301
Views
0
CrossRef citations to date
0
Altmetric
Research Articles

Fitting Bayesian Stochastic Differential Equation Models with Mixed Effects through a Filtering Approach

ORCID Icon, ORCID Icon, ORCID Icon & ORCID Icon

References

  • Bar-Shalom, Y., Li, X. R., & Kirubarajan, T. (2001). Estimation with applications to tracking and navigation: Theory algorithms and software. Wiley.
  • Bergstrom, A. R. (1988). The history of continuous-time econometric models. Econometric Theory, 4(3), 365–383. https://doi.org/10.1017/S0266466600013359
  • 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. https://doi.org/10.1207/S15327906MBR3703_5
  • Boker, S. M., & Graham, J. (1998). A dynamical systems analysis of adolescent substance abuse. Multivariate Behavioral Research, 33(4), 479–507. https://doi.org/10.1207/s15327906mbr3304_3
  • Boker, S. M., & Nesselroade, J. R. (2002). A method for modeling the intrinsic dynamics of intraindividual variability: Recovering the parameters of simulated oscillators in multi-wave panel data. Multivariate Behavioral Research, 37(1), 127–160. https://doi.org/10.1207/S15327906MBR3701_06
  • Boker, S. M., Neale, M. C., & Klump, K. L. (2014). A differential equations model for the ovarian hormone cycle. In P. C. Molenaar, R. M. Lerner, & K. M. Newell (Eds.), Handbook of developmental systems theory and methodology (pp. 369–391). Guilford Publications.
  • Boker, S. M., Tiberio, S. S., & Moulder, R. G. (2018). Robustness of time delay embedding to sampling interval misspecification. In K. Van Montfort, J. H. L. Oud, & M. C. Voelkle (Eds.), Continuous time modeling in the behavioral and related sciences (pp. 239–258). Springer.
  • Butler, E. A. (2011). Temporal interpersonal emotion systems: The “ties” that form relationships. Personality and Social Psychology Review: An Official Journal of the Society for Personality and Social Psychology, Inc, 15(4), 367–393. https://doi.org/10.1177/1088868311411164
  • Cattell, R. B. (1952). The three basic factor-analytic research designs—their interrelations and derivatives. Psychological bulletin, 49(5), 499–520. https://doi.org/10.1037/h0054245
  • Chambers, M. J., McCrorie, J. R., & Thornton, M. A. (2018). Continuous time modelling based on an exact discrete time representation. In K. Van Montfort, J. H. L. Oud, & M. C. Voelkle, (Eds.), Continuous time modeling in the behavioral and related sciences (pp. 317–357). Springer.
  • Chatel-Goldman, J., Congedo, M., Jutten, C., & Schwartz, J.-L. (2014). Touch increases autonomic coupling between romantic partners. Frontiers in Behavioral Neuroscience, 8(95), 1–12. https://doi.org/10.3389/fnbeh.2014.00095
  • Chen, M., Chow, S.-M., & Hunter, M. D. (2018). Stochastic differential equation models with time-varying parameters. In K. Van Montfort, J. H. L. Oud, & M. C. Voelkle (Eds.), Continuous time modeling in the behavioral and related sciences (pp. 205–238). Springer.
  • Chen, M., Ferrer, E., & Song, H. (2023). Longitudinal models for assessing dynamics in dydic data. In R. H. Hoyle (Ed.), Handbook of strucutral equation modeling (2nd ed., pp. 640–652). The Guilford Press .
  • Chow, S. M. (2019). Practical tools and guidelines for exploring and fitting linear and nonlinear dynamical systems models. Multivariate Behavioral Research, 54(5), 690–718. https://doi.org/10.1080/00273171.2019.1566050
  • Chow, S.-M., Bendezu, J. J., Cole, P. M., & Ram, N. (2016a). A comparison of two-stage approaches for fitting nonlinear ordinary differential equation models with mixed effects. Multivariate Behavioral Research, 51(2–3), 154–184. https://doi.org/10.1080/00273171.2015.1123138
  • Chow, S.-M., Ferrer, E., & Nesselroade, J. R. (2007). An unscented kalman filter approach to the estimation of nonlinear dynamical systems models. Multivariate Behavioral Research, 42(2), 283–321. https://doi.org/10.1080/00273170701360423
  • Chow, S.-M., Lu, Z., Sherwood, A., & Zhu, H. (2016b). Fitting nonlinear ordinary differential equation models with random effects and unknown initial conditions using the stochastic approximation expectation–maximization (saem) algorithm. Psychometrika, 81(1), 102–134. https://doi.org/10.1007/s11336-014-9431-z
  • 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 (Washington, D.C.), 5(2), 208–225. https://doi.org/10.1037/1528-3542.5.2.208
  • De Haan-Rietdijk, S., Gottman, J. M., Bergeman, C. S., & Hamaker, E. L. (2016). Get over it! a multilevel threshold autoregressive model for state-dependent affect regulation. Psychometrika, 81(1), 217–241. https://doi.org/10.1007/s11336-014-9417-x
  • de Valpine, P., Turek, D., Paciorek, C. J., Anderson-Bergman, C., Lang, D. T., & Bodik, R. (2017). Programming with models: Writing statistical algorithms for general model structures with NIMBLE. Journal of Computational and Graphical Statistics, 26(2), 403–413. https://doi.org/10.1080/10618600.2016.1172487
  • Deboeck, P. R. (2010). Estimating dynamical systems: Derivative estimation hints from Sir Ronald A. fisher. Multivariate Behavioral Research, 45(4), 725–745. https://doi.org/10.1080/00273171.2010.498294
  • Donnet, S., & Samson, A. (2008). Parametric inference for mixed models defined by stochastic differential equations. ESAIM: Probability and Statistics, 12, 196–218.
  • Dormand, J. R., & Prince, P. J. (1980). A family of embedded Runge-Kutta formulae. Journal of Computational and Applied Mathematics, 6(1), 19–26. https://doi.org/10.1016/0771-050X(80)90013-3
  • Driver, C. C., & Voelkle, M. C. (2018). Hierarchical bayesian continuous time dynamic modeling. Psychological methods, 23(4), 774–799. https://doi.org/10.1037/met0000168
  • 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. https://doi.org/10.18637/jss.v077.i05
  • Felmlee, D. H., & Greenberg, D. F. (1999). A dynamic systems model of dyadic interaction. The journal of Mathematical Sociology, 23(3), 155–180. https://doi.org/10.1080/0022250X.1999.9990218
  • Ferrer, E., & Helm, J. L. (2013). Dynamical systems modeling of physiological coregulation in dyadic interactions. International journal of Psychophysiology: Official Journal of the International Organization of Psychophysiology, 88(3), 296–308. https://doi.org/10.1016/j.ijpsycho.2012.10.013
  • Ferrer, E., & Steele, J. (2014). Differential equations for evaluating theoretical models of dyadic interactions. In P. C. M. Molenaar, R. M. Lerner, & K. M. Newell (Eds.), Handbook of developmental systems theory and methodology (pp. 345–368). Psychology Press.
  • Gelman, A., & Hill, J. (2007). Why? Chap. 1 in Data analysis using regression and multilevel/hierarchical models (pp. 1–12). Cambridge University Press.
  • Gelman, A., & Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7(4), 457–472. https://doi.org/10.1214/ss/1177011136
  • Geman, S., & Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(6), 721–741. ):https://doi.org/10.1109/tpami.1984.4767596
  • Gianino, A., & Tronick, E. Z. (1988). The mutual regulation model: The infant’s self and interactive regulation and coping and defensive capacities. In T. Field, P. M. McCabe, & N. Schneiderman (Eds.), Stress and coping across development (pp. 47–68). Lawrence Erlbaum Associates, Inc.
  • Giuliani, N. R., McRae, K., & Gross, J. J. (2008). The up- and down-regulation of amusement: Experiential, behavioral, and autonomic consequences. Emotion (Washington, D.C.), 8(5), 714–719. https://doi.org/10.1037/a0013236
  • Graham, J. W., Taylor, B. J., Olchowski, A. E., & Cumsille, P. E. (2006). Planned missing data designs in psychological research. Psychological methods, 11(4), 323–343. https://doi.org/10.1037/1082-989X.11.4.323
  • Gross, J. J. (2015). Emotion regulation: Current status and future prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781
  • Hamilton, J. D. (1994). Time series analysis. Princeton University Press.
  • Harvey, A. C. (1990). Forecasting, structural time series models and the Kalman filter. Cambridge University Press.
  • Hastings, W. K. (1970). Monte Carlo sampling methods using Markov Chains and their applications. Biometrika, 57(1), 97–109. https://doi.org/10.1093/biomet/57.1.97
  • Haykin, S. (2004). Kalman filtering and neural networksKalman filtering and neural networks. John Wiley & Sons.
  • Hoffman, M. D., & Gelman, A. (2014). The no-u-turn sampler: Adaptively setting path lengths in hamiltonian monte carlo. Journal of Machine Learning Research, 15(1), 1593–1623.
  • Hu, Y., & Treinen, R. (2019). A one-step method for modelling longitudinal data with differential equations. The British Journal of Mathematical and Statistical Psychology, 72(1), 38–60. https://doi.org/10.1111/bmsp.12135
  • Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. Journal of Fluids Engineering, 82(Series D), 35–45.
  • Kloeden, P. E., Platen, E., & Schurz, H. (2012). Numerical solution of SDE through computer experiments. Springer Science & Business Media.
  • Kruschke, J. K. (2015). Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan (2nd ed.). Academic Press is an imprint of Elsevier.
  • Kuiper, R. M., & Ryan, O. (2018). Drawing conclusions from cross-lagged relationships: Re-considering the role of the time-interval. Structural Equation Modeling, 25(5), 809–823. https://doi.org/10.1080/10705511.2018.1431046
  • Kulikov, G., & Kulikova, M. V. (2014). Accurate numerical implementation of the continuous-discrete extended Kalman filter. Automatic Control, IEEE Transactions on, 59(1), 273–279. https://doi.org/10.1109/TAC.2013.2272136
  • Liu, S. (2017). Person-specific versus multilevel autoregressive models: Accuracy in parameter estimates at the population and individual levels. The British Journal of Mathematical and Statistical Psychology, 70(3), 480–498. https://doi.org/10.1111/bmsp.12096
  • Lotka, A. J. (1925). Elements of physical biology. Williams & Wilkins.
  • Lu, Z. H., Chow, S. M., Sherwood, A., & Zhu, H. (2015). Bayesian analysis of ambulatory blood pressure dynamics with application to irregularly spaced sparse data. The annals of Applied Statistics, 9(3), 1601–1620. https://doi.org/10.1214/15-aoas846
  • Lunkenheimer, E., Tiberio, S. S., Buss, K. A., Lucas-Thompson, R. G., Boker, S. M., & Timpe, Z. C. (2015). Coregulation of respiratory sinus arrhythmia between parents and preschoolers: Differences by children’s externalizing problems. Developmental Psychobiology, 57(8), 994–1003. https://doi.org/10.1002/dev.21323
  • Lunn, D., Spiegelhalter, D., Thomas, A., & Best, N. (2009). The BUGS project: Evolution, critique and future directions. Statistics in Medicine, 28(25), 3049–3067. https://doi.org/10.1002/sim.3680
  • Marci, C. D., Ham, J., Moran, E., & Orr, S. P. (2007). Physiologic correlates of perceived therapist empathy and social-emotional process during psychotherapy. The Journal of Nervous and Mental Disease, 195(2), 103–111. https://doi.org/10.1097/01.nmd.0000253731.71025.fc
  • Mauss, I. B., & Robinson, M. D. (2009). Measures of emotion: A review. Cognition & Emotion, 23(2), 209–237. https://doi.org/10.1080/02699930802204677
  • McArdle, J. J. (1994). Structural factor analysis experiments with incomplete data. Multivariate Behavioral Research, 29(4), 409–454. https://doi.org/10.1207/s15327906mbr2904_5
  • Messina, I., Palmieri, A., Sambin, M., Kleinbub, J. R., Voci, A., & Calvo, V. (2013). Somatic underpinnings of perceived empathy: The importance of psychotherapy training. Psychotherapy Research : Journal of the Society for Psychotherapy Research, 23(2), 169–177. https://doi.org/10.1080/10503307.2012.748940
  • Metropolis, N., & Ulam, S. (1949). The Monte Carlo method. Journal of the American Statistical Association, 44(247), 335–341. https://doi.org/10.1080/01621459.1949.10483310
  • Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953). Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6), 1087–1092. https://doi.org/10.1063/1.1699114
  • Mortensen, S. B., Klim, S., Dammann, B., Kristensen, N. R., Madsen, H., & Overgaard, R. V. (2007). A matlab framework for estimation of nlme models using stochastic differential equations. Journal of Pharmacokinetics and Pharmacodynamics, 34(5), 623–642. https://doi.org/10.1007/s10928-007-9062-4
  • Neal, R. M. (2003). Slice sampling. Annals of Statistics, 31(3), 705–741.
  • Nunan, D., Sandercock, G. R. H., & Brodie, D. A. (2010). A quantitative systematic review of normal values for short-term heart rate variability in healthy adults. Pacing and Clinical Electrophysiology : PACE, 33(11), 1407–1417. https://doi.org/10.1111/j.1540-8159.2010.02841.x
  • Oravecz, Z., Tuerlinckx, F., & Vandekerckhove, J. (2009). A hierarchical ornstein–uhlenbeck model for continuous repeated measurement data. Psychometrika, 74(3), 395–418. https://doi.org/10.1007/s11336-008-9106-8
  • Oud, J. H., & Jansen, R. A. (2000). Continuous time state space modeling of panel data by means of SEM. Psychometrika, 65(2), 199–215. https://doi.org/10.1007/BF02294374
  • Palumbo, R. V., Marraccini, M. E., Weyandt, L. L., Wilder-Smith, O., McGee, H. A., Liu, S., & Goodwin, M. S. (2017). Interpersonal autonomic physiology: A systematic review of the literature. Personality and Social Psychology Review : An Official Journal of the Society for Personality and Social Psychology, Inc, 21(2), 99–141. https://doi.org/10.1177/1088868316628405
  • 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, (Vol. 124, pp. 1–10). Vienna, Austria.
  • Russell, J. A. (1980). A circumplex model of affect. Journal of Personality and Social Psychology, 39(6), 1161–1178. https://doi.org/10.1037/h0077714
  • Schuurman, N. K. (2016). Multilevel autoregressive modeling in psychology: Snags and solutions [PhD thesis, Utrecht University].
  • Schweppe, F. (1965). Evaluation of likelihood functions for Gaussian signals. IEEE Transactions on Information Theory, 11(1), 61–70. https://doi.org/10.1109/TIT.1965.1053737
  • Shumway, R. H. (2000). Dynamic mixed models for irregularly observed time series. Resenhas do Instituto de Matemática e Estatística da Universidade de São Paulo, 4, 433–456.
  • Singer, H. (2012). SEM modeling with singular moment matrices part II: ML-Estimation of sampled stochastic differential equations. The Journal of Mathematical Sociology, 36(1), 22–43. https://doi.org/10.1080/0022250X.2010.532259
  • Stan Development Team. (2022a). 8.1 Overview of Stan’s program blocks. In Stan modeling language users guide and reference manual (2.29 ed.). https://mc-stan.org/docs/2_29/reference-manual/overview-of-stans-program-blocks.html.
  • Stan Development Team. (2022b). Stan modeling language users guide and reference manual (2.29 ed.).
  • Suveg, C., Shaffer, A., & Davis, M. (2016). Family stress moderates relations between physiological and behavioral synchrony and child self-regulation in mother–preschooler dyads. Developmental Psychobiology, 58(1), 83–97. https://doi.org/10.1002/dev.21358
  • Tarvainen, M. P., Georgiadis, S., Lipponen, J. A., Hakkarainen, M., & Karjalainen, P. A. (2009). Time-varying spectrum estimation of heart rate variability signals with Kalman Smoother algorithm. In Proceedings of the 31st Annual International Conference of the IEEE Engineering in Medicine and Biology Society: Engineering the Future of Biomedicine, EMBC 2009 (pp. 1–4).
  • Voelkle, M. C., & Oud, J. H. (2013). Continuous time modelling with individually varying time intervals for oscillating and non-oscillating processes. British Journal of Mathematical and Statistical Psychology, 66, 103–126. https://doi.org/10.1111/j.2044-8317.2012.02043.x
  • Volterra, V. (1926). Fluctuations in the abundance of a species considered mathematically. Nature, 118, 558–560.
  • Williams, L. E., Bargh, J. A., Nocera, C. C., & Gray, J. R. (2009). The unconscious regulation of emotion: Nonconscious reappraisal goals modulate emotional reactivity. Emotion (Washington, D.C.), 9(6), 847–854. https://doi.org/10.1037/a0017745

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.