Optimized Gaussian Mixture Models for Upper Limb Motion

Proceedings of the 26th Annual International Conference of the IEEE EMBS San Francisco, CA, USA • September 1-5, 2004

Optimized Gaussian Mixture Models for Upper Limb Motion Classification Y. Huang1,2, K.B. Englehart1,2, B. Hudgins1,2, A.D.C. Chan3

1 Institute of Biomedical Engineering, University of New Brunswick, NB, Canada Department of Electrical and Computer Engineering, University of New Brunswick, NB, Canada 3 Department of Systems and Computer Engineering, Carleton University, ON, Canada

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and optimize the approach for the multiple limb motion classification using steady-state MES associated with a constant contraction. The GMM has been shown to generally produce the best performance in speaker identification [4] and verification [7]. In this paper, a GMM-based pattern recognition system is presented to obtain an effective way to classify various limb motions by using the MES from limb muscles. The experiments demonstrated the proposed approach not only achieves high classification accuracy but also has a low computational cost.

Abstract—This paper introduces the use of Gaussian mixture models (GMM) for discriminating multiple classes of limb motions using continuous myoelectric signals (MES). The purpose of this work is to investigate an optimum configuration of a GMM-based limb motion classification scheme. For this effort, a complete experimental evaluation of the Gaussian mixture motion model is conducted on a 12-subject database. The experiments examine algorithmic issues of the GMM including the model order selection and variance limiting. The final classification performance of this GMM system has been compared with that of three other classifiers (a linear discriminant analysis (LDA), a linear perceptron neural network (LP) and a multilayer perceptron (MLP) neural network) . The Gaussian mixture motion model attains 96.3% classification accuracy using four channel MES for distinguishing six limb motions and is shown to outperform the other motion modeling techniques on an identical six limb motion task.

II. THE GAUSSIAN MIXTURE MOTION MODEL A. GMM Model Description A Gaussian mixture density is a weighted sum of M r component densities. For a feature vector denoted as x , the mixture density for the Nth model is defined as the equation M r r p ( x | λ N ) = ∑ w iN p iN ( x )

Keywords—Gaussian mixture model, myoelectric signals, pattern recognition, prosthesis, EMG

i =1

I. INTRODUCTION

where M is the mixture order; wiN , i=1,…,M, are the mixture r weights; p iN ( x ) , i=1,…,M, are multi-variate Gaussian

Myoelectric signals (MES) collected at the skin surface using electrodes are capable of providing information about neuromuscular activity in a noninvasive manner, and have become an important and effective control input for powered prostheses. Pattern recognition of the MES plays a key role in advanced control of powered prostheses for individuals with amputations or congenitally deficient upper limbs. The success of a myoelectric control scheme depends greatly on the classification accuracy. Attempts to increase the classification accuracy to discriminate the desired classes of limb activations have been made by proposing a variety of pattern recognition methods. Hudgins developed a control scheme of the transient MES based upon simple time-domain (TD) statistics and a multilayer perceptron (MLP) neural network classifier with an error rate of around 10% for four types of limb motions [5]. Englehart investigated time-frequency transforms based representations on the transient MES classification and 6.25% error performance was achieved for four classes of limb motions using linear discriminant analysis (LDA) [6][8]. Englehart developed a real-time continuous control scheme to discriminate four classes of limb motions using a TD feature set and a MLP classifier with an error rate of 6.75% [3]. Chan showed the potential of the Gaussian mixture model (GMM) on MES classification with an error rate of 6% for a six-class problem [1]. This research will extend this preliminary work

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density functions. The complete Gaussian mixture density is parameterized by the mixture weights, mean vectors and covariance matrices from all components densities and r notated as λ N = { w iN , µ iN , C iN } , i=1,…,M. In this work, the parameters of the GMM are estimated using the expectation-maximization (EM) algorithm. B. Motivation of Using a GMM in MES Classification The GMM has been shown to be an exceptionally powerful approach to text-independent speaker identification and verification. The motivation of applying the GMM to MES classification problem is that the problem of speaker verification and identification bears a strong analogy to the task of motion classification for MES. The individual component densities of a Gaussian mixture density may model some underlying set of motion classes and a linear combination of Gaussian basis functions is capable of representing a large class of sample distributions. C. GMM Motion Model Interpretations This research uses pattern recognition to process four channels of MES with the task of discriminating six classes of limb motions. For MES identification, each motion is represented by a GMM and is referred to by its model λ as shown in Fig.1. For motion identification, a group of

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MV decision includes the previous m sample and the next m sample. In this case, m=8. The value of the MV decision is simply the class with the greatest number of occurrences in this 2m+1=17 point window of the decision stream.

motions N={1,2,…,6} are represented by GMM’s λ1, λ2 ,…, λ6. The goal of the training is to estimate the parameters of the GMM. The Gaussian mixture density for each GMM is Pλ1, Pλ2 ,…, Pλ6. The objective of the test is to find the motion class which has the maximum a posteriori probability Pmax for a given observed sequence.

Feature Set

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IV. EXPERIMENTAL EVALUATION A. Data Description

Pmax

The data used in this work are the same as the data presented in [1][3]. Data were collected from 12 normally limbed subjects. In each session, subjects performed six limb motions, holding the contraction for five seconds. In each session, each limb motion was performed twice. Four sessions were conducted for each subject. Subjects performed each motion in a predetermined sequence: wrist flexion, wrist extension, supination, pronation, hand open, and hand closed in the first and third sessions and in random order in the second and fourth sessions. In the following experiments, the first, second and third sessions were used as the training sessions and the fourth session was used as the test session unless indicated otherwise.

Motion Class

Fig.1 The GMM-based motion classification system

The limb motion classification was performed using six GMMs (λ1- λ6). In the training session, for the given training MES data from each motion, the EM algorithm was used to estimate the parameters of each GMM. In the test session, each test pattern was provided to six GMMs. The class i associated with the GMM λi that has the highest Gaussian mixture likelihood function was chosen for a given feature set.

B. Evaluation Criteria The evaluation of the GMM-based limb motion classification system was conducted in the following manner as shown in Fig 2. The analysis window is 256ms and the decision increment is 32ms. Six GMMs are used to model six limb motions. Six probabilities were computed and compared at each frame, i.e. Pij, i=1,…, 6 and j=1,…,8. The maximum probability was chosen for each frame, i.e. Pmax1… Pmax8, and then labeled as the predicted class of limb motion at each frame. The final decision on a 256ms analysis window is made using a majority vote technique of the ensemble of Pmax decisions.

III. CLASSIFICATION PROCESS A. Front-End Processing MES processing is performed on 256 ms analysis windows. It was evident that most of the classification errors occurred at the transitions between classes in a typical test scenario [3]. This is expected because the MES is in an undetermined state between different contraction types. In order to avoid transitory data, the 256ms sample data that overlapped a class transition point and the two 256ms adjacent windows to the overlapping record were removed from both of the training and test data. The training data were segmented into disjoint and adjacent 256ms windows while the test data were segmented into overlapping 256ms windows with 32ms apart [2].

256ms

32ms

P11 P21 P31 P41 P51 P61 Pmax1

B. Feature Extraction The data were subject to classification using 6th order autoregressive (AR) coefficients and the root mean square (RMS) value computed on each window. For each 256 ms analysis window, a 7-dimensional feature vector (the 6thorder AR+RMS) was extracted on each channel. After concatenating the feature sets of four channels, the final multi-dimensional feature vector was then provided to the pattern classifier.

P12 P22 P32 P42 P52 P62 Pmax2

P13 P23 P33 P43 P53 P63 Pmax3

P14 P24 P34 P44 P54 P64 Pmax4

P15 P25 P35 P45 P55 P65 Pmax5

P16 P26 P36 P46 P56 P66 Pmax6

P17 P27 P37 P47 P57 P67 Pmax7

P18 P28 P38 P48 P58 P68 Pmax8

Fig. 2. Data segmentation for the performance evaluation of the GMMbased limb motion classification system.

The identified motion of each frame was stored in a prediction vector and it was then compared to the actual motion class vector. The number of frames which were incorrectly classified was tabulated. The final performance evaluation was then computed as the following formula:

C. Post Processing Since decisions are made more frequently than the required response time of a prosthesis, a majority vote (MV) technique was used in the post processing. It is noted that a MV algorithm helps to significantly eliminate spurious misclassifications [1][2][3]. For a given decision point, the

% error =

number of errors × 100 total number of decisions

While there are variations among the individual subject’s performance, the aim of the evaluation measure was to track

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training set were evaluated in this experiment to show the effect of the training set size on the optimal mixture component.

the average performance of the system for a roster of 12 subjects, allowing a common basis of comparison. On the other hand, since EM algorithm used to estimate the GMM parameters begins with random initialization, the performance for each subject was averaged over 10 trials to avoid the deviation of different trials.

Classification Error(%)

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C. Algorithmic Issue 1– Model Order For limb motion classification, the objective is to choose the best mixture components to achieve high discriminate accuracy. Theoretically, too few mixture components can produce a GMM motion model which doesn’t accurately model the distinguishing characteristics of a motion’s distribution. However, too many components can reduce performance when there are a large number of model parameters relative to the available training data and can also result in excessive computational complexity [4].

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Fig.3. Averaged GMM classification performance for varying amounts of training data set

Fig.3 illustrates the GMM classification performance averaged across 12 subjects for one session training set, two session training set and three session training set. Clearly, performance improves with larger training sets. For threesession training, the optimal mixture order of the model is M=3; for two-session training, the optimal mixture order is M=2; for one-session training, the optimal mixture order is M=1. The results demonstrate that the optimal mixture number increases when the amount of training data increases. This is reasonable because a large number of mixture components can reduce performance when there are a large number of model parameters relative to the available training data. Thus it indicates that we should consider larger mixture component when we have larger amount of training data.

The first experiment was conducted using a fixed amount of training data to investigate the motion classification performance of the GMM with respect to the number of component densities. The average classification performance across all subjects for each mixture number is shown in Table I. TABLE I AVERAGED GMM CLASSIFICATION PERFORMANCE FOR EACH MODEL ORDER Classification Error (%) 4.37 4.35 4.32 4.49 4.46 4.62 4.81 4.80 4.66 4.89

The Influence of Large Mixture Order The third experiment of the mixture model selection is to explore the effect of some large mixture number on the performance. The mixture numbers M=1,10,12,14,15,20,30, 40,60,80 are examined. These large mixture numbers caused very high computational expense but did not seem to yield good performance for the small amount of available training data. It demonstrates that the mixture component selection is limited by the amount of training data available. It is also evident that model order selection becomes more important with smaller amount of training data.

One can see the optimal mixture number for this case is M=3. Table II shows the optimal mixture number for individual subject. It demonstrates that the optimal mixture number of the model is subject-dependent. There is no theoretical way to estimate it a priori. TABLE II THE OPTIMAL MIXTURE NUMBER OF THE GMM FOR EACH SUBJECT Subject # Optimal Mixture #

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The Influence of Model Order on Performance

Model Order 1 2 3 4 5 6 7 8 9 10

Three Session Training Set Two Session Training Set One Session Training Set

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D. Algorithmic Issue 2– Variance Limiting (VL)

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When training a GMM, it is common that that the variance elements can be quite small. These small variances produce a singularity in the model’s likelihood function and can degrade performance [6]. In order to avoid these spurious errors, a VL constraint was applied. For an arbitrary element of mixture component i’s variance vector, σi2, and a minimum variance value, σmin2, the constraint of the VL is shown as follows: 2  σ 2 if σ i2 > σ min σ i2 =  2i 2 2 σ min if σ i ≤ σ min

Training Set Size vs. Model Order In another experiment, we examine the performance of the GMM motion classification system for different model orders using varying amounts of training data. A one session training set, a two session training set and a three session

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significantly if we could select the optimal mixture number for each subject. This implies that the better performance could be achieved if we could have a validation data set for selecting the optimal mixture number for each subject.

This constraint of a minimum variance value was placed on elements of all variance vectors after each EM iteration for each GMM. An empirical investigation was undertaken to determine the minimum variance value. If it is too high, the component variances are masked to the same value which would overly constrain the model and hence degrade performance. If the value is set too low, it may not perform the desired limiting at all. Therefore we experimented on 12 subjects with the mixture number M=1, M=2 and M=3 and found that a variance limit between 0.01 to 0.1 provided the best classification performance.

V. CONCLUSION This paper has evaluated the use of the GMM for limb motion classification. The primary focus of this work is to explore the optimum configuration of the GMM-based limb motion pattern recognition scheme. The experimental evaluation investigated several aspects of using GMM for limb motion classification. Some conclusions from this work are as follows: • The model order selection of the GMM is crucial for achieving good classification performance. • Variance limiting of the GMM is important to avoid model singularities. • The GMM outperforms the LDA, LP and MLP classifiers. Some suggestion for future work includes: • A larger model order should be considered when larger amounts of training data are available. • In order to achieve even better performance, a validation data set for selecting the optimal mixture number for each subject is necessary. These results indicate that the GMM provides robust motion discrimination for the task of motion classification. The models are computationally inexpensive and easily implemented on a real-time platform.

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Fig.4. The effect of the variance limiting on averaged classification performance

An experiment was conducted to examine the effect of VL on the average classification performance using VL=0.05. Fig.4 depicts the effect of VL on the average classification performance. It shows that the GMM performance is sensitive to VL and VL decreases the classification error by roughly 1.6%. This result proves that VL can be an algorithm factor of the GMM with regard to the performance.

REFERENCES [1] A.D.C Chan, K.B. Englehart, “Continuous classification of myoelectric signals for powered prosthesis using Gaussian mixture models,” in 25th Engineering in Medicine and Biology Society International Conference, Cancun, Mexico, Sept. 2003. [2] K. Englehart and B. Hudgins, “A Robust, Real-Time Control Scheme for Multifunction Myoelectric Control,” IEEE Trans Biomed Eng, 50(7):848-54, July, 2003. [3] K. Englehart, B. Hudgins and A.D.C Chan, “Continuous Multifunction Myoelectric Control using Pattern Recognition,” Technology and Disability, Volume 15, No.2, pp.95-103,2003. [4] D. A. Reynolds, “Robust Text-Independent Speaker Identification Using Gaussian Mixture Speaker Models,” IEEE Transactions on speech and audio processing, 3(1), Jan. 1995. [5] B. Hudgins, P.A. Parker, RN. Scott, “A new strategy for multifunction myoelectric control,” IEEE Trans Biomed Eng. 40(1):82-94, Jan. 1993. [6] K. Englehart, B. Hudgins, P. A. Parker and M. Stevenson, “Classification of the myoelectric signal using time-frequency based representations,” Medical Engineering and Physics, v 21, n 6-7, p 431-438, July, 1999. [7] D. A. Reynolds, T. F. Quatieri, and R. B. Dunn, “Speaker Verification Using Adapted Gaussian Mixture Models,” Digital Signal Processing, Vol. 10, No. 1, pp. 19-41, Jan. 2000. [8] K. Englehart, B. Hudgins and P. A. Parker, “A Wavelet-Based Continuous Classification Scheme for Multifunction Myoelectric Control,” IEEE Transactions on Biomedical Engineering, v 48, n 3, p 302-311, 2001.

E. Comparison to other Limb Motion Classifiers The aim of this experiment is to compare the performance of the GMM with the LDA, linear perceptron (LP) and MLP using the same data and the feature set. TABLE III AVERAGED CLASSIFICATION PERFORMANCE FOR SELECTED MODELS Model GMM (Selected M) GMM(M=3) LDA LP MLP

Classification Error (%) 3.72 4.32 4.42 4.53 4.61

Table III describes the average classification error percentages for various classification techniques. It is observed that the average classification accuracy of the GMM with the optimal mixture number M=3 is slightly higher than that of the LDA, LP and MLP. The GMM with the selected mixture number for each subject significantly decreases the performance error to 3.72% which is much lower than other models. The result indicates that the averaged GMM classification accuracy will increase

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