- Email: [email protected]
Nature of variability in rhythmical movement
__ __ l!!B
Jss
ELSEVIER
HlJMAN Human Movement Science 14 (1995) 371-384
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Nature of variability in rhythmical movement Norimasa Yamada * Laboratory of Human Movement Sciences, Faculty of Education, Hokkaido University, Kita II, Nishi 7, Kita-ky Sapporo 060, Japan
Abstract
If you look back at your footprints on a snowy road, you will notice that you never move in the same way. Learning theory (Hebb, 1949) can explain the decrease of variability in human movement, but cannot explain the nature of the variability. Feedfoward motor control by the cerebellum (Shidara et al., 1993) can explain the precise movement of the limbs to objects, but cannot explain error in movement from objects. In past studies (e.g., Schijner et al., 1986; Kelso et al., 1987; Kay, 19881, the cause of errors was assumed to be noise originating in the movement system. In other words, human movement has been considered to be the result of a combination of deterministic and stochastic mechanisms. Hurst (1965) proposed an empirical law of variability for natural phenomena from observation of the rise and fall of water in rivers. Variability in stereotyped movements such as walking and running, that have been considered as repetition of the same movement, was analyzed using Hurst’s empirical law. The result suggests that variation in stereotyped human movement can not be modeled as Brownian motion or noise separated from a deterministic movement system.
1. Introduction A notable feature of stereotyped movements is reduced variability, which has been studied by many researchers. For example, Shapiro et al. (1981) reported that relative times which divide one step of walking or running
* E-mail: [email protected],
Tel./Fax:
+ 81 11 706-5442.
0167-9457/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDZ 0167-9457(95)00018-6
372
N. Yamada /Human
Movement Science 14 (1995) 371-384
Normalized angular position Fig. 1. Typical phase portrait of the finger’s rhythmical movement. angular position and the ordinate is the finger’s angular velocity.
The ahsissa
represents
the finger’s
into four phases were almost constant under conditions of change in movement speed. On the other hand, the results obtained from the experiment carried out by Kelso (1984), in which subjects were instructed to rotate both index fingers about the first joints rhythmically, showed that the amplitude and frequency of the movement of the index fingers varied slightly for every movement. This is shown more clearly in phase space, i.e., plotting the finger’s angular position versus angular velocity (Fig. 1). As can be seen in Fig. 1, rhythmical movement does not produce a clean, single orbit in phase space. Kay (1988) modeled this variation observed in rhythmical movement as a stochastic process. Thus, rhythmical movement was modeled as the behavior of a dynamical system governed by a limit cycle attractor with noise. The model was based on the concept of human movement being the result of a combination of deterministic and stochastic mechanisms. Variability of successive intervals during finger tapping has been used as an index of variability in timing in rhythmical movement (e.g., Stevens, 1886; Wing and Kristofferson, 1973; Ivry et al., 1988). Wing and Kristofferson (1973) developed a model which is based on the idea that temporal inaccuracy in periodic tapping could be partitioned into two independent sources which are due to the lack of precision of a hypothetical internal timekeeper (central), and to temporal noise in the execution motor system (peripheral), respectively. They postulated that the timekeeper intervals and the motor delays are sequences of independent random variables with normal variances. In this model, all timekeeper and motor delay intervals are assumed to be mutually independent.
N. Yamada /Human
Movement Science 14 (1995) 371-384
373
Randomness is inherent in all natural and physiological phenomena. The concept of randomness in human movement has been adopted as a noise or independent random process. The models in this framework are essentially linear, with noise added to a certain system. There is good evidence that many natural and physiological shapes and time series can best be described as fractals (Mandelbrot, 1982). Time series that can be described as fractals are known as fractional Brownian motion. An important feature of fractal Brownian motion is that past increments in a particle’s displacement are correlated with future increments, which means each observation is not independent. Thus, if fractals are to be useful in the description of the nature of variability in rhythmical movement, we must develop the concept of randomness from observation of human movement. The purpose of this study is to test whether variation observed from rhythmical movement is an independent random process, using a new statistical method called resealed range analysis (R/S analysis). 2. Methods 2.1, Experimental methods
The experimental setup is depicted schematically in Fig. 2. Five healthy subjects, aged 21-34 years, were used in the study. The subjects had no evidence or known history of a skeletal disorder. Each subject was in-
A/D Converter
EWS Computer
Fig. 2. Experimental setup. Five subjects tapped an accelerometer with their index finger 800 times in constant rhythm. Acceleration signals were digitized to a resolution of 12 bits, sampled at 1 KHz. The time intervals between peaks of the digitized acceleration data were used as an index of the variation of rhythmical movement.
374
N. Yamada /Human
Movement Science 14 (1995) 371-384
strutted to sit a chair with his right arm on a table. In the experiment, the subjects tapped an accelerometer with their index finger 800 times in constant rhythm with no paced stimulus. Before the experiment, synchronization trials were not conducted. Thus, each subject was instructed to produce their own natural rhythmic motion, as there was no rhythmic stimulus for them to synchronize their motions. Acceleration signals were digitized to a resolution of 12 bits, sampled at 1 KHz. Time intervals between peaks of the digitized acceleration data were calculated, and the data were used as an index of variation in rhythmical movements. 2.2. Hurst exponent Hurst (1965) proposed a new statistical method called resealed range analysis (R/S analysis). He found that most natural phenomena, including river discharges, temperatures, rainfall, and sunspots, do not follow random processes. This method was expanded to include the concept of fractals by Mandelbrot (1982). Consider the time series of t numbers as g+(t). The average of the r numbers data is: W7 = ;$(t).
(1)
Let X(t) be the accumulated G%
X(t, 4 =
c {5(u)-
departure
of the data t(t)
from the mean
(t-)7}.
u=l
The difference between the maximum and the minimum data X is the range R. The explicit expression for R is R(r) = r’=p:,X(‘~ T) - rn&r7X(t, 7)
(2) accumulated (3)
where t is a discrete integer-valued time and 7 is the time-span considered. Hurst (1965) found that the observed resealed range R/S, where S is the standard deviation, for many records in time can be very well described by following the empirical relation: R/S = (a& (4) where a is a constant value and the exponent H is called the Hurst exponent. As can be seen in Eq. (4), the Hurst exponent is expressed as the
N. Yamada /Human
Movement
Science 14 (1995) 371-384
375
t ( iteration )
T-----
2.5
3
log,, t Fig. 3. Random time series generated analysis of the data (bottom).
by statistically
independent
processes
(top), and the result of R/S
slope of a double logarithmic plot of R/S as a function of lag 7. The case H = l/2 is a special case of an independent random process with finite variance. Fig. 3 shows the case of H = l/2, where the data were generated by statistically independent processes (random function in the software of Mathmatica was used). The upper panel shows the random time series, and the bottom panel shows R/S as a function of lag 7. As the Hurst exponent is expressed as the slope of this graph, the data were fitted to a linear equation and the exponent was estimated. As can be seen in Fig. 3, the exponent is almost equal to l/2. However, in the case where H is not
376
N. Yamada /Human
Movement Science 14 (1995) 371-384
600[.
1
J
500.
h
;
v 7
400.
1
0
0
200
400 t (
600
800
tapping number )
2.5 I
Fig. 4. Typical rhythmical movement analysis of the data (bottom).
time intervals
of the index finger
(top),
and the result
of R/S
equal to l/2 (1 > H > l/2), increasing or decreasing trends in the past generally imply increasing or decreasing trends in the future, which leads to persistence. This type of behavior is known as fractional Brownian motion. By applying R/S analysis to analysis of variability of successive intervals during rhythmical finger tapping, the nature of variability can be analyzed. 3. Results The upper panel of Fig. 4 shows the typical rhythmical movement time intervals of the index finger, calculated by digitized acceleration data. As
N. Yamada /Human Table 1 Means, standard deviations, intervals for all subjects
and
Hurst
Movement Science 14 (1995) 371-384
exponents
of the time series
of rhythmical
Subjects
Mean (s)
SD (s)
Hurst exponent
A B C D E
0.436 0.443 0.524 0.481 0.452
0.0194 0.0190 0.0296 0.0230 0.0219
0.898 0.828 0.612 0.730 0.700
377
movement
time
this figure shows, rhythmical movement time intervals of the index finger varied slightly every time. The Hurst exponent was calculated using the data to test weather the variation was an independent random process. A typical result of R/S analysis is shown in the lower figure in Fig. 4, and the estimated Hurst exponents for the subjects are summarized in Table 1 along with means and standard deviations of the data. The Hurst exponents for all subjects were much greater than l/2.
4. Discussion 4.1. Validity of the estimated value of H Even if a value significantly different than l/2 is found for H, there may be some doubt as to whether the estimate is valid. The validity of the estimated value of H can be tested (Scheinkman and LeBaron, 1989; Peters, 1991) by randomly scrambling the data so that the order of the observations is completely different from that of the original time series. The calculation of the Hurst exponent was repeated on the scrambled data. If there is a long memory effect (persistence) in place, the order of the data is important. By scrambling the data, the structure of the observed data should be destroyed, and then the Hurst exponent should be much lower, and closer to l/2. Fig. 5 shows typical results of R/S analysis of the scrambled and unscrambled series. The Hurst exponents of both data series for the subjects are summarized in Table 2. A qualitative difference can be seen. The original series gave a value of H greater than l/2, while the scrambled series gave a value of approximately l/2 for H. This drop in the value of H shows that the long memory process in the original time
378
N. Yamada /Human
Fig. 5. Typical results of R/S analysis rhythmical movement time intervals.
Movement Science 14 (1995) 371-384
of scrambled
and
unscrambled
time
series
of the finger’s
series was destroyed by the scrambling process. This means that there were long memory processes in the original data, in which case the estimated value of H can be considered to be larger than l/2. 4.2. Nature of the variation Randomness is inherent in all natural and physiological phenomena. The concept of randomness in human movement has been adopted as a noise or independent random process (e.g., SchGner et al., 1986; Kelso et al., 1987; Kay, 1988). In the present study, R/S analysis was performed to test whether the variation observed from rhythmical movement was an indepen-
Table 2 Hurst exponents all subjects
of unscrambled
and scrambled
time series of rhythmical
movement
Subjects
Hurst exponent Unscrambled
Scrambled
A B C D E
0.898 0.828 0.612 0.730 0.700
0.506 0.568 0.510 0.505 0.470
time intervals
for
A? Yamada /Human
Movement
Science 14 (1995) 311-384
379
dent random process. Two kinds of variation in rhythmical movement can be considered: one is the variation in time (frequency of the rhythmical movement) and the other is the variation in space (amplitude of the rhythmical movement). However, in this analysis only variation in time was used. As can be seen in Table 1, the Hurst exponents for all subjects were much greater than l/2. This suggests that variation in stereotyped human movement can not be modeled as Brownian motion and each observation is not independent. The correlation function of future increments with past increments, which is time-independent, is given by an expression using the Hurst exponent (Feder, 1988): c = 2(22H-’ - 1).
(5) As can be seen from Eq. (51, in the case of H > l/2 (my results), C is greater than 0. This means that increasing trends in the past generally imply increasing trends in the future. Thus, the observations were not independent. Each observation carries a memory of all the events that preceded it, which leads to persistence (bias). The strength of persistence, or the biased behavior, increases as H approaches to 1.0, or 100 percent correlation in Eq. (5). A tendency of change in the time series with increases of H can be seen using a simulation technique. Feder (1988) produced a formula for creating simulated fractal Brownian motion:
rl-H “,(‘)
= I’( H + l/2) n(M- 1) +
c i=l
((n
i(i)H-‘/*5(1
i i= 1 +i)H-1’2
-
+n(iM+t)-i)
(i)“-“‘)t(l
+ n(M-
1 + t) - i) , i
(6) where (ti} is a set of Gaussian random variables with unit variance and zero mean, H is the Hurst exponent, and r(n) is the gamma function. t is an integer time step, usually one period, which is split into II intervals to approximate a continuous integral. The effect of increasing n is to give a more precise approximation to the short-term behavior of B,(t), n = 8 was chosen for the simulation here. M is the number of periods for which the long memory effect is generated. Theoretically, it should be infinite, but, for the purpose of simulation, a large value of M will suffice. M = 700 was chosen for this simulation.
380
N. Yamada /Human
-4I.
Movement Science 14 (1995) 371-384
I
I
0
200
‘loo
600
800
1000
0
200
400
600
800
1000
-4I
I
f ( iteration ) Fig. 6. Numerical simulation of increments of the fractal Brownian motion. (a) Ordinary Brownian increments for H = l/Z. (b) Fractal increments for H = 0.7. (c) Fractal increments for H = 0.9.
Fig. 6 shows a simulated series for H = 0.5, 0.7, 0.9. There is no dramatic change in the high-frequency components as H is increased. However, as H increases, the low-frequency components increase and generate a large amplitude excursion compared with the high-frequency components. The
N. Yamada /Human
‘“102
Movement Science 14 (1995) 371-384
10-l
loo
381
101
Frequency (Hz) Fig. 7. Typical power spectra of rhythmical movement time intervals of the index finger.
low-frequency components are equivalent to the persistence or bias. Thus, different value than l/2 for H is related to the degree of persistence. In the model proposed by Wing and Kristofferson (1973) for self-paced periodic tapping, each time interval is assumed to be the sum of three events. Two of the events are attributed to a delay in executing the motor system, and the third is the hypothetical internal timekeeper. Additional assumptions of the model are that the timekeeper intervals and the motor delays are sequences of independent random variables with normal variances. These assumptions imply that the lag one autocorrelation between inter-tap intervals is negative. This means that large intervals tend to be followed by short ones, and vice versa. This has received empirical support in a number of studies (reviewed in Wing, 1980). This relation between adjacent intervals appears in high frequency components in spectrum analysis of time series of successive inter-tap intervals. Fig. 7 shows the typical power spectra of inter-tap intervals. As can be seen in Fig. 7, the spectra of the inter-tap intervals contain a broadband spectrum of not only high frequency components. A remarkable feature of fractal Brownian motion, in the case of H > l/2, is persistence which is shown as low frequency components in spectrum analysis. Wing and Kristofferson (1973) explained the relation between adjacent intervals using the autocorrelation function (ACF). They and other researchers (e.g. Stevens, 1886) have found the persistence in the time series of inter-tap intervals. However, the
382
N. Yamada /Human
Movement Science 14 (1995) 371-384
principle behind the persistence in the human rhythmical movement has not been investigated fully. Mandelbrot (1972) showed that the ACF does not work well for longmemory processes, like fractal Brownian motion, and recommended that R/S analysis should be used for the analysis of the such time series. 4.3. Physiological interpretations Muscles in the finger are activated by efferent signals from the central nervous system (CNS), and at the same time, finger movement activate specific sets of sense organs in the finger proprioceptors, such as golgi tendon organs or muscle spindles which produce afferent traffic of nerve impulses towards the CNS. Therefore the neural structure of the movement system is influenced successively by the movement itself (Varela, 1979). In recent studies about human movement control (e.g. Kelso and Ding, 1993), the neural structure of the movement system is considered to be a dynamical system. Dynamical systems are systems that change over time. Here, randomness is viewed as part of the system’s dynamic and not a superimposed component. In the present study, time variation of inter-tap intervals is described as fractal Brownian motion which is produced by long term-memory process. Each tapping movement affects inside state of the movement system, and the system changes successively over time. Thus, variation of inter-tap intervals is consider to be part of the intrinsic dynamics of the movement system which is consider to be a dynamic system. It is thought that fatigue of the muscles in the finger influence variation of the rhythmical movement. When H is different from l/2, shorter range
Table 3 Means and standard deviations of the movement time intervals for all subjects Subjects
A B C D E
first
half
First half of the data
and
latter
half Latter
of the time
series
of rhythmical
half of the data
Mean (s)
SD kJ
Mean (s)
SD (s)
0.437 0.443 0.526 0.490 0.453
0.0176 0.0194 0.0289 0.0207 0.0226
0.435 0.443 0.521 0.473 0.452
0.0210 0.0186 0.0301 0.0221 0.0211
N. Yamada /Human
Movement Science 14 (1995) 371-384
383
and longer range (biased time series) must have occurred non-periodically (see Fig. 6). However, it is almost impossible to remove the effect of fatigue from bias. Therefore, a simple test was carried out to ascertain the degree of the effect of fatigue on the bias. The time series of the rhythmical movement time intervals were divided into two parts; the first half and the latter half of the data. The means and standard deviations of the data were calculated independently. If the influence of fatigue is remarkable, time intervals of the latter half of the data should be slower than that of the first half of the data. In other words, the latter half of the data should be biased to a slower range. The results in Table 3 show that the means and standard deviations for the two parts are similar. Thus, the effect of fatigue on bias was minimal in this experiment.
5. Conclusion
Variation in rhythmical movement can not be modeled as an independent random process (noise), and is consider to be part of the intrinsic dynamics of a movement system, which is consider to be a dynamical system.
Acknowledgements
This work was supported by a special grant-in-aid promotion of education and science in Hokkaido University provided by the Ministry of Education, Science and Culture.
References Feder, J., 1988. Fractals. New York: Plenum Press. Hebb, D.O., 1949. The organization of behavior. New York: Wiley. Hurst, H.E., 1965. Long-term storage: An experimental study. London: Constable. Ivry, R.B., S.W.Keele and H.C. Diener, 1988. Dissociation of the lateral and medial cerebellum in movement timing and movement execution. Experimental Brain Research 73, 167-180. Kay, B.A., 1988. The dimensional&y of movement trajectories and the degrees of freedom problem: A tutorial. Human Movement Science 7, 343-364. Kelso, J.A.S., 1984. Phase transition and critical behavior in human biannual coordination. American Journal of Physiology: Regulatory, Integrative and Comparative Physiology 15, RlOOO-RlfKtO4.
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Movement Science 14 (1995) 371-384
Kelso, J.A.S. and M. Ding, 1993. ‘Fluctuations, Intermittency and Controllable chaos in biological coordination’. In: K.M. Newell and D.M. Corcos (eds.), Variability and motor control (pp. 291-316). Human Kinetics Publishers. Kelso, J.A.S., G.S. Schoner, J.P. Scholz and H. Haken, 1987. Phase-locked modes, phase transitions, and component oscillators in biological motion. Physica Scripta 35, 79-87. Mandelbrot, B.B., 1972. Statistical methodology for non-periodic cycles: From the covariance to R/S analysis. Annals of Economic and Social Measurement 1, 259-290. Mandelbrot, B.B., 1982. The fractal geometry of nature. New York: Freeman. Stevens, L.T. 1886. On the time-sense, Mind 11, 393-404. Scheinkman, J.A. and B. LeBaron, 1989. Nonlinear dynamics and stock returns. Journal of Business 62, 311-337. Shapiro, D.C., R.F. Zernicke, R.J. Gregor and J.D. Distel, 1981. Evidence for generalized motor programs using gait pattern analysis. Journal of Motor Behavior 13, 33-47. Shidara, M., K. Kawano, H. Gomi and M. Kawato, 1993. Inverse-dynamics model eye movement control by Purkinje cells in the cerebellum. Nature 364, 50-52 SchGner,G.S., H. Haken and J.A.S.Kelso, 1986. Stochastic theory phase transitions in human hand movement. Biological Cybernetics 53, 442-452. Peters, E.E., 1991. Chaos and order in the capital markets. New York: Wiley. Varela, F., 1979. Principles of biological autonomy. Amsterdam: North-Holland. Wing,A.M.,lWO. ‘The long and short of timing in response sequences’. In: G. Stlemach and J. Requin teds.), Tutorials in motor behavior (pp. 469-486). Amsterdam: North-Holland. Wing, A.M. and A.B. Kristofferson, 1973. Response delays and the timing of discrete motor responses. Perception and Psychophysics 14, 5-12.
Jss
ELSEVIER
HlJMAN Human Movement Science 14 (1995) 371-384
!#l!!NT
Nature of variability in rhythmical movement Norimasa Yamada * Laboratory of Human Movement Sciences, Faculty of Education, Hokkaido University, Kita II, Nishi 7, Kita-ky Sapporo 060, Japan
Abstract
If you look back at your footprints on a snowy road, you will notice that you never move in the same way. Learning theory (Hebb, 1949) can explain the decrease of variability in human movement, but cannot explain the nature of the variability. Feedfoward motor control by the cerebellum (Shidara et al., 1993) can explain the precise movement of the limbs to objects, but cannot explain error in movement from objects. In past studies (e.g., Schijner et al., 1986; Kelso et al., 1987; Kay, 19881, the cause of errors was assumed to be noise originating in the movement system. In other words, human movement has been considered to be the result of a combination of deterministic and stochastic mechanisms. Hurst (1965) proposed an empirical law of variability for natural phenomena from observation of the rise and fall of water in rivers. Variability in stereotyped movements such as walking and running, that have been considered as repetition of the same movement, was analyzed using Hurst’s empirical law. The result suggests that variation in stereotyped human movement can not be modeled as Brownian motion or noise separated from a deterministic movement system.
1. Introduction A notable feature of stereotyped movements is reduced variability, which has been studied by many researchers. For example, Shapiro et al. (1981) reported that relative times which divide one step of walking or running
* E-mail: [email protected],
Tel./Fax:
+ 81 11 706-5442.
0167-9457/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDZ 0167-9457(95)00018-6
372
N. Yamada /Human
Movement Science 14 (1995) 371-384
Normalized angular position Fig. 1. Typical phase portrait of the finger’s rhythmical movement. angular position and the ordinate is the finger’s angular velocity.
The ahsissa
represents
the finger’s
into four phases were almost constant under conditions of change in movement speed. On the other hand, the results obtained from the experiment carried out by Kelso (1984), in which subjects were instructed to rotate both index fingers about the first joints rhythmically, showed that the amplitude and frequency of the movement of the index fingers varied slightly for every movement. This is shown more clearly in phase space, i.e., plotting the finger’s angular position versus angular velocity (Fig. 1). As can be seen in Fig. 1, rhythmical movement does not produce a clean, single orbit in phase space. Kay (1988) modeled this variation observed in rhythmical movement as a stochastic process. Thus, rhythmical movement was modeled as the behavior of a dynamical system governed by a limit cycle attractor with noise. The model was based on the concept of human movement being the result of a combination of deterministic and stochastic mechanisms. Variability of successive intervals during finger tapping has been used as an index of variability in timing in rhythmical movement (e.g., Stevens, 1886; Wing and Kristofferson, 1973; Ivry et al., 1988). Wing and Kristofferson (1973) developed a model which is based on the idea that temporal inaccuracy in periodic tapping could be partitioned into two independent sources which are due to the lack of precision of a hypothetical internal timekeeper (central), and to temporal noise in the execution motor system (peripheral), respectively. They postulated that the timekeeper intervals and the motor delays are sequences of independent random variables with normal variances. In this model, all timekeeper and motor delay intervals are assumed to be mutually independent.
N. Yamada /Human
Movement Science 14 (1995) 371-384
373
Randomness is inherent in all natural and physiological phenomena. The concept of randomness in human movement has been adopted as a noise or independent random process. The models in this framework are essentially linear, with noise added to a certain system. There is good evidence that many natural and physiological shapes and time series can best be described as fractals (Mandelbrot, 1982). Time series that can be described as fractals are known as fractional Brownian motion. An important feature of fractal Brownian motion is that past increments in a particle’s displacement are correlated with future increments, which means each observation is not independent. Thus, if fractals are to be useful in the description of the nature of variability in rhythmical movement, we must develop the concept of randomness from observation of human movement. The purpose of this study is to test whether variation observed from rhythmical movement is an independent random process, using a new statistical method called resealed range analysis (R/S analysis). 2. Methods 2.1, Experimental methods
The experimental setup is depicted schematically in Fig. 2. Five healthy subjects, aged 21-34 years, were used in the study. The subjects had no evidence or known history of a skeletal disorder. Each subject was in-
A/D Converter
EWS Computer
Fig. 2. Experimental setup. Five subjects tapped an accelerometer with their index finger 800 times in constant rhythm. Acceleration signals were digitized to a resolution of 12 bits, sampled at 1 KHz. The time intervals between peaks of the digitized acceleration data were used as an index of the variation of rhythmical movement.
374
N. Yamada /Human
Movement Science 14 (1995) 371-384
strutted to sit a chair with his right arm on a table. In the experiment, the subjects tapped an accelerometer with their index finger 800 times in constant rhythm with no paced stimulus. Before the experiment, synchronization trials were not conducted. Thus, each subject was instructed to produce their own natural rhythmic motion, as there was no rhythmic stimulus for them to synchronize their motions. Acceleration signals were digitized to a resolution of 12 bits, sampled at 1 KHz. Time intervals between peaks of the digitized acceleration data were calculated, and the data were used as an index of variation in rhythmical movements. 2.2. Hurst exponent Hurst (1965) proposed a new statistical method called resealed range analysis (R/S analysis). He found that most natural phenomena, including river discharges, temperatures, rainfall, and sunspots, do not follow random processes. This method was expanded to include the concept of fractals by Mandelbrot (1982). Consider the time series of t numbers as g+(t). The average of the r numbers data is: W7 = ;$(t).
(1)
Let X(t) be the accumulated G%
X(t, 4 =
c {5(u)-
departure
of the data t(t)
from the mean
(t-)7}.
u=l
The difference between the maximum and the minimum data X is the range R. The explicit expression for R is R(r) = r’=p:,X(‘~ T) - rn&r7X(t, 7)
(2) accumulated (3)
where t is a discrete integer-valued time and 7 is the time-span considered. Hurst (1965) found that the observed resealed range R/S, where S is the standard deviation, for many records in time can be very well described by following the empirical relation: R/S = (a& (4) where a is a constant value and the exponent H is called the Hurst exponent. As can be seen in Eq. (4), the Hurst exponent is expressed as the
N. Yamada /Human
Movement
Science 14 (1995) 371-384
375
t ( iteration )
T-----
2.5
3
log,, t Fig. 3. Random time series generated analysis of the data (bottom).
by statistically
independent
processes
(top), and the result of R/S
slope of a double logarithmic plot of R/S as a function of lag 7. The case H = l/2 is a special case of an independent random process with finite variance. Fig. 3 shows the case of H = l/2, where the data were generated by statistically independent processes (random function in the software of Mathmatica was used). The upper panel shows the random time series, and the bottom panel shows R/S as a function of lag 7. As the Hurst exponent is expressed as the slope of this graph, the data were fitted to a linear equation and the exponent was estimated. As can be seen in Fig. 3, the exponent is almost equal to l/2. However, in the case where H is not
376
N. Yamada /Human
Movement Science 14 (1995) 371-384
600[.
1
J
500.
h
;
v 7
400.
1
0
0
200
400 t (
600
800
tapping number )
2.5 I
Fig. 4. Typical rhythmical movement analysis of the data (bottom).
time intervals
of the index finger
(top),
and the result
of R/S
equal to l/2 (1 > H > l/2), increasing or decreasing trends in the past generally imply increasing or decreasing trends in the future, which leads to persistence. This type of behavior is known as fractional Brownian motion. By applying R/S analysis to analysis of variability of successive intervals during rhythmical finger tapping, the nature of variability can be analyzed. 3. Results The upper panel of Fig. 4 shows the typical rhythmical movement time intervals of the index finger, calculated by digitized acceleration data. As
N. Yamada /Human Table 1 Means, standard deviations, intervals for all subjects
and
Hurst
Movement Science 14 (1995) 371-384
exponents
of the time series
of rhythmical
Subjects
Mean (s)
SD (s)
Hurst exponent
A B C D E
0.436 0.443 0.524 0.481 0.452
0.0194 0.0190 0.0296 0.0230 0.0219
0.898 0.828 0.612 0.730 0.700
377
movement
time
this figure shows, rhythmical movement time intervals of the index finger varied slightly every time. The Hurst exponent was calculated using the data to test weather the variation was an independent random process. A typical result of R/S analysis is shown in the lower figure in Fig. 4, and the estimated Hurst exponents for the subjects are summarized in Table 1 along with means and standard deviations of the data. The Hurst exponents for all subjects were much greater than l/2.
4. Discussion 4.1. Validity of the estimated value of H Even if a value significantly different than l/2 is found for H, there may be some doubt as to whether the estimate is valid. The validity of the estimated value of H can be tested (Scheinkman and LeBaron, 1989; Peters, 1991) by randomly scrambling the data so that the order of the observations is completely different from that of the original time series. The calculation of the Hurst exponent was repeated on the scrambled data. If there is a long memory effect (persistence) in place, the order of the data is important. By scrambling the data, the structure of the observed data should be destroyed, and then the Hurst exponent should be much lower, and closer to l/2. Fig. 5 shows typical results of R/S analysis of the scrambled and unscrambled series. The Hurst exponents of both data series for the subjects are summarized in Table 2. A qualitative difference can be seen. The original series gave a value of H greater than l/2, while the scrambled series gave a value of approximately l/2 for H. This drop in the value of H shows that the long memory process in the original time
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Fig. 5. Typical results of R/S analysis rhythmical movement time intervals.
Movement Science 14 (1995) 371-384
of scrambled
and
unscrambled
time
series
of the finger’s
series was destroyed by the scrambling process. This means that there were long memory processes in the original data, in which case the estimated value of H can be considered to be larger than l/2. 4.2. Nature of the variation Randomness is inherent in all natural and physiological phenomena. The concept of randomness in human movement has been adopted as a noise or independent random process (e.g., SchGner et al., 1986; Kelso et al., 1987; Kay, 1988). In the present study, R/S analysis was performed to test whether the variation observed from rhythmical movement was an indepen-
Table 2 Hurst exponents all subjects
of unscrambled
and scrambled
time series of rhythmical
movement
Subjects
Hurst exponent Unscrambled
Scrambled
A B C D E
0.898 0.828 0.612 0.730 0.700
0.506 0.568 0.510 0.505 0.470
time intervals
for
A? Yamada /Human
Movement
Science 14 (1995) 311-384
379
dent random process. Two kinds of variation in rhythmical movement can be considered: one is the variation in time (frequency of the rhythmical movement) and the other is the variation in space (amplitude of the rhythmical movement). However, in this analysis only variation in time was used. As can be seen in Table 1, the Hurst exponents for all subjects were much greater than l/2. This suggests that variation in stereotyped human movement can not be modeled as Brownian motion and each observation is not independent. The correlation function of future increments with past increments, which is time-independent, is given by an expression using the Hurst exponent (Feder, 1988): c = 2(22H-’ - 1).
(5) As can be seen from Eq. (51, in the case of H > l/2 (my results), C is greater than 0. This means that increasing trends in the past generally imply increasing trends in the future. Thus, the observations were not independent. Each observation carries a memory of all the events that preceded it, which leads to persistence (bias). The strength of persistence, or the biased behavior, increases as H approaches to 1.0, or 100 percent correlation in Eq. (5). A tendency of change in the time series with increases of H can be seen using a simulation technique. Feder (1988) produced a formula for creating simulated fractal Brownian motion:
rl-H “,(‘)
= I’( H + l/2) n(M- 1) +
c i=l
((n
i(i)H-‘/*5(1
i i= 1 +i)H-1’2
-
+n(iM+t)-i)
(i)“-“‘)t(l
+ n(M-
1 + t) - i) , i
(6) where (ti} is a set of Gaussian random variables with unit variance and zero mean, H is the Hurst exponent, and r(n) is the gamma function. t is an integer time step, usually one period, which is split into II intervals to approximate a continuous integral. The effect of increasing n is to give a more precise approximation to the short-term behavior of B,(t), n = 8 was chosen for the simulation here. M is the number of periods for which the long memory effect is generated. Theoretically, it should be infinite, but, for the purpose of simulation, a large value of M will suffice. M = 700 was chosen for this simulation.
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N. Yamada /Human
-4I.
Movement Science 14 (1995) 371-384
I
I
0
200
‘loo
600
800
1000
0
200
400
600
800
1000
-4I
I
f ( iteration ) Fig. 6. Numerical simulation of increments of the fractal Brownian motion. (a) Ordinary Brownian increments for H = l/Z. (b) Fractal increments for H = 0.7. (c) Fractal increments for H = 0.9.
Fig. 6 shows a simulated series for H = 0.5, 0.7, 0.9. There is no dramatic change in the high-frequency components as H is increased. However, as H increases, the low-frequency components increase and generate a large amplitude excursion compared with the high-frequency components. The
N. Yamada /Human
‘“102
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10-l
loo
381
101
Frequency (Hz) Fig. 7. Typical power spectra of rhythmical movement time intervals of the index finger.
low-frequency components are equivalent to the persistence or bias. Thus, different value than l/2 for H is related to the degree of persistence. In the model proposed by Wing and Kristofferson (1973) for self-paced periodic tapping, each time interval is assumed to be the sum of three events. Two of the events are attributed to a delay in executing the motor system, and the third is the hypothetical internal timekeeper. Additional assumptions of the model are that the timekeeper intervals and the motor delays are sequences of independent random variables with normal variances. These assumptions imply that the lag one autocorrelation between inter-tap intervals is negative. This means that large intervals tend to be followed by short ones, and vice versa. This has received empirical support in a number of studies (reviewed in Wing, 1980). This relation between adjacent intervals appears in high frequency components in spectrum analysis of time series of successive inter-tap intervals. Fig. 7 shows the typical power spectra of inter-tap intervals. As can be seen in Fig. 7, the spectra of the inter-tap intervals contain a broadband spectrum of not only high frequency components. A remarkable feature of fractal Brownian motion, in the case of H > l/2, is persistence which is shown as low frequency components in spectrum analysis. Wing and Kristofferson (1973) explained the relation between adjacent intervals using the autocorrelation function (ACF). They and other researchers (e.g. Stevens, 1886) have found the persistence in the time series of inter-tap intervals. However, the
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principle behind the persistence in the human rhythmical movement has not been investigated fully. Mandelbrot (1972) showed that the ACF does not work well for longmemory processes, like fractal Brownian motion, and recommended that R/S analysis should be used for the analysis of the such time series. 4.3. Physiological interpretations Muscles in the finger are activated by efferent signals from the central nervous system (CNS), and at the same time, finger movement activate specific sets of sense organs in the finger proprioceptors, such as golgi tendon organs or muscle spindles which produce afferent traffic of nerve impulses towards the CNS. Therefore the neural structure of the movement system is influenced successively by the movement itself (Varela, 1979). In recent studies about human movement control (e.g. Kelso and Ding, 1993), the neural structure of the movement system is considered to be a dynamical system. Dynamical systems are systems that change over time. Here, randomness is viewed as part of the system’s dynamic and not a superimposed component. In the present study, time variation of inter-tap intervals is described as fractal Brownian motion which is produced by long term-memory process. Each tapping movement affects inside state of the movement system, and the system changes successively over time. Thus, variation of inter-tap intervals is consider to be part of the intrinsic dynamics of the movement system which is consider to be a dynamic system. It is thought that fatigue of the muscles in the finger influence variation of the rhythmical movement. When H is different from l/2, shorter range
Table 3 Means and standard deviations of the movement time intervals for all subjects Subjects
A B C D E
first
half
First half of the data
and
latter
half Latter
of the time
series
of rhythmical
half of the data
Mean (s)
SD kJ
Mean (s)
SD (s)
0.437 0.443 0.526 0.490 0.453
0.0176 0.0194 0.0289 0.0207 0.0226
0.435 0.443 0.521 0.473 0.452
0.0210 0.0186 0.0301 0.0221 0.0211
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383
and longer range (biased time series) must have occurred non-periodically (see Fig. 6). However, it is almost impossible to remove the effect of fatigue from bias. Therefore, a simple test was carried out to ascertain the degree of the effect of fatigue on the bias. The time series of the rhythmical movement time intervals were divided into two parts; the first half and the latter half of the data. The means and standard deviations of the data were calculated independently. If the influence of fatigue is remarkable, time intervals of the latter half of the data should be slower than that of the first half of the data. In other words, the latter half of the data should be biased to a slower range. The results in Table 3 show that the means and standard deviations for the two parts are similar. Thus, the effect of fatigue on bias was minimal in this experiment.
5. Conclusion
Variation in rhythmical movement can not be modeled as an independent random process (noise), and is consider to be part of the intrinsic dynamics of a movement system, which is consider to be a dynamical system.
Acknowledgements
This work was supported by a special grant-in-aid promotion of education and science in Hokkaido University provided by the Ministry of Education, Science and Culture.
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Kelso, J.A.S. and M. Ding, 1993. ‘Fluctuations, Intermittency and Controllable chaos in biological coordination’. In: K.M. Newell and D.M. Corcos (eds.), Variability and motor control (pp. 291-316). Human Kinetics Publishers. Kelso, J.A.S., G.S. Schoner, J.P. Scholz and H. Haken, 1987. Phase-locked modes, phase transitions, and component oscillators in biological motion. Physica Scripta 35, 79-87. Mandelbrot, B.B., 1972. Statistical methodology for non-periodic cycles: From the covariance to R/S analysis. Annals of Economic and Social Measurement 1, 259-290. Mandelbrot, B.B., 1982. The fractal geometry of nature. New York: Freeman. Stevens, L.T. 1886. On the time-sense, Mind 11, 393-404. Scheinkman, J.A. and B. LeBaron, 1989. Nonlinear dynamics and stock returns. Journal of Business 62, 311-337. Shapiro, D.C., R.F. Zernicke, R.J. Gregor and J.D. Distel, 1981. Evidence for generalized motor programs using gait pattern analysis. Journal of Motor Behavior 13, 33-47. Shidara, M., K. Kawano, H. Gomi and M. Kawato, 1993. Inverse-dynamics model eye movement control by Purkinje cells in the cerebellum. Nature 364, 50-52 SchGner,G.S., H. Haken and J.A.S.Kelso, 1986. Stochastic theory phase transitions in human hand movement. Biological Cybernetics 53, 442-452. Peters, E.E., 1991. Chaos and order in the capital markets. New York: Wiley. Varela, F., 1979. Principles of biological autonomy. Amsterdam: North-Holland. Wing,A.M.,lWO. ‘The long and short of timing in response sequences’. In: G. Stlemach and J. Requin teds.), Tutorials in motor behavior (pp. 469-486). Amsterdam: North-Holland. Wing, A.M. and A.B. Kristofferson, 1973. Response delays and the timing of discrete motor responses. Perception and Psychophysics 14, 5-12.