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Nonlinear Regime-Switching State-Space (RSSS) Models

Published online by Cambridge University Press:  01 January 2025

Sy-Miin Chow  and
Guangjian Zhang
Sy-Miin Chow*
Affiliation:
The Pennsylvania State University
Guangjian Zhang
Affiliation:
University of Notre Dame
*
Requests for reprints should be sent to Sy-Miin Chow, The Pennsylvania State University, 422 Biobehavioral Health Building, University Park, PA 16801, USA. E-mail: symiin@psu.edu
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Abstract

Nonlinear dynamic factor analysis models extend standard linear dynamic factor analysis models by allowing time series processes to be nonlinear at the latent level (e.g., involving interaction between two latent processes). In practice, it is often of interest to identify the phases—namely, latent “regimes” or classes—during which a system is characterized by distinctly different dynamics. We propose a new class of models, termed nonlinear regime-switching state-space (RSSS) models, which subsumes regime-switching nonlinear dynamic factor analysis models as a special case. In nonlinear RSSS models, the change processes within regimes, represented using a state-space model, are allowed to be nonlinear. An estimation procedure obtained by combining the extended Kalman filter and the Kim filter is proposed as a way to estimate nonlinear RSSS models. We illustrate the utility of nonlinear RSSS models by fitting a nonlinear dynamic factor analysis model with regime-specific cross-regression parameters to a set of experience sampling affect data. The parallels between nonlinear RSSS models and other well-known discrete change models in the literature are discussed briefly.

Keywords

regime-switchingstate-spacenonlinear latent variable modelsdynamic factor analysisKim filter

Information

Type
Original Paper
Information
Psychometrika , Volume 78 , Issue 4 , October 2013 , pp. 740 - 768
Copyright
Copyright © 2013 The Psychometric Society

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References

Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Petrov, B.N., & Csaki, F. (Eds.), Second international symposium on information theory (pp. 267–281). Budapest: Akademiai Kiado.Google Scholar
Anderson, B.D.O., & Moore, J.B. (1979). Optimal filtering. Englewood Cliffs: Prentice Hall.Google Scholar
Aptech Systems Inc. (2009). GAUSS (version 10) [computer software manual]. Black Diamond: Aptech Systems Inc.Google Scholar
Bar-Shalom, Y., Li, X.R., & Kirubarajan, T. (2001). Estimation with applications to tracking and navigation: theory algorithms and software. New York: Wiley.Google Scholar
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.CrossRefGoogle ScholarPubMed
Browne, M.W., & Nesselroade, J.R. (2005). Representing psychological processes with dynamic factor models: some promising uses and extensions of autoregressive moving average time series models. In Maydeu-Olivares, A., & McArdle, J.J. (Eds.), Contemporary psychometrics: a festschrift for Roderick P. McDonald (pp. 415–452). Mahwah: Erlbaum.Google Scholar
Browne, M.W., & Zhang, G. (2007). Developments in the factor analysis of individual time series. In Cudeck, R., & MacCallum, R.C. (Eds.), Factor analysis at 100: historical developments and future directions (pp. 265–291). Mahwah: Erlbaum.Google Scholar
Cacioppo, J.T., & Berntson, G.G. (1999). The affect system: architecture and operating characteristics. Current Directions in Psychological Science, 8, 133–137.CrossRefGoogle Scholar
Cattell, R., & Barton, K. (1974). Changes in psychological state measures and time of day. Psychological Reports, 35, 219–222.Google Scholar
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
Chow, S.-M., Nesselroade, J.R., Shifren, K., & McArdle, J. (2004). Dynamic structure of emotions among individuals with Parkinson’s disease. Structural Equation Modeling, 11, 560–582.CrossRefGoogle Scholar
Chow, S.-M., Tang, N., Yuan, Y., Song, X., & Zhu, H. (2011). Bayesian estimation of semiparametric dynamic latent variable models using the Dirichlet process prior. British Journal of Mathematical & Statistical Psychology, 64(1), 69–106.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Cohen, S., Kamarck, T., & Mermelstein, R. (1983). A global measure of perceived stress. Journal of Health and Social Behavior, 24(4), 385–396.CrossRefGoogle ScholarPubMed
Collins, L.M., & Wugalter, S.E. (1992). Latent class models for stage-sequential dynamic latent variables. Multivariate Behavioral Research, 28, 131–157.CrossRefGoogle Scholar
De Jong, P. (1991). The diffuse Kalman filter. The Annals of Statistics, 19, 1073–1083.Google Scholar
Dolan, C.V. (2009). Structural equation mixture modeling. In Millsap, R.E., & Maydeu-Olivares, A. (Eds.), The SAGE handbook of quantitative methods in psychology (pp. 568–592). Thousand Oaks: Sage.CrossRefGoogle Scholar
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
Dolan, C.V., & Van der Maas, H.L.J. (1998). Fitting multivariate normal finite mixtures subject to structural equation modeling. Psychometrika, 63(3), 227–253.CrossRefGoogle Scholar
Doornik, J.A. (1998). Object-oriented matrix programming using Ox 2.0. London: Timberlake Consultants Press.Google Scholar
Du Toit, S.H.C., & Browne, M.W. (2007). Structural equation modeling of multivariate time series. Multivariate Behavioral Research, 42, 67–101.CrossRefGoogle ScholarPubMed
Durbin, J., & Koopman, S.J. (2001). Time series analysis by state space methods. New York: Oxford University Press.Google Scholar
Elliott, R.J., Aggoun, L., & Moore, J. (1995). Hidden Markov models: estimation and control. New York: Springer.Google Scholar
Emotions and Dynamic Systems Laboratory (2010). The affective dynamics and individual differences (ADID) study: developing non-stationary and network-based methods for modeling the perception and physiology of emotions Unpublished manual, University of North Carolina at Chapel Hill.Google Scholar
Engle, R.F., & Watson, M. (1981). A one-factor multivariate time series model of metropolitan wage rates. Journal of the American Statistical Association, 76, 774–781.CrossRefGoogle Scholar
Ferrer, E., & Nesselroade, J.R. (2003). Modeling affective processes in dyadic relations via dynamic factor analysis. Emotion, 3, 344–360.CrossRefGoogle ScholarPubMed
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
Geweke, J.F., & Singleton, K.J. (1981). Maximum likelihood confirmatory factor analysis of economic time series. International Economic Review, 22, 133–137.CrossRefGoogle Scholar
Hamaker, E.L., Dolan, C.V., & Molenaar, P.C.M. (2003). ARMA-based SEM when the number of time points T exceeds the number of cases N: raw data maximum likelihood. Structural Equation Modeling, 10, 352–379.CrossRefGoogle Scholar
Hamaker, E.L., & Grasman, R.P.P.P. (2012). Regime switching state-space models applied to psychological processes: handling missing data and making inferences. Psychometrika, 77(2), 400–422.CrossRefGoogle Scholar
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
Harvey, A.C. (2001). Forecasting, structural time series models and the Kalman filter. Cambridge: Cambridge University Press.Google Scholar
Heron, K.E., & Smyth, J.M. (2010). Ecological momentary interventions: incorporating mobile technology into psychosocial and health behavior treatments. British Journal of Health Psychology, 15, 1–39.CrossRefGoogle Scholar
Hosmer, D.W. (1974). Maximum likelihood estimates of parameters of a mixture of two regression lines. Communications in Statistics. Theory and Methods, 3, 995–1006.Google Scholar
Jackson, C.H. (2011). Multi-state models for panel data: the msm package for R. Journal of Statistical Software, 38(8), 1–29. Available from http://www.jstatsoft.org/v38/i08/.CrossRefGoogle Scholar
Kenny, D.A., & Judd, C.M. (1984). Estimating the nonlinear and interactive effects of latent variables. Psychological Bulletin, 96, 201–210.CrossRefGoogle Scholar
Kim, C.-J., & Nelson, C.R. (1999). State-space models with regime switching: classical and Gibbs-sampling approaches with applications. Cambridge: MIT Press.CrossRefGoogle Scholar
Kishton, J.M., & Widaman, K.F. (1994). Unidimensional versus domain representative parceling of questionnaire items: an empirical example. Educational and Psychological Measurement, 54, 757–765.CrossRefGoogle Scholar
Koopman, S.J. (1997). Exact initial Kalman filtering and smoothing for nonstationary time series models. Journal of the American Statistical Association, 92, 1630–1638.CrossRefGoogle Scholar
Kuppens, P., Allen, N.B., & Sheeber, L.B. (2010). Emotional inertia and psychological adjustment. Psychological Science, 21, 984–991.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Larsen, R.J., & Diener, E. (1992). Promises and problems with the circumplex model of emotion. Review of Personality and Social Psychology, 13, 25–59.Google Scholar
Li, F., Duncan, T.E., & Acock, A. (2000). Modeling interaction effects in latent growth curve models. Structural Equation Modeling, 7(4), 497–533.CrossRefGoogle Scholar
Little, R.J.A., & Rubin, D.B. (2002). Statistical analysis with missing data. New York: Wiley.CrossRefGoogle Scholar
Marsh, W.H., Wen, Z.L., & Hau, J.-T. (2004). Structural equation models of latent interactions: evaluation of alternative estimation strategies and indicator construction. Psychological Methods, 9, 275–300.CrossRefGoogle ScholarPubMed
Molenaar, P.C.M. (1985). A dynamic factor model for the analysis of multivariate time series. Psychometrika, 50, 181–202.CrossRefGoogle Scholar
Molenaar, P.C.M. (1994a). Dynamic factor analysis of psychophysiological signals. In Jennings, J.R., Ackles, P.K., & Coles, M.G.H. (Eds.), Advances in psychophysiology: a research annual (Vol. 5, pp. 229–302). London: Jessica Kingsley Publishers.Google Scholar
Molenaar, P.C.M. (1994). Dynamic latent variable models in developmental psychology. In von Eye, A., & Clogg, C. (Eds.), Latent variables analysis: applications for developmental research (pp. 155–180). Thousand Oaks: Sage.Google Scholar
Molenaar, P.C.M., & Nesselroade, J.R. (1998). A comparison of pseudo-maximum likelihood and asymptotically distribution-free dynamic factor analysis parameter estimation in fitting covariance-structure models to block-Toeplitz matrices representing single-subject multivariate time series. Multivariate Behavioral Research, 33, 313–342.CrossRefGoogle ScholarPubMed
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 & Statistical Psychology, 56, 199–214.CrossRefGoogle ScholarPubMed
Muthén, B.O., & Asparouhov, T. (2011). LTA in Mplus: transition probabilities influenced by covariates (Mplus Web Notes: No. 13).Google Scholar
Muthén, L.K., & Muthén, B.O. (2001). Mplus: the comprehensive modeling program for applied researchers: user’s guide. Los Angeles: Muthén & Muthén. 1998–2001.Google Scholar
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 Moskowitz, D.S., & Hershberger, S.L. (Eds.), Modeling intraindividual variability with repeated measures data: methods and applications (pp. 235–265). Mahwah: Erlbaum.Google Scholar
Nylund-Gibson, K., Muthen, B., Nishina, A., Bellmore, A., & Graham, S. (2013, under review). Stability and instability of peer victimization during middle school: using latent transition analysis with covariates, distal outcomes, and modeling extensions.Google Scholar
Piaget, J., & Inhelder, B. (1969). The psychology of the child. New York: Basic Books.Google Scholar
R Development Core Team (2009). R: a language and environment for statistical computing [computer software manual]. Vienna, Austria: R Foundation for Statistical Computing. ISBN 3-900051-07-0. Available from. http://www.R-project.org.Google Scholar
Russell, J.A. (1980). A circumplex model of affect. Journal of Personality and Social Psychology, 39 1–sup–61161–1178.CrossRefGoogle Scholar
SAS Institute Inc. (2008). SAS 9.2 help and documentation [computer software manual]. Cary: SAS Institute Inc.Google Scholar
Sbarra, D.A., & Ferrer, E. (2006). The structure and process of emotional experience following non-marital relationship dissolution: dynamic factor analyses of love, anger, and sadness. Emotion, 6, 224–238.CrossRefGoogle Scholar
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 ScholarPubMed
Schumacker, R.E., & Marcoulides, G.A. (Eds.) (1998). Interaction and nonlinear effects in structural equation modeling. Mahwah: Erlbaum.Google Scholar
Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
Schweppe, F. (1965). Evaluation of likelihood functions for Gaussian signals. IEEE Transactions on Information Theory, 11, 61–70.CrossRefGoogle Scholar
Tiao, G.C., & Tsay, R.S. (1994). Some advances in non-linear and adaptive modelling in time-series. Journal of Forecasting, 13, 109–131.CrossRefGoogle Scholar
Tong, H., & Lim, K.S. (1980). Threshold autoregression, limit cycles and cyclical data. Journal of the Royal Statistical Society. Series B, 42, 245–292.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Van Dijk, M., & Van Geert, P. (2007). Wobbles, humps and sudden jumps: a case study of continuity, discontinuity and variability in early language development. Infant and Child Development, 16(1), 7–33.CrossRefGoogle Scholar
Visser, I. (2007). Depmix: an R-package for fitting mixture models on mixed multivariate data with Markov dependencies (Tech. Rep.). University of Amsterdam. Available from http://cran.r-project.org.Google Scholar
Watson, D., Clark, L.A., & Tellegen, A. (1988). Development and validation of brief measures of positive and negative affect: the PANAS scale. Journal of Personality and Social Psychology, 54(6), 1063–1070.CrossRefGoogle Scholar
Wen, Z., Marsh, H.W., & Hau, K.-T. (2002). Interaction effects in growth modeling: a full model. Structural Equation Modeling, 9(1), 20–39.CrossRefGoogle Scholar
Yang, M., & Chow, S.-M. (2010). Using state-space model with regime switching to represent the dynamics of facial electromyography (EMG) data. Psychometrika, 74(4), 744–771. Application and Case Studies.CrossRefGoogle Scholar
Yung, Y.F. (1997). Finite mixtures in confirmatory factor-analysis models. Psychometrika, 62, 297–330.CrossRefGoogle Scholar
Zautra, A.J., Potter, P.T., & Reich, J.W. (1997). The independence of affect is context-dependent: an integrative model of the relationship between positive and negative affect. In Lawton, M.P., Schaie, K.W., & Lawton, M.P. (Eds.), Focus on adult development (pp. 75–103). New York: Springer.Google Scholar
Zautra, A.J., Reich, J.W., Davis, M.C., Potter, P.T., & Nicolson, N.A. (2000). The role of stressful events in the relationship between positive and negative affects: evidence from field and experimental studies. Journal of Personality, 68, 927–951.CrossRefGoogle ScholarPubMed
Zhang, Z., Hamaker, E.L., & Nesselroade, J.R. (2008). Comparisons of four methods for estimating a dynamic factor model. Structural Equation Modeling, 15, 377–402.CrossRefGoogle Scholar
Zhang, Z., & Nesselroade, J.R. (2007). Bayesian estimation of categorical dynamic factor models. Multivariate Behavioral Research, 42, 729–756.CrossRefGoogle Scholar