A Method of Correcting Estimation Failure in Latent Differential Equations with Comparisons to Kalman Filtering

References

• Bisconti, T. L., Bergeman, C. S., & Boker, S. M. (2004). Emotional well-being in recently bereaved widows: A dynamical systems approach. The Journals of Gerontology. Series B, Psychological Sciences and Social Sciences, 59(4), 158–167 doi: 10.1093/geronb/59.4.P158.
   Web of Science ®Google Scholar
• Boker, S., Deboeck, P. R., Schiller, C. E., & Keel, P. K. (2010). Generalized local linear approximation of derivatives from time series. In Statistical methods for modeling human dynamics: An interdisciplinary dialogue, Notre Dame Series on quantitative methodologies. New York: Routledge. doi: 10.4324/9780203864746.
   Google Scholar
• Boker, S., Neale, M., & Rausch, J. (2004). Latent differential equation modeling with multivariate multi-occasion indicators. In Kees van Montfort, Johan Oud, & Albert Satorra (Eds.), Recent developments on structural equation models (pp. 151–174). Dordrecht, Netherlands: Springer.
   Google Scholar
• Boker, S. M. (2002). Consequences of continuity: The hunt for intrinsic properties within parameters of dynamics in psychological processes. Multivariate Behavioral Research, 37(3), 405–422. doi: 10.1207/S15327906MBR3703_5
   PubMed Web of Science ®Google Scholar
• Boker, S. M. (2012). Dynamical systems and differential equation models of change. In H. Cooper, A. Panter, P. Camic, R. Gonzalez, D. Long, & K. Sher (Eds.), APA handbook of research methods in psychology (pp. 323–333). Washington, DC: American Psychological Association.
   Google Scholar
• Boker, S. M. & Laurenceau, J.-P. (2006). Dynamical systems modeling: An application to the regulation of intimacy and disclosure in marriage. In Joseph L. Schafer, & Theodore A. Walls (Eds.), Models for intensive longitudinal data. New York, NY: Oxford University Press.
   Google Scholar
• Bollen, K. A. (1989). Structural equations with latent variables. Wiley series in probability and mathematical statistics. Applied probability and statistics. New York: Wiley.
   Google Scholar
• Box, G. E. P., & Jenkins, G. M. (1976). Time series analysis: Forecasting and control (Rev. ed.). San Francisco, CA: Holden-Day.
   Google Scholar
• Chow, S.-M., Ho, M.-H. R., Hamaker, E. L., & Dolan, C. V. (2010). Equivalence and differences between structural equation modeling and state-space modeling techniques. Structural Equation Modeling: A Multidisciplinary Journal, 17(2), 303–332. doi: 10.1080/10705511003661553.
   Web of Science ®Google Scholar
• Chow, S.-M., Ram, N., Boker, S. M., Fujita, F., & Clore, G. (2005). Emotion as a thermostat: Representing emotion regulation using a damped oscillator model. Emotion, 5(2), 208. doi: 10.1037/1528-3542.5.2.208.
   PubMed Web of Science ®Google Scholar
• Deboeck, P., & Boker, S. (2010). Modeling noisy data with differential equations using observed and expected matrices. Psychometrika, 75(3), 420–437. doi: 10.1007/s11336-010-9168-2.
   Web of Science ®Google Scholar
• Driver, C. C., Oud, J. H. L., & Voelkle, M. C. (2017). Continuous time structural equation modeling with R package ctsem. Journal of Statistical Software, 77(5), 1–35. doi: 10.18637/jss.v077.i05.
   Web of Science ®Google Scholar
• Gasimova, F., Robitzsch, A., Wilhelm, O., Boker, S. M., Hu, Y., & Hülür, G. (2014). Dynamical systems analysis applied to working memory data. Frontiers in Psychology, 5, 687–687. doi: 10.3389/fpsyg.2014.00687.
   PubMed Web of Science ®Google Scholar
• Helm, J. L., Sbarra, D., & Ferrer, E. (2012). Assessing cross-partner associations in physiological responses via coupled oscillator models. Emotion, 12(4), 748–762. doi: 10.1037/a0025036.
   PubMed Web of Science ®Google Scholar
• Hu, Y., Boker, S., Neale, M., & Klump, K. L. (2014). Coupled latent differential equation with moderators: Simulation and application. Psychological Methods, 19(1), 56. doi: 10.1037/a0032476.
   PubMed Web of Science ®Google Scholar
• Hunter, M. D. (2018). State space modeling in an open source, modular, structural equation modeling environment. Structural Equation Modeling: A Multidisciplinary Journal, 25(2), 307–324. doi: 10.1080/10705511.2017.1369354.
   Web of Science ®Google Scholar
• Kalman, R. E. (1960a). Contributions to the theory of optimal control. Boletín de la Sociedad Matemática Mexicana, 5(2), 102–119.
   Google Scholar
• Kalman, R. E. (1960b). A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 82(1), 35–45. doi: 10.1115/1.3662552.
   Google Scholar
• Kalman, R. E., & Bucy, R. S. (1961). New results in linear filtering and prediction theory. Journal of Basic Engineering, 83(1), 95–108. doi: 10.1115/1.3658902.
   Google Scholar
• Mahalanobis, P. C. (1936). On the generalized distance in statistics. Calcutta, India: National Institute of Science of India.
   Google Scholar
• McKee, K. L., Rappaport, L. M., Boker, S. M., Moskowitz, D. S., & Neale, M. C. (2018). Adaptive equilibrium regulation: Modeling individual dynamics on multiple timescales. Structural Equation Modeling: A Multidisciplinary Journal, 25(6), 1–18. doi: 10.1080/10705511.2018.1442224.
   Web of Science ®Google Scholar
• Mullen, K. M., Ardia, D., Gil, D. L., Windover, D., & Cline, J. (2011). Deoptim: An R package for global optimization by differential evolution. Journal of Statistical Software, 40(6), 1–26. doi: 10.18637/jss.v040.i06.
   Web of Science ®Google Scholar
• Neale, M. C., Hunter, M. D., Pritikin, J. N., Zahery, M., Brick, T. R., Kirkpatrick, R. M., … Boker, S. M. (2016). OpenMx 2.0: Extended structural equation and statistical modeling. Psychometrika, 81(2), 535–549. doi: 10.1007/s11336-014-9435-8.
   PubMed Web of Science ®Google Scholar
• Nelder, J. A., & Mead, R. (1965). A simplex method for function minimization. The Computer Journal, 7(4), 308–313. doi: 10.1093/comjnl/7.4.308.
   Web of Science ®Google Scholar
• Nicholson, J. S., Deboeck, P. R., Farris, J. R., Boker, S. M., & Borkowski, J. G. (2011). Maternal depressive symptomatology and child behavior: Transactional relationship with simultaneous bidirectional coupling. Developmental Psychology, 47(5), 1312–1323. doi: 10.1037/a0023912.
   PubMed Web of Science ®Google Scholar
• Oravecz, Z., Tuerlinckx, F., & Vandekerckhove, J. (2011). A hierarchical latent stochastic differential equation model for affective dynamics. Psychological Methods, 16(4), 468–490. doi: 10.1037/a0024375.
   PubMed Web of Science ®Google Scholar
• Oud, J. (2017). Comparison of four procedures to estimate the damped linear differential oscillator for panel data. In Longitudinal models in the behavioral and related sciences (pp. 19–39). New York, NY: Routledge.
   Google Scholar
• Oud, J. H. L., & Singer, H. (2008). Continuous time modeling of panel data: SEM versus filter techniques. Statistica Neerlandica, 62(1), 4–28. doi: 10.1111/j.1467-9574.2007.00376.x.
   Web of Science ®Google Scholar
• Preacher, K. J. (2008). Latent growth curve modeling. Quantitative Applications in the Social Sciences. Los Angeles, CA/London: Sage.
   Google Scholar
• R Core Team. (2017). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing.
   Google Scholar
• Ramsay, J. O. A. (2005). Functional data analysis. Springer Series in Statistics (2nd ed.). New York, NY: Springer.
   Google Scholar
• Santos, D. A., & Duarte, M. (2016). A public data set of human balance evaluations. PeerJ, 4, e2648. doi: 10.7717/peerj.2648.
   PubMed Web of Science ®Google Scholar
• Steele, J. S., & Ferrer, E. (2011). Latent differential equation modeling of self-regulatory and coregulatory affective processes. Multivariate Behavioral Research, 46(6), 956–984. doi: 10.1080/00273171.2011.625305.
   PubMed Web of Science ®Google Scholar
• Strogatz, S. H. (2000). Nonlinear dynamics and chaos. Cambridge, MA: Perseus Books.
   Google Scholar
• Tuma, N. B., & Hannan, M. T. (1984). Social dynamics models and methods. Orlando, FL: Elsevier.
   Google Scholar
• Wan, E., & Van Der Merwe, R. (2000). The unscented Kalman filter for nonlinear estimation (pp. 153–158). Piscataway, NJ: IEEE Publishing.
   Google Scholar
• Wei, W. W. S. (2006). Time series analysis: Univariate and multivariate methods (2nd ed.). Boston, MA: Pearson Addison-Wesley.
   Google Scholar
• Zentall, S. R., Boker, S. M., & Braungart-Rieker, J. M. (2006). Mother-infant synchrony: A dynamical systems approach. Proceedings of the Fifth International Conference on Development and Learning, Bloomington, IN
   Google Scholar