, Volume 83, Issue 2, pp 476–510 | Cite as

Representing Sudden Shifts in Intensive Dyadic Interaction Data Using Differential Equation Models with Regime Switching

  • Sy-Miin ChowEmail author
  • Lu Ou
  • Arridhana Ciptadi
  • Emily B. Prince
  • Dongjun You
  • Michael D. Hunter
  • James M. Rehg
  • Agata Rozga
  • Daniel S. Messinger


A growing number of social scientists have turned to differential equations as a tool for capturing the dynamic interdependence among a system of variables. Current tools for fitting differential equation models do not provide a straightforward mechanism for diagnosing evidence for qualitative shifts in dynamics, nor do they provide ways of identifying the timing and possible determinants of such shifts. In this paper, we discuss regime-switching differential equation models, a novel modeling framework for representing abrupt changes in a system of differential equation models. Estimation was performed by combining the Kim filter (Kim and Nelson State-space models with regime switching: classical and Gibbs-sampling approaches with applications, MIT Press, Cambridge, 1999) and a numerical differential equation solver that can handle both ordinary and stochastic differential equations. The proposed approach was motivated by the need to represent discrete shifts in the movement dynamics of \(n= 29\) mother–infant dyads during the Strange Situation Procedure (SSP), a behavioral assessment where the infant is separated from and reunited with the mother twice. We illustrate the utility of a novel regime-switching differential equation model in representing children’s tendency to exhibit shifts between the goal of staying close to their mothers and intermittent interest in moving away from their mothers to explore the room during the SSP. Results from empirical model fitting were supplemented with a Monte Carlo simulation study to evaluate the use of information criterion measures to diagnose sudden shifts in dynamics.


differential equation regime switching hidden Markov dynamic dyadic strange situation 

Supplementary material

Open image in new window
11336_2018_9605_MOESM1_ESM.txt (1.2 mb)
Supplementary material 1 (txt 1203 KB)
11336_2018_9605_MOESM2_ESM.r (4 kb)
Supplementary material 2 (R 3 KB)
11336_2018_9605_MOESM3_ESM.pdf (183 kb)
Supplementary material 3 (pdf 183 KB)


  1. Ainsworth, M. D. S., Blehar, M. C., Waters, E., & Wall, S. (1978). Patterns of attachment: A psychological study of the strange situation. Oxford: Lawrence Erlbaum.Google Scholar
  2. Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In B. N. Petrov & F. Csaki (Eds.), Second international symposium on information theory (pp. 267–281). Budapest: Akademiai Kiado.Google Scholar
  3. Aptech Systems Inc. (2009). GAUSS (Version 10). WA: Black Diamond.Google Scholar
  4. Arnold, L. (1974). Stochastic differential equations. New York: Wiley.Google Scholar
  5. Bar-Shalom, Y., Li, X. R., & Kirubarajan, T. (2001). Estimation with applications to tracking and navigation: Theory algorithms and software. New York: Wiley.CrossRefGoogle Scholar
  6. Behrens, K. Y., Parker, A. C., & Haltigan, J. D. (2011). Maternal sensitivity assessed during the strange situation procedure predicts child’s attachment quality and reunion behaviors. Infant Behavior and Development, 34(2), 378–381.CrossRefPubMedGoogle Scholar
  7. Beskos, A., Papaspiliopoulos, O., & Roberts, G. (2009). Monte carlo maximum likelihood estimation for discretely observed diffusion processes. The Annals of Statistics, 37(1), 223–245.CrossRefGoogle Scholar
  8. Boker, S. M., Deboeck, P. R., Edler, C., & Keel, P. K. (2010). Generalized local linear approximation of derivatives from time series. In S. Chow, E. Ferrer, & F. Hsieh (Eds.), Statistical methods for modeling human dynamics: An interdisciplinary dialogue (pp. 161–178). New York: Taylor & Francis.Google Scholar
  9. Boker, S. M., & Graham, J. (1998). A dynamical systems analysis of adolescent substance abuse. Multivariate Behavioral Research, 33, 479–507.CrossRefPubMedGoogle Scholar
  10. Boker, S. M., Neale, H., Maes, H., Wilde, M., Spiegel, M., Brick, T., et al. (2011). Openmx: An open source extended structural equation modeling framework. Psychometrika, 76(2), 306–317.CrossRefPubMedPubMedCentralGoogle Scholar
  11. Boker, S. M., Neale, M. C., & Rausch, J. (2008). 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). Amsterdam: Kluwer.Google Scholar
  12. Bolger, N., Davis, A., & Rafaeli, E. (2003). Diary methods: Capturing life as it is lived. Annual Review of Psychology, 54, 579–616.CrossRefPubMedGoogle Scholar
  13. Bowlby, J. (1973). Separation: Anxiety & anger. Attachment and Loss. (International psycho-analytical library no. 95) (Vol. 2). London: Hogarth Press.Google Scholar
  14. Bowlby, J. (1982). Attachment (2nd ed., Vol. 1). New York: Basic Books.Google Scholar
  15. Casella, G., & Berger, R. L. (2001). Statistical inference (2nd ed.). Pacific Grove: Duxbury Press.Google Scholar
  16. Chow, S.-M., Bendezú, J. J., Cole, P. M., & Ram, N. (2016a). A comparison of two- stage approaches for fitting nonlinear ordinary differential equation (ode) models with mixed effects. Multivariate Behavioral Research, 51(2–3), 154–184.CrossRefPubMedPubMedCentralGoogle Scholar
  17. 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.CrossRefPubMedGoogle Scholar
  18. Chow, S.-M., Ho, M.-H. R., Hamaker, E. J., & Dolan, C. V. (2010). Equivalences and differences between structural equation and state-space modeling frameworks. Structural Equation Modeling, 17, 303–332.CrossRefGoogle Scholar
  19. 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, 102–134.CrossRefPubMedGoogle Scholar
  20. Chow, S.-M., & Zhang, G. (2013). Nonlinear regime-switching state-space (RSSS) models. Psychometrika: Application Reviews and Case Studies, 78(4), 740–768.CrossRefGoogle Scholar
  21. Chow, S.-M., Zu, J., Shifren, K., & Zhang, G. (2011). Dynamic factor analysis models with time-varying parameters. Multivariate Behavioral Research, 46(2), 303–339.CrossRefPubMedGoogle Scholar
  22. Collins, L. M., & Wugalter, S. E. (1992). Latent class models for stage-sequential dynamic latent variables. Multivariate Behavioral Research, 28, 131–157.CrossRefGoogle Scholar
  23. Deboeck, P. R. (2010). Estimating dynamical systems: Derivative estimation hints from Sir Ronald A Fisher. Multivariate Behavioral Research, 45, 725–745.CrossRefPubMedGoogle Scholar
  24. Dijkstra, T. K. (1992). On statistical inference with parameter estimates on the boundary of the parameter space. British Journal of Mathematical and Statistical Psychology, 45(2), 289–309.CrossRefGoogle Scholar
  25. Dolan, C. V. (2009). Structural equation mixture modeling. In R. E. Millsap & A. Maydeu-Olivares (Eds.), The SAGE handbook of quantitative methods in psychology (pp. 568–592). Thousand Oaks: Sage.CrossRefGoogle Scholar
  26. Dolan, C. V., & Molenaar, P. C. M. (1991). A note on the calculation of latent trajectories in the quasi Markov simplex model by means of regression method and the discrete kalman filter. Kwantitatieve Methoden, 38, 29–44.Google Scholar
  27. Dolan, C. V., Schmittmann, V. D., Lubke, G. H., & Neale, M. C. (2005). Regime switching in the latent growth curve mixture model. Structural Equation Modeling, 12(1), 94–119.CrossRefGoogle Scholar
  28. Doornik, J. A. (1998). Object-oriented matrix programming using Ox 2.0. London: Timberlake Consultants Press.Google Scholar
  29. Durbin, J., & Koopman, S. J. (2001). Time series analysis by state space methods. New York: Oxford University Press.Google Scholar
  30. Elliott, R. J., Aggoun, L., & Moore, J. (1995). Hidden markov models: Estimation and control. New York: Springer.Google Scholar
  31. Enders, C., & Tofighi, D. (2008). The impact of misspecifying class-specific residual variances in growth mixture models. Structural Equation Modeling: A Multidisciplinary Journal, 15, 75–95.CrossRefGoogle Scholar
  32. Fahrmeir, L., & Tutz, G. (2001). Multivariate statistical modelling based on generalized linear models. Berlin: Springer.CrossRefGoogle Scholar
  33. Fukuda, K., & Ishihara, K. (1997). Development of human sleep and wakefulness rhythm during the first six months of life: Discontinuous changes at the 7th and 12th week after birth. Biological Rhythm Research, 28, 94–103.CrossRefGoogle Scholar
  34. Gates, K. M., & Molenaar, P. C. M. (2012). Group search algorithm recovers effective connectivity maps for individuals in homogeneous and heterogeneous samples. Neuroimage, 63, 310–319.CrossRefPubMedGoogle Scholar
  35. Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica, 57, 357–384.CrossRefGoogle Scholar
  36. Harvey, A. C. (2001). Forecasting, structural time series models and the Kalman filter. Cambridge: Cambridge University Press.Google Scholar
  37. Jackson, C. H. (2011). Multi-state models for panel data: The msm package for R. Journal of Statistical Software, 38(8), 1–29.CrossRefGoogle Scholar
  38. Jazwinski, A. H. (1970). Stochastic processes and filtering theory. New York: Academic Press.Google Scholar
  39. Jones, R. H. (1993). Longitudinal data with serial correlation: A state-space approach. Boca Raton: Chapman & Hall/CRC.CrossRefGoogle Scholar
  40. Kalman, R. E., & Bucy, R. S. (1961). New results in linear filtering and prediction theory. In Trans. ASME, Ser. D, J. Basic Eng, p 109.Google Scholar
  41. Kim, C.-J., & Nelson, C. R. (1999). State-space models with regime switching: Classical and Gibbs-sampling approaches with applications. Cambridge: MIT Press.Google Scholar
  42. Kulikov, G., & Kulikova, M. (2014). Accurate numerical implementation of the continuous-discrete extended Kalman filter. IEEE Transactions on Automatic Control, 59(1), 273–279.CrossRefGoogle Scholar
  43. Kulikova, M. V., & Kulikov, G. Y. (2014). Adaptive ODE solvers in extended Kalman filtering algorithms. Journal of Computational and Applied Mathematics, 262, 205–216.CrossRefGoogle Scholar
  44. Lanza, S. T., & Collins, L. M. (2008). A new SAS procedure for latent transition analysis: Transitions in dating and sexual risk behavior. Developmental Psychology, 44(2), 446–456.CrossRefPubMedPubMedCentralGoogle Scholar
  45. Lawley, D. N., & Maxwell, M. A. (1971). Factor analysis as a statistical method (2nd ed.). London: Butterworths.Google Scholar
  46. Lu, Z.-H., Chow, S.-M., Sherwood, A., & Zhu, H. (2015). Bayesian analysis of ambulatory cardiovascular dynamics with application to irregularly spaced sparse data. Annals of Applied Statistics, 9, 1601–1620.CrossRefPubMedPubMedCentralGoogle Scholar
  47. Mbalawata, I. S., Särkkä, S., & Haario, H. (2013). Parameter estimation in stochastic differential equations with markov chain monte carlo and non-linear Kalman filtering. Computational Statistics, 28(3), 1195–1223.CrossRefGoogle Scholar
  48. Miao, H., Xin, X., Perelson, A. S., & Wu, H. (2011). On identifiability of nonlinear ODE models and applications in viral dynamics. SIAM Review, 53(1), 3–39.CrossRefPubMedPubMedCentralGoogle Scholar
  49. Molenaar, P. C. M. (2004). A manifesto on psychology as idiographic science: Bringing the person back into scientific pyschology-this time forever. Measurement: Interdisciplinary Research and Perspectives, 2, 201–218.Google Scholar
  50. Molenaar, P. C. M., & Newell, K. M. (2003). Direct fit of a theoretical model of phase transition in oscillatory finger motions. British Journal of Mathematical and Statistical Psychology, 56, 199–214.CrossRefPubMedGoogle Scholar
  51. Muthén, B. O., & Asparouhov, T. (2011). LTA in Mplus: Transition probabilities influenced by covariates. Mplus web notes: No. 13. Available at Accessed 23 July 2016.
  52. Muthén, L. K., & Muthén, B. O. (2001). Mplus: The comprehensive modeling program for applied researchers: User’s guide (1998–2001). Los Angeles: Muthén & Muthén.Google Scholar
  53. Nylund, K. L., Asparouhov, T., & Muthén, B. O. (2007). Deciding on the number of classes in latent class analysis and growth mixture modeling: A Monte Carlo simulation study. Structural Equation Modeling, 14(4), 535–569.CrossRefGoogle Scholar
  54. Nylund-Gibson, K., Muthén, B. O., Nishina, A., Bellmore, A., & Graham, S. (2013). Stability and instability of peer victimization during middle school: Using latent transition analysis with covariates, distal outcomes, and modeling extensions. Manuscript submitted for publication.Google Scholar
  55. Oravecz, Z., Tuerlinckx, F., & Vandekerckhove, J. (2009). A hierarchical Ornstein-Uhlenbeck model for continuous repeated measurement data. Psychometrika, 74, 395–418.CrossRefGoogle Scholar
  56. Oravecz, Z., Tuerlinckx, F., & Vandekerckhove, J. (2011). A hierarchical latent stochastic differential equation model for affective dynamics. Psychological Methods, 16, 468–490.CrossRefPubMedGoogle Scholar
  57. Ou, L., Hunter, M. D., & Chow, S.-M. (2016). dynr: Dynamic Modeling in R. R package version 0.1.7-22.Google Scholar
  58. Oud, J. H. L., & Jansen, R. A. R. G. (2000). Continuous time state space modeling of panel data by means of SEM. Psychometrika, 65(2), 199–215.CrossRefGoogle Scholar
  59. Oud, J. H. L., & Singer, H., (Eds.) (2008). Special issue: Continuous time modeling of panel data, Vol. 62.Google Scholar
  60. Oud, J. H. L., van den Bercken, J. H., & Essers, R. J. (1990). Longitudinal factor score estimation using the Kalman filter. Applied psychological measurement, 14, 395–418.CrossRefGoogle Scholar
  61. Piaget, J., & Inhelder, B. (1969). The psychology of the child. New York: Basic Books.Google Scholar
  62. Pincus, S., Gladstone, I., & Ehrenkranz, R. (1991). A regularity statistic for medical data analysis. Journal of Clinical Monitoring, 7, 335–345.CrossRefPubMedGoogle Scholar
  63. Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (2002). Numerical recipes in C. Cambridge: Cambridge University Press.Google Scholar
  64. R Development Core Team (2009). R: A language and environment for statistical computing. Vienna: Austria. ISBN: 3-900051-07-0.Google Scholar
  65. Ramsay, J. O., Hooker, G., Campbell, D., & Cao, J. (2007). Parameter estimation for differential equations: a generalized smoothing approach (with discussion). Journal of Royal Statistical Society: Series B, 69(5), 741–796.CrossRefGoogle Scholar
  66. Särkkä, S. (2013). Bayesian Filtering and Smoothing. Hillsdale: Cambridge University Press.CrossRefGoogle Scholar
  67. SAS Institute Inc. (2008). SAS 9.2 help and documentation. Cary, NC.Google Scholar
  68. Schmittmann, V. D., Dolan, C. V., van der Maas, H., & Neale, M. C. (2005). Discrete latent Markov models for normally distributed response data. Multivariate Behavioral Research, 40(2), 207–233.CrossRefGoogle Scholar
  69. Schwartz, J. E., & Stone, A. A. (1998). Strategies for analyzing ecological momentary assessment data. Health Psychology, 17, 6–16.CrossRefPubMedGoogle Scholar
  70. Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
  71. Schweppe, F. (1965). Evaluation of likelihood functions for Gaussian signals. IEEE Transactions on Information Theory, 11, 61–70.CrossRefGoogle Scholar
  72. Shapiro, A. (1985). Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrics, 72(1), 133–44.CrossRefGoogle Scholar
  73. Shapiro, A. (1986). Asymptotic theory or overparameterized structural models. American Statistical Association Journal, 81(393), 142–149.CrossRefGoogle Scholar
  74. Sherwood, A., Thurston, R., Steffen, P., Blumenthal, J. A., Waugh, R. A., & Hinderliter, A. L. (2001). Blunted nighttime blood pressure dipping in postmenopausal women. American Journal of Hypertension, 14, 749–754.CrossRefPubMedGoogle Scholar
  75. Singer, H. (2002). Parameter estimation of nonlinear stochastic differential equations: Simulated maximum likelihood vs. extended Kaman filter and itô-Taylor expansion. Journal of Computational and Graphical Statistics, 11, 972–995.CrossRefGoogle Scholar
  76. Singer, H. (2003). Nonlinear continuous discrete filtering using kernel density estimates and functional integrals. Journal of Mathematical Sociology, 27, 1–28.CrossRefGoogle Scholar
  77. Singer, H. (2010). Sem modeling with singular moment matrices part i: Ml-estimation of time series. The Journal of Mathematical Sociology, 34(4), 301–320.CrossRefGoogle Scholar
  78. 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.CrossRefGoogle Scholar
  79. Sivalingam, R., Cherian, A., Fasching, J., Walczak, N., Bird, N., Morellas, V., Murphy, B., Cullen, K., Lim, K., Sapiro, G., & Papanikolopoulos, N. (2012). A multi-sensor visual tracking system for behavior monitoring of at-risk children. In IEEE international conference on robotics and automation (ICRA).Google Scholar
  80. Tofghi, D., & Enders, C. K. (2007). Identifying the correct number of classes in mixture models. In G. R. Hancock & K. M. Samulelsen (Eds.), Advances in latent variable mixture models (pp. 317–341). Greenwich: Information Age.Google Scholar
  81. Trail, J. B., Collins, L. M., Rivera, D. E., Li, R., Piper, M. E., & Baker, T. B. (2013). Functional data analysis for dynamical system identification of behavioral processes. Psychological Methods, 19, 175–182.CrossRefPubMedPubMedCentralGoogle Scholar
  82. van der Maas, H. L. J., & Molenaar, P. C. M. (1992). Stagewise cognitive development: An application of catastrophe theory. Psychological Review, 99(3), 395–417.CrossRefPubMedGoogle Scholar
  83. Van Geert, P. (2000). The dynamics of general developmental mechanisms: From Piaget and Vygotsky to dynamic systems models. Current Directions in Psychological Science, 9(2), 64–68.CrossRefGoogle Scholar
  84. Vecchiato, W. (1997). New models for irregularly spaced time series analysis with applications to high frequency financial data. In Proceedings of the IEEE/IAFE 1997 computational intelligence for financial engineering (CIFEr), pp. 144–149.Google Scholar
  85. Visser, I. (2007). Depmix: An R-package for fitting mixture models on mixed multivariate data with markov dependencies. Technical report, University of Amsterdam.Google Scholar
  86. Voelkle, M. C., Oud, J. H. L., Davidov, E., & Schmidt, P. (2012). An sem approach to continuous time modeling of panel data: relating authoritarianism and anomia. Psychological Methods, 17, 176–192.CrossRefPubMedGoogle Scholar
  87. Waters, E. (2002). Comments on strange situation classification. Retrieved on May 1, 2012 from
  88. Yang, M., & Chow, S.-M. (2010). Using state-space model with regime switching to represent the dynamics of facial electromyography (EMG) data. Psychometrika: Application and Case Studies, 74(4), 744–771.CrossRefGoogle Scholar

Copyright information

© The Psychometric Society 2018

Authors and Affiliations

  1. 1.Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Georgia Institute of TechnologyAtlantaUSA
  3. 3.University of MiamiCoral GablesUSA
  4. 4.University of Oklahoma Health Sciences CenterOklahoma CityUSA

Personalised recommendations