An Automatic Sequential Smoothing Method for Processing Biomechanical Kinematic Signals

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1 An Automatic Sequential Smoothing Method or Processing Biomechanical Kinematic Signals F.J. ALONSO, F. ROMERO, D.R. SALGADO Department o Mechanical, Energetic and Material Engineering University o Extremadura Escuela de Ingenierías Industriales, Avda. de Elvas s/n, 66, Badajoz SPAIN jas@unex.es Abstract: - he combined application o Singular Spectrum Analysis (SSA) and Cluster Analysis to the automatic smoothing o raw kinematic signals is an alternative to the use o traditional digital iltering and spline based methods. SSA is a non parametric technique that decomposes original time series into a number o additive time series each o which can be easily identiied as being part o the noise present in the acquired signal. Nevertheless, the smoothing automation is not a trivial task. his work presents a heuristic automatic smoothing procedure or processing kinematic biomechanical signals based in sequential SSA. Cluster analysis is used to group the SSA decomposition in order to obtain several independent components in the requency domain. he procedure eliminates iteratively the noise present in the signal in a simple and intuitive way. he new method is applied to several signals to demonstrate its perormance. Key-Words: - Signal processing, Smoothing, Noise removal, Singular Spectrum Analysis, Signal dierentiation, Automatic smoothing, Cluster analysis. Introduction Motion capture systems used in biomechanical analysis introduce introduce systematic and random measurement errors that appear in the orm o highrequency noise in the recorded displacement signals. Raw displacement signals must be iltered or smoothed prior to dierentiation to avoid noise ampliication due to the ill-posed derivative estimation process [-3]. he iltering o displacement signals to obtain noiseless velocities and accelerations has been extensively treated in the literature. raditional iltering techniques include Digital Butterworth ilters, splines, and ilters based on spectral analysis [-5]. In order to ilter non-stationary signals, advanced iltering techniques like Discrete Wavelet ransorms [6], the Wigner Function [7-8], nonstationary Butterworth ilter [9] and Singular Spectrum Analysis (SSA) [] have been used. Nonetheless, the drawback in these cases is the complexity o devising an automatic and systematic procedure. A mother wavelet unction must be selected when using Discrete Wavelet ransorms, the iltering unction parameters must be chosen when using the Wigner Function, window length and grouping strategy must be selected when using SSA []. he goal o this paper is to demonstrate the advantages o automatic smoothing methods based on sequential Singular Spectrum Analysis (SSA) and cluster analysis techniques. SSA is a non-parametric technique that decomposes original time series into a number o additive time series each o which can be easily identiied as being part o the noise present in the acquired signal. his work presents a heuristic automatic smoothing procedure or processing kinematic biomechanical signals based on sequential SSA. Cluster analysis is used to group the SSA decomposition in order to obtain several independent components in the requency domain. he procedure then applies sequential SSA to eliminate iteratively the noise present in the signal in a simple and intuitive way. Singular Spectrum Analysis Singular spectrum analysis is a novel nonparametric technique o time series analysis based on principles o multivariate statistics. It decomposes a given time series into an additive set o independent time series. he set o series resulting rom the decomposition can be interpreted as consisting o a trend representing the signal mean at each instant, a set o periodic series, and an aperiodic noise []. he original application o SSA was to extract trends rom climatic and geophysical time ISBN:

2 series and to identiy periodic motion in complex dynamical systems. he SSA method builds a Hankel matrix, called the trajectory matrix, rom the original time series in a process called embedding. his matrix consists o vectors obtained by means o a sliding window that traverses the series. he trajectory matrix is then subjected to a singular value decomposition (SVD). he SVD decomposes the trajectory matrix into a sum o unit-rank matrices known as elementary matrices. Each o these matrices can be transormed into a reconstructed time series. Elementary matrices are no longer Hankel matrices, but an approximate time series may be recovered by taking the average o the diagonals (diagonal averaging). he resulting time series are called principal components []. he sum o all the principal components is equal to the original time series. he objective is to obtain a requency decomposition o the original signal in which the latent low-requency signal can be detected in a simple ashion. he SSA decomposition algorithm will be described in the ollowing. A more detailed explanation may be ound in Golyandina et al. []. he above description o SSA may be expressed in ormal terms as ollows: Step. Embedding Let F = (,,, ) be the length N time series N representing the original signal. Let L be the window length, with < L < N and L an integer. Each column X j o the Hankel matrix corresponds to the "snapshot" taken by the sliding window: X j = ( j, j,, j+ L ), j =,,, K where K = N L + is the number o columns, i.e., the number o dierent possible positions o the said window. he matrix X = (X, X,, XK ) is a Hankel matrix since all elements on the diagonal i + j = c are equal. his matrix is sometimes reerred to as the trajectory matrix. he orm o this matrix is: = L L L 3 L N L N N N L+ X () Step. Singular value decomposition (SVD) o the trajectory matrix. It can be proven that the trajectory matrix (or any matrix, or that matter) may be expressed as the summation o d rank-one elementary matrices X = E + E + + E d, where d is the number o non-zero eigenvalues o the L L matrix S = X X. he elementary matrices are given by Ei = λ i Ui Vi, i =,,, d, where λ, λ,, λd are the non-zero eigenvalues o S = X X in decreasing order, U, U,, Ud are the corresponding eigenvectors, and the vectors V i are obtained rom Vi = X Ui / λi. he norm o elementary matrix E i equals λ i, so that, the contribution o the irst matrices to the norm o X is much higher than the contribution o the last matrices. hereore, it is likely that these last matrices represent noise in the signal. he plot o the eigenvalues in decreasing order is called the singular spectrum, and gives the method its name. Reconstruction (diagonal averaging) At this step, each elementary matrix E i is transormed into a principal component o length N by applying a linear transormation known as diagonal averaging or Hankelization. he elementary matrices are not themselves Hankel matrices, so that to reconstruct each principal component one calculates the average along the diagonals i + j = c. he diagonal averaging algorithm [] is as ollows: Let Y be any o the elementary matrices E i o dimension L K, the elements o which are y ij, i L, j K. he time series G = g, g,, gn (the principal component) corresponding to this elementary matrix is given by: k + < + ym, k m+ or k L k m= L g k = y, + < m k m or L k K () L m= N K + < ym, k m+ or K k N N k m= k K + Where L = min( L, K), K = max( L, K), and the length N = L + K. It can be shown that the squared norm o each elementary matrix equals the ISBN:

3 corresponding eigenvalue, and that the squared norm o the trajectory matrix is the sum o the squared norms o the elementary matrices []. hus the ratio: d λ / λ (3) i i represents the contribution o the elementary matrix E i in the expansion o the trajectory matrix X = E + E + + E d. he largest eigenvalues in the singular spectrum represent the large amplitude components in the decomposition. Contrariwise, the lowamplitude components o the signal are represented in the singular spectrum by the smallest eigenvalues.. Separability he obtained SSA decomposition is a unction o the window length choice. his choice thereore conditions whether the components obtained will be correlated to a greater or lesser degree in the requency domain. o study whether these components are mutually independent, one deines the ollowing necessary (but not suicient) separability condition []: wo principal components F and F obtained rom the elementary matrices E and E are separable (w-orthogonal) i the inner product o series F and F is null, i.e.: N, = w i = ( F ) w ( i) ( i) F (4) where the weights wi are deined by i + or i L w i = L or L i < K (5) N i or K i N and L = min( L, K), and K = max( L, K) I the original series F is decomposed using SSA into a sum o separable components F, F,, F L that match equation (4) then this sum can be interpreted as an expansion o the original signal F with respect to a certain w-orthogonal basis generated by the original series itsel []. he weights in the inner product have the orm o a trapezium and reduce the inluence o data close to the end-points o the components with respect to the central terms in the series, particularly or large window lengths. Real-lie signals do not match equation (4). In practice, one speaks only about approximate separability. o quantiy the quality o separation between two components, one deines the weighted correlation or w-correlation o two components F and F : w ( ) ( ) (, ) i i i w F F w ρ = = (6) F F N N w w w ( i) ( i) w ( i) ( i) i N i his coeicient is a measure o the deviation o two series, F and F, rom w- orthogonality. I the absolute value o the w- correlation is near zero, then the two series are separable. I it is large (near ), the two components are badly separable. It is desirable to obtain a decomposition such that the principal components are mutually independent in order to extract the noise present in the displacement signal. Figure shows the SSA decomposition o a displacement signal (described in the results section) using a window length L =. Figure shows the decomposition obtained together with the original signal (see igure caption or details). Figure (b) shows a graphical representation o the w- correlation matrix corresponding to the decomposition obtained. Cell ij represents the correlation between components i and j, gray-scale w w coded rom black or ρ = to white or ρ = ij Fig.. (a) Singular spectrum. (b) w-correlation matrix. (c) Original time series and principal components obtained in the time domain. ij ISBN:

4 3 Cluster Analysis As was discussed above, the statistical independence o the principal components depends on the size o the window length chosen. With large values ( L 5) while the separation is good, the number o principal components obtained or the representation o the trend signal is impractically large. In order to reduce the number o principal components with which to extract the trend signal and ensure their statistical independence, we apply a cluster analysis procedure. Namely, we perorm K -means clustering on the w-correlation matrix in order to obtain K groups o independent components. he K -means algorithm classiies a given data set through a certain number K o clusters ixed a priori []. he main idea is to deine K centroids, one or each cluster. Since dierent locations o these centroids cause dierent results, the best choice is to place them as ar as possible rom each other. he next step is to take each point belonging to a given data set and associate it to the nearest centroid. When no point is let, the irst step is completed and a putative grouping is made. At this point, one needs to recalculate K new centroids as barycenters o the clusters resulting rom the previous step. Ater one has these K new centroids, a new binding procedure has to be carried out between the same data set points and the nearest new centroid. A loop has thus been generated. As a result o this loop, the K centroids change their location step by step until no more changes are made, i.e., the centroids do not move any more. Finally, the algorithm aims at minimizing an objective unction, in this case the ollowing squared error unction: k n ( j) J = x i c j (7) j= ( j) Where i c j x, a distance measure ( j) between a data point x i and the cluster centre c j, is an indicator o the distance o the n data points rom their respective cluster centers. his simple version o the K -means procedure can be viewed as a greedy algorithm or partitioning the n samples into K clusters so as to minimize the sum o the squared distances to the cluster centers. Unortunately there is no general theoretical solution to ind the optimal number o clusters or any given data set. A simple and systematic heuristic approach to obtain an optimal number o clusters is to compare the obtained error unction (Equation 7) o multiples runs with dierent K classes and chose the number o clusters that minimize this unction. 4 Smoothing Automation he SSA smoothing procedure presented in Alonso et al. [] is based on the act that raw-displacement acquired signals present a very large signal-to-noise ratio. In this situation, the contribution o the irst matrices to the norm o X is much higher than the contribution o the last matrices, that represent noise. o eliminate the noise present in the displacement signal it is suicient to choose the leading eigenvalues that represent a large percentage o the entire singular spectrum. As it has been pointed out [-], one o the drawbacks o SSA application is the lack o general rules or selecting the values o the parameters L and r that arise in the SSA algorithm. Moreover, certain window lengths and grouping strategy choices produce a poor separation between signal trend and noise. In other words, trend components would be mixed with noise components in the reconstruction o the signal. A way to overcome the uncertainty in the choice o the truncation value r is to apply sequential SSA. his means that we extract some components o the initial series by the standard SSA and then extract the components o interest by applying SSA smoothing to an already smoothed record. Such a recursive SSA application produces a gradual elimination o the noise present in the signal. o ensure that any signiicant part o the noise is eliminated, the number o eigenvalues L r to eliminate in each iteration was chosen to satisy the ollowing criteria: log < log λ L λ λ r λ r L (8) his criterion ensures that the eigenvalues whose λr logarithmic dierence log is lower than the λr average logarithmic range o the entire singular λ spectrum: log are eliminated. his criterion L λ L ensures that we eliminate eigenvalues in a zone where the singular spectrum has suicient latness. he convergence o the sequential procedure may be measured by means o the percentage root mean square (RMS) dierence between the current and previous acceleration signals obtained in each iteration. he algorithm stops when this dierence ISBN:

5 is suiciently small, namely the stop criterion at iteration i is: RMS ( G i Gi ) E = < ε RMS( G ) = (9) i Where G i is the smoothed acceleration reconstructed at iteration i. he automatic iltering procedure is summarized in the ollowing way: Choose an arbitrary window length L. Perorm 4 -means clustering on the w-correlation matrix in order to obtain 3 groups o independent components and extract the trend signal. Apply sequential SSA to the extracted trend signal (truncation value r is ixed to account or the grouping criterion, Equation 8). Calculate the acceleration signal numerically in each iteration. Stop the procedure according to the stop criterion (Equation 9). Fig.. (a) Experimental layout. (b). Noisy vertical displacement signal acquired. Figure 3 shows the w-correlation matrices obtained using L = 5, L =, L = 5 and L =. It is clear rom Figure 3 that the smaller window lengths do not adequately separate the high and low requency components. he greater window lengths, however, yield a good separation o the high and low requencies. 5 Results In order to study the perormance o the iltering procedure three signals were tested. wo reerence acceleration signals were taken rom the literature. A double dierentiation is perormed on each signal in order to quantiy the eect o smoothing. First order central inite dierences are used to calculate the higher derivatives. he sequential SSA algorithm stops when the percentage dierence between the current and previous values o the RMS acceleration is smaller than % in order to prevent excessive smoothing. he irst signal is the motion o a vertical slider moved by hand to obtain a non-stationary mono-dimensional motion o biomechanical origin. A subject was asked to move the slider randomly with ast upward and downward movements. he vertical position (Figure ) was acquired with the use o a marker attached to the slider and the Qualisys camera system (Qualisys Medical AB). he second derivative o this record, or o the record obtained ater smoothing, is compared to the acceleration obtained directly rom an accelerometer attached to the slider. Displacement and acceleration were sampled at Hz during 5.9 seconds, obtaining records o 8 elements each. SSA decomposition was attained using a window length L = (see Figure ). Various window lengths were tested in order to study the inluence o this parameter on the obtained decomposition. Fig. 3. w-correlation matrices. (a) L = 5. (b) L =. (c) L = 5. (d) L =. Ater the SSA decomposition, a 4-means cluster analysis was perormed in order to obtain our requency independent components or each vibration signal. he principal components orming the low-requency signal are I = [,,3]. It is important to stress that our components would also have obtained using a window length o L = 4 in a SSA decomposition o the original series, but without perorming the cluster analysis, these our components would not, however, have been independent in requency, which is the undamental point o this smoothing method. In order to eliminate the noise present in the obtained reconstruction sequential SSA-cluster analysis was applied. Such a recursive SSA application produces a gradual elimination o the ISBN:

6 noise present in the reconstructed signal. Figure 4 shows the results obtained or two iterations o the method using L =. Figure 4 (top) shows the acceleration calculated rom original raw displacement data (dotted line) and acceleration measured by the accelerometer (continuous line). Figure 4 (middle) shows the acceleration calculated ater having been passed through a 4 Hz cut-o requency Butterworth ilter (dotted line) and acceleration measured by the accelerometer (continuous line). his igure shows the large endpoint errors associated to the usage o this ilter. Finally, Figure 4 (bottom) represents the acceleration calculated ater smoothing the raw displacement signal using sequential SSA-cluster analysis ( L = ) and acceleration measured by the accelerometer (continuous line). Eigenvalue 6 Singular spectrum Eigenvalue number Fig. 5. Singular Spectrum evolution or the irst signal able summarizes the results obtained using several window lengths. It can be concluded that the results are robust to variations o the window length L and number o clusters. L RMSE N. iterations able. Summary o the results or several window lengths. Fig. 4. Signal results. See descriptions in the text. Figure 5 shows the singular spectrum evolution in each o the two iterations required to achieve the convergence criterion. It is clear rom the igure that the proposed procedure sequentially reduces the noise amplitude in each iteration. Comparison with traditional and advanced iltering techniques is quantiied by taking the root mean square o the error signal (RMSE) o the acceleration. he error signal is the dierence between the measured reerence signal (accelerometer) and the signal obtained ater iltering and dierentiating the raw data. he errors in terms o RMSE are 9.54 m / s or the Butterworth ilter and.8 m / s using SSA-cluster analysis respectively. he second signal corresponds to the woman ile rom GAILAB (Vaughan et al., 99). he signal measures the lateral displacement o a marker attached on the right tibial tubercle. Data was acquired or.94 seconds using a sampling requency o 5 Hz. he reerence signal is taken as that obtained ater iltering the raw data at 6.5 Hz, but dierent amplitude white noises are added in order to test smoothing methods. An amplitude, time generated, white noise was used or comparisons, see Giakas and Baltzopulos, 997 [5]. For signal, using L = the automatic procedure achieved RMSE = mm / s versus RMSE = 4 mm / s obtained by Giakas and Baltzopoulos, [5] using Power Spectrum Assessment (PSA) and RMSE = mm / s using a SSA non-automatic approach. he accuracy o the obtained acceleration may also be appreciated in Figure 6 (bottom), where the acceleration obtained rom the reerence signal is plotted along with the ISBN:

7 acceleration calculated rom SSA smoothed displacement data. he SSA algorithm decomposes the original signal into independent additive components o decreasing weight. his act allows the method to successully extract the latent trend in the signal rom the random noise inherent to the motion capture system. An automatic heuristic procedure based in sequential SSA and cluster analysis to extract the trend signal has been presented. Fig. 6. (op) Singular Spectrum evolution. (Bottom) Acceleration obtained rom reerence signal (continuous line) and acceleration calculated rom SSA-smoothed displacement data (dotted line). he third signal is a measure o the angular coordinate o a pendulum impacting against a compliant wall [4]. he angular acceleration obtained rom the motion capture system is compared to that obtained directly (ater dividing by pendulum length) rom accelerometers. hree accelerometers were used in order to average their measurements to reduce noise. he average signal, logged at a sampling rate o 5 Hz is used as the acceleration reerence signal. he same procedure was perormed on the third signal. Using L = 5 the automatic procedure achieved RMSE = 4.37 rad / s (Figure 7). he result is similar to the value RMSE = 3.6 rad / s obtained by Giakas et al. [8] with the help o the Wigner distribution. he slight loss o accuracy in the SSA method is compensated by the ease with which the method is applied and the act that one must not extend the ends o the record in order to eliminate end point errors. Moreover, the automatic iltering procedure obtain similar results or a reasonable range o window lengths. 6 Conclusions he present work has studied the applicability o the sequential SSA method and cluster analysis to automatic smoothing o biomechanical kinematic signals. Fig. 7. (op) Singular Spectrum evolution. (Bottom) Acceleration obtained rom reerence signal 3 (continuous line) and acceleration calculated rom SSA-smoothed displacement data (dotted line). he method does not use any inormation rom the reerence acceleration signal (which is not available in practice) to perorm the smoothing. One o the main advantages to the method is the act that the algorithm requires the selection o just one parameter. Namely, the window length L, moreover, the results are robust to variations o the window length. In conclusion, we believe that this new automatic smoothing technique, that has proven its eectiveness with complex signals, will help to improve the accuracy o raw kinematic data processing in biomechanical analysis. Reerences: [] C.L. Vaughan, Smoothing and dierentiation o displacement-time data: an application o splines and digital iltering. International Journal o Bio-Medical Computing, No. 3, 98, pp [] H. Hatze, he undamental problem o myoskeletal inverse dynamics and its implications, Journal o Biomechanics, No. 35,, pp [3] L. Chiari, U. Della Croce, Leardini, A. Cappozzo A., Human movement analysis using stereophotogrammetry Part : Instrumental ISBN:

8 errors, Gait and Posture, No., 5, pp.97-. [4] J. Dowling, A modelling strategy or the smoothing o biomechanical data. In: Johnsson, B. (Ed.), Biomechanics, Vol. XB, Human Kinetics, Champaign, IL, 63 67, 985. [5] G. Giakas, V. Baltzopoulos, A comparison o automatic iltering techniques applied to biomechanical walking data, Journal o Biomechanics, No. 3, 997, pp [6] R.I. Adham, S.A. Shihab, Discrete wavelet transorm: a tool in smoothing kinematic data, Journal o Biomechanics, No.3, 999, pp [7] A. Georgiakis, L.K. Stergioulas, G. Giakas, Wigner iltering with smooth roll-o boundary or dierentiation o noisy non-stationary signals, Signal Processing, No. 8,, pp [8] G. Giakas, L.K. Stergioulas, A. Vourdas, ime-requency analysis and iltering o kinematic signals with impacts using the Wigner unction: accurate estimation o the second derivative, Journal o Biomechanics, No.33,, pp [9] K.S. Erer, Adaptive usage o the Butterworth digital ilter, Journal o Biomechanics, No. 4, 7, pp [] F. J. Alonso, J. M. Del Castillo, P. Pintado, Application o singular spectrum analysis to the smoothing o raw kinematic signals, Journal o Biomechanics, No.38, 5, pp [] N. Golyandina, V. Nekrutkin, A. Zhigljavsky, Analisys o time series structure - SSA and related techniques, Chapman & Hall/CRC, Boca Raton, Florida,. [] J.B. MacQueen, Some methods or classiication and analysis o multivariate observations, in: Proceedings o the Fith Berkeley Symposium on Mathematical Statistics and Probability, vol.. Berkeley, CA, 967, pp ISBN: