Fitting Nonlinear Ordinary Differential Equation Models with Random Effects and Unknown Initial Conditions Using the Stochastic Approximation Expectation–Maximization (SAEM) Algorithm
详细信息 查看全文
- 作者：Sy-Miin Chow ; Zhaohua Lu ; Andrew Sherwood ; Hongtu Zhu
- 关键词：differential equation ; dynamic ; nonlinear ; stochastic EM ; longitudinal
- 刊名：Psychometrika
- 出版年：2016
- 出版时间：March 2016
- 年：2016
- 卷：81
- 期：1
- 页码：102-134
- 全文大小：1,791 KB
- 参考文献：Ait-Sahalia, Y. (2008). Closed-form likelihood expansions for multivariate diffusions. The Annals of Statistics, 36(2), 906–937.CrossRef
Anderson, T. W. (2003). An introduction to multivariate statistical analysis (3rd ed.)., Probability and Statistics New York, NY: Wiley.
Arminger, G. (1986). Linear stochastic differential equation models for panel data with unobserved variables. In N. Tuma (Ed.), Sociological methodology (pp. 187–212). San Francisco: Jossey-Bass.
Bereiter, C. (1963). Some persisting dilemmas in the measurement of change. In C. W. Harris (Ed.), Problems in measuring change (pp. 3–20). Madison, WI: University of Wisconsin Press.
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.CrossRef
Beskos, A., Papaspiliopoulos, O., Roberts, G., & Fearnhead, P. (2006). Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion). Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(3), 333–382.CrossRef
Boker, S. M., & Graham, J. (1998). A dynamical systems analysis of adolescent substance abuse. Multivariate Behavioral Research, 33, 479–507.CrossRef PubMed
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, 127–160.CrossRef PubMed
Bolger, N., Davis, A., & Rafaeli, E. (2003). Diary methods: Capturing life as it is lived. Annual Review of Psychology, 54, 579–616.CrossRef PubMed
Brown, E. N., & Luithardt, H. (1999). Statistical model building and model criticism for human circadian data. Journal of Biological Rhythms, 14, 609–616.CrossRef PubMed
Brown, E. N., Luithardt, H., & Czeisler, C. A. (2000). A statistical model of the human coretemperature circadian rhythm. American Journal of Physiology, Endocrinology and Metabolism, 279, 669–683.
Browne, M. W., & du Toit, H. C. (1991). Models for learning data. In L. M. Collins & J. L. Horn (Eds.), Best methods for the analysis of change: Recent advances, unanswered questions, future directions (pp. 47–68). Washington, D.C.: American Psychological Association.CrossRef
Cao, J., Huang, J. Z., & Wu, H. (2012). Penalized nonlinear least squares estimation of time-varying parameters in ordinary differential equations. Journal of Computational and Graphical Statistics, 21(1), 42–56. doi:10.1198/jcgs.2011.10021 .
Carels, R. A., Blumenthal, J. A., & Sherwood, A. (2000). Emotional responsivity during daily life: Relationship to psychosocial functioning and ambulatory blood pressure. International Journal of Psychophysiology, 36, 25–33.CrossRef PubMed
Carlin, B. P., Gelfand, A., & Smith, A. (1992). Hierarchical bayesian analysis of changepoints problems. Applied Statistics, 41, 389–405.CrossRef
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.CrossRef PubMed
Chow, S.-M., Grimm, K. J., Guillaume, F., Dolan, C. V., & McArdle, J. J. (2013). Regime switching bivariate dual change score model. Multivariate Behavioral Research, 48(4), 463–502.CrossRef PubMed
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).
Chow, S.-M., & Nesselroade, J. R. (2004). General slowing or decreased inhibition? Mathematical models of age differences in cognitive functioning. Journals of Gerontology Series B—Psychological Sciences & Social Sciences, 59B(3), 101–109.CrossRef
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 and Statistical Psychology, 64(1), 69–106.PubMedCentral CrossRef PubMed
Chow, S.-M., & Zhang, G. (2013). Nonlinear regime-switching state-space (RSSS) models. Psychometrika: Application Reviews and Case Studies, 78(4), 740–768.CrossRef
Cronbach, L. J., & Furby, L. (1970). How should we measure “change”—or should we? Psychological Bulletin, 74(1), 68–80.CrossRef
Cudeck, R., & Klebe, K. J. (2002). Multiphase mixed-effects models for repeated measures data. Psychological Methods, 7(1), 41–6.CrossRef PubMed
Dembo, A., & Zeitouni, O. (1986). Parameter estimation of partially observed continuous time stochastic processes via the EM algorithm. Stochastic Processes and Their Applications, 23, 91–113.CrossRef
Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39(1), 1–38.
Diebolt, J., & Celeux, G. (1993). Asymptotic properties of a stochastic EM algorithm for estimating mixing proportions. Communications in Statistics B—Stochastic Models, 9(4), 599–613.CrossRef
Donnet, S., & Samson, A. (2007). Estimation of parameters in incomplete data models defined by dynamical systems. Journal of Statistical Planning and Inference, 137, 2815–2831.CrossRef
Du Toit, S. H. C., & Browne, M. W. (2001). The covariance structure of a vector ARMA time series. Structural equation modeling: Present and future (pp. 279–314). Chicago: Scientific Software International.
Duncan, T. E., Duncan, S. C., Strycker, L. A., Li, F., & Alpert, A. (1999). An introduction to latent variable growth curve modeling: Concepts, issues, and applications. Mahwah, NJ: Lawrence Erlbaum Associates, Publishers.
Durbin, J., & Koopman, S. J. (2001). Time series analysis by state space methods. New York, NY: Oxford University Press.
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.CrossRef PubMed
Geweke, J., & Tanizaki, H. (2001). Bayesian estimation of state-space models using the Metropolis–Hastings algorithm within Gibbs sampling. Computational Statistics & Data Analysis, 37, 151–170.CrossRef
Gordon, N. J., Salmond, D. J., & Smith, A. F. M. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEEE Proceedings-F, Radar and Signal Processing, 140(2), 107–113.CrossRef
Gu, M. G., & Zhu, H. T. (2001). Maximum likelihood estimation for spatial models by Markov chain Monte Carlo stochastic approximation. Journal of the Royal Statistical Society, Series B, 63, 339–355.CrossRef
Hairer, M., Stuart, A. M., Voss, J., & Wiberg, P. (2005). Analysis of spdes arising in path sampling. part i: The gaussian case. Communications in Mathematical Sciences, 3(4), 587–603.
Hale, J. K., & Koçak, H. (1991). Dynamics and bifurcation. New York, NY: Springer.CrossRef
Harris, C. W. (Ed.). (1963). Problems in measuring change. Madison, WI: University of Wisconsin Press.
Harvey, A. C., & Souza, R. C. (1987). Assessing and modelling the cyclical behaviour of rainfall in northeast Brazil. Journal of Climate and Applied Meteorology, 26, 1317–1322.CrossRef
Hürzeler, M., & Künsch, H. (1998). Monte carlo approximations for general state-space models. Journal of Computational and Graphical Statistics, 7, 175–193.
Jones, R. H. (1984). Fitting multivariate models to unequally spaced data. In E. Parzen (Ed.), Time series analysis of irregularly observed data (Vol. 25, p. 158–188). New York, NY: Springer.
Jones, R. H. (1993). Longitudinal data with serial correlation: A state-space approach. Boca Raton, FL: Chapman & Hall/CRC.CrossRef
Kaplan, D., & Glass, L. (1995). Understanding nonlinear dynamics. New York, NY: Springer.CrossRef
Kenny, D. A., & Judd, C. M. (1984). Estimating the nonlinear and interactive effects of latent variables. Psychological Bulletin, 96, 201–210.CrossRef
Kincanon, E., & Powel, W. (1995). Chaotic analysis in psychology and psychoanalysis. The Journal of Psychology, 129, 495–505.CrossRef PubMed
Kitagawa, G. (1998). A self-organizing state-space model. Journal of the American Statistical Association, 93(443), 1203–1215.
Klein, A. G., & Muthén, B. O. (2007). Quasi maximum likelihood estimation of structural equation models with multiple interaction and quadratic effects. Multivariate Behavioral Research, 42(4), 647–673.CrossRef
Kuhn, E., & Lavielle, M. (2005). Maximum likelihood estimation in nonlinear mixed effects models. Computational Statistics & Data Analysis, 49, 1020–1038.CrossRef
Kulikov, G., & Kulikova, M. (2014). Accurate numerical implementation of the continuous-discrete extended Kalman filter. IEEE Transactions on Automatic Control, 59(1), 273–279. doi:10.1109/TAC.2013.2272136 .CrossRef
Lee, S., & Song, X. (2003). Maximum likelihood estimation and model comparison for mixtures of structural equation models with ignorable missing data. Journal of Classification, 20(2), 221–255. doi:10.1007/s00357-003-0013-5 .CrossRef
Li, F., Duncan, T. E., & Acock, A. (2000). Modeling interaction effects in latent growth curve models. Structural Equation Modeling, 7(4), 497–533.CrossRef
Liang, H., Miao, H., & Wu, H. (2010). Estimation of constant and time-varying dynamic parameters of HIV infection in a nonlinear differential equation model. Annals of Applied Statistics, 4(1), 460–483.PubMedCentral CrossRef PubMed
Longstaff, M. G., & Heath, R. A. (1999). A nonlinear analysis of the temporal characteristics of handwriting. Human Movement Science, 18, 485–524.CrossRef
Losardo, D. (2012). An examination of initial condition specification in the structural equation modeling framework. Unpublished doctoral dissertation, University of North Carolina, Chapel Hill, NC.
Louis, T. A. (1982). Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society, Series B, 44, 190–200.
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.CrossRef PubMed
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.CrossRef
McArdle, J. J., & Hamagami, F. (2001). Latent difference score structural models for linear dynamic analysis with incomplete longitudinal data. In L. Collins & A. Sayer (Eds.), New methods for the analysis of change (pp. 139–175). Washington, DC: American Psychological Association.CrossRef
Mcardle, J. J., & Hamagami, F. (2003). Structural equation models for evaluating dynamic concepts within longitudinal twin analyses. Behavior Genetics, 33(2), 137–159. doi:10.1023/A:1022553901851 .CrossRef PubMed
Meredith, W., & Tisak, J. (1990). Latent curve analysis. Psychometrika, 55, 107–122.CrossRef
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.PubMedCentral CrossRef PubMed
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.
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. doi:10.1348/000711003770480002 .CrossRef PubMed
Ortega, J. (1990). Numerical analysis: A second course. Philadelphia, PA: Society for Industrial and Academic Press.CrossRef
Oud, J. H. L. (2007). Comparison of four procedures to estimate the damped linear differential oscillator for panel data. In J. Oud & A. Satorra (Eds.), Longitudinal models in the behavioral and related sciences. Mahwah, NJ: Lawrence Erlbaum Associates.
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.CrossRef
Oud, J. H. L., & Singer, H. (Eds.). (2010). Special issue: Continuous time modeling of panel data, 62 (1).
Pickering, T. G., Shimbo, D., & Haas, D. (2006). Ambulatory blood-pressure monitoring. The New England Journal of Medicine, 354, 2368–2374.CrossRef PubMed
Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (2002). Numerical recipes in C. Cambridge: Cambridge University Press.
R Development Core Team. (2009). R: A language and environment for statistical computing [Computer software manual]. Vienna, Austria: R Foundation for Statistical Computing. Retrieved April, 2014, from http://www.R-project.org (ISBN: 3-900051-07-0).
Ralston, A., & Rabinowitz, P. (2001). A first course in numerical analysis (2nd ed.). Mineola, NY: Dover.
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.CrossRef
Raudenbush, S. W., & Liu, X.-F. (2001). Effects of study duration, frequency of observation, and sample size on power in studies of group differences in polynomial change. Psychological Methods, 6(4), 387–401.CrossRef PubMed
Särkkä, S. (2013). Bayesian filtering and smoothing. Hillsdale, NJ: Cambridge University Press.CrossRef
SAS Institute Inc. (2008). SAS 9.2 Help and Documentation (Computer software manual). Cary, NC: SAS Institute Inc.
Sherwood, A., Steffen, P., Blumenthal, J., Kuhn, C., & Hinderliter, A. L. (2002). Nighttime blood pressure dipping: The role of the sympathetic nervous system. American Journal of Hypertension, 15, 111–118.CrossRef PubMed
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.CrossRef PubMed
Singer, H. (1992). The aliasing-phenomenon in visual terms. Journal of Mathematical Sociology, 14(1), 39–49.CrossRef
Singer, H. (1995). Analytical score function for irregularly sampled continuous time stochastic processes with control variables and missing values. Econometric Theory, 11, 721–735. doi:10.1017/S0266466600009701 .CrossRef
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.CrossRef
Singer, H. (2007). Stochastic differential equation models with sampled data. In K. van Montfort, J. H. L. Oud, & A. Satorra (Eds.), Longitudinal models in the behavioral and related sciences (pp. 73–106). Mahwah, NJ: Lawrence Erlbaum Associates.
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. doi:10.1080/0022250X.2010.532259 .CrossRef
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. doi:10.1080/0022250X.2010.532259 .CrossRef
Stone, A. A., & Shiffman, S. (1994). Ecological momentary assessment (ema) in behavioral medicine. Annals of Behavioral Medicine, 16(3), 199–202.
Strogatz, S. H. (1994). Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering. Cambridge, MA: Westview.
Stuart, A. M., Voss, J., & Wilberg, P. (2004). Conditional path sampling of sdes and the langevin mcmc method. Communications in Mathematical Sciences, 2(4), 685–697.
Tanizaki, H. (1996). Nonlinear filters: Estimation and applications (2nd ed.). Berlin: Springer.CrossRef
Thatcher, R. W. (1998). A predator–prey model of human cerebral development. In K. M. Newell & P. C. M. Molenaar (Eds.), Applications of nonlinear dynamics to developmental process modeling (pp. 87–128). Mahwah, NJ: Lawrence Erlbaum.
Wen, Z., Marsh, H. W., & Hau, K.-T. (2002). Interaction effects in growth modeling: A full model. Structural Equation Modeling, 9(1), 20–39.CrossRef
Wu, H. (2005). Statistical methods for HIV dynamic studies in AIDS clinical trials. Statistical Methods in Medical Research, 14, 171–192.CrossRef PubMed
Zhu, H., Gu, M., & Peterson, B. (2007). Maximum likelihood from spatial random effects models via the stochastic approximation expectation maximization algorithm. Statistics and Computing Archive, 17(2), 163–177.CrossRef
Zhu, H. T., & Zhang, H. P. (2006). Generalized score test of homogeneity for mixed effects models. Annals of Statistics, 34, 1545–1569.CrossRef - 作者单位：Sy-Miin Chow (1)
Zhaohua Lu (2)
Andrew Sherwood (3)
Hongtu Zhu (2)
1. The Pennsylvania State University, 413 Biobehavioral Health Building, University Park, PA, 16802 , USA
2. University of North Carolina at Chapel Hill, Chapel Hill, USA
3. Duke University, Durham, USA - 刊物主题：Psychometrics; Assessment, Testing and Evaluation; Statistics for Social Science, Behavorial Science, Education, Public Policy, and Law; Statistical Theory and Methods;
- 出版者：Springer US
- ISSN：1860-0980
文摘
The past decade has evidenced the increased prevalence of irregularly spaced longitudinal data in social sciences. Clearly lacking, however, are modeling tools that allow researchers to fit dynamic models to irregularly spaced data, particularly data that show nonlinearity and heterogeneity in dynamical structures. We consider the issue of fitting multivariate nonlinear differential equation models with random effects and unknown initial conditions to irregularly spaced data. A stochastic approximation expectation–maximization algorithm is proposed and its performance is evaluated using a benchmark nonlinear dynamical systems model, namely, the Van der Pol oscillator equations. The empirical utility of the proposed technique is illustrated using a set of 24-h ambulatory cardiovascular data from 168 men and women. Pertinent methodological challenges and unresolved issues are discussed.