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Endpoint error in smoothing and differentiating raw kinematic data: An evaluation of four popular methods
Pergamon
PII: S0021-9290(96)00079-6
J. Biomechanics,Vol.29, No. 12, pp. 1637 1642,1996 Copyright/~ 1996ElsevierScienceLtd. All rightsreserved Printedin GreatBritain 0021-9290/96 $15.00+ .00
TECHNICAL NOTE
ENDPOINT KINEMATIC
ERROR IN SMOOTHING DATA: AN EVALUATION
AND DIFFERENTIATING RAW OF FOUR POPULAR METHODS
Peter F. Vint a n d Richard N. Hinrichs Exercise and Sport Research Institute, Arizona State University, Tempe, AZ 85287-0404, U.S.A. Abstraet--'Endpoint error' describes the erratic behavior at the beginning and end of the computed acceleration data which is commonly observed after smoothing and differentiating raw displacement data. To evaluate endpoint error produced by four popular smoothing and differentiating techniques, Lanshammar's (1982, J. Biomechanics 15, 99-105) modification of the Pezzack et al. (1977, J. Biomechanics, 10, 377-382) raw angular displacement data set was truncated at three different locations corresponding to the major peaks in the criterion acceleration curve. Also, for each data subset, three padding conditions were applied. Each data subset was smoothed and differentiated using the Butterworth digital filter, cubic spline, quintic spline, and Fourier series to obtain acceleration values. RMS residual errors were calculated between the computed and criterion accelerations in the endpoint regions. Although no method completely eliminated endpoint error, the results demonstrated clear superiority of the quintic spline over the other three methods in producing accurate acceleration values close to the endpoints of the modified Pezzack et al. (1977) data set. In fact, the quintic spline performed best with non-padded data (cumulative error = 48.0 rad s-2). Conversely, when applied to non-padded data, the Butterworth digital filter produced wildly deviating values beginning more than the 10 points from the terminal data point (cumulative error = 226.6 rad s-2). Each of the four methods performed better when applied to data subsets padded by linear extrapolation (average cumulative error = 68.8 fads -2) than when applied to analogous subsets padded by reflection (average cumulative error = 86.1 rads-2). Copyright © 1996 Elsevier Science Ltd. Keywords: Endpoint error; Smoothing; Differentiation; Data padding.
INTRODUCTION The complete and accurate description of human motion typically necessitates that the investigator obtain measures of the displacement, velocity, and acceleration of the body of interest. However, in the presence of noise incurred during experimentation, differentiation of raw displacement data will yield grossly inaccurate velocity and acceleration values (Pezzack et al., 1977). In order to obtain reasonable derivatives, raw displacement data are typically subjected to some form of smoothing prior to differentiation. Several methods of smoothing have been introduced in the literature over the past three decades, each with varying degrees of success. The Butterworth digital filter (Pezzack et al., 1977; Winter et al., 1974), cubic spline (McLaughlin et al., 1977; Soudan and Dierckx, 1979), quintic spline (Wood and Jennings, 1979), and Fourier series (Anderssen and Bloomfield, 1974) include four of the most popular smoothing and differentiating routines as made evident from a survey of the recent biomechanics literature and the reviews of Wood (1982), Vaughan (1982), Woltring (1985, 1995) and Hatze (1990). These methods have also been incorporated into many of the commercially available motion analysis software packages. Although each of these four methods has yielded acceptable results on criterion test data sets (e.g. the data provided by Pezzack et al., 1977), pros and cons have been associated with the implementation of each of these procedures (Wood, 1982). One problem,
Received in final form 24 April 1996. Author to whom correspondence should be addressed: Richard N. Hinrichs, Ph.D., Exercise and Sport Research Institute, Department of Exercise Science and Physical Education, Arizona State University, Tempe, AZ 85287-0404, U.S.A.
however, which has been largely overlooked in the literature is that of 'endpoint error'. Throughout the rest of this manuscript, endpoint error will be used to describe the erratic behavior at the beginning and end of the computed acceleration data which is commonly observed after smoothing and differentiating raw displacement data.* Owing to mathematical constraints, the Butterworth digital filter, cubic spline, and Fourier series are each susceptible to endpoint error in higher derivative data. Although one can usually avoid endpoint error by collecting additional data on either side of the data of interest, there are some situations in which this cannot be done (e.g. impacts). Consider, for instance, a baseball bat hitting a ball. The frequency content of the impact will be substantially higher than the rest of the swing. If a researcher inputs coordinate data corresponding to the entire swing, the impact will tend to be oversmoothed thereby elongating its duration and attenuating the peak deceleration of the bat. The researcher will consequently be led to believe that the bat started to slow down before the impact actually occurred. A subsequent analysis of the net joint moments would find (falsely) that the musculature reversed its activity as it strived to slow down the bat before impact. Instead, the researcher would be advised to smooth pre-impact data only. In this case it would be impossible to collect additional data which could be discarded to avoid endpoint error as every data point up to impact would be of interest. As an alternative to collecting additional data, however, several studies have demonstrated that augmenting or
*Endpoint errors have also been referred to in the literature as 'end-point problems' (Phillips and Roberts, 1983; Vaughan, 1982), 'boundary effects' (Woltring, 1985) and 'edge effects' (D'Amico and Ferrigno, 1990).
1637
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Technical Note
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Sample Number Fig. 1. Digitized replica of criterion[analog)angular acceleration courtesy of Dr Graeme Wood, University of Western Australia. Points 95, 104, and 118 correspond to the three major peaks in the criterion acceleration curve. 'padding' the original data set with 5 20 points on either side of the terminal data points can reduce endpoint error when using either the cubic spline (McLaughlin et al., 1977; Phillips and Roberts, 1983; Zernicke et al., 1976) or the Butterworth digital filter (Smith, 1989). Endpoint error resulting from the Fourier series is attributable to a property known as Gibbs phenomenon (Carslaw, 1930; Cartwright, 1990) and can be minimized by odd periodic extension of the original data (Anderssen and Bloomfield, 1974). The purpose of this study was to evaluate the accuracy of various smoothing and differentiating techniques for computing accelerations near the endpoints of a data set. Four popular smoothing and differentiating methods were tested: (1) Butterworth digital filter (DF) followed by finite difference differentiation. (2) Cubic spline (CS) followed by differentiation of spline coefficients. (3) Quintic spline (QS) followed by differentiation of spline coefficients. (4) Fourier series (FS) followed by differentiation of the Fourier series coefficients.
MATERIALS AND METHODS Lanshammar's (1982) modification of the Pezzack et al. raw angular displacement data set (142 points) was truncated at three different locations corresponding to the major peaks in the criterion acceleration curve (see the appendix).* These peaks corresponded to points 95, 104, and 118 (Fig. 1). Thus, three data subsets were created: the first contained points 1-95; the second contained points 1-104; and the third contained points 1-118. Prior to smoothing and differentiation, each of the three data subsets were padded using each of the following schemes: no padding; padding via linear extrapolation; and padding via
+The original analog acceleration data have never been made available in tabular form (J. Pezzack, personal communication, 6 September 1994). The digitized replica of the Pezzack et al. analog acceleration data has been provided courtesy of Dr Graeme Wood, University of Western Australia.
reflection (Smith, 1989). Data subsets padded by the linear extrapolation and reflection schemes were padded by 20 points. Each data subset was smoothed and differentiated using the software package 'SMOOTH' (Wood, 1991). This package contains the programs used for comparing smoothing methods in Wood's 1982 review paper. Wood (1991) provides programs for the Butterworth digital filter, cubic spline, quintic spline, and Fourier series. However, Wood's implementation of the Butterworth digital filter automatically pads the data with 10 extra points at the beginning and end of the data set using linear extrapolation. Therefore, the Butterworth digital filtering program was adapted and separate programs were written to pad the data sets as desired. In each test using the Butterworth digital filter, finite difference methods were used to compute the second derivatives. While a three-point central difference method was used for the Butterworth digital filter on each of the padded data sets, forward and backward difference methods were also required to calculate the acceleration values at the terminal points (i = 1 and i = n) when the Butterworth digital filter was applied to the non-padded data set. The following finite difference equations were derived from standard numerical analysis techniques (Burden et al., 1981): for i = 1 (forward difference) al = (5xx -- l l x 2 + 7x 3 -- Xg)/(4At2), for i = 2 to n - 1 (central difference) ai = (xi+ x -- 2xi + x i - 1 ) / A t 2,
for i = n (backward difference) a, = (5x, - l l x , _ , + 7x,-2 - x,-3)/(4At2). When using the Butterworth digital filter, cubic spline, and quintic spline routines, the investigator must select the degree of smoothing which will be applied to the data. For the Butterworth digital filter, the extent of smoothing is determined by the cutoff frequency, fc, while for each of the spline routines, this parameter is determined by the smoothing factor, S. During each test, the appropriate cutoff frequency or smoothing factor was determined by minimizing the root mean square (RMS) of the residuals between the calculated acceleration and the
Technical Note
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Smoothing Method Fig. 2. Composite error scores based on RMS residual analysis between criterion and computed acceleration values in the endpoint regions of each data subset. R M S error values for each method across the last 10 points of each data subset were added to yield a composite endpoint error score. The sum of the R M S error values of all three data subsets is represented by the total height of a given bar. Individual RMS error scores are represented by the difference between the top and bottom of a particular band on a given bar (refer to legend). Key: D F = Butterworth digital filter; QS = quintic spline; CS = cubic spline; FS = Fourier series; and subscripts E = data padded by linear extrapolation; R = data padded by reflection; N = unpadded data. 80
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Sample Number Fig. 3. Criterion angular acceleration and accelerations computed from the first truncated region of the Pezzack et al. data set (points 1-95). For each method tested, endpoint error is evident more than 10 points before the terminal data point and worsens as the actual endpoint is approached. Key: ( ) criterion acceleration; (11--11) accelerations computed from data padded by linear extrapolation; (e---e) accelerations computed from data padded by reflection; ( x - - × ) accelerations computed from unpadded data.
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Technical Note
criterion acceleration [equation (1)]: RMS = ~ / ~ = 1(calculat~-' - criteri°n~)2
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To avoid the confounding effects of the endpoint error and to maintain the integrity of the majority of the body of the data set, residual analyses for cutoff frequency and smoothing factor determinations were carried out across all but the first and last 10 points of the original truncated data sets (i.e. for the 1-95 point set, the RMS residual analysis was performed over points
11-85). For the Fourier series, Wood (1991) employed the method of Anderssen and Bloomfield (1974) which computes the optimal filtering window for each derivative based upon the spectral analysis equivalent of the regulation procedure of Cullum (1971). Optimization of the regulating parameter, ~, is achieved via the principle of maximum likelihood (Anderssen and Bloomfield, 1974). The usual Fourier series requirement of n = 2p data points is ensured by odd periodic extension of the time series. This method of Fourier series data extension minimizes endpoint bias and has been recommended over extension by zero padding (Anderssen and Bloomfield, 1974). Each data subset was then smoothed and differentiated using each of the four methods tested. To quantify the accuracy of the computed accelerations near the terminal endpoints, an RMS residual analysis between the computed and criterion acceleration values was carried out over the last 10 points of the data
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subset (i.e. for the 1-95 point set, the RMS residual analysis was performed over points 86-95). RMS error values for each method across each data subset were combined to yield a composite endpoint error score. This composite score represents the summation of endpoint RMS residual errors which were computed for each of the three data subsets.
RESULTS The quintic spline (without padding) yielded the lowest cumulative endpoint error (48.0 rad s-2) while, when applied to nonpadded data, the Butterworth digital filter resulted in the highest cumulative error (226.6 rad s-2). When applied to the data subsets padded by linear extrapolation, the Butterworth digital filter (69.7 rad s-2), quintic spline (67.0 rad s-2), cubic spline (69.7 rads-2), and Fourier series (69.0 rads -2) all resulted in comparable endpoint error scores which were bettered only by the quintic spline (without padding). For each method tested, data subsets padded by linear extrapolation demonstrated 20% lower composite endpoint error than did analogous subsets padded by reflection (mean composite scores were 68.8 and 86.1 tad s-2 for data subsets padded by linear extrapolation and reflection, respectively). This result was anticipated as the reflection method created an inflection point in the displacement data which subsequently yielded zero acceleration at the endpoint of the data set (Figs 2-5).
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DISCUSSION The purpose of this study was to evaluate the accuracy of four popular smoothing and differentiating techniques for computing accelerations near the endpoints of a data set. Although no method completely eliminated endpoint error, the results of this investigation are consistent with the findings of Vaughan (1982), who had demonstrated that the quintic spline was noticeably better than either the cubic spline or the Butterworth digital filter at predicting the airborne acceleration of a dropped golfball, especially near the endpoints of the data. It should be noted, however, that Vaughan's data set is atypical of those usually found in studies of human movement. It is possible to optimize the quintic spline to fit a perfect parabola to the airborne displacement data of the falling projectile and thereby eliminate endpoint error altogether. With the exception of the quintic spline, the results of this study provide additional support for the practice of padding the data set before smoothing and differentiation and, hence, are consistent with the findings of Smith (1989), Phillips and Roberts (1989), Zernicke et al. (1976), and McLaughlin et al. (1977). However, whereas Smith (1989) found the reflection method to be somewhat superior to the linear extrapolation method when using the Butterworth digital filter, in the present study, the linear extrapolation method produced more accurate results when the magnitude of the terminal acceleration was relatively large. This result was primarily attributable to the nature of the
reflection algorithm which created an inflection point at the endpoint of the displacement data. An inflection point, by definition, will yield a second derivative equal to zero. Padding by reflection, while not as effective as linear extrapolation in any of the smoothing methods used in this study, was still superior to no padding at all (Fig. 2). The methods tested in this investigation included four of the most popular smoothing and differentiating routines as made evident from a survey of the recent biomechanics literature and the reviews of Wood (1982), Vaughan (1982), Woltring (1985, 1995) and Hatze (1990). These methods have also been incorporated into many of the commercially available motion analysis software packages. There are certainly many other methods which are available. The LAMBDA algorithm (D'Amico and Ferrigno, 1990), for example, offers a great deal of promise as a method which allows for the automatic and objective selection of the optimal smoothing parameter. In a recent paper (D'Amico and Ferrigno, 1992), its developers have claimed that LAMBDA outperforms the optimally regularized Fourier series algorithm (Hatze, 1981) and the generalized cross-validated quintic spline (Woltring, 1985, 1986) - - e v e n in the endpoint regions. Unfortunately, LAMBDA is only available through a proprietary motion analysis software package. It should be stated that the results of this study are specific to subsets of the modified Pezzack et al. (1977) data. Clearly, additional research is needed to determine if the results of this study can be generalized and applied to real biomechanical data.
1642
Technical Note
Herein lies the problem of this line of research. Acceleration data which are derived under artificial test conditions (e.g. Pezzack et al., 1977) may offer true criterion values but are typically unrepresentative of real biomechanical data (Smith, 1989). As others have done before us (e.g. Hatze, 1981; Lanshammar, 1982; Wood, 1982), we have added noise to the Pezzack et al. data to facilitate a more rigorous evaluation of the smoothing and differentiating methods which were tested. It is acknowledged that it would have been best to test these methods on real experimental data. However, it would have been difficult, if not impossible, to do this because true acceleration values are typically not known and hence cannot be used as criteria. Researchers should be aware of this dilemma when considering the merits of a particular smoothing algorithm. In conclusion, of the four popular methods evaluated in the current investigation, the quintic spline appeared to be best for smoothing raw kinematic data and for computing accurate accelerations in the endpoint regions of the modified Pezzack et al. (1977) data set. In fact, the quintic spline worked best when applied to a non-padded data set. When implemented without padding the endpoints of the data set, the commonly used Butterworth digital filter produced wildly deviating values beginning more than 10 points from the terminal data point. When applied to data subsets padded by linear extrapolation, the Butterworth digital filter, quintic spline, cubic spline, and Fourier series all resulted in comparable endpoint error scores which were bettered only by the quintic spline (without padding). For each method tested, data subsets padded by linear extrapolation demonstrated lower endpoint error than did analogous subsets padded by reflection.
REFERENCES
A nderssen, R. S. and Bloomfield, P. (1974) Numerical differentiation procedures for non-exact data. Numer. Math. 22, 1157 1182. Burden, R. L., Faires, J. D. and Reynolds, A. C. (1981) Numerical Analysis (2nd Edn.). PWS Publishers, Boston. Carslaw, H. S. (1930) Introduction to the Theory of Fourier's Series and Integrals (3rd Edn.) Dover, New York. Cartwright, M. (1990) Fourier Series for Mathematicians, Scientists and Engineers. Ellis Horwood, New York. Cullum, J. (1971) Numerical differentiation and regulation. SIAM J. Numer. Anal. 8, 254-265. D'Amico, M. and Ferrigno, G. (1990). Technique for the evaluation of derivatives from noisy biomechanical displacement data using a model-based bandwidth-selection procedure. Med. Biol. Engng Comput. 28, 407-415. D'Amico, M. and Ferrigno, G. (1992) Comparison between the more recent techniques for smoothing and derivative assessment in biomechanics. Med. Biol. Engng Comput. 30, 193 204. Hatze, H. (1981) The use of optimally regularized Fourier series for estimating higher-order derivatives of noisy biomechanical data. J. Biomechanics 14, 13 18. Hatze, H. (1990) Data conditioning and differentiation techniques. In Biomechanics of Human Movement: Applications in Rehabilitation, Sports and Ergonomics (Edited by Berme, N. and Cappozzo, A), pp. 237-248. Bertec Corporation, Worthington, OH. Lanshammar, H. (1982) On practical evaluation of differentiation techniques for human gait analysis. J. Biomechanics 15, 99-105. McLaughlin, T. M., Diltman, C. J. and Lardner, T. J. (1977) Biomechanical analysis with cubic spline functions. Res. Q. 48, 569 582. Pezzack, J. C., Norman, R. W. and Winter, D. A. (1977) An assessment of derivative determining techniques used for motion analysis. J. Biomechanics 10, 377-382. Phillips, S. J. and Roberts, E. M. (1983) Sptine solution to terminal zero acceleration problems in biomechanical data. Med. Sci. Sports Exerc 15, 382-387.
Smith, G. (1989) Padding point extrapolation techniques for the Butterworth digital filter. J. Biomechanics 22, 967-971. Soudan, K. and Dierckx, P. (1979) Calculation of derivatives and Fourier-coefficients of human motion data, while using spline functions. J. Biomechanics 12, 21-26. Vaughan, C. L. (1982) Smoothing and differentiation of displacement-time data: an application of splines and digital filtering. Int. J. Bio-Med. Comp. 13, 375 396. Winter, D. A., Sidwell, H. G. and Hobson, D. A. (1974). Measurement and reduction of noise in kinematics of locomotion. J. Biomechanics 7, 157-159. Woltring, H. J. (1985) On optimal smoothing and derivative estimation form noisy displacement data in biomechanics. Hum. Mvt Sci. 4, 229-245. Woltring, H. J. (1986) A F O R T R A N package for generalized cross-validatory spline smoothing and differentiation. Adv. Engn 9 Software 8, 104-113. Woltring, H. J. (1995) Smoothing and differentiation techniques applied to 3-D data. In Three-dimensional Analysis of Human Movement (Edited by Allard, P., Stokes, I. A. F., and Blanchi, J.-P.), pp. 79-99 Human Kinetics, Champaign, IL. Wood, G. A. (1982) Data smoothing and differentiation procedures in biomechanics. In R. L. Terjung (Ed.), Exercise and Sport Science Reviews, Vol. 10 (Edited by Terjung, R. L.), pp. 308 362. Franklin Institute Press, Philadelphia. Wood, G. A. (1991) Laboratory exercise in data smoothing and differentiation. Paper Presented at the 3rd National Symposium on Teaching Kinesiology and Biomechanics in Sports. Iowa State University, Ames, IA. Wood, G. A. and Jennings, L. S. (1979) On the use of spline functions for data smoothing. J. Biomechanics 12, 477-479. Zernicke, R. F., Caldwell, G. and Roberts, E. M, (1976) Fitting biomechanical data with cubic spline functions. Res. Q. 47, 9-19.
APPENDIX
Digitized reproduction of Pezzack et al. (1977) analog acceleration data. Courtesy of Dr Graeme Wood, University of Western Australia. Read across (n = 142; At = 0.0201 s; units = rad s - 2). 0.0000 11.2559 6.9002 12.0132 2.1317 6.7036 - 0.8388 - 5.1941 --12.7041 - 16.8098 - 12.8567 - 12.0719 - 9.0157 - 1.5880 5.2150 5.6382 8.7372 7.9724 36.8281 63.7493 -- 46.4395 --72.8745 - 18.7420 14.8536 21.8410 14.0482 2.4879 -2.7355 1.7317
0.0000 10.3661 11.8463 5.5654 3.7727 6.9231 - 3.3770 - 5.4091 -14.2113 - 17.3823 - 11.7014 - 11.6754 - 7.0292 0.1245 6.0321 6.8504 8.0801 9.9837 45.9046 48.4565 - 67.9201 -53.2223 - 19.8545 37.1942 18.1663 9.5020 0.8712 -4.2197 4.7378
3.0221 8.3941 13.1885 0.0069 6.7221 7.8175 - 4.3438 -- 5.8552 -15.7212 - 18.0002 - 12.5057 - 11.5053 -- 5.9946 2.6354 7.6032 7.3479 8.6690 13.8321 55.9493 18.9879 - 85.8038 -38.1749 - 7.2523 40.9521 11.2460 8.1188 - 1.0786 -4.5081
7.1397 5.4586 13.6488 0.3641 9.1319 5.9401 -4.5612 -6.8748 -16.8054 -16.0863 -12.3400 -11.6477 -4.8061 5.0069 6.5306 9.3425 8.7642 18.3173 63.1218 -5.7848 -96.0019 -24.1167 7.1091 30.4765 11.1952 6.0769 -1.9539 -3.9986
10.0329 3.9121 14.0208 1.5719 8.2096 2.2294 -4.7513 -8.7070 -17.0763 -13.5466 -12.0316 - 11.0592 -3.1786 5.4394 6.0740 9.8927 7.9082 24.7846 68.3515 -20.4538 -91.5508 -18.0062 3.5716 22.1798 13.1521 3.6119 -2.3031 -1.3109
PII: S0021-9290(96)00079-6
J. Biomechanics,Vol.29, No. 12, pp. 1637 1642,1996 Copyright/~ 1996ElsevierScienceLtd. All rightsreserved Printedin GreatBritain 0021-9290/96 $15.00+ .00
TECHNICAL NOTE
ENDPOINT KINEMATIC
ERROR IN SMOOTHING DATA: AN EVALUATION
AND DIFFERENTIATING RAW OF FOUR POPULAR METHODS
Peter F. Vint a n d Richard N. Hinrichs Exercise and Sport Research Institute, Arizona State University, Tempe, AZ 85287-0404, U.S.A. Abstraet--'Endpoint error' describes the erratic behavior at the beginning and end of the computed acceleration data which is commonly observed after smoothing and differentiating raw displacement data. To evaluate endpoint error produced by four popular smoothing and differentiating techniques, Lanshammar's (1982, J. Biomechanics 15, 99-105) modification of the Pezzack et al. (1977, J. Biomechanics, 10, 377-382) raw angular displacement data set was truncated at three different locations corresponding to the major peaks in the criterion acceleration curve. Also, for each data subset, three padding conditions were applied. Each data subset was smoothed and differentiated using the Butterworth digital filter, cubic spline, quintic spline, and Fourier series to obtain acceleration values. RMS residual errors were calculated between the computed and criterion accelerations in the endpoint regions. Although no method completely eliminated endpoint error, the results demonstrated clear superiority of the quintic spline over the other three methods in producing accurate acceleration values close to the endpoints of the modified Pezzack et al. (1977) data set. In fact, the quintic spline performed best with non-padded data (cumulative error = 48.0 rad s-2). Conversely, when applied to non-padded data, the Butterworth digital filter produced wildly deviating values beginning more than the 10 points from the terminal data point (cumulative error = 226.6 rad s-2). Each of the four methods performed better when applied to data subsets padded by linear extrapolation (average cumulative error = 68.8 fads -2) than when applied to analogous subsets padded by reflection (average cumulative error = 86.1 rads-2). Copyright © 1996 Elsevier Science Ltd. Keywords: Endpoint error; Smoothing; Differentiation; Data padding.
INTRODUCTION The complete and accurate description of human motion typically necessitates that the investigator obtain measures of the displacement, velocity, and acceleration of the body of interest. However, in the presence of noise incurred during experimentation, differentiation of raw displacement data will yield grossly inaccurate velocity and acceleration values (Pezzack et al., 1977). In order to obtain reasonable derivatives, raw displacement data are typically subjected to some form of smoothing prior to differentiation. Several methods of smoothing have been introduced in the literature over the past three decades, each with varying degrees of success. The Butterworth digital filter (Pezzack et al., 1977; Winter et al., 1974), cubic spline (McLaughlin et al., 1977; Soudan and Dierckx, 1979), quintic spline (Wood and Jennings, 1979), and Fourier series (Anderssen and Bloomfield, 1974) include four of the most popular smoothing and differentiating routines as made evident from a survey of the recent biomechanics literature and the reviews of Wood (1982), Vaughan (1982), Woltring (1985, 1995) and Hatze (1990). These methods have also been incorporated into many of the commercially available motion analysis software packages. Although each of these four methods has yielded acceptable results on criterion test data sets (e.g. the data provided by Pezzack et al., 1977), pros and cons have been associated with the implementation of each of these procedures (Wood, 1982). One problem,
Received in final form 24 April 1996. Author to whom correspondence should be addressed: Richard N. Hinrichs, Ph.D., Exercise and Sport Research Institute, Department of Exercise Science and Physical Education, Arizona State University, Tempe, AZ 85287-0404, U.S.A.
however, which has been largely overlooked in the literature is that of 'endpoint error'. Throughout the rest of this manuscript, endpoint error will be used to describe the erratic behavior at the beginning and end of the computed acceleration data which is commonly observed after smoothing and differentiating raw displacement data.* Owing to mathematical constraints, the Butterworth digital filter, cubic spline, and Fourier series are each susceptible to endpoint error in higher derivative data. Although one can usually avoid endpoint error by collecting additional data on either side of the data of interest, there are some situations in which this cannot be done (e.g. impacts). Consider, for instance, a baseball bat hitting a ball. The frequency content of the impact will be substantially higher than the rest of the swing. If a researcher inputs coordinate data corresponding to the entire swing, the impact will tend to be oversmoothed thereby elongating its duration and attenuating the peak deceleration of the bat. The researcher will consequently be led to believe that the bat started to slow down before the impact actually occurred. A subsequent analysis of the net joint moments would find (falsely) that the musculature reversed its activity as it strived to slow down the bat before impact. Instead, the researcher would be advised to smooth pre-impact data only. In this case it would be impossible to collect additional data which could be discarded to avoid endpoint error as every data point up to impact would be of interest. As an alternative to collecting additional data, however, several studies have demonstrated that augmenting or
*Endpoint errors have also been referred to in the literature as 'end-point problems' (Phillips and Roberts, 1983; Vaughan, 1982), 'boundary effects' (Woltring, 1985) and 'edge effects' (D'Amico and Ferrigno, 1990).
1637
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Sample Number Fig. 1. Digitized replica of criterion[analog)angular acceleration courtesy of Dr Graeme Wood, University of Western Australia. Points 95, 104, and 118 correspond to the three major peaks in the criterion acceleration curve. 'padding' the original data set with 5 20 points on either side of the terminal data points can reduce endpoint error when using either the cubic spline (McLaughlin et al., 1977; Phillips and Roberts, 1983; Zernicke et al., 1976) or the Butterworth digital filter (Smith, 1989). Endpoint error resulting from the Fourier series is attributable to a property known as Gibbs phenomenon (Carslaw, 1930; Cartwright, 1990) and can be minimized by odd periodic extension of the original data (Anderssen and Bloomfield, 1974). The purpose of this study was to evaluate the accuracy of various smoothing and differentiating techniques for computing accelerations near the endpoints of a data set. Four popular smoothing and differentiating methods were tested: (1) Butterworth digital filter (DF) followed by finite difference differentiation. (2) Cubic spline (CS) followed by differentiation of spline coefficients. (3) Quintic spline (QS) followed by differentiation of spline coefficients. (4) Fourier series (FS) followed by differentiation of the Fourier series coefficients.
MATERIALS AND METHODS Lanshammar's (1982) modification of the Pezzack et al. raw angular displacement data set (142 points) was truncated at three different locations corresponding to the major peaks in the criterion acceleration curve (see the appendix).* These peaks corresponded to points 95, 104, and 118 (Fig. 1). Thus, three data subsets were created: the first contained points 1-95; the second contained points 1-104; and the third contained points 1-118. Prior to smoothing and differentiation, each of the three data subsets were padded using each of the following schemes: no padding; padding via linear extrapolation; and padding via
+The original analog acceleration data have never been made available in tabular form (J. Pezzack, personal communication, 6 September 1994). The digitized replica of the Pezzack et al. analog acceleration data has been provided courtesy of Dr Graeme Wood, University of Western Australia.
reflection (Smith, 1989). Data subsets padded by the linear extrapolation and reflection schemes were padded by 20 points. Each data subset was smoothed and differentiated using the software package 'SMOOTH' (Wood, 1991). This package contains the programs used for comparing smoothing methods in Wood's 1982 review paper. Wood (1991) provides programs for the Butterworth digital filter, cubic spline, quintic spline, and Fourier series. However, Wood's implementation of the Butterworth digital filter automatically pads the data with 10 extra points at the beginning and end of the data set using linear extrapolation. Therefore, the Butterworth digital filtering program was adapted and separate programs were written to pad the data sets as desired. In each test using the Butterworth digital filter, finite difference methods were used to compute the second derivatives. While a three-point central difference method was used for the Butterworth digital filter on each of the padded data sets, forward and backward difference methods were also required to calculate the acceleration values at the terminal points (i = 1 and i = n) when the Butterworth digital filter was applied to the non-padded data set. The following finite difference equations were derived from standard numerical analysis techniques (Burden et al., 1981): for i = 1 (forward difference) al = (5xx -- l l x 2 + 7x 3 -- Xg)/(4At2), for i = 2 to n - 1 (central difference) ai = (xi+ x -- 2xi + x i - 1 ) / A t 2,
for i = n (backward difference) a, = (5x, - l l x , _ , + 7x,-2 - x,-3)/(4At2). When using the Butterworth digital filter, cubic spline, and quintic spline routines, the investigator must select the degree of smoothing which will be applied to the data. For the Butterworth digital filter, the extent of smoothing is determined by the cutoff frequency, fc, while for each of the spline routines, this parameter is determined by the smoothing factor, S. During each test, the appropriate cutoff frequency or smoothing factor was determined by minimizing the root mean square (RMS) of the residuals between the calculated acceleration and the
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Smoothing Method Fig. 2. Composite error scores based on RMS residual analysis between criterion and computed acceleration values in the endpoint regions of each data subset. R M S error values for each method across the last 10 points of each data subset were added to yield a composite endpoint error score. The sum of the R M S error values of all three data subsets is represented by the total height of a given bar. Individual RMS error scores are represented by the difference between the top and bottom of a particular band on a given bar (refer to legend). Key: D F = Butterworth digital filter; QS = quintic spline; CS = cubic spline; FS = Fourier series; and subscripts E = data padded by linear extrapolation; R = data padded by reflection; N = unpadded data. 80
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To avoid the confounding effects of the endpoint error and to maintain the integrity of the majority of the body of the data set, residual analyses for cutoff frequency and smoothing factor determinations were carried out across all but the first and last 10 points of the original truncated data sets (i.e. for the 1-95 point set, the RMS residual analysis was performed over points
11-85). For the Fourier series, Wood (1991) employed the method of Anderssen and Bloomfield (1974) which computes the optimal filtering window for each derivative based upon the spectral analysis equivalent of the regulation procedure of Cullum (1971). Optimization of the regulating parameter, ~, is achieved via the principle of maximum likelihood (Anderssen and Bloomfield, 1974). The usual Fourier series requirement of n = 2p data points is ensured by odd periodic extension of the time series. This method of Fourier series data extension minimizes endpoint bias and has been recommended over extension by zero padding (Anderssen and Bloomfield, 1974). Each data subset was then smoothed and differentiated using each of the four methods tested. To quantify the accuracy of the computed accelerations near the terminal endpoints, an RMS residual analysis between the computed and criterion acceleration values was carried out over the last 10 points of the data
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RESULTS The quintic spline (without padding) yielded the lowest cumulative endpoint error (48.0 rad s-2) while, when applied to nonpadded data, the Butterworth digital filter resulted in the highest cumulative error (226.6 rad s-2). When applied to the data subsets padded by linear extrapolation, the Butterworth digital filter (69.7 rad s-2), quintic spline (67.0 rad s-2), cubic spline (69.7 rads-2), and Fourier series (69.0 rads -2) all resulted in comparable endpoint error scores which were bettered only by the quintic spline (without padding). For each method tested, data subsets padded by linear extrapolation demonstrated 20% lower composite endpoint error than did analogous subsets padded by reflection (mean composite scores were 68.8 and 86.1 tad s-2 for data subsets padded by linear extrapolation and reflection, respectively). This result was anticipated as the reflection method created an inflection point in the displacement data which subsequently yielded zero acceleration at the endpoint of the data set (Figs 2-5).
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DISCUSSION The purpose of this study was to evaluate the accuracy of four popular smoothing and differentiating techniques for computing accelerations near the endpoints of a data set. Although no method completely eliminated endpoint error, the results of this investigation are consistent with the findings of Vaughan (1982), who had demonstrated that the quintic spline was noticeably better than either the cubic spline or the Butterworth digital filter at predicting the airborne acceleration of a dropped golfball, especially near the endpoints of the data. It should be noted, however, that Vaughan's data set is atypical of those usually found in studies of human movement. It is possible to optimize the quintic spline to fit a perfect parabola to the airborne displacement data of the falling projectile and thereby eliminate endpoint error altogether. With the exception of the quintic spline, the results of this study provide additional support for the practice of padding the data set before smoothing and differentiation and, hence, are consistent with the findings of Smith (1989), Phillips and Roberts (1989), Zernicke et al. (1976), and McLaughlin et al. (1977). However, whereas Smith (1989) found the reflection method to be somewhat superior to the linear extrapolation method when using the Butterworth digital filter, in the present study, the linear extrapolation method produced more accurate results when the magnitude of the terminal acceleration was relatively large. This result was primarily attributable to the nature of the
reflection algorithm which created an inflection point at the endpoint of the displacement data. An inflection point, by definition, will yield a second derivative equal to zero. Padding by reflection, while not as effective as linear extrapolation in any of the smoothing methods used in this study, was still superior to no padding at all (Fig. 2). The methods tested in this investigation included four of the most popular smoothing and differentiating routines as made evident from a survey of the recent biomechanics literature and the reviews of Wood (1982), Vaughan (1982), Woltring (1985, 1995) and Hatze (1990). These methods have also been incorporated into many of the commercially available motion analysis software packages. There are certainly many other methods which are available. The LAMBDA algorithm (D'Amico and Ferrigno, 1990), for example, offers a great deal of promise as a method which allows for the automatic and objective selection of the optimal smoothing parameter. In a recent paper (D'Amico and Ferrigno, 1992), its developers have claimed that LAMBDA outperforms the optimally regularized Fourier series algorithm (Hatze, 1981) and the generalized cross-validated quintic spline (Woltring, 1985, 1986) - - e v e n in the endpoint regions. Unfortunately, LAMBDA is only available through a proprietary motion analysis software package. It should be stated that the results of this study are specific to subsets of the modified Pezzack et al. (1977) data. Clearly, additional research is needed to determine if the results of this study can be generalized and applied to real biomechanical data.
1642
Technical Note
Herein lies the problem of this line of research. Acceleration data which are derived under artificial test conditions (e.g. Pezzack et al., 1977) may offer true criterion values but are typically unrepresentative of real biomechanical data (Smith, 1989). As others have done before us (e.g. Hatze, 1981; Lanshammar, 1982; Wood, 1982), we have added noise to the Pezzack et al. data to facilitate a more rigorous evaluation of the smoothing and differentiating methods which were tested. It is acknowledged that it would have been best to test these methods on real experimental data. However, it would have been difficult, if not impossible, to do this because true acceleration values are typically not known and hence cannot be used as criteria. Researchers should be aware of this dilemma when considering the merits of a particular smoothing algorithm. In conclusion, of the four popular methods evaluated in the current investigation, the quintic spline appeared to be best for smoothing raw kinematic data and for computing accurate accelerations in the endpoint regions of the modified Pezzack et al. (1977) data set. In fact, the quintic spline worked best when applied to a non-padded data set. When implemented without padding the endpoints of the data set, the commonly used Butterworth digital filter produced wildly deviating values beginning more than 10 points from the terminal data point. When applied to data subsets padded by linear extrapolation, the Butterworth digital filter, quintic spline, cubic spline, and Fourier series all resulted in comparable endpoint error scores which were bettered only by the quintic spline (without padding). For each method tested, data subsets padded by linear extrapolation demonstrated lower endpoint error than did analogous subsets padded by reflection.
REFERENCES
A nderssen, R. S. and Bloomfield, P. (1974) Numerical differentiation procedures for non-exact data. Numer. Math. 22, 1157 1182. Burden, R. L., Faires, J. D. and Reynolds, A. C. (1981) Numerical Analysis (2nd Edn.). PWS Publishers, Boston. Carslaw, H. S. (1930) Introduction to the Theory of Fourier's Series and Integrals (3rd Edn.) Dover, New York. Cartwright, M. (1990) Fourier Series for Mathematicians, Scientists and Engineers. Ellis Horwood, New York. Cullum, J. (1971) Numerical differentiation and regulation. SIAM J. Numer. Anal. 8, 254-265. D'Amico, M. and Ferrigno, G. (1990). Technique for the evaluation of derivatives from noisy biomechanical displacement data using a model-based bandwidth-selection procedure. Med. Biol. Engng Comput. 28, 407-415. D'Amico, M. and Ferrigno, G. (1992) Comparison between the more recent techniques for smoothing and derivative assessment in biomechanics. Med. Biol. Engng Comput. 30, 193 204. Hatze, H. (1981) The use of optimally regularized Fourier series for estimating higher-order derivatives of noisy biomechanical data. J. Biomechanics 14, 13 18. Hatze, H. (1990) Data conditioning and differentiation techniques. In Biomechanics of Human Movement: Applications in Rehabilitation, Sports and Ergonomics (Edited by Berme, N. and Cappozzo, A), pp. 237-248. Bertec Corporation, Worthington, OH. Lanshammar, H. (1982) On practical evaluation of differentiation techniques for human gait analysis. J. Biomechanics 15, 99-105. McLaughlin, T. M., Diltman, C. J. and Lardner, T. J. (1977) Biomechanical analysis with cubic spline functions. Res. Q. 48, 569 582. Pezzack, J. C., Norman, R. W. and Winter, D. A. (1977) An assessment of derivative determining techniques used for motion analysis. J. Biomechanics 10, 377-382. Phillips, S. J. and Roberts, E. M. (1983) Sptine solution to terminal zero acceleration problems in biomechanical data. Med. Sci. Sports Exerc 15, 382-387.
Smith, G. (1989) Padding point extrapolation techniques for the Butterworth digital filter. J. Biomechanics 22, 967-971. Soudan, K. and Dierckx, P. (1979) Calculation of derivatives and Fourier-coefficients of human motion data, while using spline functions. J. Biomechanics 12, 21-26. Vaughan, C. L. (1982) Smoothing and differentiation of displacement-time data: an application of splines and digital filtering. Int. J. Bio-Med. Comp. 13, 375 396. Winter, D. A., Sidwell, H. G. and Hobson, D. A. (1974). Measurement and reduction of noise in kinematics of locomotion. J. Biomechanics 7, 157-159. Woltring, H. J. (1985) On optimal smoothing and derivative estimation form noisy displacement data in biomechanics. Hum. Mvt Sci. 4, 229-245. Woltring, H. J. (1986) A F O R T R A N package for generalized cross-validatory spline smoothing and differentiation. Adv. Engn 9 Software 8, 104-113. Woltring, H. J. (1995) Smoothing and differentiation techniques applied to 3-D data. In Three-dimensional Analysis of Human Movement (Edited by Allard, P., Stokes, I. A. F., and Blanchi, J.-P.), pp. 79-99 Human Kinetics, Champaign, IL. Wood, G. A. (1982) Data smoothing and differentiation procedures in biomechanics. In R. L. Terjung (Ed.), Exercise and Sport Science Reviews, Vol. 10 (Edited by Terjung, R. L.), pp. 308 362. Franklin Institute Press, Philadelphia. Wood, G. A. (1991) Laboratory exercise in data smoothing and differentiation. Paper Presented at the 3rd National Symposium on Teaching Kinesiology and Biomechanics in Sports. Iowa State University, Ames, IA. Wood, G. A. and Jennings, L. S. (1979) On the use of spline functions for data smoothing. J. Biomechanics 12, 477-479. Zernicke, R. F., Caldwell, G. and Roberts, E. M, (1976) Fitting biomechanical data with cubic spline functions. Res. Q. 47, 9-19.
APPENDIX
Digitized reproduction of Pezzack et al. (1977) analog acceleration data. Courtesy of Dr Graeme Wood, University of Western Australia. Read across (n = 142; At = 0.0201 s; units = rad s - 2). 0.0000 11.2559 6.9002 12.0132 2.1317 6.7036 - 0.8388 - 5.1941 --12.7041 - 16.8098 - 12.8567 - 12.0719 - 9.0157 - 1.5880 5.2150 5.6382 8.7372 7.9724 36.8281 63.7493 -- 46.4395 --72.8745 - 18.7420 14.8536 21.8410 14.0482 2.4879 -2.7355 1.7317
0.0000 10.3661 11.8463 5.5654 3.7727 6.9231 - 3.3770 - 5.4091 -14.2113 - 17.3823 - 11.7014 - 11.6754 - 7.0292 0.1245 6.0321 6.8504 8.0801 9.9837 45.9046 48.4565 - 67.9201 -53.2223 - 19.8545 37.1942 18.1663 9.5020 0.8712 -4.2197 4.7378
3.0221 8.3941 13.1885 0.0069 6.7221 7.8175 - 4.3438 -- 5.8552 -15.7212 - 18.0002 - 12.5057 - 11.5053 -- 5.9946 2.6354 7.6032 7.3479 8.6690 13.8321 55.9493 18.9879 - 85.8038 -38.1749 - 7.2523 40.9521 11.2460 8.1188 - 1.0786 -4.5081
7.1397 5.4586 13.6488 0.3641 9.1319 5.9401 -4.5612 -6.8748 -16.8054 -16.0863 -12.3400 -11.6477 -4.8061 5.0069 6.5306 9.3425 8.7642 18.3173 63.1218 -5.7848 -96.0019 -24.1167 7.1091 30.4765 11.1952 6.0769 -1.9539 -3.9986
10.0329 3.9121 14.0208 1.5719 8.2096 2.2294 -4.7513 -8.7070 -17.0763 -13.5466 -12.0316 - 11.0592 -3.1786 5.4394 6.0740 9.8927 7.9082 24.7846 68.3515 -20.4538 -91.5508 -18.0062 3.5716 22.1798 13.1521 3.6119 -2.3031 -1.3109