD and 16 healthy controls. The same approach for the evaluation is applied here, selecting half of the data for training and the rest for the test. The Specificity, the Sensitivity and the Accuracy of the classification are also measured in this case. The results have been shown in Table 3. 1.5 1 0.5
W1
0 −0.5 −1 −1.5 −2 20
30
40
50
60
70
Time (s) ^ 1 in one training pattern. Fig. 7. Partial parameter convergence of W
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Fig. 8. Classification result of patient als1 using smallest error principle.
Table 2 Performance of the proposed classification approach evaluated by the all-training-all-testing and leave-one-out cross-validation methods. Subjects were categorized into the healthy control (CO) and ALS groups. Evaluation methods
All-training-all-testing Leave-one-out
Predicted groups
ALS CO ALS CO
Actual groups ALS
CO
11 0 12 2
2 16 1 14
Sensitivity (%)
Specificity (%)
Accuracy (%)
84.62
100
93.1
92.31
87.5
89.66
4.3. Classification of HD As we mentioned before, the data are collected from 20 subjects with HD. In this section, we use gait dynamics extracted from these subjects and we evaluate them against the data from the healthy control subjects (16 cases) in our database. The same approach for the evaluation is also applied here, selecting half of the data for training and the rest for the test. The Specificity, the Sensitivity and the Accuracy of the classification are also measured in this case. The results have been shown in Table 4. 4.4. Classification of NDDs In the last experiment we put together the data from all three groups of NDDs including HD, PD and ALS. We evaluate the Accuracy, the Sensitivity and the Specificity of the classification using the data from the healthy control subjects and all the NDDs. The results have been summarized in Table 5. We obtain the accuracy of 93.75% with the proposed method. Table 3 Performance of the proposed classification approach evaluated by the all-training-all-testing and leave-one-out cross-validation methods. Subjects were categorized into the healthy control (CO) and PD groups. Evaluation methods
Predicted groups
All-training-all-testing
PD CO PD CO
Leave-one-out
Actual groups PD
CO
15 0 13 2
0 16 2 14
Sensitivity (%)
Specificity (%)
Accuracy (%)
100
100
100
86.67
87.5
87.1
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Table 4 Performance of the proposed classification approach evaluated by the all-training-all-testing and leave-one-out cross-validation methods. Subjects were categorized into the healthy control (CO) and HD groups. Evaluation methods
All-training-all-testing Leave-one-out
Predicted groups
HD CO HD CO
Actual groups HD
CO
20 0 17 3
0 16 3 13
Sensitivity (%)
Specificity (%)
Accuracy (%)
100
100
100
85
81.25
83.33
Table 5 Performance of the proposed classification approach evaluated by the all-training-all-testing method. Subjects were categorized into the healthy control (CO) and three NDDs altogether. Evaluation methods
All-training-all-testing
Predicted groups
NDDs CO
Actual groups NDDs
CO
44 0
4 16
Sensitivity (%)
Specificity (%)
Accuracy (%)
91.67
100
93.75
5. Discussion and conclusion In the present study, by analyzing the SCPG model in which the unperturbed inner frequency of the nonlinear forced Van der Pol oscillator can mimic the gait cycle, we prove that the functions containing the walking time series states of human gait represent gait dynamics. The swing and stance intervals of the left and right feet are extracted as gait features for the classification. They indicate that gait dynamics between NDDs and the healthy controls are significantly different, and the gait symmetry of NDDs is significantly perturbed. Then, the RBF networks can be used to approximate gait dynamics, and the difference of gait dynamics between the NDDs and healthy control subjects will be used for the classification of NDDs with effective classification performance on the 64 subjects studied (PD, HD, ALS and CO). Our study contributes the accuracy improvements to the classification of NDDs. By using the same all-training-all-testing evaluation method, the correct classification rate obtained by the Elman’s recurrent neural network method is reported to be 91.7% for ALS patients in the study of [10], which is a bit lower than the 93:1% accuracy in the present study. In [41], the reported classification accuracy for ALS patients is 92:3%, which is also a bit lower than the accuracy of the present study. By using the same all-training-all-testing evaluation method, we also obtain the accuracy of 100%, 100% and 93.75% for the classification of PD, HD and three neuro-degenerative diseases altogether, respectively. The results are also a bit higher than the reported accuracy of 89.33%, 90.23% and 90.63%, respectively, in the study of [7]. Compared with the classification results in [41,7,10], our classification approach achieves superior performance. The results indicate that the proposed system can be effective for the classification of NDDs using gait dynamics. In the current database, the three groups of patients with NDDs are at different stages. They are measured with disease severity or duration. For the subjects with Parkinson’s disease, this is the Hohn and Yahr score. For the subjects with Huntington’s disease, this is the total functional capacity measure. For the subjects with amyotrophic lateral sclerosis, the number here is the time in months since the diagnosis of the disease. In the experiments, we have shown that patients with NDDs, including those at an early stage, can be correctly classified. This verified the efficiency of the proposed method in early detection. Nevertheless, further clinical research and statistical analysis using bigger sample sizes are needed to confirm the efficiency and applicability of the method in early detection. Our future research work may be carried out in the following aspects. (1) The size of the current database is small, which may limit the test of the generalization ability of the proposed approach. In future, we will collect data of larger population of neurodegenerative diseased patients for a better classification performance evaluation, and will study the effect of age, gender and medication. (2) Other gait features, such as joint angles/angular velocities and silhouette features, can also reflect the difference of gait dynamics between patients with NDDs and healthy controls. They will be useful for evaluation and classification of the two groups. In future, different kinds of gait features may be fused to improve the accuracy of the classification. (3) It would also be interesting to study in the further work progression of NDDs and the effects of intervention therapies. We may recruit a larger number of neurodegenerative patients at different disease stages and the matched healthy subjects, and also demand to compare gait dynamics in NDDs before and after the intervention therapies.
Conflict of interest There is no conflict of interest.
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Acknowledgments The authors sincerely appreciate the Editor, Associate Editor and anonymous reviewers for their constructive feedback and insight comment which greatly improve the presentation of this paper. This work was supported by the National Natural Science Foundation of China (Grant No. 61304084), by the National Science Fund for Distinguished Young Scholars (Grant No. 61225014), by the Educational and Scientific Research Project for Middle-aged and Young Teachers of Fujian Province of China (Grant No. JA14298), and by the Science and Technology Project of Longyan University (Grants Nos. LG2014005, LC2014005). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
Classification of neurodegenerative diseases using gait dynamics via deterministic learning Wei Zeng a,b,⇑, Cong Wang c a
School of Mechanical & Electrical Engineering, Longyan University, Longyan 364012, China School of Mechanical & Automotive Engineering, South China University of Technology, Guangzhou 510640, China c School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China b
a r t i c l e
i n f o
Article history: Received 15 December 2014 Received in revised form 21 April 2015 Accepted 24 April 2015 Available online 2 May 2015 Keywords: Gait analysis Deterministic learning Neurodegenerative diseases Gait dynamics Movement disorders
a b s t r a c t Neurodegenerative diseases (NDDs), such as Parkinson’s disease (PD), Huntington’s disease (HD) and amyotrophic lateral sclerosis (ALS), create serious gait abnormalities. They lead to altered gait rhythm and gait dynamics which can be reflected by a time series of stride-to-stride measures of footfall contact times. The temporal fluctuations in gait dynamics provide us with a non-invasive technique to evaluate the effects of neurological impairments on gait and its variations with diseases. In this paper, we present a new method using gait dynamics to classify (diagnose) NDDs via deterministic learning theory. The classification approach consists of two phases: a training phase and a classification phase. In the training phase, gait features representing gait dynamics are derived from the time series of swing intervals and stance intervals of the left and right feet. Gait dynamics underlying gait patterns of healthy controls and NDDs subjects are locally accurately approximated by radial basis function (RBF) neural networks. The obtained knowledge of approximated gait dynamics is stored in constant RBF networks. Gait patterns of healthy controls and NDDs subjects constitute a training set. In the classification phase, a bank of dynamical estimators is constructed for all the training gait patterns. Prior knowledge of gait dynamics represented by the constant RBF networks is embedded in the estimators. By comparing the set of estimators with a test NDDs gait pattern to be classified, a set of test errors are generated. The average L1 norms of the errors are taken as the classification measure between the dynamics of the training gait patterns and the dynamics of the test NDDs gait pattern according to the smallest error principle. Finally, experiments are carried out to demonstrate that the proposed method can effectively separate the gait patterns between the groups of healthy controls and neurodegenerative patients. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Neurodegenerative diseases (NDDs), including Parkinson’s disease (PD), Huntington’s disease (HD), and amyotrophic lateral sclerosis (ALS), produce changes in altered neuromuscular control. Since flexion and extension motions of two lower limbs are regulated by the central nervous system, the gait of a patient with a neurodegenerative disorder would become abnormal due to deterioration of motor neurons. PD and HD are typical disorders of the basal ganglia and are associated with
⇑ Corresponding author at: School of Mechanical & Electrical Engineering, Longyan University, Longyan 364012, China. Tel./fax: +86 597 2799753. E-mail address: [email protected] (W. Zeng). http://dx.doi.org/10.1016/j.ins.2015.04.047 0020-0255/Ó 2015 Elsevier Inc. All rights reserved.
W. Zeng, C. Wang / Information Sciences 317 (2015) 246–258
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characteristic changes in gait rhythm [4]. ALS is a disorder primarily affecting the motoneurons of the cerebral cortex, brain stem, and spinal cord [2]. Because the neurons of the basal ganglia likely play an important role in regulating muscular motor control such as balance and sequencing of movements, it is reasonable to expect that the stride-to-stride dynamics, as well as the gait cycle duration, are affected by these NDDs. Gait information has been widely used for the movement studies in healthy controls and also in subjects with different types of diseases. Analysis of temporal gait parameters is very useful for a better understanding of the mechanisms of movement disorders, and also has the high potential in presenting automatic non-invasive method based on gait dynamics for the classification of NDDs [2]. Gait dynamics are regulated by a complex nervous system. It is believed that the regulatory feedback loops of a physiologic system need to operate across multiple spatial and temporal scales to be able to adapt to an ever changing environment [30]. Thus, the time series of stride, stance or swing intervals are likely to exhibit fluctuations across multiple spatial and temporal scales [15]. In recent related studies, computer-aided tools have been utilized to measure the gait interval parameters in healthy adults, and also to describe the distinct characteristics of the gait in NDDs [8,9,16,19,22,20,24,27–29,32,3 7,40,43,31]. Aziz and Arif [2] converted the stride time series into a kind of symbol sequence, then applied a threshold dependent symbolic entropy method in the analysis of gait complexity. They observed that the normalized corrected Shannon entropy of the symbolic stride sequences is much lower in ALS at different short thresholds. Scafetta et al. [29] used the supercentral pattern generator (SCPG) model to simulate human gait dynamics, and also discussed the stochastic and fractal properties of the gait in PD, HD and ALS. Wu and Shi [41] proposed a statistical analysis method for the classification of gait cadence in subjects with ALS and healthy controls. In their approach, the probability density functions of gait cadence were estimated using Parzen-window method and then the Kullback–Leibler divergence was derived. With this method it was able to classify the stride patterns of the ALS and the control subjects with the accuracy rate of 82.8%. In their another similar study [39], it was hypothesized that the swing interval turns count (SWITC) of the ALS patients might be different from that of healthy subjects. The ALS gait patterns, characterized by the SWITC parameter and the known ASI feature, could be distinguished from the gait patterns of healthy subjects through the linear and nonlinear classifiers. Using the SWITC parameter and the average stride interval could reach to the classification rate of 89.66% for the gaits in subjects with ALS and gaits in healthy control subjects. Daliri [7] presented an approach for the diagnosis of NDDs based on gait dynamics. The proposed method used information from a time series of stride intervals, swing intervals, stance intervals and double support intervals of stride-to-stride measures. The support vector machines using different kernels were examined for the diagnosis. Based on the findings in [3], a reasonable approach to investigate the effects of neurological impairments on patients’ ability to mediate the locomotion of two lower limbs was proposed. It compared the left and right stance-interval series in terms of their regularities at multi-resolution levels. Liao et al. [23] investigated gait asymmetry in NDDs using the multi-resolution entropy analysis of stance time fluctuations. Their results showed that gait symmetry is significantly disturbed in subjects with PD, HD, and ALS, and the degree of disturbance is more prominent in the subjects with ALS. A number of scientists showed that NDDs patients present altered fractal dynamics of gait characterized by reduced stride-interval correlations, that is, the walking of these individuals becomes more random [30,18,21]. The randomness increases with the severity of the neurodegenerative impairment. Our primary interest in this paper is to investigate modeling of gait dynamics using the time series of the left and right feet. Specifically, we would like to know whether the three types of neurological disease (PD, HD and ALS) would impair the patient’s ability to regulate the locomotion of two feet and whether the difference of gait dynamics is related to the particular type of disease that the patient has. Nevertheless, research on the temporal variability of gait dynamics, such as step to step, is limited. It has been postulated that the ability to maintain a steady gait (i.e. low stride-to-stride variability of gait cycle timing and its sub-phases) would be diminished in NDDs [17]. Irregular timing of steps in NDDs suggests a disturbance of rhythmic locomotor activity generation and gait dynamics [11]. The classification (diagnosis) of NDDs based on the difference of gait dynamics between NDDs and healthy controls is what we want to attempt. In this paper, we present a new method using gait dynamics for the classification of NDDs via deterministic learning theory. The time series of swing and stance intervals of the left and right feet are used to model gait dynamics of NDDs and healthy control subjects according to the SCPG model. Gait dynamics underlying gait patterns of healthy controls and NDDs subjects are locally accurately approximated by radial basis function (RBF) neural networks. The obtained knowledge of approximated gait dynamics is stored in constant RBF networks. The gait patterns of healthy controls and NDDs subjects constitute a training set. In the classification phase, a bank of dynamical estimators is constructed for all the training gait patterns. Prior knowledge of gait dynamics represented by the constant RBF networks is embedded in the estimators. By comparing the set of estimators with a test NDDs gait pattern to be classified, a set of test errors are generated. The average L1 norms of the errors are taken as the classification measure between the dynamics of the training gait patterns and the dynamics of the test NDDs gait pattern to be classified according to the smallest error principle. The proposed method can effectively separate the gait patterns between the groups of healthy controls and neurodegenerative patients. Compared with other recently reported results in [41,7,10], our method achieves superior classification performance. The rest of the paper is organized as follows. Section 2 introduces preliminary knowledge about deterministic learning theory and problem formulation. Section 3 describes the proposed method. This includes the data description, feature extraction and selection, learning and classification mechanisms. Section 4 presents experimental results. Section 5 gives some discussions and conclusions.
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2. Preliminaries and problem formulation 2.1. Deterministic learning theory In deterministic learning theory, identification of system dynamics of general nonlinear systems is achieved according to the following elements: (i) employment of localized RBF networks; (ii) satisfaction of a partial persistence of excitation (PE) condition; (iii) exponential stability of the adaptive system along the periodic or recurrent orbit; (iv) locally accurate neural network (NN) approximation of the unknown system dynamics [34,35]. P The RBF networks can be described by f nn ðZÞ ¼ Ni¼1 wi si ðZÞ ¼ W T SðZÞ, where Z 2 XZ Rp is the input vector, W ¼ ½w1 ; . . . ; wN T 2 RN is the weight vector, N is the NN node number, and SðZÞ ¼ ½s1 ðkZ l1 kÞ; . . . ; sN ðkZ lN kÞT , with si ðÞ being a radial basis function, and li ði ¼ 1; . . . ; NÞ being distinct points in state space. The Gaussian function T
si ðkZ li kÞ ¼ exp½ðZligÞ2 ðZli Þ is one of the most commonly used radial basis functions, where i
li ¼ ½li1 ; li2 ; . . . ; liN T is the
center of the receptive field and gi is the width of the receptive field. The Gaussian function belongs to the class of localized radial basis functions in the sense that si ðkZ li kÞ ! 0 as kZk ! 1. It has been shown in [26] that for any continuous function f ðZÞ : XZ ! R where XZ Rp is a compact set, and for the NN approximator, where the node number N is sufficiently large, there exists an ideal constant weight vector W , such that for each > 0; f ðZÞ ¼ W T SðZÞ þ ðZÞ; 8Z 2 XZ , where jðZÞ < j is the approximation error. Moreover, for any bounded trajectory Z f ðtÞ within the compact set XZ ; f ðZÞ can be approximated by using a limited number of neurons located in a local region T Nf along the trajectory: f ðZÞ ¼ W T f Sf ðZÞ þ f , where Sf ðZÞ ¼ ½sj1 ðZÞ; . . . ; sjf ðZÞ 2 R , with N f < N; jsji j > iðji ¼ j1 ; . . . ; jf Þ; i > 0 is a
small positive constant, W f ¼ ½wj1 ; . . . ; wjf T , and
f
is the approximation error, with jjf j jjj being small.
Based on previous results on the PE property of RBF networks [13], it is shown in [36] that for a localized RBF network W T SðZÞ whose centers are placed on a regular lattice, almost any recurrent trajectory ZðtÞ can lead to the satisfaction of the PE condition of the regressor subvector Sf ðZÞ. 2.2. Problem formulation Consider a general nonlinear human gait dynamical system in the following form:
x_ ¼ Fðx; pÞ;
xðt0 Þ ¼ x0
ð1Þ
where x ¼ ½x1 ; . . . ; xn T 2 Rn is the state of the system representing the gait features, p is a constant vector of system parameters (different p will in general generate different gait patterns of healthy controls and NDDs subjects). T
Fðx; pÞ ¼ ½f 1 ðx; pÞ; . . . ; f n ðx; pÞ is a smooth but unknown nonlinear vector field. Assumption 1. The gait system state x remains uniformly bounded. Moreover, the system trajectory starting from x0 , denoted as uf ðx0 Þ, is in either a periodic or periodic-like (recurrent) motion. For the human gait system, Murray et al. [25] suggested that human gait is a form of periodic or quasi-periodic motion. Thus, Assumption 1 is reasonable. Our objective is to choose suitable time series of gait features satisfying Assumption 1 and design a dynamic RBF network to identify and approximate the unknown vector Fðx; pÞ for healthy controls and NDDs subjects. The approximation result can be used to represent gait dynamics which will be stored and used for the classification of NDDs. 3. Method In this section, we propose a method for the classification of NDDs using the information obtained from gait dynamics. Firstly, several gait features are extracted from the time series of the left and right feet as the feature vectors, which represent the variations of gait dynamics. Then, the RBF networks are used to approximate gait dynamics through the supercentral pattern generator (SCPG) model. The difference of gait dynamics will be applied to distinguish the healthy controls from the group of subjects with NDDs. The outline of the proposed method has been shown in Fig. 1. 3.1. Data description Data sets for analysis and classification are taken from gait time series in neurodegenerative database contributed by [18], which is available online in the PhysioNet (http://www.physionet.org/physiobank/database/gaitndd). The database contains 64 recordings of gait from 16 healthy controls, 13 subjects with ALS, 20 subjects with HD and 15 subjects with PD. Healthy control group consists of subjects with age range 20–74, ALS group consists of subjects with age range 36–70, HD group consists of subjects with age range 29–71 and PD group consists of subjects with age range 44–80. Subjects were instructed to walk at their normal pace along a 77 m long hallway for 5 min. Force sensitive switches were placed in the subjects’ shoes, output of these switches provided a force applied on the floor [19]. A 12-bit on-board analog-to-digital converter sampled the output
W. Zeng, C. Wang / Information Sciences 317 (2015) 246–258
249
Fig. 1. Overview of the proposed method for the (classification) diagnosis of NDDs using gait dynamics.
of foot switches at 300 Hz. From the recorded force applied to the ground during walking, the time series of the left (right) stride time, left (right) stance time and left (right) swing time were derived. To minimize startup effects the first 20 s recorded data were excluded and median filter was used to remove data points (outliers) that were far away from the median value [17]. These outliers were mainly due to the turns at the end of the hallway.
3.2. Feature extraction and selection For each time series, we extract four different features to create our feature vectors for the classification. These features including the left and right swing intervals, the left and right stance intervals are employed to create a feature vector of 4 dimensions for each subject in the database.
Fig. 2. An example of sequence of stride times from different groups of subjects including a subject with PD, a subject with HD, a subject with ALS disease and a healthy control (CO) subject.
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The raw data are obtained using force-sensitive resistors, with the output roughly proportional to the force under the foot. Stride-to-stride measures of footfall contact times are derived from these signals and have been shown in Fig. 2. Some time series data are derived from the signals in the left and right feet. These include left and right swing intervals, left and right stance intervals. Samples of these time series for subjects with PD, HD, ALS and a healthy control subject have been shown in Figs. 3–6. Here, in Table 1, we give the values of measures of the fluctuation magnitude of swing, stance and stride intervals in ALS, PD, HD and healthy control subjects. It is observed from the data sets that: (1) the average swing, stance and stride intervals of patients with ALS are significantly longer than those of healthy controls or of patients with PD or HD; (2) the values of two measures of the fluctuation magnitude of swing, stance and stride intervals in ALS, i.e., the standard deviation (SD) and coefficient of variation (CV = standard deviation/mean 100) of the detrended interval time series, are larger than those of healthy controls, PD and HD patients, respectively; (3) the fluctuation of the stride interval can be reflected by the swing and stance intervals since stride interval equals to swing interval plus stance interval; (4) the fluctuation of the time series between the left and right legs of healthy controls is smaller than other NDDs patients, which shows the good gait symmetry of healthy controls. In other words, gait symmetry is significantly disturbed in subjects with ALS, PD and HD. It is found that gait symmetry in subjects with PD and HD is disturbed less than in subjects with ALS. It is seen from Table 1 that the time series of four groups of subjects are obviously different, which means gait dynamics of the four groups of subjects represented by the time series are different. That is why we attempt the classification of NDDs based on gait dynamics. Hence the left and right swing intervals, the left and right stance intervals are selected as gait features for the classification of NDDs. 3.3. Training and learning mechanism based on selected features In this subsection, we present a scheme for modeling and identification of gait dynamics of healthy controls and NDDs subjects using the above mentioned gait features. Human locomotion is known to be a voluntary process. However, it is also regulated through a network of neurons called a central pattern generator (CPG), which is capable of producing a syncopated output [5]. The early nonlinear dynamical models of CPGs for gait assumed that a single nonlinear oscillator be used for each limb participating in the locomotion process [6,38]. More recent dynamical models, using the property of synchronization of nonlinear dynamical systems, allowed for neurons within an assembly to become enslaved to a single rhythmic muscular activity. Thus, rather than having a separate nonlinear oscillator for each limb, it is possible to have a single CPG to determine how human walk. More recently, West and Scafetta [37] developed a supercentral pattern generator (SCPG) model that reproduced both fractal and multifractal properties of gait dynamics. It has been tested its effectiveness on the dataset available on the web site (http://www.physionet.org/). The potential utility of the model is that it is capable of reproducing some aspects of walking dynamics. Stride time interval sequences of healthy controls and patients with NDDs are characterized by different fractal and multifractal exponents. These exponents can be used as a measure of the degree of maturation or degeneration of the neural network that regulates human movement. In the SCPG model, a forced Van der Pol oscillator is used to mimic human gait (including the healthy controls and NDDs subjects) which is defined by the following equation:
€x þ lðx2 P2 Þx_ þ ð2pf j Þ2 x ¼ A sinð2pf 0 tÞ
ð2Þ
where parameter P controls the amplitude of the oscillations, l controls the degree of nonlinearity of the oscillator, f j is the inner virtual frequency of the oscillator during the jth cycle which is related to the intensity of the jth neural fired impulse, A and f 0 are, respectively, the strength and frequency of the external driver. The frequency of the oscillator would be f ¼ f j if A ¼ 0. The complex gait system can be modeled by assuming that the amplitude of the impulses of the correlated firing neural centers regulates only the unperturbed inner frequency of the nonlinear forced Van der Pol oscillator that mimics the gait cycle. The stride interval is assumed to coincide with the actual period of the Van der Pol oscillator. In this way the gait frequency may differ slightly from the potential frequency induced by the neural firing activity. In fact, the chaotic behavior of nonlinear oscillators, such as the Van der Pol oscillator, allows a more complex behavior that may be controlled also by a constraint that forces the oscillator to follow a particular fixed frequency [37]. A Van der Pol oscillator is herein adopted because it is a prototypical nonlinear oscillator capable of producing stable oscillations (known as limit cycles) which describe the quasiperiodic gait dynamics very well. The observed stride interval is assumed to coincide with the actual period of each cycle of the Van der Pol oscillator, a period that depends on the unperturbed inner frequency of the oscillator, the amplitude, and the frequency of the forcing function. Since the frequency of stepping increases in proportion to the amplitude of the electric stimulation, we can assume that the time series of the intensity of the impulses fired by the neural centers is associated with a time series of virtual frequencies f j . This same mechanism may describe the fractal transition for patients with NDDs as well [30]. Let
x1 ¼ x x2 ¼ x_
ð3Þ
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0.7
right swing interval (s)
left swing interval (s)
0.7 0.6 0.5 0.4 0.3 0.2
0
50
100
0.6 0.5 0.4 0.3 0.2
150
0
50
Time (s) right stance interval (s)
left stance interval (s)
150
2
2
1.5
1
0.5
100
Time (s)
0
50
100
150
1.5
1
0.5
0
50
Time (s)
100
150
Time (s)
Fig. 3. A sample of time series derived from the signals in the left and right feet for a subject with PD.
1
right swing interval (s)
left swing interval (s)
1 0.8 0.6 0.4 0.2
0
50
100
0.8 0.6 0.4 0.2
150
0
50
Time (s)
100
150
Time (s) right stance interval (s)
left stance interval (s)
1.5 1
0.5
0
0
50
100
Time (s)
150
1
0.5
0
0
50
100
150
Time (s)
Fig. 4. A sample of time series derived from the signals in the left and right feet for a subject with HD.
Then, Eq. (2) can be converted into the following form:
(
x_ 1 ¼ x2 2 x_ 2 ¼ lðx21 P2 Þx2 ð2pf j Þ x1 þ A sinð2pf 0 tÞ
ð4Þ
It is seen from [14] that x can be expressed as the function of f i . Then, the gait system dynamics can be represented by the 2
function Fðx1 ; x2 Þ ¼ lðx21 P2 Þx2 ð2pf j Þ x1 þ A sinð2pf 0 tÞ ¼ Fðf i Þ. In order to more accurately describe the human walking, gait dynamics can be modeled as the following form:
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1
right swing interval (s)
left swing interval (s)
1 0.8 0.6 0.4 0.2
0
50
100
0.8 0.6 0.4 0.2
150
0
50
Time (s)
150
1.6
right stance interval (s)
left stance interval (s)
2
1.5
1
0.5
100
Time (s)
0
50
100
150
1.4 1.2 1 0.8
0
50
Time (s)
100
150
Time (s)
Fig. 5. A sample of time series derived from the signals in the left and right feet for a subject with ALS.
0.5
right swing interval (s)
left swing interval (s)
0.5 0.4 0.3 0.2 0.1
0
50
100
0.4 0.3 0.2 0.1
150
0
50
Time (s) right stance interval (s)
left stance interval (s)
0.9 0.8 0.7 0.6 0.5
0
50
100
Time (s)
100
150
Time (s)
150
0.9 0.8 0.7 0.6 0.5
0
50
100
150
Time (s)
Fig. 6. A sample of time series derived from the signals in the left and right feet for a healthy control (CO) subject.
x_ ¼ Fðx; pÞ þ v ðx; pÞ
ð5Þ
where x ¼ ½x1 ; . . . ; xn T 2 Rn are the states of system (5) which represent the time series features (including the left and right swing intervals, the left and right stance intervals) of human gait, p is a constant vector of system parameters. T
Fðx; pÞ ¼ ½f 1 ðx; pÞ; . . . ; f n ðx; pÞ is a smooth but unknown nonlinear vector representing the gait system dynamics, v ðx; pÞ is the modeling uncertainty. The system trajectory starting from initial condition x0 , is denoted as uf ðx0 Þ. Since the modeling uncertainty v ðx; pÞ and the gait system dynamics Fðx; pÞ cannot be decoupled from each other, we consider the two terms together as an undivided term, and define /ðx; pÞ :¼ Fðx; pÞ þ v ðx; pÞ as the general gait system dynamics.
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Table 1 Mean, SD and CV values of the swing, stance and stride intervals for 16 healthy control (CO) subjects, 13 patients with ALS, 15 patients with PD and 20 patients with HD. Gait rhythm (sec)
Healthy control (CO) mean ± SD (CV)
ALS mean ± SD (CV)
Left swing interval Right swing interval Left stance interval Right stance interval Left stride interval Right stride interval
0.3969 ± 0.0400 0.3907 ± 0.0415 0.7007 ± 0.0603 0.7063 ± 0.0589 1.0976 ± 0.0924 1.0970 ± 0.0923
0.4515 ± 0.0733 0.4491 ± 0.0827 1.0187 ± 0.3101 1.0198 ± 0.3020 1.4687 ± 0.3619 1.4704 ± 0.3701
(10.08) (10.62) (8.61) (8.34) (8.42) (8.41)
PD mean ± SD (CV) (16.23) (18.41) (30.44) (29.61) (24.64) (25.17)
0.3744 ± 0.0386 0.3690 ± 0.0555 0.7677 ± 0.0989 0.7681 ± 0.0962 1.1421 ± 0.1145 1.1371 ± 0.1099
HD mean ± SD (CV) (10.31) (15.04) (12.88) (12.52) (10.03) (9.66)
0.4009 ± 0.0670 0.3827 ± 0.0611 0.7596 ± 0.1194 0.7789 ± 0.1352 1.1605 ± 0.1679 1.1616 ± 0.1725
(16.71) (15.97) (15.72) (17.36) (14.47) (14.85)
The objective of the training or learning phase is to identify or approximate the general gait system dynamics /ðx; pÞ to a desired accuracy via deterministic learning. Based on deterministic learning theory, the following dynamical RBF networks are employed to identify the gait system dynamics /ðx; pÞ ¼ ½/1 ðx; pÞ; . . . ; /n ðx; pÞT :
^ T SðxÞ ^x_ ¼ Að^x xÞ þ W
ð6Þ
where ^ x ¼ ½^ x1 ; . . . ; ^ xn is the state vector of the dynamical RBF networks, A ¼ diag½a1 ; . . . ; an is a diagonal matrix, with ai > 0 ^ T SðxÞ ¼ ½W ^ T S1 ðxÞ; . . . ; W ^ T Sn ðxÞT are used to approximate the unknown being design constants, localized RBF networks W 1
n
/ðx; pÞ. The employment of RBF NN is due to its associated properties, including the function approximation ability, the spatially localized structure and a property concerning the PE condition [34,13,36]. The NN weight updating law is given by:
_ ^_ i ¼ W ^i f W xi ri Ci W i ¼ Ci SðxÞ~
ð7Þ
fi ¼ W ^ i W ; W is the ideal constant weight vector such that / ðx; pÞ ¼ where ~ xi ¼ ^ xi xi ; W i i i
T W i SðxÞ
þ i ðxÞ; i ðxÞ < is the
T i
NN approximation error, Ci ¼ C > 0, and ri > 0 is a small value. With Eqs. (5) and (6), the derivative of the state estimation error ~ xi satisfies
^ T SðxÞ / ðx; pÞ ¼ ai ~xi þ W f T SðxÞ i ~x_ i ¼ ai ~xi þ W i i i
ð8Þ
By using the local approximation property of RBF networks, the overall system consisting of dynamical model (8) and the NN weight updating law (7) can be summarized into the following form in the region Xf
"
x~_ i _ f fi W
#
" ¼
ai
Sfi ðxÞT
Cfi Sfi ðxÞ
0
#"
~xi f fi W
#
" þ
fi
# ð9Þ
^ fi ri Cfi W
and
_ f ^_ ¼ W ^ W xi ri Cfi W fi ¼ Cfi Sfi ðxÞ~ fi fi
ð10Þ
f T S ðxÞ. The subscripts ðÞ and ðÞ are used to stand for terms related to the regions close to and far away where fi ¼ i W f f fi f from the trajectory uf ðx0 Þ. The region close to the trajectory is defined as Xf :¼ ZjdistðZ; uf Þ 6 di , where Z ¼ x; di > 0 is a constant satisfying sðdi Þ > i; sðÞ is the RBF used in the network, i is a small positive constant. The related subvectors are given as: Sf ðxÞ ¼ ½sj1 ðxÞ; . . . ; sjf ðxÞT 2 RNf , with the neurons centered in the local region Xf , and W f ¼ ½wj1 ; . . . ; wjf T 2 RNf is the corref T S ðxÞj is small, so fi ¼ Oði Þ. sponding weight subvector, with N f < N. For localized RBF networks, j W fi f
^ fi . For the human gait sysThe nominal part of system (9) is referred to as system (9) without the terms fi and ri Cfi W tem, Murray et al. [25] suggested that human gait is a form of periodic or quasi-periodic motion. In SubSection 3.2, we have shown that the selected gait features are quasi-periodic signals generated from the time series. Hence, the NN input x ¼[left swing interval, right swing interval, left stance interval, right stance interval]T is quasi-periodic. According to Theorem 1 in [33], the regression subvector Sfi ðxÞ satisfies PE condition almost always. This will lead to expof fi Þ ¼ 0 of the nominal part of system (9) [12]. Based on the analysis results given in [33], the NN xi ; W nential stability of ð~ f fi converges to small neighborhoods of zero, with the sizes of the neighborhoods being determined weight estimate error W ^ T SðxÞ can approximate the by fi and kri Cfi W k, both of which are small values. This means that the entire RBF network W fi
i
unknown /i ðx; pÞ along the trajectory uf , and
^ T SðxÞ þ i1 /i ðx; pÞ ¼ W i where
i1 ¼ Oðfi Þ.
ð11Þ
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By the convergence result, we can obtain a constant vector of neural weights according to
^ i ðtÞ W i ¼ meant2½ta ;tb W
ð12Þ
where tb > ta > 0 represent a time segment after the transient process. Therefore, we conclude that accurate identification of the function /i ðx; pÞ is obtained along the trajectory uf ðx0 Þ by using W Ti Si ðxÞ, i.e.,
/i ðx; pÞ ¼ W Ti SðxÞ þ i2 where
ð13Þ
i2 ¼ Oði1 Þ and subsequently i2 ¼ Oð Þ.
3.4. Classification mechanism In this subsection, we present a scheme for the classification of NDDs using the learned gait dynamics. Consider a training dataset containing dynamical gait patterns ukf of healthy controls and NDDs subjects, k ¼ 1; . . . ; M, with the kth gait training pattern ukf generated from
x_ ¼ F k ðx; pk Þ þ v k ðx; pk Þ;
xðt0 Þ ¼ xf0
where F k ðx; pk Þ denotes the gait system dynamics, vector.
ð14Þ
v k ðx; pk Þ denotes the modeling uncertainty, pk
is the system parameter
As shown in SubSection 3.3, the general gait system dynamics /k ðx; pk Þ :¼ F k ðx; pk Þ þ v k ðx; pk Þ can be accurately identified T
and stored in constant RBF networks W k SðxÞ. By utilizing the learned knowledge obtained in the training phase, a bank of M estimators is first constructed for the trained gait systems as follows: T
v_ k ¼ Bðv k xÞ þ W k SðxÞ
ð15Þ kn T
k ¼ ½v k1 ; . . . ; v is the state of the estimator, B ¼ diag½b1 ; . . . ; bn is a where k ¼ 1; . . . ; M is used to stand for the kth estimator, v diagonal matrix which is kept the same for all estimators, x is the state of an input test pattern generated from Eq. (5). In the classification phase, by comparing the test gait pattern (standing for a certain NDDs subject) generated from human gait system (5) with the set of M estimators (15), we obtain the following test error systems: T
v~_ ki ¼ bi v~ ki þ W ki Si ðxÞ /i ðx; pÞ; i ¼ 1; . . . ; n; k ¼ 1; . . . ; M where
ð16Þ
v ¼ v xi is the state estimation (or synchronization) error. We compute the average L1 norm of the error v ~ ki
ki
~ ki ðtÞk1 ¼ kv
~ ki ðtÞ
1 Tc
Z
t
tTc
~ ki ðsÞjds; jv
t P Tc
ð17Þ
where Tc is the cycle of human gait. The fundamental idea of the NDDs classification is that if a test gait pattern generated from a certain NDDs subject to be T
classified is similar to the trained gait pattern sðs 2 f1; . . . ; kgÞ, the constant RBF network W si Si ðxÞ embedded in the matched estimator s will quickly recall the learned knowledge by providing accurate approximation to gait dynamics. Thus, the cor~ si ðtÞk1 will become the smallest among all the errors kv ~ ki ðtÞk1 . Based on the smallest error principle, the responding error kv appearing NDDs pattern can be classified. We have the following classification scheme. Neurodegenerative diseases classification scheme: If there exists some finite time t s ; s 2 f1; . . . ; kg and some ~ si ðtÞk1 < kv ~ ki ðtÞk1 for all t > t s , then the appearing NDDs subject can be classified. i 2 f1; . . . ; ng such that kv 4. Experimental results Experiments are implemented using matlab software and tested on an Intel Corei5 3.5 GHz computer with 4 GB RAM. The computation time is related to the number of the neurons and the size of the feature data employed. We assign feature vector sequences for all the 64 healthy controls and NDDs subjects. The time series for each subject has been divided into two subseries, one for training and the other for test. That is, each subject contains one sequence for training and one sequence for test, respectively. Based on the method described in SubSection 3.2, we extract all the 64 subjects’ gait features through time series, which means the input of the RBF networks x ¼[left swing interval, right swing interval, left stance interval, right stance interval]T. In order to eliminate the data difference between different gait features, all the gait feature data are normalized to ½1; 1. Here we evaluate our proposed method for the classification of NDDs using gait dynamics. Several experiments are conducted to test the ability of the proposed method. First we evaluate each group of NDDs subjects against the healthy control subjects. Then we evaluate all groups of NDDs against the healthy control subjects. For the evaluation, three measurements are used including the Sensitivity, the Specificity and the Accuracy. These measurements are defined as follows [1,42]:
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Sensitivity ¼
TP TP þ FN
ð18Þ
Specificity ¼
TN TN þ FP
ð19Þ
Accuracy ¼
TP þ TN TP þ TN þ FN þ FP
ð20Þ
where TP is the number of true positives, FN is the number of false negatives, TN is the number of true negatives and FP is the number of false positives. The classification results of the NDDs in the all-training-all-test and leave-one-out training–testing styles are presented in the following tables. 4.1. Classification of ALS One group of NDDs considered in our study is the subjects with ALS. The data are available from 13 subjects with ALS disease (numbered ‘als1’, ‘als2’, . . ., ‘als13’) and from 16 healthy control subjects (numbered ‘control1’, ‘control2’, . . ., ‘control16’). The data are divided into the training and test subsets using the strategy explained above, the proposed approach is applied in the selection of the data and the Sensitivity, the Specificity and the Accuracy are measured for the test sets. Figs. 7 and 8 show an example of the training and classification of the patient with ALS numbered ‘als1’ in the 29-person ^ T Si ðxÞ is constructed in a regular lattice, with nodes N ¼ 83521, the centers dataset. In the training phase, the RBF network W i
li evenly spaced on ½1; 1 ½1; 1 ½1; 1 ½1; 1, and the widths g ¼ 0:15. The weights of the RBF networks are updated ^ i ð0Þ ¼ 0. The design parameters for (6) and (7) are ai ¼ 0:5; according to Eq. (7). The initial weights W
C ¼ diagf1:5; 1:5; 1:5; 1:5g; ri ¼ 10; ði ¼ 1; . . . ; 4Þ. The convergence of neural weights is shown in Fig. 7, which demonstrates partial parameter convergence, that is, only the weight estimates of some neurons whose centers close to the orbit are activated and updated. These weights converge to their optimal values W i . Based on the deterministic learning theory, gait T
dynamics /ki ðx; pk Þ can be locally accurately approximated by W ki Si ðxÞ; ðk ¼ 1; . . . ; 29Þ along the quasi-periodic system trajectories, then these constant weights are stored for each training pattern. T
In the classification phase, by using the constant networks W ki Si ðxÞ; ðk ¼ 1; . . . ; 29Þ, 29 RBF network estimators are constructed based on (15). The parameters in (15) and (17) are bi ¼ 500ði ¼ 1; . . . ; 4Þ; T c ¼ 1:1s. Consider classification of the test patient ‘als1’ by 29 training patterns. The average L1 norms of the synchronization errors, that is, k~ xki ðtÞk1 ðk ¼ 1; . . . ; 29Þ are shown in Fig. 8. It is obvious that after certain time, the average L1 norm generated by the training pattern ‘als1’ becomes smaller than the others. Then, the test pattern can be effectively classified. The classification results have been shown in Table 2. 4.2. Classification of PD We also consider the data collected from the subjects with Parkinson’s disorder. In this case we have the data from 15 subjects with PD and 16 healthy controls. The same approach for the evaluation is applied here, selecting half of the data for training and the rest for the test. The Specificity, the Sensitivity and the Accuracy of the classification are also measured in this case. The results have been shown in Table 3. 1.5 1 0.5
W1
0 −0.5 −1 −1.5 −2 20
30
40
50
60
70
Time (s) ^ 1 in one training pattern. Fig. 7. Partial parameter convergence of W
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Fig. 8. Classification result of patient als1 using smallest error principle.
Table 2 Performance of the proposed classification approach evaluated by the all-training-all-testing and leave-one-out cross-validation methods. Subjects were categorized into the healthy control (CO) and ALS groups. Evaluation methods
All-training-all-testing Leave-one-out
Predicted groups
ALS CO ALS CO
Actual groups ALS
CO
11 0 12 2
2 16 1 14
Sensitivity (%)
Specificity (%)
Accuracy (%)
84.62
100
93.1
92.31
87.5
89.66
4.3. Classification of HD As we mentioned before, the data are collected from 20 subjects with HD. In this section, we use gait dynamics extracted from these subjects and we evaluate them against the data from the healthy control subjects (16 cases) in our database. The same approach for the evaluation is also applied here, selecting half of the data for training and the rest for the test. The Specificity, the Sensitivity and the Accuracy of the classification are also measured in this case. The results have been shown in Table 4. 4.4. Classification of NDDs In the last experiment we put together the data from all three groups of NDDs including HD, PD and ALS. We evaluate the Accuracy, the Sensitivity and the Specificity of the classification using the data from the healthy control subjects and all the NDDs. The results have been summarized in Table 5. We obtain the accuracy of 93.75% with the proposed method. Table 3 Performance of the proposed classification approach evaluated by the all-training-all-testing and leave-one-out cross-validation methods. Subjects were categorized into the healthy control (CO) and PD groups. Evaluation methods
Predicted groups
All-training-all-testing
PD CO PD CO
Leave-one-out
Actual groups PD
CO
15 0 13 2
0 16 2 14
Sensitivity (%)
Specificity (%)
Accuracy (%)
100
100
100
86.67
87.5
87.1
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Table 4 Performance of the proposed classification approach evaluated by the all-training-all-testing and leave-one-out cross-validation methods. Subjects were categorized into the healthy control (CO) and HD groups. Evaluation methods
All-training-all-testing Leave-one-out
Predicted groups
HD CO HD CO
Actual groups HD
CO
20 0 17 3
0 16 3 13
Sensitivity (%)
Specificity (%)
Accuracy (%)
100
100
100
85
81.25
83.33
Table 5 Performance of the proposed classification approach evaluated by the all-training-all-testing method. Subjects were categorized into the healthy control (CO) and three NDDs altogether. Evaluation methods
All-training-all-testing
Predicted groups
NDDs CO
Actual groups NDDs
CO
44 0
4 16
Sensitivity (%)
Specificity (%)
Accuracy (%)
91.67
100
93.75
5. Discussion and conclusion In the present study, by analyzing the SCPG model in which the unperturbed inner frequency of the nonlinear forced Van der Pol oscillator can mimic the gait cycle, we prove that the functions containing the walking time series states of human gait represent gait dynamics. The swing and stance intervals of the left and right feet are extracted as gait features for the classification. They indicate that gait dynamics between NDDs and the healthy controls are significantly different, and the gait symmetry of NDDs is significantly perturbed. Then, the RBF networks can be used to approximate gait dynamics, and the difference of gait dynamics between the NDDs and healthy control subjects will be used for the classification of NDDs with effective classification performance on the 64 subjects studied (PD, HD, ALS and CO). Our study contributes the accuracy improvements to the classification of NDDs. By using the same all-training-all-testing evaluation method, the correct classification rate obtained by the Elman’s recurrent neural network method is reported to be 91.7% for ALS patients in the study of [10], which is a bit lower than the 93:1% accuracy in the present study. In [41], the reported classification accuracy for ALS patients is 92:3%, which is also a bit lower than the accuracy of the present study. By using the same all-training-all-testing evaluation method, we also obtain the accuracy of 100%, 100% and 93.75% for the classification of PD, HD and three neuro-degenerative diseases altogether, respectively. The results are also a bit higher than the reported accuracy of 89.33%, 90.23% and 90.63%, respectively, in the study of [7]. Compared with the classification results in [41,7,10], our classification approach achieves superior performance. The results indicate that the proposed system can be effective for the classification of NDDs using gait dynamics. In the current database, the three groups of patients with NDDs are at different stages. They are measured with disease severity or duration. For the subjects with Parkinson’s disease, this is the Hohn and Yahr score. For the subjects with Huntington’s disease, this is the total functional capacity measure. For the subjects with amyotrophic lateral sclerosis, the number here is the time in months since the diagnosis of the disease. In the experiments, we have shown that patients with NDDs, including those at an early stage, can be correctly classified. This verified the efficiency of the proposed method in early detection. Nevertheless, further clinical research and statistical analysis using bigger sample sizes are needed to confirm the efficiency and applicability of the method in early detection. Our future research work may be carried out in the following aspects. (1) The size of the current database is small, which may limit the test of the generalization ability of the proposed approach. In future, we will collect data of larger population of neurodegenerative diseased patients for a better classification performance evaluation, and will study the effect of age, gender and medication. (2) Other gait features, such as joint angles/angular velocities and silhouette features, can also reflect the difference of gait dynamics between patients with NDDs and healthy controls. They will be useful for evaluation and classification of the two groups. In future, different kinds of gait features may be fused to improve the accuracy of the classification. (3) It would also be interesting to study in the further work progression of NDDs and the effects of intervention therapies. We may recruit a larger number of neurodegenerative patients at different disease stages and the matched healthy subjects, and also demand to compare gait dynamics in NDDs before and after the intervention therapies.
Conflict of interest There is no conflict of interest.
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Acknowledgments The authors sincerely appreciate the Editor, Associate Editor and anonymous reviewers for their constructive feedback and insight comment which greatly improve the presentation of this paper. This work was supported by the National Natural Science Foundation of China (Grant No. 61304084), by the National Science Fund for Distinguished Young Scholars (Grant No. 61225014), by the Educational and Scientific Research Project for Middle-aged and Young Teachers of Fujian Province of China (Grant No. JA14298), and by the Science and Technology Project of Longyan University (Grants Nos. LG2014005, LC2014005). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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