Effects of Sampling Rate and Movement Frequency on Entropic Measures of Regularity
Douglas W. Powell1,*, Brian Szekely2, Sarah E. Blackmore1, Alexis Nelson1, Alexandra Schallert3, Deranda B. Lester1, Nicholas G. Murray2, and Melissa Puppa1
1School of Health Studies, University of Memphis, Memphis, TN, USA. 2Department of Kinesiology, University of Nevada Reno, Reno, NV, USA. 3Univerity of Tennessee Health Science Center, Memphis, TN, USA.
Variability is an inherent feature of biological systems and has been measured using a variety of methods. Sample entropy (SampEn) has been a popular tool to describe the variability of a system, but has been suggested to be sensitive to sampling rate and movement frequency. This study quantified the effects of movement frequency and sampling rate on SampEn calculations. Continuous ankle joint angles were recorded 10 healthy participants during treadmill walking. Data were resampled at 50%, 200% and 400% of the original sampling rate and the movement frequency was simulated at 70% and 130% of the original movement frequency. SampEn was calculated for each data set. Simulated sampling rates were associated with significant changes in ApEn values. Faster movement frequencies were associated with significantly greater SampEn values compared to original or slower movement frequency. Current findings demonstrate that sampling rate and movement frequency significantly alter SampEn values.
Variability | Sample Entropy | Gait | Nonlinear | Methodology Sampling Rate | Frequency
Variability is an inherent characteristic of biological systems. Several perspectives of variability have been presented in motor control literature including the Generalized Motor Program Theory (GMPT), Uncontrolled Manifold Hypothesis (UMH) and Dynamical Systems Theory (DST). Generally, GMPT posits that variability is the manifestation of error in the capacity of the motor program to predict the required parameters to accurately complete a given motor task [1-3]. UMH suggests that variability may be functional and is the result of motor redundancy, or the capacity to successfully complete a given task using a variety of strategies [4, 5]. DST views variability as a representation of the stability of a given system and suggests that variability provides important information regarding the self-organization of the motor system [6-9].
Sample entropy is a nonlinear measure of variability and quantifies the moment-to-moment regularity of a signal and the changing complexity of a time-series [9, 10]. The sample entropy technique has been applied to a variety of biological signals including heart rate, blood pressure, hormone release, electromyography and electroencephalography [11-13]. Emerging research has suggested that variability is an inherent and important component of biological signals and provides a measure by which researchers and clinicians can infer the flexibility or adaptability of a given system [10, 14].
Many studies have used approximate entropy to quantify the regularity of biological signals associated with voluntary movement in individuals of various ages and pathological disorders including cerebral palsy, Parkinsoní»s disease and peripheral artery disease [15-25]. However, approximate entropy is sensitive to changes in movement, sampling and equation parameters making it difficult to reliably compare approximate entropy values across studies. For example, Kurz and Hou (2010) investigated the effect of levodopa on lower extremity regularity during treadmill walking in individuals with Parkinsoní»s disease and reported lower extremity approximate entropy values between 0.05 and 0.20. However, Myers et al. (2010) used a similar methodology to investigate the regularity of lower extremity kinematics in young adults undergoing simulated leg claudication and reperfusion, and reported lower extremity approximate entropy values between 0.31 and 0.71. Though both of these studies investigated lower extremity regularity using similar approximate entropy calculations (m = 2, r = 0.20), the quantitative findings are very different. A limitation in comparing these two studies is that they investigated different í░pathologicalí▒ populations walking on a treadmill at different velocities. A further limitation in the comparison and interpretation of these findings is the sensitivity of the approximate entropy measure to a variety of methodological parameters.
It is possible that the unique populations investigated in the studies by Kurz and Hou (2010) and Myers et al. (2010) may underlie the different approximate entropy values reported. However, a comparison of two studies investigating the regularity of resting tremor in individuals with Parkinsoní»s disease also revealed substantial differences in calculated approximate entropy values, despite using similar parameters (m = 2, r = 0.20) for the calculation of approximate entropy [18, 23]. Vaillancourt and Newell (2000) reported approximate entropy values associated with resting tremor between 0.65 and 0.75 while Morrison et al. (2008) reported approximate entropy values nearly two-fold greater. The two studies had similar patient populations with similar disease severity and investigated the same variable, but reported vastly different approximate entropy values. However, Morrison (2008) selected a sampling rate of 100 Hz while Vaillancourt collected kinematic data at 200 Hz (2000). Therefore, it is likely that the differences observed are the manifestation of the methodological parameters used in each study, and the sensitivity of the approximate entropy calculation to those methodological differences. The sensitivity of approximate entropy to methodological differences hinders the interpretation and comparison of approximate entropy values.
Several key methodological parameters should be considered when calculating or interpreting approximate entropy. Previous research has suggested that both sampling rate and movement frequency affect calculated approximate entropy values [16, 26]. Ivkovic and Kurz (2011) investigated the effects of movement frequency on the variability of leg swing kinematics in healthy older adults and individuals with Parkinsoní»s disease. Their results suggest that greater movement frequency was associated with significantly lower approximate entropy values. However, the study used approximate entropy as a tool to measure the organization of the motor system underlying the movement task and may not have adequately elucidated the independent effects of movement frequency on approximate entropy without changes in the stability of the underlying neuromuscular pattern in the study participants. To date, no study has clearly demonstrated the effects of different sampling rates or movement frequencies on calculations of approximate entropy.
Therefore, the purpose of the current study was to elucidate some limitations of the approximate entropy calculation by quantifying the effects of altering sampling rate and movement frequency on approximate entropy calculations. As approximate entropy is commonly used to investigate the self-organization of biological signals, a commonly studied biological signal, ankle joint angle during treadmill walking, was the focus of this study. It was hypothesized that higher sampling rates would be associated with significantly lower approximate entropy values and that lower movement frequency would be associated with significantly lower approximate entropy values.
Ten individuals (6 male, 4 female) participated in this study. Subject anthropometrics are presented in Table 1. All individuals were free of musculoskeletal or neurological conditions that would limit their walking capacity. The experimental protocol was approved by the University Institutional Review Board and conducted in accordance with the Declaration of Helsinki. Study procedures were explained and all subjects read, understood and signed an approved informed consent form prior to participation in this study.
Table 1. Subjectsí» anthropometric measurements. Presented as mean (SD).
2.2 Experimental Protocol
Three-dimensional gait kinematics were collected from the right lower extremity of each subject using an 8-camera motion capture system (240 Hz, Qualisys Inc., Goteburg, Sweden). A Helen Hayes clinical gait marker set was used to define and track lower extremity segments including the shank and foot. Kinematic data were filtered using a lowpass filter with a cutoff frequency of 10 Hz. Each subject attended the laboratory on two separate sessions at least 24 hours apart. In the first session, subjects were familiarized with treadmill walking at 1.3 m/s (2.9 mph). The second session consisted of each subject performing two 3-minute treadmill walking trials while two 30-second epochs of data were collected from the second minute of each walking trial. Subjects were given two minutes of rest between treadmill walking trials to minimize the effect of fatigue. Retro-reflective markers were not removed between treadmill walking trials. Figure 1 presents continuous and average ankle joint angles during a representative treadmill walking trial.
Figure 1. Continuous (A) and ensemble (B) ankle joint angles recorded from a 30-second treadmill walking trial of a representative subject. Positive values indicate dorsiflexion while the vertical lines represent heel strike (A) and toe off (B), respectively.
2.3 Data manipulation
2.3a Sampling rate simulation
Custom software that included the resample.m subroutine was used to calculate ankle joint angles from exported segment data and to modify signal content (MatLab 2010a, MathWorks, Inc., Natick, MA). To simulate increased and decreased sampling rates, the original signal was resampled at 0.5, 2 and 4 times the original sampling rate. As the signal frequency content was not altered, and the number of points was decreased or increased by the given ratios, sampling rates of 60 Hz, 240 Hz and 480 Hz were simulated, respectively. The number of points in each time-series was increased or decreased by the conversion factor (0.5, 2 and 4 x original number of points) during the sampling rate simulation.
2.3b Movement frequency simulation
To simulate increased and decreased movement frequency, custom software (MatLab 2009a, MathWorks, Inc., Natick, MA) employed a fast Fourier transform to convert the original continuous ankle joint angle time-series to the frequency spectrum. The frequency component of the original time-series (ORIG) was then multiplied by 0.7 and 1.3 to decrease and increase the signal frequency, respectively. The signals were then converted back to the time domain using an inverse fast Fourier transform, simulating an ankle joint angle signal 30% slower (LOW) and 30% faster (HIGH) than the original treadmill walking frequency. The number of points in each time-series was not altered during the frequency simulation.
2.4 Approximate entropy calculations
ApEn was used to quantify the regularity and time-dependent structure of the original and simulated ankle joint angle time-series. ApEn has been previously used to quantify regularity of lower extremity kinematics during treadmill walking . Calculation of ApEn yields a value between 0 and 2 and reflects the predictability of future values in a time-series based on preceding values within the signal. For example, an ApEn analysis of a sinusoidal waveform would produce an ApEn value approaching zero as the time-series has a high short-term and long-term predictability and all values of the waveform can be predicted from the first cycle of the waveform. Conversely, ApEn analysis of white noise would produce an ApEn value approaching 2 as each observation within the time-series is completely independent of the preceding values and cannot be predicted from those preceding observations. In the present study, ApEn values for the original and simulated time-series were calculated using the mathematical formula denoted in Equation 1:
where m was the length of compared runs (m = 2), r was the similarity criterion between points in a time-series (r = .2), N was the number of measurements in the time-series (i.e. number of points) and C was the number of vectors defined by m based on the r criterion. The selected parameters were chosen based on previously published parameters used to investigate treadmill walking [17, 19] as well as other cyclic voluntary [16, 20] and involuntary movements [18, 23, 27]. The authors chose not to assess auto mutual information (AMI) or to employ the calculated lag in this study to highlight the possible difficulties in using the standard approximate entropy calculation without calculating AMI or lag. The authors acknowledge that employing an appropriate lag may reduce the effect of sampling rate or movement frequency. Generally, a minimum of 1000 data points are required in the application of the approximate entropy technique . In the current analysis, the sample size (N) was not controlled after the manipulation of sampling rate; however, the number of cycles completed was maintained across all conditions.
2.5 Statistical Analysis
Subject mean ApEn values for each condition were calculated as the mean of the two individual 30-second trials. A 4 x 1 analysis of variance (ANOVA) with Tukeyí»s post-hoc analysis was used to determine the effect of sampling rate on ApEn values. To determine the effect of movement frequency on calculations of ApEn, a 3 x 1 ANOVA followed by Tukeyí»s post-hoc was conducted. Significance was defined as p < 0.05. A follow up correlation was conducted to quantify the nonlinear relationship between sampling rate and ApEn values.
Figure 2. Comparison of mean ApEn values calculated from the original continuous ankle joint angle time-series (120 Hz) and when the sampling rates were simulated at 60 Hz, 240 Hz and 480 Hz. Error bars are shown as standard deviation. a denotes a significant difference from the 60 Hz sampling rate; b denotes a significant difference from the 120 Hz sampling rate.
3.1 Effect of sampling rate
Figure 2 presents mean ApEn values calculated from the original and signals resampled at different sampling rates, while Figure 3 presents the correlation analysis between sampling rate and approximate entropy values. Sampling rate significantly affected ApEn values (F = 58.55, p < 0.001) of the continuous ankle angle time-series. The post-hoc analysis revealed that the 60 Hz sampling rate was associated with significantly greater ApEn values than the 120 Hz (p < 0.001), 240 Hz (p < 0.001) or 480 Hz sampling rates (p<0.001). Further, the 120 Hz was associated with significantly larger ApEn values than the 240 Hz (p = 0.009) and 480 Hz sampling rates (p < 0.001). No significant difference was observed between the ApEn values calculated from the 240 Hz and 480 Hz sampling rates (p = 0.444). The correlation analysis revealed that sampling rate had a strong exponential relationship with ApEn values as evidenced by a correlation coefficient of r = 0.998. The exponential regression equation associated with sampling rate and approximate entropy was y = 0.477e-0.005x.
Figure 3. A strong exponential correlation was observed between sampling rates and ApEn values calculated from kinematics resampled at 60 Hz, 120 Hz, 240 Hz and 480 Hz (r = 0.998).
3.2 Effect of movement frequency
The relationship between movement frequency and ApEn values is presented in Figure 4. The ANOVA revealed a significant effect of movement frequency on ApEn calculations (F = 11.572, p < 0.001). The Tukeyí»s post-hoc analysis showed that there were no differences between the LOW condition and ORIG condition (p = 0.157). However, the ORIG condition was associated with significantly lower ApEn values than the HIGH condition (p = 0.022). The LOW condition was associated with significantly smaller ApEn values than the HIGH condition (p < 0.001).
Figure 4. Mean ApEn values calculated from the original continuous ankle joint angle time-series (Original) and when the movement frequency was decreased (LOW) and increased (HIGH) by 30%. Error bars are shown as standard deviation. a denote a significant difference from the LOW condition; b denotes a significant difference from the ORIG condition.
4.1 Sampling Rate
Many research studies have used approximate entropy as a measure of regularity and variability of biological signals including heart rate, center of pressure and lower extremity kinematics [11, 12, 17, 19-21, 27]. As would be expected with the variety of biological signals examined using this technique, the selected sampling rates have also varied from 60 Hz to over 1000 Hz [17-19, 23, 28]. While this may be a function of the plethora of signals being investigated, several studies with similar variables of interest have also used different sampling rates which may give rise to unique approximate entropy values despite the similarities in movement. The findings of the current study suggest that the selection of different sampling rates may underlie the observed differences between multiple studies of a given variable. In this study, it is clearly demonstrated that higher sampling rates result in significantly reduced approximate entropy values. Moreover, the greatest differences in approximate entropy were present at lower sampling rates. For example, the mean approximate entropy value at 60 Hz was 0.482 í└ 0.133 and at 120 Hz the mean approximate entropy value was 0.225 í└ 0.080; however, approximate entropy values at the 240 Hz and 480 Hz conditions were 0.104 í└ 0.032 and 0.050 í└ 0.015, respectively. Further, the difference in approximate entropy between the 240 Hz and 480 Hz was not statistically significant (p = 0.444). These data suggest that as the sampling rate increases, the effect of sampling rate on approximate entropy calculations is diminished. Figure 3 demonstrates this trend graphically and revealed a very strong correlation between sampling rate and approximate entropy calculations (r = 0.998).
The findings of this study provide a method by which to compare approximate entropy values from different research studies that were collected at different sampling rates. For example, Vaillancourt et al. (2001) reported an approximate entropy value associated with resting tremor of 0.72 while Morrison et al. (2008) reported approximate entropy value of 1.2 when investigating the same variable. The respective sampling rates for these two studies were 200 Hz  and 100 Hz , suggesting that differences in approximate entropy values may be the manifestation of the different sampling rates. Using the regression equation determined in this study, one can adjust the presented approximate entropy values for comparison. In this example, when Vaillancourtí»s (2001) approximate entropy values are adjusted based on sampling rate, the resultant approximate entropy value is 1.26. This resultant value is comparable to the values presented by Morrison (2008) and may allow for better interpretation of approximate entropy results. It should be noted that this comparison after adjustment for sampling rate is only valid because the approximate entropy equation parameters (m and r) were the same between these two studies [18, 23].
Sampling rate is a parameter of importance in the design of research studies focusing on kinematics and kinetics of motion. The findings of this study clearly demonstrate that the sampling rate selected for a given study will affect calculations of approximate entropy. Thus, sampling rate should be carefully considered during study design as comparisons with previously existing literature may be limited if different sampling rates are used. However, as sampling rate is usually maintained within a given study, comparisons made between groups within a study will not be altered by sampling rate.
The results of the current study also demonstrate that as the sampling rate continues to increase, the effects of sampling rate on approximate entropy calculations are minimized. This finding suggests that while most studies using the approximate entropy technique in kinematics have limited their sampling rates to 60 Hz or 120 Hz [3, 17-19, 22, 27], the use of lower sampling rates may not be required and may limit the accuracy of the underlying kinematic data. For example, it has been suggested that at slower sampling rates information may be lost at initial contact of walking and running movements. Thus, higher sampling rates are prescribed to diminish the loss of kinematic data at initial contact. The findings of this study demonstrate that though approximate entropy values may be affected, comparisons within the study will maintain their reliability while the accuracy of kinematic measures will be enhanced.
4.2 Movement Frequency
The current study presents data demonstrating the approximate entropy calculations are affected by movement frequency. These findings support previous research data that demonstrated a significant effect of movement frequency on approximate entropy calculations during a uniaxial leg swinging task . In the study by Ivkovic and Kurz (2011), a collection of healthy young and older adults as well as a group of individuals with Parkinsoní»s disease performed a uniaxial leg swinging task for two minutes at the legí»s pendular frequency (PEND), 80% of the legí»s pendular frequency (SLOW) and 120% of the legí»s pendular frequency (FAST). The frequency of leg swing was maintained using a metronome. Similar to the findings of this study, approximate entropy values were significantly greater at the SLOW and PEND frequencies than the FAST frequency; however, no differences were observed between the SLOW and PEND frequencies . While their findings clearly demonstrate that approximate entropy values are affected by movement frequency, the data were collected from a heterogeneous population as a measure of the regularity of the neuromuscular system underlying the given movements. In the present study, the signal frequency was artificially adjusted and is a better reflection of the implications of movement frequency on approximate entropy calculations.
The current findings have many implications on study design. To adequately compare approximate entropy values between a healthy group and a pathological group, the research must either constrain movement frequency within the healthy population or note as a limitation that the two groups performed the given task at unique movement frequencies limiting the applicability of the approximate entropy findings. For example, Kurz and Hou (2010) investigated the effects of levodopa on lower extremity regularity during gait in individuals with Parkinsoní»s disease. Their findings revealed that levodopa significantly decreased approximate entropy values at the ankle; however, no data representing healthy controls was reported. A similar study reported lower extremity regularity in individuals with Parkinsoní»s disease during a treadmill walking task using approximate entropy . In this study, the authors compared lower extremity approximate entropy values in individuals with Parkinsoní»s disease with and without deep brain stimulation to healthy, age-matched controls. Their results demonstrated that individuals with Parkinsoní»s disease had significantly greater approximate entropy values at the ankle and hip when walking with and without deep brain stimulation treatment. However, a limitation of this study was that healthy controls walked with a significantly slower cadence than individuals with Parkinsoní»s disease limiting the comparison of the healthy controls to individuals with Parkinsoní»s disease. The findings of Powell et al. (2012) demonstrate that when comparing approximate entropy values between a healthy control group and individuals with pathology, care must be taken in the study design. To allow for direct comparisons to be made between individuals with Parkinsoní»s disease and healthy controls in the study by Powell et al. (2012), the cadence of age-matched controls should have been matched to that of the Parkinsoní»s group. The effect of movement frequency on approximate entropy calculations should be considered during study design and weighed against the possible negative effects of constraining voluntary movement on motor control.
The authors acknowledge several limitations of the present study. First and foremost, the present study does not consider auto mutual information nor does the approximate entropy calculation used in this study use an appropriate lag to offset the effect of redundant data with increased sampling rate or decreased movement frequency. It is possible that using a lag calculated using AMI may reduce the strong effects of sampling rate and/or movement frequency on approximate entropy values which may improve interpretation and comparison of signal regularity. A second limitation is that the authors did not alter the length of the simulated ankle joint angle signals after modifying the signal sampling rate. Approximate entropy has been shown to be sensitive to the length of the continuous time-series . However, it should be noted that a longer time-series has also been suggested to improve the accuracy of the approximate entropy calculation.
The findings of this study clearly demonstrate that both sampling rate and movement frequency significantly alter calculations of approximate entropy. While sampling rate may affect approximate entropy values, the internal reliability of a study conducted with a given sampling rate will be maintained, though comparisons to previously published data may be difficult. Conversely, the decreasing approximate entropy values associated with higher sampling rates may reduce the ability of the approximate entropy measure to differentiate between signals with unique moment-to-moment regularities. It is highly recommended that an appropriate lag, based on the calculation of auto mutual information, be used to limit the deleterious effects of sampling rate and movement frequency.
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Conflict of Interest: No conflicts declared.
* Corresponding Author. Douglas Powell, Ph.D., CSCS, TSAC-F, FACSM, FAHA. Co-Director | Exercise Neuroscience Research Laboratory. Assistant Professor | School of Health Sciences
University of Memphis
Memphis, TN 38016, USA.
Phone: (402) 715 - 0450
© 2018 by the Authors | Journal of Nature and Science (JNSCI).