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North American Congress on Biomechanics Canadian Society for Biomechanics - American Society of Biomechanics University of Waterloo Waterloo, Ontario, Canada August 14-18, 1998 |
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Patterns of time series (EMG, angles or moments, etc.) are often evaluated by using their temporal characteristics such as onsets, offsets or time to peak values. The subjectivity and inconsistency involved in the evaluation challenges researchers whom attempt to compare different time series. In this paper we present an alternative method, which uses the 95% confidence interval of the cross correlation and its corresponding values in the time domain to identify phase shifting in time series data.
The temporal characteristics of mechanical and neuromuscular function during human movement have been studied with different time dependant variables, such as electromyographic (EMG) recordings, force measurements or joint moment calculations. Often researchers wish to compare patterns between subjects or movement conditions. For example, the onset and offset times that identify the duration of EMG bursts have been used to compare patterns and identify pattern change in rhythmic human motions such as running (McClay et al., 1990) and cycling (Gregor et al., 1991). However, there is a lack of agreement on the appropriate threshold criteria to determine these onsets and offsets (Hodges and Bui, 1996), and, there is often a large degree of subjectivity involved in this determination. The purpose of this paper is to present an alternative method that can be used to compare between patterns and detect alterations objectively and statistically.
Theoretical derivation The similarity in the pattern of two time series x and y, each with N data points, can be calculated by using the coefficient of cross correlation rxy(k), where k is a variable that designates a time (or phase) shift of one series with respect to the other (Chatfield, 1984). The correlation rxy(0) will give an indication of pattern similarity of the two sets of data without any time shifting. An objective measure of the actual time shift between two similar patterns is the k=m at which the rxy(k)= rxy(m) is maximized . The 95% confidence interval of rxy(m) (Devore, 1990) can be used to evaluate the statistical significance of the time shift between the patterns. The low 95% confidence interval of rxy(m) will identify two other correlation values rxy(m-l) and rxy(m+n), which can then be converted to values in the time domain, m-l and m+n (Figure 1). The phase shift between x and y is significant if the period between the two 95% rxy times does not include zero.
Data collection and analysis As an example, five trials of EMG data were collected at 1000 Hz from gluteus maximus (GM) during cycling at low (65 rpm) and high (95 rpm) pedaling cadences. Raw EMG data were converted to linear envelopes by rectification and smoothing (low pass digital filter with cutoff of 20Hz), and averaged to produce ensemble curves. Details on these procedures can be found in Li and Caldwell (1998).
Figure 2 shows the ensemble EMG linear envelopes as a function of one complete crank revolution. Each ensemble curve was scaled to its respective maximal value. Also shown are arbitrary thresholds of 25% and 10% of the maximum EMG. With the use of these two different threshold criteria, not only were the onset and offset times of the two curves changed, but the sequence of activity offsets was also reversed, making it difficult to ascertain whether a shift in pattern exists. The subjectivity of threshold selection is inherent with this method. On the other hand, there is no subjectivity used in the cross correlation calculation. The highest rxy(k) value of the EMG activity between the low and high cadence conditions was 0.972, which corresponded to a crank angle shift of 20° (high cadence signal shifted towards 0° crank position). The lower boundary of the 95% confidence interval of this coefficient was 0.959. The corresponding angle of rxy = 0.959 on both sides of rxy(20°) was 10° and 30° respectively. This indicates a significant shift in the EMG pattern between conditions, as 0° is not contained between 10° and 30°. For identification of pattern shifting, this method is preferred to the use of discrete events because of its clarity and objectivity. One unique feature of the cycling GM EMG profile illustrated the difficulty of identification with the discrete points (i.e., onset and offset of the EMG bursts). Especially in the high cadence condition, it was difficult to determine onset with 25% maximum as threshold, or offset with 10% maximum as threshold. The onset and offset of the EMG activity was also not adequate to evaluate the overall pattern alteration between conditions. The coefficient rxy(k) provides not only clear information about the direction of the phase shift, but also a measure of similarity in shape of the two time series (rxy(0) = 0.972). In summary, this method can be used to determine phase shift based on the entire profile of the time series without the subjective judgement of the investigator.
Chatfield, C. The analysis of time series. Chapman and Hall, 1984.
Devore, J.L. Probability and statistics for engineering and the sciences. Brooks / Cole, 1990.
Gregor, R.J. et al. Exerc. Sport Sci. Rev. 19, 127-169, 1991.
Hodges, P.W., Bui, B.H. Electroen. Clin. Neurophysiol. 101, 511-519, 1996.
Li, L., Caldwell, G.E. J. Appl. Physiol., submitted, 1998.
McClay, I.S., et al, In: Biomechanics of distance running, ed. P. R. Cavanagh, 165-186. Human Kinetics, 1990.
