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A statistical approach to the use of control entropy identifies differences in constraints of gait in highly trained versus untrained runners
1.  Department of Mathematics & Computer Science, Clarkson University, Potsdam, NY 13676, United States, United States, United States 
2.  Applied Physiology Laboratory, Eastern Michigan University, Ypsilanti, MI 48197, United States, United States 
References:
[1] 
W. Aziz and M. Arif, Complexity analysis of stride interval time series by threshold dependent symbolic entropy,, Eur. J. Appl. Physiol., 98 (2006), 30. doi: 10.1007/s0042100602265. Google Scholar 
[2] 
O. Beauchet, V. Dubost, F. R. Herrmann and R. W. Kressig, Stridetostride variability while backward counting among healthy young adults,, J. Neuroeng. Rehabil., 2 (2005). doi: 10.1186/17430003226. Google Scholar 
[3] 
B. R. Bloem, V. V. Valkenburg, M. Slabbekoorn and M. D. Willemsen, The multiple tasks test: Development and normal strategies,, Gait Posture, 14 (2001), 191. Google Scholar 
[4] 
E. M. Bollt, J. D. Skufca and S. J. McGregor, Control entropy: A complexity measure for nonstationary signals,, Mathematical Biosciences and Engineering, 6 (2009), 1. doi: 10.3934/mbe.2009.6.1. Google Scholar 
[5] 
U. H. Buzzi and B. D. Ulrich, Dynamic stability of gait cycles as a function of speed and system constraints,, Motor Control, 8 (2004), 241. Google Scholar 
[6] 
P. Cavanagh, The mechanics of distance running: A historical perspective,, in, (1990). Google Scholar 
[7] 
T. M. Cover and J. A. Thomas, "Elements of Information Theory,", Wiley Series in Telecommunications, (1991). doi: 10.1002/0471200611. Google Scholar 
[8] 
K. Davids, S. Bennett and K. M. Newell, "Movement System Variability,", Human Kinetics, (2006). Google Scholar 
[9] 
D. P. Ferris, G. S. Sawicki and M. A. Daley, A physiologist's perspective on robotic exoskeletons for human locomotion,, Int. J. HR, 4 (2007), 507. doi: 10.1142/S0219843607001138. Google Scholar 
[10] 
A. D. Georgoulis, C. Moraiti, S. Ristanis and N. Stergiou, A novel approach to measure variability in the anterior cruciate ligament deficient knee during walking: The use of the approximate entropy in orthopaedics,, J. Clin. Monit. Comput., 20 (2006), 11. doi: 10.1007/s1087700610327. Google Scholar 
[11] 
P. S. Glazier and K. Davids, Constraints on the complete optimization of human motion,, Sports Med., 39 (2009), 15. doi: 10.2165/0000725620093901000002. Google Scholar 
[12] 
G. H. Golub and C. F. Van Loan, "Matrix Computations,", The Johns Hopkins University Press, (1996). Google Scholar 
[13] 
P. Grassberger and I. Procaccia, Estimation of the Kolmogorov entropy from a chaotic signal,, Physical Review A, 28 (1983), 2591. doi: 10.1103/PhysRevA.28.2591. Google Scholar 
[14] 
J. M. Hausdorff, Gait dynamics, fractals and falls: Finding meaning in the stridetostride fluctuations of human walking,, Hum. Mov. Sci., 26 (2007), 555. Google Scholar 
[15] 
H. Kantz and T. Schreiber, "Nonlinear Time Series Analysis," Second edition,, Cambridge University Press, (2004). Google Scholar 
[16] 
C. K. Karmakar, A. H. Khandoker, R. K. Begg, M. Palaniswami and S. Taylor, Understanding ageing effects by approximate entropy analysis of gait variability,, Conf. Proc. IEEE Eng. Med. Biol. Soc., (2007), 1965. doi: 10.1109/IEMBS.2007.4352703. Google Scholar 
[17] 
A. H. Khandoker, M. Palaniswami and R. K. Begg, A comparative study on approximate entropy measure and poincaire plot indexes of minimum foot clearance variability in the elderly during walking,, J. Neuroeng. Rehabil., 5 (2008). doi: 10.1186/1743000354. Google Scholar 
[18] 
M. J. Kurz and N. Stergiou, The aging human neuromuscular system expresses less certainty for selecting joint kinematics during gait,, Neurosci. Lett., 348 (2003), 155. doi: 10.1016/S03043940(03)007365. Google Scholar 
[19] 
F. Liao, J. Wang and P. He, Multiresolution entropy analysis of gait symmetry in neurological degenerative diseases and amyotrophic lateral sclerosis,, Med. Eng. Phys., 30 (2008), 299. doi: 10.1016/j.medengphy.2007.04.014. Google Scholar 
[20] 
K. V. Mardia, J. T. Kent and J. M. Bibby, "Multivariate Analysis,", Probability and Mathematical Statistics: A Series of Monographs and Textbooks, (1979). Google Scholar 
[21] 
S. J. McGregor, M. A. Busa, J. D. Skufca, J. A. Yaggie and E. M. Bollt, Control entropy identifies differential changes in complexity of walking and running gait patterns with increasing speed in highly trained runners,, Chaos, 19 (2009). Google Scholar 
[22] 
S. J. McGregor, M. A. Busa, J. A. Yaggie and E. M. Bollt, High resolution MEMS accelerometers to estimate VO2 and compare running mechanics between highly trained intercollegiate and untrained runners,, PLoS One, 4 (2009). doi: 10.1371/journal.pone.0007355. Google Scholar 
[23] 
S. P. Messier, C. Legault, C. R. Schoenlank, J. J. Newman, D. F. Martin and P. Devita, Risk factors and mechanisms of knee injury in runners,, Med. Sci. Sports Exerc., 40 (2008), 1873. doi: 10.1249/MSS.0b013e31817ed272. Google Scholar 
[24] 
D. J. Miller, N. Stergiou and M. J. Kurz, An improved surrogate method for detecting the presence of chaos in gait,, J. Biomech., 39 (2006), 2873. doi: 10.1016/j.jbiomech.2005.10.019. Google Scholar 
[25] 
R. MoeNilssen, A new method for evaluating motor control in gait under reallife environmental conditions. Part 2: Gait analysis,, Clin. Biomech. (Bristol, 13 (1998), 328. doi: 10.1016/S02680033(98)000904. Google Scholar 
[26] 
K. M. Newell, Constraints on the development of coordination,, in, (1986), 341. Google Scholar 
[27] 
K. M. Newell and D. E. Vaillancourt, Dimensional change in motor learning,, Hum. Mov. Sci., 20 (2001), 695. doi: 10.1016/S01679457(01)000732. Google Scholar 
[28] 
S. M. Pincus, Approximate entropy as a measure of system complexity,, Proceedings of the National Academy of Sciences of the United States of America, 88 (1991), 2297. Google Scholar 
[29] 
S. M. Pincus, Assessing serial irregularity and its implications for health,, Annals of the New York Academy of Sciences, 954 (2001). doi: 10.1111/j.17496632.2001.tb02755.x. Google Scholar 
[30] 
A. Renyi, On measures of entropy and information,, Proceedings of the 4th Berkeley Sympo sium on Mathematical Statistics and Probability, 1 (1961), 547. Google Scholar 
[31] 
J. S. Richman and J. R. Moorman, Physiological timeseries analysis using approximate en tropy and sample entropy,, American Journal of Physiology Heart and Circulatory Physiology, 278 (2000), 2039. Google Scholar 
[32] 
C. Robinson, "Infinite Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDE and the Theory of Global Attractors,", 2^{nd} edition, (2001). Google Scholar 
[33] 
L. A. Schrodt, V. S. Mercer, C. A. Giuliani and M. Hartman, Characteristics of stepping over an obstacle in community dwelling older adults under dualtask conditions,, Gait Posture, 19 (2004), 279. doi: 10.1016/S09666362(03)000675. Google Scholar 
[34] 
C. E. Shannon and W. Weaver, "The Mathematical Theory of Information,", Uni versity of Illinois Press, 97 (1949). Google Scholar 
[35] 
J. S. Slawinski and V. L. Billat, Difference in mechanical and energy cost between highly, well, and nontrained runners,, Med. Sci. Sports Exerc., 36 (2004), 1440. doi: 10.1249/01.MSS.0000135785.68760.96. Google Scholar 
[36] 
G. YogevSeligmann, J. M. Hausdorff and N. Giladi, The role of executive function and attention in gait,, Mov. Disord., 23 (2008), 329. doi: 10.1002/mds.21720. Google Scholar 
[37] 
M. Joyner and E. Coyle, Endurance exercise performance: The physiology of champions,, The Journal of Physiology, 586 (2008), 35. Google Scholar 
[38] 
J. Lin, E. Keogh, S. Lonardi and B. Chiu, A symbolic representation of time series, with implications for streaming algorithms,, Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery, (2003), 2. Google Scholar 
[39] 
Y. Nakayama, K. Kudo and T. Ohtsuki, Variability and fluctuation in running gait cycle of trained runners and nonrunners,, Gait Posture, 31 (2009), 331. doi: 10.1016/j.gaitpost.2009.12.003. Google Scholar 
[40] 
K. Jordan and K. M. Newell, The structure of variability in human walking and running is speeddependent,, Exerc. Sport Sci. Rev., 36 (2008), 200. doi: 10.1097/JES.0b013e3181877d71. Google Scholar 
show all references
References:
[1] 
W. Aziz and M. Arif, Complexity analysis of stride interval time series by threshold dependent symbolic entropy,, Eur. J. Appl. Physiol., 98 (2006), 30. doi: 10.1007/s0042100602265. Google Scholar 
[2] 
O. Beauchet, V. Dubost, F. R. Herrmann and R. W. Kressig, Stridetostride variability while backward counting among healthy young adults,, J. Neuroeng. Rehabil., 2 (2005). doi: 10.1186/17430003226. Google Scholar 
[3] 
B. R. Bloem, V. V. Valkenburg, M. Slabbekoorn and M. D. Willemsen, The multiple tasks test: Development and normal strategies,, Gait Posture, 14 (2001), 191. Google Scholar 
[4] 
E. M. Bollt, J. D. Skufca and S. J. McGregor, Control entropy: A complexity measure for nonstationary signals,, Mathematical Biosciences and Engineering, 6 (2009), 1. doi: 10.3934/mbe.2009.6.1. Google Scholar 
[5] 
U. H. Buzzi and B. D. Ulrich, Dynamic stability of gait cycles as a function of speed and system constraints,, Motor Control, 8 (2004), 241. Google Scholar 
[6] 
P. Cavanagh, The mechanics of distance running: A historical perspective,, in, (1990). Google Scholar 
[7] 
T. M. Cover and J. A. Thomas, "Elements of Information Theory,", Wiley Series in Telecommunications, (1991). doi: 10.1002/0471200611. Google Scholar 
[8] 
K. Davids, S. Bennett and K. M. Newell, "Movement System Variability,", Human Kinetics, (2006). Google Scholar 
[9] 
D. P. Ferris, G. S. Sawicki and M. A. Daley, A physiologist's perspective on robotic exoskeletons for human locomotion,, Int. J. HR, 4 (2007), 507. doi: 10.1142/S0219843607001138. Google Scholar 
[10] 
A. D. Georgoulis, C. Moraiti, S. Ristanis and N. Stergiou, A novel approach to measure variability in the anterior cruciate ligament deficient knee during walking: The use of the approximate entropy in orthopaedics,, J. Clin. Monit. Comput., 20 (2006), 11. doi: 10.1007/s1087700610327. Google Scholar 
[11] 
P. S. Glazier and K. Davids, Constraints on the complete optimization of human motion,, Sports Med., 39 (2009), 15. doi: 10.2165/0000725620093901000002. Google Scholar 
[12] 
G. H. Golub and C. F. Van Loan, "Matrix Computations,", The Johns Hopkins University Press, (1996). Google Scholar 
[13] 
P. Grassberger and I. Procaccia, Estimation of the Kolmogorov entropy from a chaotic signal,, Physical Review A, 28 (1983), 2591. doi: 10.1103/PhysRevA.28.2591. Google Scholar 
[14] 
J. M. Hausdorff, Gait dynamics, fractals and falls: Finding meaning in the stridetostride fluctuations of human walking,, Hum. Mov. Sci., 26 (2007), 555. Google Scholar 
[15] 
H. Kantz and T. Schreiber, "Nonlinear Time Series Analysis," Second edition,, Cambridge University Press, (2004). Google Scholar 
[16] 
C. K. Karmakar, A. H. Khandoker, R. K. Begg, M. Palaniswami and S. Taylor, Understanding ageing effects by approximate entropy analysis of gait variability,, Conf. Proc. IEEE Eng. Med. Biol. Soc., (2007), 1965. doi: 10.1109/IEMBS.2007.4352703. Google Scholar 
[17] 
A. H. Khandoker, M. Palaniswami and R. K. Begg, A comparative study on approximate entropy measure and poincaire plot indexes of minimum foot clearance variability in the elderly during walking,, J. Neuroeng. Rehabil., 5 (2008). doi: 10.1186/1743000354. Google Scholar 
[18] 
M. J. Kurz and N. Stergiou, The aging human neuromuscular system expresses less certainty for selecting joint kinematics during gait,, Neurosci. Lett., 348 (2003), 155. doi: 10.1016/S03043940(03)007365. Google Scholar 
[19] 
F. Liao, J. Wang and P. He, Multiresolution entropy analysis of gait symmetry in neurological degenerative diseases and amyotrophic lateral sclerosis,, Med. Eng. Phys., 30 (2008), 299. doi: 10.1016/j.medengphy.2007.04.014. Google Scholar 
[20] 
K. V. Mardia, J. T. Kent and J. M. Bibby, "Multivariate Analysis,", Probability and Mathematical Statistics: A Series of Monographs and Textbooks, (1979). Google Scholar 
[21] 
S. J. McGregor, M. A. Busa, J. D. Skufca, J. A. Yaggie and E. M. Bollt, Control entropy identifies differential changes in complexity of walking and running gait patterns with increasing speed in highly trained runners,, Chaos, 19 (2009). Google Scholar 
[22] 
S. J. McGregor, M. A. Busa, J. A. Yaggie and E. M. Bollt, High resolution MEMS accelerometers to estimate VO2 and compare running mechanics between highly trained intercollegiate and untrained runners,, PLoS One, 4 (2009). doi: 10.1371/journal.pone.0007355. Google Scholar 
[23] 
S. P. Messier, C. Legault, C. R. Schoenlank, J. J. Newman, D. F. Martin and P. Devita, Risk factors and mechanisms of knee injury in runners,, Med. Sci. Sports Exerc., 40 (2008), 1873. doi: 10.1249/MSS.0b013e31817ed272. Google Scholar 
[24] 
D. J. Miller, N. Stergiou and M. J. Kurz, An improved surrogate method for detecting the presence of chaos in gait,, J. Biomech., 39 (2006), 2873. doi: 10.1016/j.jbiomech.2005.10.019. Google Scholar 
[25] 
R. MoeNilssen, A new method for evaluating motor control in gait under reallife environmental conditions. Part 2: Gait analysis,, Clin. Biomech. (Bristol, 13 (1998), 328. doi: 10.1016/S02680033(98)000904. Google Scholar 
[26] 
K. M. Newell, Constraints on the development of coordination,, in, (1986), 341. Google Scholar 
[27] 
K. M. Newell and D. E. Vaillancourt, Dimensional change in motor learning,, Hum. Mov. Sci., 20 (2001), 695. doi: 10.1016/S01679457(01)000732. Google Scholar 
[28] 
S. M. Pincus, Approximate entropy as a measure of system complexity,, Proceedings of the National Academy of Sciences of the United States of America, 88 (1991), 2297. Google Scholar 
[29] 
S. M. Pincus, Assessing serial irregularity and its implications for health,, Annals of the New York Academy of Sciences, 954 (2001). doi: 10.1111/j.17496632.2001.tb02755.x. Google Scholar 
[30] 
A. Renyi, On measures of entropy and information,, Proceedings of the 4th Berkeley Sympo sium on Mathematical Statistics and Probability, 1 (1961), 547. Google Scholar 
[31] 
J. S. Richman and J. R. Moorman, Physiological timeseries analysis using approximate en tropy and sample entropy,, American Journal of Physiology Heart and Circulatory Physiology, 278 (2000), 2039. Google Scholar 
[32] 
C. Robinson, "Infinite Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDE and the Theory of Global Attractors,", 2^{nd} edition, (2001). Google Scholar 
[33] 
L. A. Schrodt, V. S. Mercer, C. A. Giuliani and M. Hartman, Characteristics of stepping over an obstacle in community dwelling older adults under dualtask conditions,, Gait Posture, 19 (2004), 279. doi: 10.1016/S09666362(03)000675. Google Scholar 
[34] 
C. E. Shannon and W. Weaver, "The Mathematical Theory of Information,", Uni versity of Illinois Press, 97 (1949). Google Scholar 
[35] 
J. S. Slawinski and V. L. Billat, Difference in mechanical and energy cost between highly, well, and nontrained runners,, Med. Sci. Sports Exerc., 36 (2004), 1440. doi: 10.1249/01.MSS.0000135785.68760.96. Google Scholar 
[36] 
G. YogevSeligmann, J. M. Hausdorff and N. Giladi, The role of executive function and attention in gait,, Mov. Disord., 23 (2008), 329. doi: 10.1002/mds.21720. Google Scholar 
[37] 
M. Joyner and E. Coyle, Endurance exercise performance: The physiology of champions,, The Journal of Physiology, 586 (2008), 35. Google Scholar 
[38] 
J. Lin, E. Keogh, S. Lonardi and B. Chiu, A symbolic representation of time series, with implications for streaming algorithms,, Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery, (2003), 2. Google Scholar 
[39] 
Y. Nakayama, K. Kudo and T. Ohtsuki, Variability and fluctuation in running gait cycle of trained runners and nonrunners,, Gait Posture, 31 (2009), 331. doi: 10.1016/j.gaitpost.2009.12.003. Google Scholar 
[40] 
K. Jordan and K. M. Newell, The structure of variability in human walking and running is speeddependent,, Exerc. Sport Sci. Rev., 36 (2008), 200. doi: 10.1097/JES.0b013e3181877d71. Google Scholar 
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