Journal of Orthopaedic Research 8:241-258 Raven Press, Ltd., New York 0 1990 Orthopaedic Research Society

Dynamic Electromyography . I. Numerical Representation Using Principal Component Analysis M. E. Wootten, M. P. Kadaba, and G. V. B. Cochran Orthopaedic Engineering and Research Center, Helen Hayes Hospital, West Haverstraw, New York, U.S.A.

Summary: A complete description of human gait requires consideration of linear and temporal gait parameters such as velocity, cadence, and stride length, as well as graphic waveforms such as limb rotations, forces, and moments at the joints and phasic activity of muscles. This results in a large number of interactive parameters, making interpretation of gait data extremely difficult. Statistical pattern recognition techniques can simplify this problem. For this approach to be successful, first it is necessary to reduce the number of interactive parameters to a manageable set. In this study, we present an application of principal component analysis as a means for representing graphic waveforms in a parsimonious manner. In particular, we concentrate on representing the phasic muscle activity recorded using surface electrodes from ten major muscles of the lower extremity of 35 normal subjects during level walking. A 32 point vector is created in which each point of the vector represents the normalized area under the curve of a portion of rectified and smoothed electromyographic signal, expressed as a function of gait cycle. Principal components are computed and the first few weighting coefficients are retained as features to represent the original EMG data. We show that the corresponding basis vectors span parts of the gait cycle where the most variability between individual subjects exists. We also show that the basis vectors can be used to represent the EMG data of subjects not originally used to generate the basis vectors. Key Words: Gait analysis-Electromyographic data-Principal component analysis-Numerical representation-Pattern recognition.

Quantitative gait analysis using computer-aided video motion analysis, force plates, and electromyography is of recognized value in the assessment of gait disabilities and in quantitative evaluation of treatment. However, despite dramatic improvements in techniques, gait analysis still lacks widespread clinical utility. One problem relates to the management of large quantities of data generated by measurements of various kinetic, kinematic, and

electromyographic parameters over several gait cycles to detect clinically significant patterns of performance. Identifying clinically useful information from a gait evaluation is further complicated by the fact that gait patterns vary among patients with similar diagnoses and, to a certain extent, even among normals. Interpretation of gait data may be simplified by using statistical pattern recognition techniques. The problem of pattern recognition is that of classifying a pattern into one of several classes on the basis of certain physical measurements and observations or features that are derived from the measurements. The features selected must describe the pattern accurately and enable subgroups to be identified. Further, for a pattern recognition approach to be suc-

Received October 21, 1987; accepted April 26, 1989. Address correspondence and reprint requests to Ms. M. E. Wootten at Gait Analysis Laboratory, Helen Hayes Hospital, Route 9W, West Haverstraw, NY 10993, U.S.A. This work was presented in part at the 30th Annual Meeting of the Orthopaedic Research Society, Atlanta, Georgia, February 7-9, 1984

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cessful, the enormous quantity of data must be reduced to a parsimonious set of features that describe gait patterns accurately. Hence, the first step in the application of pattern recognition is to reduce the dimensionality of the original data set. Much of the data measured during a gait analysis (for example, phasic electromyographic activity, hip, knee and ankle motion, and the ground reaction forces) when expressed as a function of gait cycle are graphic waveforms of complex shapes. Representation of these graphic patterns in terms of a discrete set of numerical variables would make comparison between subjects or between groups more convenient. Since these gait parameters are periodic (cyclic) functions of time, the simplest way to represent numerically these waveforms is by discrete Fourier series (18). The Fourier series expansion is in terms of a set of basic waveforms (basis vectors) that are sinusoids, whose contributions to the original data are determined by the Fourier coefficients (weighting coefficients). An alternative approach , utilizing principal component analysis, derives the basis vectors from the original data set of the subject population. Principal component analysis is an optimal linear expansion in which the waveforms belonging to a group are used to develop basis vectors and weighting coefficients based on a least mean square error fit process. This process separates out the predominant waveform component that is common to all the waveforms in the set, from the small uncorrelated events in each individual waveform (6). Thus, the first basis vector represents the waveform characteristic that is common to all of the waveforms and the second basis vector represents the best fit to residuals from the first and so on. Due to the relative similarity between data waveforms in the original set, the number of basis vectors and the number of new coefficients needed to represent the observed waveforms are substantially less than the number of original data waves. The principal component analysis thus effects a reduction in dimensionality of the original set of waveforms. Individual waveforms are reconstructed using a linear combination of the basis vectors modulated by the weighting coefficients. Since the basis vectors are derived from the original data waveforms, they may be highly correlated to certain events in the gait cycle. Principal component analysis has been used extensively in the study of evoked potentials (17) and electrocardiography (5). Principal component anal-

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ysis has also been used in the evaluation of hip diseases by Yamamoto et al. (21). Shiavi and Griffin (16) employed principal component analysis to determine the population distribution of electromyographic patterns in normal subjects during level walking. The gait cycles were first divided into 16 equally spaced points and the data vectors were constructed using a composite binary criterion. The elements of the data vector represented the fraction of time that the muscle was considered to be active, and the muscle was considered to be active if the magnitude of the electromyogram (EMG) was above a threshold value. Using the weighting coefficients, they were able to identify 5 to 10 patterns of muscle activity for each of the muscles. However, the binary representation of the EMG utilized in their study did not take into consideration the relative intensity of muscular contraction. Patla (14) used principal component analysis to determine variant and invariant features in the EMG activity of eight muscles of the lower limb while the subjects walked at different speeds and stride lengths. The objective was to identify locomotion control patterns by identifying common activity patterns between seven major muscles of the lower extremity. Since the muscles were grouped together, variabilities in the pattern of activity in individual muscles were not considered. In this paper, we present an application of the principal component analysis for representing phasic electromyographic data acquired during level walking. Presently, no consistent, universal correlation exists relating muscle force to dynamic EMG data, but several investigators have studied this important relationship under static conditions (1,7,1012,19). These studies have shown that the integrated EMG is linearly correlated with isometric force. In view of these results, an area vector representation of EMG data is used in the principal component expansion. Each point in the vector represents the area under the curve of a portion of the rectified and smoothed EMG waveform, depicting both the timing and intensity of muscle activity. Thirty-two point data vectors for each muscle from each subject are used as input to the expansion algorithm. The basis vectors and weighting coefficients for each of the ten muscles are computed and the biomechanical significance of the basis vectors and weighting coefficients are described. We also show that these basis vectors can be used to represent other normal EMG data, not used in the generation of basis vectors.

249

DYNAMIC ELECTROMYOGRAPHY ANALYTICAL CONSIDERATIONS

The general matrix form of any linear expansion is given by

X

=

(1)

FY

where X is the matrix of data vectors (T X N), where the columns represent individual data vectors; F is the orthogonal matrix of basis vectors (T x 7); Y is the matrix of weighting coefficients (T x N);and T is the number of time points, and N is the number of cases (subjects). In principal component analysis, the basis vectors are the eigenvectors of the covariance or correlation matrix of X and eigenvalues represent the power contribution of the corresponding eigenvalues (2). In this application, the correlation matrix (R) is calculated using the formula

R

=

(l/N) XX’

(2)

where X’ is the transpose of X. The mean at each time point is not removed. Each column of the eigenvector matrix (F) is one eigenvector, and there are T eigenvectors each having T points. The matrix of weighting coefficients (Y) is calculated from the original data vectors and the transpose of the eigenvector matrix. In matrix notation, Y

=

F’X

vector to the total power in the set of data waveforms. Details of the principal component analysis are found in Cohen ( 2 ) , Glaser (4), Fukunaga (3), and Harmon (6).

(3)

For a complete and accurate representation of each subject’s data, all eigenvectors and weighting coefficients would be required in the reconstruction. However, a strength of this linear transformation is that the original data vectors can be adequately represented by a smaller set of basis vectors. The mean square error (MSE) in representation of X with M < T components is the sum of the eigenvalues not retained, since the magnitude of the eigenvalue hi represents the power contribution of the ith basis

MATERIALS AND METHODS

Gait analysis was performed on 35 normal adult subjects (age range of 1 7 4 0 years). Ten muscles, namely the gluteus maximus, gluteus medius, adductor longus, vastus lateralis, rectus femoris, vastus medialis, medial hamstring (semitendinosus), lateral hamstring, anterior tibialis, and gastrocnemius (medial head) were evaluated. Electrode placement was determined by the anatomical location of the motor point and the approximate locations of the electrodes are shown in Table l. Electromyographic data were recorded using surface electrodes (Iomed Inc., Salt Lake City, UT, U.S.A.). The electrode has a miniature preamplifier with three stainless steel sensing elements (13 mm in diameter spaced 5 mm apart with the center element acting as the ground plate) mounted directly on the case of the preamplifier assembly. The differential amplifier has an input impedance of lo5 MLR and a common mode rejection ratio (CMRR) of 105 dB. The electrodes were applied to the muscle or muscle group of interest using surgical tape. Foot contact patterns were recorded using foot switches taped to the heel, first and fifth metatarsal and great toe of each foot. The EMG and foot switch signals were transmitted over a hard-wired system to recording instrumentation via electrical isolation amplifiers. The EMG data were band pass filtered from 20 to 500 Hz; foot switch and electromyographic signals were sampled at 1 kHz and stored for further analysis. In order to achieve repeatable results, the subjects

TABLE 1. Placement of Electrodes for Muscles Studied Muscle Gluteus maximus Gluteus medius Adductor longus Vastus lateralis Rectus femoris Vastus medialis Medial hamstring Lateral hamstring Anterior tibialis Gastrocnemius

Position Midway between the greater trocanter and the sacrum, parallel to fibers One inch distal to the midpoint of the iliac crest, parallel to the fibers Four finger breadths along muscle distd to the pubic tubercle Lateral thigh, one hand breadth above patella Midway between the superior border of the patella and ASIS Four finger breadths proximal to the superior/medial patella Midway between the medial femoral condyle and the ischial tuberosity Midway between the fibular head and the ischial tuberosity Four finger breadths below the tibial tuberosity and one finger breadth lateral to the tibial crest One hand breadth below the popliteal crease on the medial mass of the calf

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250

were instructed to walk at their self-selected speed along a 10 m walkway. Data acquisition was controlled by two photoelectric eyes placed at the beginning and end of a 6 m portion of the walkway. A minimum of three walks were recorded for each day of testing. Each subject was tested on three different days at least 1 week apart. A minimum of 27 cycles of data were available for each subject. Data Analysis Utilizing the foot switch data, the walks were divided into separate cycles. The cycles were then rectified and smoothed, using a 32 point moving cosine window (9,13). Each gait cycle was divided into equally spaced intervals. The gait cycle was then represented by a 32 point vector, each point in the vector representing the area under the rectified, smoothed curve for each interval (Fig. 1). The data vectors were normalized to their respective maximum area element in order to eliminate any bias

introduced by the large amplitude data vectors in the computation of basis vectors. This type of normalization also reduces the intersubject and intrasubject variability (22) introduced by factors such as electrode placement, electrode-skin interface conditions, excitation voltage, etc. In order to form a representative data vector for each muscle of each subject, 27 cycles of area vector data from 3 days of testing were averaged. The 32 point area vectors were input to the algorithm for computing principal components. For each muscle, a 32 by 35 point data matrix X was formed using area vectors from all of the 35 subjects. The arrangement of X was such that the column vector represented the data vector of one subject for the specific muscle in question. Figure 2 shows the flow chart of the data analysis procedure that was followed for the principal component analysis. The correlation matrix of X was formed and the eigenvectors and eigenvalues were calculated using the EIGEN subroutine (Scientific Subroutine Package, Digital Equipment Corporation, Maynard, MA, U.S.A.). The eigenvalues were expressed as a percent of total power by dividing each by the sum of all of the eigenvalues. In this expansion, the basis vectors are the eigenvectors and the weighting coefficients were calculated using Eq. (3). The individual area vectors can be reconstructed faithfully by using a linear combination of all of the basis vectors and corresponding weighting coefficients. However, in order to reduce the dimensionality of the original data vectors, a smaller number of coefficients and basis vectors can be used provided that the original data vector is represented accurately. The average percent of power represented by the reduced set of vectors and coefficients is equal to the sum of the eigenvalues retained divided by the sum of all of the eigenvalues. An underlying hypothesis in this application of principal component analysis is that the basis vectors calculated can be used to represent effectively the muscle activity of not only the original 35 subjects but also of any normal subject walking at his/ her self-selected walking speed. In order to illustrate the efficacy of this representation, muscle activity data from five additional subjects were used to calculate the weighting coefficients [Eq. (3), using the same set of basis vectors]. The percent MSE between the original data vector (x) and its reconstruction (a) for each muscle from each subject in this group was calculated as follows:

1 Rectified

1 Lpt

Area Vector

1

0

Percent of Gait Cycle

100 3 2 pt. Vector

(.05..10..05..06..16,.29,.39.99.88,.66,.68.1.00..96..80..28..05. .02..00..03..05..03..04..04..03..0

1 ..O 1 ..02..02..02..02..05..04)

FIG. 1. Electromyographic data are divided into gait cycles and expressed as a 32 point area vector normalized to the maximum value.

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DYNAMIC ELECTROMYOGRAPHY Initial Processing

251

Principal Component Representation

Calculate Correlation Matrix

Rectify and Smooth

10 Channel EMG

Matrix

Divide into Cycles

Foot SwitchData

analysis sequence for computing weighting coefficients and basis vectors. N, Number of subjects; T, Number of time

ElGEN

Calculate Eigenvalues

% Power

Eigenvectors

Eigenvalues

Calculate T Point Area Vector

=,I

r

3 days

Number of Features Retained

Matrix

Reprc ientative

4 +

Weighting Coefficients

I

-

(M)

*+

Reconstructed Vectors

Cycles

X

RESULTS The percentage of cumulative power that can be represented by the basis vectors corresponding to the first six eigenvalues (features) is shown in Table

E r i x

Mean Square Error In Representation

2. The criterion for selecting the number of features for adequate representation of muscle activity was based on whether or not retaining an additional feature resulted in a substantial increase in percent of cumulative power. Specifically, if the increase was less than 1%, the feature was judged to be unnecessary in the representation of the muscle activity. For example, 97.4% of power for the gluteus medius was represented by four features and 98.0%by five features; therefore, only four features were re-

TABLE 2. Cumulative percent of power represented by eigenvalues ~~

Muscle Gluteus maximus Gluteus medius Adductor longus Vastus lateralis Rectus femoris Vastus medialis Medial hamstring Lateral hamstring Anterior tibialis Gastrocnemius

~

1 92.8 91.8 87.4 93.4 91.8 93.1 86.7 83.9 93.8 92.2

2

95.8 94.5 91.3 95.8 94.5 95.3 91.3 90.0 95.5 95.1

3

4

5

6

Features

97.0 96.3 94.2 97.2 96.4 97.0 94.5 93.7 96.8 97.1

97.8 97.4 95.7 98.4 97.6 98.0 96.3 95.4 97.8 98.1

98.4 98.0 96.8 98.8 98.2 98.5 97.6 96.7 98.3 98.9

98.8 98.6 97.6 99.0 98.7 98.8 98.4 97.5 98.7 99.2

3 4 5

4 4 4 5

5 4 4

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Gluteus Medius

Gluteus Maximus

i'----------./ -i

I

I I

I

FIG. 3. Three basis vectors for the gluteus maximus and gluteus rnedius; the dashed line indicates the toe-off point.

> -1

0

PERCENT OF CYCLE

8

I08

PUlCWT OF iYUt

tained in the representation of this muscle. Based upon this criterion, the number of features used to represent each muscle is shown in Table 2. For the adductor and medial and lateral hamstring muscles, the percentages of cumulative power represented by the first basis vector were 87,87, and 84, respectively, values which were lower than for the rest of the muscles evaluated in this study. The lower cumulative power indicates that, for these muscles, the patterns of activity were more variable between the subjects. Basis Vectors The basis vectors for the muscles evaluated in this study are shown in Figs. 3-7. Since the first

188

basis vector of the correlation matrix for each muscle is the best fit to the data vectors from this group of 35 normal subjects, it was similar to the average pattern of muscle activity. Further, since the second and third basis vectors are the best fit to the residual variance not accounted for by the first and second vectors, respectively, they tended to modify the amount of activity during the stance phase and/ or the swing phase of the gait cycle. The basis vectors for the gluteus maximus and medius are shown in Fig. 3. For the gluteus maximus, the second basis vector modified the timing and intensity of the stance phase activity and the end of the swing. Those with extended stance phase activity tended to have a positive weighting on the

FIG. 4. The first five basis vectors for the adductor longus muscle; the dashed line indicates the toe-off point.

e

u A

e

PERCENT OF CYME

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I ee

WlCENT OF CYUE

188

253

DYNAMIC ELECTROMYOGRAPHY Vastus Medialis

-i

Rectus Femoris

- . JZ

I I

I I

j p : J I

I

0

0

PERCENT OF CYCLE

PERCENT OF CYCLE

I08

0

PERCENT OF CYCLE

1@0

FIG. 5. The first three basis vectors for the quadricep muscles.

The first five basis vectors for the adductor muscle group are shown in Fig. 4. For this muscle group, the surface electrode was placed over the adductor longus muscle (Table 1). The adductor longus is predominantly active in terminal stance and preswing while the gracilis is active during initial contact and terminal swing (15). The first basis vector displayed a pattern similar to a combination of phasic activity reported for gracilis and adductor longus muscles. The second vector modified owing phase activity. The third vector modified terminal

second vector. Those with less stance phase activity but slightly increased terminal swing amplitude tended to have a larger negative weighting on the second vector. The first basis vector for the gluteus medius (Fig. 3) is similar to the usually depicted pattern for the gluteus medius. The second basis vector affected the amount of muscle activity during the stance phase. Additionally, the second and third vectors in combination contributed primarily to data vectors that displayed two peaks of muscle activity during stance.

La t era1 H a m s t r i n g

Medial Hamstring

-j i

I

I

Y

-1

FIG. 6. The first three basis vectors for the medial and lateral hamstring muscles.

I I

v I I

0

PERCENT OF CYCLE

I@$

a

F€RCENT OF CYCLE

188

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254

G a s t rocnernius

A n t e r i o r Tibialis

nt

4

FIG. 7. The first three basis vectors for the anterior tibialis and gastrocnemius muscles.

7

n

I

PERCENT OF CYCLE

PERCENT OF CYCLE

0

188

the highest of the ten muscles evaluated in this study (8). The basis vectors for the quadriceps group are shown in Fig. 5. The basis vectors for vastus later-

swing through loading response activity while the fourth vector contributed to representation of both terminal stance and mid-swing activity. For this muscle group, the intrasubject variability was GLU. MAX.

VAS. MED.

I .YX

GLU. MU).

1.1%

HAM.

MU).

I

& I 0

ADD. LON

188

2.1% I

2.5% I I

FIG. 8. Reconstruction of the ten muscles for subject 21, one of the subjects used to generate the basis vectors. The % MSE is shown to the right of the muscle name. VAS. LAT.

0.6%

ANT. TIB.

I .3%

REC. FEM.

I I%

GASTROC .

I 0%

'

__ ORIGINAL

P E R C E N ~OF CYCLE

-

RECONSTRUCTED

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100

0

'

P E R C E N OF ~ CYCLE

'

188

Subject 2 1

255

DYNAMIC ELECTROMYOGRAPHY

principal components analysis. The first three basis vectors represented an average of 97% of the total power in this group of subjects (Fig. 7). In about 25% of the subjects who displayed two peaks of activity during the stance phase, a large positive weighting on the third vector accounted for this pattern of muscle activity. The reconstructed vectors (and the original vectors) for two representative subjects are shown in Figs. 8 and 9. For each muscle, the reconstructions were done using the number of features listed in the last column of Table 2. The mean square error between the data waveform and its reconstruction are also shown in the figures. To determine whether the basis vectors could be used to represent the muscle activity patterns of any normal subject, the weighting coefficients of five additional subjects not included in the original group were calculated using Eq. (3). The weighting coefficients were used to reconstruct data vectors and the percent MSE in reconstruction is shown in Table 3. Figure 10 shows the original and reconstructed vectors for a representative subject from this second group.

alis and vastus medialis were similar. For the rectus femoris, the second vector contributed mainly to muscle activity in the terminal stance and initial swing phase of the gait cycle. The basis vectors for the medial and lateral hamstring muscles are shown in Fig. 6. The major difference in the first basis vector of these two muscles was the amount of activity at the end of swing. For the lateral hamstring, the variability in the amount of activity during the stance phase was accounted for primarily by the second basis vector. The basis vectors for the anterior tibialis and gastrocnemius are shown in Fig. 7. For the anterior tibialis, the intersubject variability of the phasic activity pattern was small for the subjects evaluated in this study. The second and third basis vectors (Fig. 7) contributed to the variability in the timing and intensity of EMG activity during the swing and early stance. They also contributed to phasic activity during the middle to terminal stance, which is of lesser intensity and is normally present when surface electrodes are used to record anterior tibialis activity. The EMG activity of the gastrocnemius muscle was most efficiently represented by the

3.4%

VAS. MED.

I .3x

GLU.

m.

I .3%

MED. HAM.

1.a

AM).

LON.

2.8%

UT. HAH.

4.m

VAS. LAT.

2.w

ANT. TIB.

4.a

REC. FEM.

I .7X

GASTROC ,

I .4X

1 FIG. 9. Reconstruction of the ten muscles for subject 25, one of the subjects used to generate the basis vectors. The % MSE is shown to the right of the muscle name.

1

8

'

__ ORIGINAL

PERCENT OF CYCLE

-

'

RECONSTRUCTED

I BE

8

PERCENT a CYCLE

189

Subject 25

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TABLE 3 . Percent MSE in representation using principal components Subject Muscle Gluteus maximus Gluteus medius Adductor longus Vastus lateralis Rectus femoris Vastus medialis Medial hamstring Lateral hamstring Anterior tibialis Gastrocnemius

1

2

3

4

5

1.6 0.7 4.0 0.8 1.0 1.5 1.3 3.8 1.1 0.9

3.8 2.0 1.8 6.8 9.9 3.1 2.4 8.6 2.1 0.9

3.4 3.9 6.2 4.5 7.5 5.8 3.3 6.4 6.3 1.4

7.7 3.9 7.2 1.8 2.7 4.1 6.0 3.6 3.3 5.6

3.9 4.9 4.2 2.8 2.6 1.5 2.1 6.1 3.2 0.5

tors. The percent MSE between the original data and its reconstruction is also shown. For these two subjects, the maximum error in reconstruction was less than 5.5%.

A separate test of the universality of the basis vectors was also performed by leaving out some subjects and recalculating the basis vectors. The 40 subjects were divided into four groups of 10 subjects. Using three groups at a time, the basis vectors were recalculated four different times with data from 30 subjects in each set of input vectors. Figure 11 shows the reconstruction of phasic activity for the adductor muscle group from two representative subjects, using the four different sets of basis vec-

DISCUSSION In this paper,we have demonstrated that principal component analysis can be very effective in numerical representation of phasic electromyographic

GLU. MAX.

3.9%

VAS. MED.

I .5%

GLU.

4.9%

MED. HAM.

2. I %

4.3

LAT. HAM.

MU).

ADD. LON.

6. I % I

I

VAS. LAT.

REC. FEM.

0

-_

'

ORIGINAL

2.6%

PERCEN~OF CYCLE '

-

RECONSTRUCTED

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108

0

FIG. 10. Reconstruction of the ten muscles for a subject not included in the construction of the basis vectors.

ANT. TIB.

3.2%

GASTROC .

8.5%

'

PERCENi OF CYCLE '

188

S u b j e c t 38

257

DYNAMIC ELECTROMYOGRAPHY

FIG. 11. Reconstruction of the adductor muscle using four different sets of basis vectors generated by using different subsets of the subjects.

__

ADD. LON.

1.4%

ADD. LON.

2.9%

ADD. LON.

4.2%

ADD. LON.

I .5%

ADD. LON.

3.1%

ADD. LON.

4.1%

ADD. LON.

3.0%

ADD. LON.

5.5%

ORIGINAL

-

RECONSTRUCTED

data. We have also shown that the basis vectors are related to specific events in the gait cycle and therefore the corresponding weighting coefficients are directly interpretable. For the muscles evaluated in this study, the first basis vector of each muscle represented phasic activity common to all of the subjects. The weighting coefficients corresponding to the second and third basis vectors represented the distinguishing characteristic (or features) of electromyographic data of each subject. Feature selection forms the first step in developing a pattern recognition approach to analyzing gait data. Therefore, these features may be used for distinguishing one subject’s data from the other, in classification schemes similar to those used by Yamamoto (21) and Wong (20). The number of weighting coefficients and corresponding basis vectors required to reconstruct effectively or represent data waveforms depends on the variability among the waveforms of input data. The percent of cumulative power represented by the first three eigenvectors was significantly greater than that reported by Shiavi and Griffin (16). This may be due to the fact that they calculated the eigenvalues/eigenvectors utilizing the covariance matrix as opposed to the correlation matrix utilized

in this study. Additionally, the area vector representation may also have resulted in lower variability between subjects compared to the composite binary criterion used by Shiavi and Griffin (16) to represent phasic electromyographic data. In the creation of the average vectors for input into the expansion algorithm, no attempt was made to normalize the data separately with respect to the stance and swing phase of the gait cycle. In this group of normal subjects, the intrasubject and intersubject variability of the swing to stance ratio was small. This may not be true for patient groups, especially children with cerebral palsy. In that instance, the gait cycles will have to be separated and normalized with respect to stance and swing phases, and principal component analysis applied to an area vector normalized with respect to both amplitude and phase. This normalization may better represent the muscle activity as it occurs in a phase of the gait cycle and may reduce the variations due to stancekwing differences between subjects. For a given level of mean square error, principal component analysis is very efficient since only a few coefficients are necessary in the reconstruction of original data waveforms. If the purpose of numerical representation is only to reduce data stor-

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age requirements, principal component analysis may not be the most efficient method for representing small data sets, since the basis vectors are also needed for the reconstruction of data waveforms. Further, it should also be noted that in a general application of principal component analysis, the basis vectors may only be efficient in representing the original data set since they are derived from that data set. However, our results on the group of normal subjects demonstrate that the basis vectors can be used to represent other normal subjects, i.e., subjects not included in the calculation of the basis vectors. The technique becomes extremely powerful if the original data set is large enough to include all of the variants within the population. Numerical representation using principal component analysis is important for two reasons. First, it is a parsimonious representation of cyclic waveform data. Second, it may be very useful in identifying and classifying homogeneous subgroups within a larger patient population. Additional work needs to be done to understand the synergy of muscle activity within groups of muscles that perform similar functions in relation to the motion that occurs. For this reason, a pattern recognition approach seems appropriate, where numerical representations of EMG waveforms may be combined with similar representations of motion, force, moment, and temporalldistance parameters to develop a more holistic approach to gait evaluations. Extension of this analysis to electromyographic data from a group of cerebral palsy patients to determine its utility in clinical application is currently underway. Acknowledgment: This work was supported by NIH Grant AR 34886 and the New York State Department of Health. The authors wish to acknowledge the technical assistance of George Gorton and Janet Gainey and secretarial assistance of Mary Mistrulli.

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