On Fitting a Continuous-Time Stochastic Process Model in the Bayesian Framework

  • Zita OraveczEmail author
  • Julie Wood
  • Nilam Ram


Process models can be viewed as mathematical tools that allow researchers to formulate and test theories on the data-generating mechanism underlying observed data. In this chapter we highlight the advantages of this approach by proposing a multilevel, continuous-time stochastic process model to capture the dynamical homeostatic process that underlies observed intensive longitudinal data. Within the multilevel framework, we also link the dynamical processes parameters to time-varying and time-invariant covariates. However, estimating all model parameters (e.g., process model parameters and regression coefficients) simultaneously requires custom-made implementation of the parameter estimation; therefore we advocate the use of a Bayesian statistical framework for fitting these complex process models. We illustrate application to data on self-reported affective states collected in an ecological momentary assessment setting.



The research reported in this paper was sponsored by grant #48192 from the John Templeton Foundation, the National Institutes of Health (R01 HD076994, UL TR000127), and the Penn State Social Science Research Institute.


  1. Blackwell, P. G. (1997). Random diffusion models for animal movements. Ecological Modelling, 100, 87–102. CrossRefGoogle Scholar
  2. Bolger, N., & Laurenceau, J. (2013). Intensive longitudinal methods: An introduction to diary and experience sampling research. New York, NY: Guilford Press.Google Scholar
  3. Chow, S.-M., Haltigan, J. D., & Messinger, D. S. (2010). Dynamic infant-parent affect coupling during the face-to-face/still-face. Emotion, 10(1), 101–114. CrossRefGoogle Scholar
  4. 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, 208–225. CrossRefGoogle Scholar
  5. Deboeck, P. R., & Bergeman, C. (2013). The reservoir model: A differential equation model of psychological regulation. Psychological Methods, 18(2), 237–256. CrossRefGoogle Scholar
  6. Driver, C. C., Oud, J. H. L., & Voelkle, M. (2017). Continuous time structural equation modeling with R package ctsem. Journal of Statistical Software, 77(5), 1–35. CrossRefGoogle Scholar
  7. Driver, C. C., & Voelkle, M. (2018). Hierarchical Bayesian continuous time dynamic modeling. Psychological Methods.
  8. Dunn, J. E., & Gipson, P. S. (1977). Analysis of radio telemetry data in studies of home range. Biometrics, 33, 85–101. CrossRefGoogle Scholar
  9. Barrett, L. F. (2017). The theory of constructed emotion: An active inference account of interoception and categorization. Social Cognitive and Affective Neuroscience, 12(1), 1–23. Google Scholar
  10. Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian data analysis (3rd ed.). Boca Raton, FL: Chapman & Hall/CRC.zbMATHGoogle Scholar
  11. Gelman, A., & Hill, J. (2007). Data analysis using regression and multilevel/hierarchical models. Cambridge: Cambridge University Press.Google Scholar
  12. Gross, J. J. (2002). Emotion regulation: Affective, cognitive, and social consequences. Psychophysiology, 39(03), 281–291. CrossRefGoogle Scholar
  13. Hills, T. T., Jones, M. N., & Todd, P. M. (2012). Optimal foraging in semantic memory. Psychological Review, 119(2), 431–440. CrossRefGoogle Scholar
  14. Kuppens, P., Oravecz, Z., & Tuerlinckx, F. (2010). Feelings change: Accounting for individual differences in the temporal dynamics of affect. Journal of Personality and Social Psychology, 99, 1042–1060. CrossRefGoogle Scholar
  15. Oravecz, Z., Faust, K., & Batchelder, W. (2014). An extended cultural consensus theory model to account for cognitive processes for decision making in social surveys. Sociological Methodology, 44, 185–228. CrossRefGoogle Scholar
  16. Oravecz, Z., Tuerlinckx, F., & Vandekerckhove, J. (2011). A hierarchical latent stochastic differential equation model for affective dynamics. Psychological Methods, 16, 468–490. CrossRefGoogle Scholar
  17. Oravecz, Z., Tuerlinckx, F., & Vandekerckhove, J. (2016). Bayesian data analysis with the bivariate hierarchical Ornstein-Uhlenbeck process model. Multivariate Behavioral Research, 51, 106–119. CrossRefGoogle Scholar
  18. Oud, J. H. L. (2002). Continuous time modeling of the crossed-lagged panel design. Kwantitatieve Methoden, 69, 1–26.Google Scholar
  19. Oud, J. H. L., & Voelkle, M. C. (2014a). Continuous time analysis. In A. C. E. Michalos (Ed.), Encyclopedia of quality of life research (pp. 1270–1273). Dordrecht: Springer. CrossRefGoogle Scholar
  20. Oud, J. H. L., & Voelkle, M. C. (2014b). Do missing values exist? Incomplete data handling in cross-national longitudinal studies by means of continuous time modeling. Quality & Quantity, 48(6), 3271–3288. CrossRefGoogle Scholar
  21. Plummer, M. (2003). JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. In Proceedings of the 3rd International Workshop on Distributed Statistical Computing (DSC 2003) (pp. 20–22).Google Scholar
  22. Ram, N., & Grimm, K. J. (2015). Growth curve modeling and longitudinal factor analysis. In R. M. Lerner (Ed.), Handbook of child psychology and developmental science (pp. 1–31). Hoboken, NJ: John Wiley & Sons, Inc.Google Scholar
  23. Ram, N., Shiyko, M., Lunkenheimer, E. S., Doerksen, S., & Conroy, D. (2014). Families as coordinated symbiotic systems: Making use of nonlinear dynamic models. In S. M. McHale, P. Amato, & A. Booth (Eds.), Emerging methods in family research (pp. 19–37). Cham: Springer International Publishing. CrossRefGoogle Scholar
  24. Ratcliff, R., & Rouder, J. N. (1998). Modeling response times for two-choice decisions. Psychological Science, 9, 347–356. CrossRefGoogle Scholar
  25. Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods. Newbury Park, CA: Sage.Google Scholar
  26. Robert, C. P., & Casella, G. (2004). Monte Carlo statistical methods. New York, NY: Springer. CrossRefGoogle Scholar
  27. Russell, J. A. (2003). Core affect and the psychological construction of emotion. Psychological Review, 110, 145–172. CrossRefGoogle Scholar
  28. Shiffman, S., Stone, A. A., & Hufford, M. R. (2008). Ecological momentary assessment. Annual Review of Clinical Psychology, 4(1), 1–32. (PMID: 18509902).CrossRefGoogle Scholar
  29. Stone, A. A., & Shiffman, S. (1994). Ecological momentary assessment (EMA) in behavioral medicine. Annals of Behavioral Medicine, 16(3), 199–202.CrossRefGoogle Scholar
  30. Thomas, E. A., & Martin, J. A. (1976). Analyses of parent-infant interaction. Psychological Review, 83(2), 141–156. CrossRefGoogle Scholar
  31. Uhlenbeck, G. E., & Ornstein, L. S. (1930). On the theory of Brownian motion. Physical Review, 36, 823–841. CrossRefGoogle Scholar
  32. Walls, T. A., & Schafer, J. L. (2006). Models for intensive longitudinal data. New York, NY: Oxford University Press. zbMATHGoogle Scholar
  33. Ware, I. E., Snow, K. K., Kosinski, M., & Gandek, B. (1993). SF-36 Health Survey. Manual and interpretation guide. Boston, MA: Nimrod Press.Google Scholar
  34. Wetzels, R., Vandekerckhove, J., Tuerlinckx, F., & Wagenmakers, E.-J. (2010). Bayesian parameter estimation in the Expectancy Valence model of the Iowa gambling task. Journal of Mathematical Psychology, 54, 14–27. MathSciNetCrossRefGoogle Scholar
  35. Wiener, N. (1923). Differential-space. Journal of Mathematics and Physics, 2(1–4), 131–174. CrossRefGoogle Scholar
  36. Wood, J., Oravecz, Z., Vogel, N., Benson, L., Chow, S.-M., Cole, P., …Ram, N. (2017). Modeling intraindividual dynamics using stochastic differential equations: Age differences in affect regulation. Journals of Gerontology: Psychological Sciences, 73(1), 171–184. CrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The Pennsylvania State UniversityState CollegeUSA

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