Abstract
This paper investigates the precision of parameters estimated from local samples of time dependent functions. We find that time delay embedding, i.e., structuring data prior to analysis by constructing a data matrix of overlapping samples, increases the precision of parameter estimates and in turn statistical power compared to standard independent rows of panel data. We show that the reason for this effect is that the sign of estimation bias depends on the position of a misplaced data point if there is no a priori knowledge about initial conditions of the time dependent function. Hence, we reason that the advantage of time delayed embedding is likely to hold true for a wide variety of functions. We support these conclusions both by mathematical analysis and two simulations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abarbanel, H.D.I. (1996). Analysis of observed chaotic data. New York: Springer.
Baltes, P.B., Reese, H.W., & Nesselroade, J.R. (1977). Life–span developmental psychology: introduction to research methods. Pacific Grove: Brooks/Cole.
Bisconti, T.L., Bergeman, C.S., & Boker, S.M. (2006). Social support as a predictor of variability: an examination of the adjustment trajectories of recent widows. Psychology and Aging, 21(3), 590–599.
Boker, S.M., Neale, M.C., & Rausch, J. (2004). Latent differential equation modeling with multivariate multi-occasion indicators. In K. van Montfort, H. Oud, & A. Satorra (Eds.), Recent developments on structural equation models: theory and applications (pp. 151–174). Dordrecht: Kluwer Academic.
Bollen, K.A. (1989). Structural equations with latent variables. New York: Wiley.
Butner, J., Amazeen, P.G., & Mulvey, G.M. (2005). Multilevel modeling to two cyclical processes: extending differential structural equation modeling to nonlinear coupled systems. Psychological Methods, 10(2), 159–177.
Chow, S.M., Ram, N., Boker, S.M., Fujita, F., & Clore, G. (2005). Capturing weekly fluctuation in emotion using a latent differential structural approach. Emotion, 5(2), 208–225.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Hillsdale: Erlbaum.
Deboeck, P.R., Boker, S.M., & Bergeman, C.S. (2009). Modeling individual damped linear oscillator processes with differential equations: using surrogate data analysis to estimate the smoothing parameter. Multivariate Behavioral Research, 43(4), 497–523.
Grassberger, P., & Procaccia, I. (1983). Measuring the strangeness of strange attractors. Physica D, 9, 189–208.
Hotelling, H. (1927). Differential equations subject to error, and population estimates. Journal of the American Statistical Association, 22(159), 283–314.
Hox, J.J. (2002). Multilevel analysis: techniques and applications. Hillsdale: Erlbaum.
Kantz, H., & Schreiber, T. (1997). Nonlinear time series analysis. Cambridge: Cambridge University Press.
McArdle, J.J., & McDonald, R.P. (1984). Some algebraic properties of the Reticular Action Model for moment structures. British Journal of Mathematical and Statistical Psychology, 87, 234–251.
Molenaar, P.C.M. (1985). A dynamic factor model for the analysis of multivariate time series. Psychometrika, 50, 181–202.
Neale, M.C., Boker, S.M., Xie, G., & Maes, H.H. (2003). Mx: statistical modeling (6th edn.). Richmond: VCU, Department of Psychiatry.
Nesselroade, J.R. (1991). The warp and woof of the developmental fabric. In R. Downs, L. Liben, & D.S. Palermo (Eds.), Visions of aesthetics, the environment, and development: the legacy of Joachim F. Wohlwill (pp. 213–240). Hillsdale: Erlbaum.
Nesselroade, J.R., & Ram, N. (2004). Studying intraindividual variability: what we have learned that will help us understand lives in context. Research in Human Development, 1(1–2), 9–29.
Nesselroade, J.R., McArdle, J.J., Aggen, S.H., & Meyers, J.M. (2002). Dynamic factor analysis models for representing process in multivariate time-series. In D. Moskowitz, & S. Hershberger (Eds.), Modeling intraindividual variability with repeated measures data: methods and applications (pp. 235–265). Hillsdale: Lawrence Erlbaum Associates.
Oud, J.H.L., & Jansen, R.A.R.G. (2000). Continuous time state space modeling of panel data by means of SEM. Psychometrica, 65(2), 199–215.
Pinheiro, J.C., & Bates, D.M. (2000). Mixed-effects models in S and S-plus. New York: Springer.
Ramsay, J.O., Hooker, G., Campbell, D., & Cao, J. (2007). Parameter estimation for differential equations: a generalized smoothing approach. Journal of the Royal Statistical Society B, 69(5), 774–796.
Sauer, T., Yorke, J., & Casdagli, M. (1991). Embedology. Journal of Statistical Physics, 65(3–4), 95–116.
Shannon, C.E., & Weaver, W. (1949). The mathematical theory of communication. Champaign: University of Illinois Press.
Snijders, T.A.B., & Bosker, R.J. (1999). Multilevel analysis: an introduction to basic and advanced multilevel modeling. Newbury Park: Sage.
Takens, F. (1985). Detecting strange attractors in turbulence. In A. Dold & B. Eckman (Eds.), Lecture notes in mathematics : Vol. 1125. Dynamical systems and bifurcations (pp. 99–106). Berlin: Springer.
Whitney, H. (1936). Differentiable manifolds. Annals of Mathematics, 37, 645–680.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
von Oertzen, T., Boker, S.M. Time Delay Embedding Increases Estimation Precision of Models of Intraindividual Variability. Psychometrika 75, 158–175 (2010). https://doi.org/10.1007/s11336-009-9137-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11336-009-9137-9