PHYS THER
Vol. 88, No. 9, September 2008, pp. 1037-1038
DOI: 10.2522/ptj.20070171.ar
Author Response
Shohei Ohgi,
Satoru Morita,
Kek Khee Loo and
Chihiro Mizuike
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Introduction
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We thank Fetters and Scholz for sharing their insights.1 Their expertise in the analysis of infant movements and their contributions to the field are widely recognized. Our study2 was of an exploratory nature, as time series analysis of movements in young infants using data recorded from a tri-axial accelerometer is relatively novel. As such, it was not meant to compare the merits of nonlinear and linear methods, and readers should not interpret the findings for support of methodological principles. Fetters and Scholz expressed the assumption that "[t]he premise of the investigation is that such movements have properties of chaotic systems, requiring nonlinear methods to fully appreciate these properties as well as to distinguish between movements of infants with and without BI [brain injury]." We found value in the use of nonlinear methods in our study, but we made no suppositions about the requirement for nonlinear methods to be able to fully appreciate infants’ movement properties.
Similarly, Fetters and Scholz state that "although the spectra do not differ for the 2 exemplary infants with and without BI, this is insufficient to reject the usefulness of a linear approach in general." We would be remiss within the scope of our study to comment on rejecting or disproving the usefulness of a linear approach. However, the positive findings in our study were important, and these findings were noted through nonlinear methods. As we comment below, other linear methods beyond spectral analysis will be useful to complement the analyses presented in our article. In the following paragraphs, we address the more specific issues raised in the commentary.
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Methods of Nonlinear Time Series Analysis and Deterministic Chaos
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Spontaneous movements in premature infants cannot be regarded completely as the result of deterministic chaos because these infants are always perturbed by environmental noise and internal noise. We assumed that the spontaneous movements are generated by nonlinear dynamics with stochastic perturbation (that is, a nonlinear stochastic system).3 In general, such an assumption is justifiable, especially in the context of biological systems. When the effect from the stochastic randomness is stronger than that from nonlinearity, the nonlinear time series analysis does not work well. If the stochastic noise is sufficiently strong, the movement can be regarded as a linear random process. A system can be chaotic even when it is affected by random noise. From this perspective, chaos and randomness are not always inconsistent. Thus, we performed just a quantitative assessment and could not define whether the system is chaotic or random.
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Principal Component Analysis (PCA) and Nonlinear Analysis
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Principal component analysis might be useful to identify the number of modes of joint coupling required to explain the majority of joint variance.4 Principal component analysis originally was based on the assumption that the system is described by a linear stochastic model. From the standpoint of nonlinear dynamics theory, this assumption corresponds to the situation that the embedding dimension is infinity. Thus, the degrees of freedom in PCA and the embedding dimension in nonlinear time series analysis are quite different concepts. Complex systems studies5–8 have shown that the system tends to have higher flexibility if the dimension and the Lyapunov exponent are smaller. The findings in our study did not contradict previous reports in this respect. We pointed out that disorganization of spontaneous movements in infants with BI has been noted previously in observational and kinematic studies, including those of the commentators, but understandably, only general themes can be drawn from studies using completely different designs.
In our linear analysis, we analyzed only the power spectra. We agree that more sophisticated linear analysis methods, such as PCA, may be better used to measure the voluntary movements of adults with impairment and to plan their rehabilitation. However, we believe that nonlinear analysis is needed to clarify the mechanism that generates the spontaneous movements in infants. The reason is that theoretical insights for motor control of spontaneous movements are better derived from knowledge of nonlinear dynamics than from statistical linear approaches. Developmental changes in motor behaviors are proposed to emerge out of the social, cognitive, and perceptual-motor aspects of past experiences as well as the current context and task requirements.9 Moreover, our method of measuring the movement is relatively straightforward and is not time-consuming. As noted in the commentary, the difference between the infant groups in our study was quantitative rather than qualitative. We expect that combining our method and PCA will produce more useful explanations and clearer interpretations. This will be left for future work.
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Lyapunov Exponent
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It would be unreasonable to hold an expectation to establish a clinically meaningful difference in Lyapunov exponents between infant groups from an exploratory study, but we need to start from somewhere, and a statistically significant difference can be a starting point. Clinically, it appears that the infants at higher risk for long-term motor impairment had higher maximal Lyapunov exponent and false-nearest-neighbor (FNN) values. Infants with higher maximal Lyapunov exponent and FNN values might have greater problems with self-organization as a function of the coordination of subsystems.
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Surrogate Data
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In the surrogate analysis, if nonlinear prediction works well (the prediction error is significantly smaller than the surrogate data), then we accept the hypothesis that the time series demonstrates characteristics of nonlinear dynamics.10 If the noise effect is too large, the null hypothesis cannot be rejected. This does not mean that the time series is derived from a linear random process. We used the term "randomness" to denote the situation that characteristics of nonlinear dynamics are not recognized. As noted in the commentary, there are exceptional cases (y-axis data for BI group infants 1 and 5) that seem not to be random. We analyzed acceleration data for 3 axes separately and tried to integrate the results. Although we also performed nonlinear time series analysis for the 3 time series simultaneously, the result was not shown because there was no obvious difference.
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References
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- Fetters L, Scholz JP. Commentary on "Time series analysis of spontaneous upper-extremity movements of premature infants with brain injuries." Phys Ther. 2008;88:1034–1036.[Free Full Text]
- Ohgi S, Morita S, Loo KK, Mizuike C. Time series analysis of spontaneous upper-extremity movements of premature infants with brain injuries. Phys Ther. 2008;88:1022–1033.[Abstract/Free Full Text]
- Ohgi S, Morita S, Loo KK, et al. A dynamical systems analysis of spontaneous movements in newborn infants. J Mot Behav. 2007;39:203–214.[CrossRef][Web of Science][Medline]
- Fetters L, Chen YP, Jonsdottir J, Tronick EZ. Kicking coordination captures differences between full-term and premature infants with white matter disorder. Hum Mov Sci. 2004;22:729–748.[CrossRef][Medline]
- Crutchfield JP, Young K. Inferring statistical complexity. Phys Rev Lett. 1990;63;105–108.[Web of Science]
- Langton CG. Computation at the edge of chaos: phase transitions and emergent computation. Physica D. 1990;42:12–37.[CrossRef]
- Adami C. What is complexity? BioEssays. 2002;24:1085–1094.[CrossRef][Web of Science][Medline]
- Kantz H, Schreiber T. Nonlinear Time Series Analysis. Cambridge, United Kingdom: Cambridge University Press; 2004.
- Thelen E, Smith LB. A Dynamic Systems Approach to the Development of Cognition and Action. Cambridge, MA: MIT Press; 1993. Bradford Books Series in Cognitive Psychology.
- Schreiber T, Schmitz A. Improved surrogate data for nonlinearity tests. Phys Rev Lett. 1996;22:635–638.
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