Non-linear tools and methodological concerns measuring human movement variability: an overview

Authors

  • Carla Caballero
  • David Barbado
  • Francisco Javier Moreno

Abstract

In recent years, several works have explored variability using different approaches, trying to describe the variations in motor movement. Traditionally, movement variability was regarded as a system error due to noise of neuromuscular mechanisms, but alternative theories suggest that motor variability seems to reflect a functional behaviour improving motor control and enhancing learning. Controversial results have been reported about variability characteristics and its role in motor control and learning, and several works suggest that the main difficulty lies in how to measure this variability. In this work, we have outlined the most used non-linear tools to assess human variability, their applications, advantages and disadvantages. We have also suggested different methods about how to achieve a multidimensional approximation to motor variability. Finally, we have called attention to some methodological issues frequently reported as important aspects to take into account when measuring human movement variability.

Downloads

Download data is not yet available.

Metrics

Metrics Loading ...

References

Abe, T., Chen, Y., & Pham, T.D. (2014). Chaos Analysis of Brain MRI for Studying Mental Disorders Biomedical Informatics and Technology (pp. 257-270): Springer.

Adami, C., Ofria, C., & Collier, T.C. (2000). Evolution of biological complexity. Proceedings of the National Academy of Sciences, 97(9), 4463-4468.

Adjerid, K., Fino, P., Habib, M., Rezaei, A., Ross, S., & Lockhart, T. (2014). Comparing Postural Stability Entropy Analyses to Differentiate Fallers and Non-fallers.

Ahmad, S., Ramsay, T., Huebsch, L., Flanagan, S., McDiarmid, S., Batkin, I., . . . Shamji, F.M. (2009). Continuous multi-parameter heart rate variability analysis heralds onset of sepsis in adults. PLoS One, 4(8), e6642.

Ball, K.A., & Best, R.J. (2007). Different centre of pressure patterns within the golf stroke I: Cluster analysis. Journal of sports sciences, 25(7), 757-770.

Bandt, C., & Pompe, B. (2002). Permutation entropy: a natural complexity measure for time series. Phys Rev Lett, 88(17), 174102.

Barbado, D., Sabido, R., Vera-Garcia, F.J., Gusi, N., & Moreno, F.J. (2012). Effect of increasing difficulty in standing balance tasks with visual feedback on postural sway and EMG: complexity and performance. Human movement science, 31(5), 1224-1237.

Bashan, A., Bartsch, R., Kantelhardt, J.W., & Havlin, S. (2008). Comparison of detrending methods for fluctuation analysis. Physica A: Statistical Mechanics and its Applications, 387(21), 5080-5090.

Bauer, H-U., & Schöllhorn, W.l. (1997). Self-organizing maps for the analysis of complex movement patterns. Neural Processing Letters, 5(3), 193-199.

Bernstein, N.A. (1967). Co-ordination and regulation of movements.

Bian, C., Qin, C., Ma, Q. DY., & Shen, Q. (2012). Modified permutation-entropy analysis of heartbeat dynamics. Physical Review E, 85(2), 021906.

Blashfield, R.K., & Aldenderfer, M.S. (1978). The literature on cluster analysis. Multivariate Behavioral Research, 13(3), 271-295.

Borg, F.G., & Laxaback, G. (2010). Entropy of balance--some recent results. Journal of neuroengineering and rehabilitation, 7, 38.

Bruijn, S.M., Bregman, D.JJ., Meijer, O.G., Beek, P.J., & van Dieën, J.H. (2012). Maximum Lyapunov exponents as predictors of global gait stability: a modelling approach. Medical engineering & physics, 34(4), 428-436.

Buzzi, U.H., Stergiou, N., Kurz, M.J., Hageman, P.A., & Heidel, J. (2003). Nonlinear dynamics indicates aging affects variability during gait. Clinical Biomechanics, 18(5), 435-443.

Byl, N.N., Nagarajan, S.S., Merzenich, M.M., Roberts, T., & Mc Kenzie, A. (2002). Correlation of clinical neuromusculoskeletal and central somatosensory performance: variability in controls and patients with severe and mild focal hand dystonia. Neural plasticity, 9(3), 177-203.

Caballero, C., Barbado, D., & Moreno, F.J. (2013). El procesado del desplazamiento del centro de presiones para el estudio de la relación complejidad/rendimiento observada en el control postural en bipedestación. Revista Andaluza de Medicina del Deporte, 6(3), 101-107.

Cao, Y., Tung, W-W, Gao, J.B., Protopopescu, V.A., & Hively, L.M. (2004). Detecting dynamical changes in time series using the permutation entropy. PHYSICAL REVIEW-SERIES E-, 70(4; PART 2), 046217-046217.

Carroll, J.P., & Freedman, W. (1993). Nonstationary properties of postural sway. Journal of biomechanics, 26(4), 409-416.

Castiglioni, P., Parati, G., Civijian, A., Quintin, L., & Di Rienzo, M. (2009). Local scale exponents of blood pressure and heart rate variability by detrended fluctuation analysis: effects of posture, exercise, and aging. Biomedical Engineering, IEEE Transactions on, 56(3), 675-684.

Cavanaugh, J.T, Guskiewicz, K.M., & Stergiou, N. (2005). A nonlinear dynamic approach for evaluating postural control. Sports Medicine, 35(11), 935-950.

Chen, W., Wang, Z., Xie, H., & Yu, W. (2007). Characterization of surface EMG signal based on fuzzy entropy. IEEE Trans Neural Syst Rehabil Eng, 15(2), 266-272. doi: 10.1109/TNSRE.2007.897025

Chen, W., Zhuang, J., Yu, W., & Wang, Z. (2009). Measuring complexity using FuzzyEn, ApEn, and SampEn. Med Eng Phys, 31(1), 61-68. doi: 10.1016/j.medengphy.2008.04.005

Chian, L.K., & Wang, C.K. (2008). Motivational Profiles of Junior College Athletes: A Cluster Analysis. JOURNAL OF APPLIED SPORT PSYCHOLOGY, 20, 137-156.

Cignetti, F., Kyvelidou, A., Harbourne, R.T., & Stergiou, N. (2011). Anterior–posterior and medial–lateral control of sway in infants during sitting acquisition does not become adult-like. Gait & posture, 33(1), 88-92.

Clark, J.E., & Phillips, S.J. (1993). A longitudinal study of intralimb coordination in the first year of independent walking: a dynamical systems analysis. Child development, 64(4), 1143-1157.

Costa, M., Goldberger, A.L., & Peng, C.K. (2002). Multiscale entropy analysis of complex physiologic time series. Phys Rev Lett, 89(6), 068102.

Costa, M., Goldberger, A.L., & Peng, C.K. (2005). Multiscale entropy analysis of biological signals. Phys Rev E Stat Nonlin Soft Matter Phys, 71(2 Pt 1), 021906.

Costa, M., Priplata, A.A., Lipsitz, L.A., Wu, Z., Huang, N.E., Goldberger, A.L., & Peng, C-K. (2007). Noise and poise: Enhancement of postural complexity in the elderly with a stochastic-resonance–based therapy. EPL (Europhysics Letters), 77(6), 68008.

Davids, K., Glazier, P., Araujo, D., & Bartlett, R. (2003). Movement systems as dynamical systems: the functional role of variability and its implications for sports medicine. Sports medicine (Auckland, N Z ), 33(4), 245-260.

Delignieres, D., Ramdani, S., Lemoine, L., Torre, K., Fortes, M., & Ninot, G. (2006). Fractal analyses for ‘short’time series: a re-assessment of classical methods. Journal of Mathematical Psychology, 50(6), 525-544.

Dingwell, J.B., & Cusumano, J.P. (2000). Nonlinear time series analysis of normal and pathological human walking. Chaos: An Interdisciplinary Journal of Nonlinear Science, 10(4), 848-863.

Dingwell, J.B., & Kang, H.G. (2007). Differences between local and orbital dynamic stability during human walking. Journal of biomechanical engineering, 129(4), 586-593.

Donker, S.F., Roerdink, M., Greven, A.J., & Beek, P.J. (2007). Regularity of center-of-pressure trajectories depends on the amount of attention invested in postural control. Exp Brain Res, 181(1), 1-11. doi: 10.1007/s00221-007-0905-4

Duarte, M., & Sternad, D. (2008). Complexity of human postural control in young and older adults during prolonged standing. Experimental Brain Research, 191(3), 265-276.

Dutta, S., Ghosh, D., & Chatterjee, S. (2013). Multifractal detrended fluctuation analysis of human gait diseases. Frontiers in physiology, 4, 274.

Eckmann, J-P, Kamphorst, S.O., & Ruelle, D. (1987). Recurrence plots of dynamical systems. EPL (Europhysics Letters), 4(9), 973.

Eckmann, J-P, & Ruelle, D. (1985). Ergodic theory of chaos and strange attractors. Reviews of modern physics, 57(3), 617.

Edelman, G. (1992). Brilliant air, bright fire: On the matter of the mind: Penguin Group.

Glass, L. (2001). Synchronization and rhythmic processes in physiology. Nature, 410(6825), 277-284.

Gnanadesikan, R., Kettenring, J.R., & Maloor, S. (2007). Better alternatives to current methods of scaling and weighting data for cluster analysis. Journal of Statistical planning and Inference, 137(11), 3483-3496.

Goldberger, A.L. (1996). Non-linear dynamics for clinicians: chaos theory, fractals, and complexity at the bedside. The Lancet, 347(9011), 1312-1314.

Goldberger, A.L., Amaral, L.AN., Hausdorff, J.M., Ivanov, P.Ch., Peng, C-K, & Stanley, H.E. (2002). Fractal dynamics in physiology: alterations with disease and aging. Proceedings of the National Academy of Sciences, 99(suppl 1), 2466-2472.

Goldberger, A.L., Peng, C-K, & Lipsitz, L.A. (2002). What is physiologic complexity and how does it change with aging and disease? Neurobiology of aging, 23(1), 23-26.

Granata, K.P., & England, S.A. (2007). Reply to the Letter to the Editor. Gait and Posture, 26, 329–330.

Guerreschi, E., Humeau-Heurtier, A., Mahe, G., Collette, M., & Leftheriotis, G. (2013). Complexity quantification of signals from the heart, the macrocirculation and the microcirculation through a multiscale entropy analysis. Biomedical Signal Processing and Control, 8(4), 341-345.

Harbourne, R.T., & Stergiou, N. (2009). Movement variability and the use of nonlinear tools: principles to guide physical therapist practice. Physical therapy, 89(3), 267-282.

Hauge, E.R., Berle, J. Ø., Oedegaard, K.J., Holsten, F., & Fasmer, O.B. (2011). Nonlinear analysis of motor activity shows differences between schizophrenia and depression: a study using Fourier analysis and sample entropy. PloS one, 6(1), e16291.

Hausdorff, J.M., Peng, C.K., Ladin, Z., Wei, J.Y., & Goldberger, A.L. (1995). Is walking a random walk? Evidence for long-range correlations in stride interval of human gait. J Appl Physiol (1985), 78(1), 349-358.

Holden, J.G. (2005). Gauging the fractal dimension of response times from cognitive tasks. Contemporary nonlinear methods for behavioral scientists: A webbook tutorial, 267-318.

Holtrop, J.L., Loucks, T.M., Sosnoff, J.J., & Sutton, B.P. (2014). Investigating Age-related changes in fine motor control across different effectors and the impact of white matter integrity. NeuroImage.

Hornero, R., Aboy, M., Abásolo, D., McNames, J., & Goldstein, B. (2005). Interpretation of approximate entropy: analysis of intracranial pressure approximate entropy during acute intracranial hypertension. Biomedical Engineering, IEEE Transactions on, 52(10), 1671-1680.

Huang, J-J, Yen, C-T, Tsao, H-W, Tsai, M-L, & Huang, C. (2014). Neuronal Oscillations in Golgi Cells and Purkinje Cells are Accompanied by Decreases in Shannon Information Entropy. The Cerebellum, 13(1), 97-108.

Jackson, J. E. (2005). A user's guide to principal components (Vol. 587): John Wiley & Sons.

Javorka, M., Trunkvalterova, Z., Tonhajzerova, I., Lazarova, Z., Javorkova, J., & Javorka, K. (2008). Recurrences in heart rate dynamics are changed in patients with diabetes mellitus. Clinical physiology and functional imaging, 28(5), 326-331.

Kamm, K., Thelen, E., & Jensen, J.L. (1990). A dynamical systems approach to motor development. Physical therapy, 70(12), 763-775.

Kang, X., Jia, X., Geocadin, R.G., Thakor, N.V., & Maybhate, A. (2009). Multiscale entropy analysis of EEG for assessment of post-cardiac arrest neurological recovery under hypothermia in rats. IEEE Trans Biomed Eng, 56(4), 1023-1031. doi: 10.1109/TBME.2008.2011917

Kelso, A. (1995). Th1 and Th2 subsets: paradigms lost? Immunology today, 16(8), 374-379.

Kirchner, M., Schubert, P., Liebherr, M., & Haas, C.T. (2014). Detrended fluctuation analysis and adaptive fractal analysis of stride time data in Parkinson's disease: stitching together short gait trials. PloS one, 9(1), e85787.

Kodba, S., Perc, M., & Marhl, M. (2005). Detecting chaos from a time series. European journal of physics, 26(1), 205.

Kreuzer, M., Kochs, E.F., Schneider, G., & Jordan, D. (2014). Non-stationarity of EEG during wakefulness and anaesthesia: advantages of EEG permutation entropy monitoring. Journal of clinical monitoring and computing, 1-8.

Lake, D.E., Richman, J.S., Griffin, M.P., & Moorman, J.R. (2002). Sample entropy analysis of neonatal heart rate variability. American Journal of Physiology-Regulatory, Integrative and Comparative Physiology, 283(3), R789-R797.

Lamoth, C.J.C., van Lummel, R.C., & Beek, P.J. (2009). Athletic skill level is reflected in body sway: a test case for accelometry in combination with stochastic dynamics. Gait & posture, 29(4), 546-551.

Latash, M.L. (1993). Control of human movement.

Li, X., Ouyang, G., & Richards, D.A. (2007). Predictability analysis of absence seizures with permutation entropy. Epilepsy research, 77(1), 70-74.

Lipsitz, L.A. (2002). Dynamics of stability: the physiologic basis of functional health and frailty. The journals of gerontology Series A, Biological sciences and medical sciences, 57(3), B115-125.

Liu, J., Zhang, C., & Zheng, C. (2010). EEG-based estimation of mental fatigue by using KPCA–HMM and complexity parameters. Biomedical Signal Processing and Control, 5(2), 124-130.

Lomax, R.G. (2007). Statistical concepts: A second course for education and the behavioral sciences. Mahwah, NJ: Lawrence Erlbaum Associates.

Manor, B., Costa, M. D., Hu, K., Newton, E., Starobinets, O., Kang, H. G., . . . Lipsitz, L. A. (2010). Physiological complexity and system adaptability: evidence from postural control dynamics of older adults. J Appl Physiol, 109(6), 1786-1791. doi: 10.1152/japplphysiol.00390.2010

Marwan, Norbert, Carmen Romano, M, Thiel, Marco, & Kurths, Jürgen. (2007). Recurrence plots for the analysis of complex systems. Physics Reports, 438(5), 237-329.

McNeill, M.C., & Wang, C.K. (2005). Psychological profiles of elite school sports players in Singapore. Psychology of Sport and Exercise, 6(1), 117-128.

Menayo, R., Encarnación, A., Gea, G.M., & Marcos, P.J. (2014). Sample Entropy-Based Analysis of Differential and Traditional Training Effects on Dynamic Balance in Healthy People. Journal of motor behavior(ahead-of-print).

Miller, David J, Stergiou, Nicholas, & Kurz, Max J. (2006). An improved surrogate method for detecting the presence of chaos in gait. Journal of biomechanics, 39(15), 2873-2876.

Moreno, F.J., & Ordoño, E.M. (2010). Aprendizaje motor y síndrome general de adaptación. Motricidad. European Journal of Human Movement, 22, 1-19.

Moreside, J.M., Quirk, D. A., & Hubley-Kozey, C.L. (2014). Temporal patterns of the trunk muscles remain altered in a low back-injured population despite subjective reports of recovery. Archives of physical medicine and rehabilitation, 95(4), 686-698.

Muskulus, M., Slats, A.M., Sterk, P.J., & Verduyn-Lunel, S. (2010). Fluctuations and determinism of respiratory impedance in asthma and chronic obstructive pulmonary disease. Journal of Applied Physiology, 109(6), 1582-1591.

Newell, K.M., & Corcos, D. M. (1993). Variability and motor control: Human Kinetics.

Newell, K.M., Slobounov, S.M., Slobounova, B.S., & Molenaar, P.C.M. (1997). Short-term non-stationarity and the development of postural control. Gait & Posture, 6(1), 56-62.

Norris, P.R., Anderson, S.M., Jenkins, J.M., Williams, A.E., & Morris Jr, J.A. (2008). Heart rate multiscale entropy at three hours predicts hospital mortality in 3,154 trauma patients. Shock, 30(1), 17-22.

Olofsen, E., Sleigh, J.W., & Dahan, A. (2008). Permutation entropy of the electroencephalogram: a measure of anaesthetic drug effect. British journal of anaesthesia, 101(6), 810-821.

Peng, C‐K, Havlin, S., Stanley, H.E., & Goldberger, A.L. (1995). Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos: An Interdisciplinary Journal of Nonlinear Science, 5(1), 82-87.

Pincus, S.M. (1991). Approximate entropy as a measure of system complexity. Proceedings of the National Academy of Sciences, 88(6), 2297-2301.

Rabinovich, M.I., & Abarbanel, H.D.I. (1998). The role of chaos in neural systems. Neuroscience, 87(1), 5-14.

Rapp, P.E., Zimmerman, I.D., Albano, A.M., Greenbaun, N.N., & Bashore, T.R. (1986). Experimental studies of chaotic neural behavior: cellular activity and electroencephalographic signals Nonlinear oscillations in biology and chemistry (pp. 175-205): Springer.

Rein, R., Button, C., Davids, K., & Summers, J. (2010). Cluster analysis of movement patterns in multiarticular actions: a tutorial. Motor control, 14(2), 211-239.

Rhea, C.K., Silver, T.A., Hong, S.L., Ryu, J.H., Studenka, B.E., Hughes, C.M.L., & Haddad, J.M. (2011). Noise and complexity in human postural control: Interpreting the different estimations of entropy. PloS one, 6(3), e17696.

Richman, J.S., & Moorman, J.R. (2000). Physiological time-series analysis using approximate entropy and sample entropy. American Journal of Physiology-Heart and Circulatory Physiology, 278(6), H2039-H2049.

Riley, M.A., Balasubramaniam, R., & Turvey, M.T. (1999). Recurrence quantification analysis of postural fluctuations. Gait & posture, 9(1), 65-78.

Riley, M.A., & Turvey, M.T. (2002). Variability and determinism in motor behavior. Journal of motor behavior, 34(2), 99-125.

Rispens, S.M., Pijnappels, M.A.G.M., van Dieën, J.H., van Schooten, K.S., Beek, P.J., & Daffertshofer, A. (2014). A benchmark test of accuracy and precision in estimating dynamical systems characteristics from a time series. Journal of biomechanics, 47(2), 470-475.

Robbins, S.M., Astephen Wilson, J.L., Rutherford, D.J., & Hubley-Kozey, C.L. (2013). Reliability of principal components and discrete parameters of knee angle and moment gait waveforms in individuals with moderate knee osteoarthritis. Gait & posture, 38(3), 421-427.

Roerdink, M., De Haart, M., Daffertshofer, A., Donker, S. F., Geurts, A. C. H., & Beek, P. J. (2006). Dynamical structure of center-of-pressure trajectories in patients recovering from stroke. Experimental brain research, 174(2), 256-269.

Rosenstein, M.T., Collins, J.J., & De Luca, C.J. (1993). A practical method for calculating largest Lyapunov exponents from small data sets. Physica D: Nonlinear Phenomena, 65(1), 117-134.

Sano, M., & Sawada, Y. (1985). Measurement of the Lyapunov spectrum from a chaotic time series. Physical review letters, 55(10), 1082.

Schmidt, R. A., Zelaznik, H., Hawkins, B., Frank, J. S., & Quinn, J. T., Jr. (1979). Motor-output variability: a theory for the accuracy of rapid motor acts. Psychological review, 47(5), 415-451.

Schmitt, D.T., Stein, P.K., & Ivanov, P.Ch. (2009). Stratification pattern of static and scale-invariant dynamic measures of heartbeat fluctuations across sleep stages in young and elderly. Biomedical Engineering, IEEE Transactions on, 56(5), 1564-1573.

Schöllhorn, W.l. (2003). Coordination dynamics and its consequences on sports. International Journal of Computer Science in Sport, 2(2), 40-46.

Seifert, L., Komar, J., Barbosa, T., Toussaint, H., Millet, G., & Davids, K. (2014). Coordination Pattern Variability Provides Functional Adaptations to Constraints in Swimming Performance. Sports Medicine, 1-13.

Seifert, L., Wattebled, L., L’Hermette, M., Bideault, G., Herault, R., & Davids, K. (2013). Skill transfer, affordances and dexterity in different climbing environments. Human movement science, 32(6), 1339-1352.

Shelhamer, M. (2006). Nonlinear dynamics in physiology: World Scientific.

Smith, B.A., Stergiou, N., & Ulrich, B.D. (2011). Patterns of gait variability across the lifespan in persons with and without down syndrome. Journal of neurologic physical therapy: JNPT, 35(4), 170.

Smith, B.A., Teulier, C., Sansom, J., Stergiou, N., & Ulrich, B.D. (2011). Approximate entropy values demonstrate impaired neuromotor control of spontaneous leg activity in infants with myelomeningocele. Pediatric physical therapy: the official publication of the Section on Pediatrics of the American Physical Therapy Association, 23(3), 241.

Stergiou, N. (2004). Innovative analyses of human movement: Human Kinetics Publishers.

Stergiou, N., Buzzi, U.H., Kurz, M.J., & Heidel, J. (2004). Nonlinear tools in human movement. Champaing, IL: Human Kinetics.

Stergiou, N., & Decker, L.M. (2011). Human movement variability, nonlinear dynamics, and pathology: is there a connection? Human movement science, 30(5), 869-888.

Stergiou, N., Harbourne, R.T., & Cavanaugh, J.T. (2006). Optimal movement variability: a new theoretical perspective for neurologic physical therapy. Journal of Neurologic Physical Therapy, 30(3), 120-129.

Stergiou, N., Yu, Y., & Kyvelidou, A. (2013). A Perspective on Human Movement Variability With Applications in Infancy Motor Development.

Theiler, J., Eubank, S., Longtin, A., Galdrikian, B., & Doyne-Farmer, J. (1992). Testing for nonlinearity in time series: the method of surrogate data. Physica D: Nonlinear Phenomena, 58(1), 77-94.

Thelen, E. (1995). Motor development: A new synthesis. American psychologist, 50(2), 79.

Thelen, E., Ulrich, B.D., & Wolff, P.H. (1991). Hidden skills: A dynamic systems analysis of treadmill stepping during the first year. Monographs of the society for research in child development, i-103.

Thuraisingham, R.A., & Gottwald, G.A. (2006). On multiscale entropy analysis for physiological data. Physica A: Statistical Mechanics and its Applications, 366, 323-332.

Toro, B., Nester, C.J., & Farren, P.C. (2007). Cluster analysis for the extraction of sagittal gait patterns in children with cerebral palsy. Gait & posture, 25(2), 157-165.

Vaillancourt, David E, & Newell, Karl M. (2002). Changing complexity in human behavior and physiology through aging and disease. Neurobiology of aging, 23(1), 1-11.

Vakharia, V., Gupta, V.K., & Kankar, P.K. (2014). A multiscale permutation entropy based approach to select wavelet for fault diagnosis of ball bearings. Journal of Vibration and Control, 1077546314520830.

Van Dieën, J.H., Koppes, L.LJ., & Twisk, J.WR. (2010). Postural sway parameters in seated balancing; their reliability and relationship with balancing performance. Gait & posture, 31(1), 42-46.

Van Orden, G.C., Kloos, H., & Wallot, S. (2011). Living in the pink: Intentionality, wellbeing, and complexity. Philosophy of complex systems. handbook of the philosophy of science, 10.

Van Schooten, K.S., Sloot, L.H., Bruijn, S.M., Kingma, H., Meijer, O.G., Pijnappels, M., & van Dieën, J.H. (2011). Sensitivity of trunk variability and stability measures to balance impairments induced by galvanic vestibular stimulation during gait. Gait & posture, 33(4), 656-660.

Vlachopoulos, S.P., Karageorghis, C.I., & Terry, P.C. (2000). Motivation profiles in sport: A self-determination theory perspective. Research Quarterly for Exercise and Sport, 71(4), 387-397.

Webber, Jr. C.L., & Zbilut, J.P. (2005). Recurrence quantification analysis of nonlinear dynamical systems. Tutorials in contemporary nonlinear methods for the behavioral sciences, 26-94.

Wijnants, M.L., Bosman, A.M., Hasselman, F., Cox, R.F., & Van Orden, G.C. (2009). 1/f scaling in movement time changes with practice in precision aiming. Nonlinear Dynamics Psychol Life Sci, 13(1), 79-98.

Wilkins, B.A., Komanduri, R., Bukkapatnam, S., Yang, H., Warta, G., & Benjamin, B.A. (2009). Recurrence quantification analysis (RQA) used for detection of ST segment deviation. The FASEB Journal.

Wolf, A., Swift, J.B., Swinney, H.L., & Vastano, J.A. (1985). Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, 16(3), 285-317.

Wu, S-D, Wu, C-W, Lin, S-G, Lee, K-Y, & Peng, C-K. (2014). Analysis of complex time series using refined composite multiscale entropy. Physics Letters A, 378(20), 1369-1374.

Xie, H-B, Guo, J-Y, & Zheng, Y-P. (2010). Fuzzy approximate entropy analysis of chaotic and natural complex systems: detecting muscle fatigue using electromyography signals. Annals of biomedical engineering, 38(4), 1483-1496.

Zanin, M., Zunino, L., Rosso, O.A., & Papo, D. (2012). Permutation entropy and its main biomedical and econophysics applications: a review. Entropy, 14(8), 1553-1577.

Zbilut, J.P., & Webber, C.L. (2006). Recurrence quantification analysis. Wiley encyclopedia of biomedical engineering.

Zbilut, J.P., Webber, C.L., Jr., Colosimo, A., & Giuliani, A. (2000). The role of hydrophobicity patterns in prion folding as revealed by recurrence quantification analysis of primary structure. Protein Eng, 13(2), 99-104.

Zeng, X., Eykholt, R., & Pielke, R.A. (1991). Estimating the Lyapunov-exponent spectrum from short time series of low precision. Physical Review Letters, 66(25), 3229.

Zunino, L., Zanin, M., Tabak, B.M., Pérez, D.G., & Rosso, O.A. (2009). Forbidden patterns, permutation entropy and stock market inefficiency. Physica A: Statistical Mechanics and its Applications, 388(14), 2854-2864.

Downloads

Published

2014-07-11

Issue

Section

Original Research

Most read articles by the same author(s)