zbMATH — the first resource for mathematics

Optimal rate of direct estimators in systems of ordinary differential equations linear in functions of the parameters. (English) Zbl 1327.62120
Summary: Many processes in biology, chemistry, physics, medicine, and engineering are modeled by a system of differential equations. Such a system is usually characterized via unknown parameters and estimating their ’true’ value is thus required. In this paper we focus on the quite common systems for which the derivatives of the states may be written as sums of products of a function of the states and a function of the parameters.
For such a system linear in functions of the unknown parameters we present a necessary and sufficient condition for identifiability of the parameters. We develop an estimation approach that bypasses the heavy computational burden of numerical integration and avoids the estimation of system states derivatives, drawbacks from which many classic estimation methods suffer. We also suggest an experimental design for which smoothing can be circumvented. The optimal rate of the proposed estimators, i.e., their \(\sqrt{n}\)-consistency, is proved and simulation results illustrate their excellent finite sample performance and compare it to other estimation approaches.

62F12 Asymptotic properties of parametric estimators
62G05 Nonparametric estimation
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
Full Text: DOI Euclid arXiv
[1] Arnold, V. (1977)., Ordinary Differential Equations . The MIT Press, Cambridge.
[2] Bellman, R. and Åström, K. (1970). On structural identifiability., Mathematical Biosciences 7 (3), 329-339.
[3] Bellman, R. and Roth, R. S. (1971). The use of splines with unknown end points in the identification of systems., Journal of Mathematical Analysis and Applications 34 (1), 26-33. · Zbl 0217.11601 · doi:10.1016/0022-247X(71)90154-5
[4] Bickel, P. J. and Ritov, Y. (2003). Nonparametric estimators which can be “plugged-in”., The Annals of Statistics 31 (4), 1033-1053. · Zbl 1058.62031 · doi:10.1214/aos/1059655904 · euclid:aos/1059655904
[5] Brewer, D., Barenco, M., Callard, R., Hubank, M., and Stark, J. (2008). Fitting ordinary differential equations to short time course data., Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366 (1865), 519-544. · Zbl 1153.37444 · doi:10.1098/rsta.2007.2108
[6] Brunel, N. J., Clairon, Q., and d’Alché Buc, F. (2014). Parametric estimation of ordinary differential equations with orthogonality conditions., Journal of the American Statistical Association 109 (505), 173-185. · Zbl 1367.62081 · doi:10.1080/01621459.2013.841583
[7] Brunel, N. J. B. (2008). Parameter estimation of ode’s via nonparametric estimators., Electronic Journal of Statistics 2 , 1242-1267. · Zbl 1320.62063 · doi:10.1214/07-EJS132 · euclid:ejs/1229975381 · arxiv:0710.4190
[8] Campbell, D. and Steele, R. J. (2012). Smooth functional tempering for nonlinear differential equation models., Statistics and Computing 22 (2), 429-443. · Zbl 1322.62011 · doi:10.1007/s11222-011-9234-3
[9] Cheng, M.-Y., Fan, J., and Marron, J. S. (1997). On automatic boundary corrections., The Annals of Statistics 25 (4), 1691-1708. · Zbl 0890.62026 · doi:10.1214/aos/1031594737
[10] Chou, I.-C. and Voit, E. O. (2009). Recent developments in parameter estimation and structure identification of biochemical and genomic systems., Mathematical biosciences 219 (2), 57. · Zbl 1168.92019 · doi:10.1016/j.mbs.2009.03.002
[11] Cobelli, C., Distefano, J. J., et al. (1980). Parameter and structural identifiability concepts and ambiguities: A critical review and analysis., American Journal of Physiology-Regulatory, Integrative and Comparative Physiology 239 (1), R7-R24.
[12] Dattner, I. (2015). A model-based initial guess for estimating parameters in systems of ordinary differential equations., Biometrics , doi: 10.1111/biom.12348. · Zbl 1419.62083
[13] Dattner, I. and Gugushvili, S. (2015). Accelerated least squares estimation for systems of ordinary differential equations.,
[14] de Bazelaire, C., Siauve, N., Fournier, L., Frouin, F., Robert, P., Clement, O., de Kerviler, E., and Cuenod, C. A. (2005). Comprehensive model for simultaneous mri determination of perfusion and permeability using a blood-pool agent in rats rhabdomyosarcoma., European radiology 15 (12), 2497-2505.
[15] Edelstein-Keshet, L. (2005)., Mathematical Models in Biology. Classics in Applied Mathematics , Volume 46. Society for Industrial and Applied Mathematics. · Zbl 1100.92001
[16] Fang, Y., Wu, H., and Zhu, L.-X. (2011). A two-stage estimation method for random coefficient differential equation models with application to longitudinal hiv dynamic data., Statistica Sinica 21 (3), 1145. · Zbl 05961047 · doi:10.5705/ss.2009.156
[17] FitzHugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane., Biophysical Journal 1 (6), 445-466.
[18] Font, J. and Fabregat, A. (1997). Testing a predictor-corrector integral method for estimating parameters in complex kinetic systems described by ordinary differential equations., Computers & Chemical Engineering 21 (7), 719-731.
[19] Goldstein, L. and Messer, K. (1992). Optimal plug-in estimators for nonparametric functional estimation., The Annals of Statistics 20 , 1306-1328. · Zbl 0763.62023 · doi:10.1214/aos/1176348770
[20] Gugushvili, S. and Klaassen, C. A. J. (2012). \(\sqrtn\)-consistent parameter estimation for systems of ordinary differential equations: bypassing numerical integration via smoothing., Bernoulli 18 , 1061-1098. · Zbl 1257.49033 · doi:10.3150/11-BEJ362 · euclid:bj/1340887014 · arxiv:1007.3880
[21] Gugushvili, S. and Spreij, P. (2012). Parametric inference for stochastic differential equations: A smooth and match approach., Latin American Journal of Probability and Mathematical Statistics 9 (2), 609-635. · Zbl 1277.62195 · alea.impa.br · arxiv:1111.1120
[22] Hall, P. and Ma, Y. (2013). Quick and easy one-step parameter estimation in differential equations., Journal of the Royal Statistical Society: Series B (Statistical Methodology) . · doi:10.1111/rssb.12040
[23] Härdle, W. and Bowman, A. W. (1988). Bootstrapping in nonparametric regression: Local adaptive smoothing and confidence bands., Journal of the American Statistical Association 83 (401), 102-110. · Zbl 0644.62047 · doi:10.2307/2288926
[24] Haynsworth, E. V. (1968). On the schur complement. Technical report, DTIC, Document. · Zbl 0155.06304
[25] He, D., Ionides, E. L., and King, A. A. (2010). Plug-and-play inference for disease dynamics: Measles in large and small populations as a case study., Journal of the Royal Society Interface 7 (43), 271-283.
[26] Himmelblau, D., Jones, C., and Bischoff, K. (1967). Determination of rate constants for complex kinetics models., Industrial & Engineering Chemistry Fundamentals 6 (4), 539-543.
[27] Hockin, M. F., Jones, K. C., Everse, S. J., and Mann, K. G. (2002). A model for the stoichiometric regulation of blood coagulation., Journal of Biological Chemistry 277 (21), 18322-18333.
[28] Hodgkin, A. L. and Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve., The Journal of Physiology 117 (4), 500.
[29] Hooker, G., Ellner, S. P., Roditi, L. D. V., and Earn, D. J. (2011). Parameterizing state-space models for infectious disease dynamics by generalized profiling: Measles in ontario., Journal of The Royal Society Interface 8 (60), 961-974.
[30] Liang, H. and Wu, H. (2008). Parameter estimation for differential equation models using a framework of measurement error in regression models., Journal of the American Statistical Association 103 (484), 1570-1583. · Zbl 1286.62039 · doi:10.1198/016214508000000797
[31] Ljung, L. and Glad, T. (1994). On global identifiability for arbitrary model parametrizations., Automatica 30 (2), 265-276. · Zbl 0795.93026 · doi:10.1016/0005-1098(94)90029-9
[32] Miao, H., Dykes, C., Demeter, L. M., Cavenaugh, J., Park, S. Y., Perelson, A. S., and Wu, H. (2008). Modeling and estimation of kinetic parameters and replicative fitness of hiv-1 from flow-cytometry-based growth competition experiments., Bulletin of Mathematical Biology 70 (6), 1749-1771. · Zbl 1166.92029 · doi:10.1007/s11538-008-9323-4
[33] Miao, H., Dykes, C., Demeter, L. M., and Wu, H. (2009). Differential equation modeling of hiv viral fitness experiments: Model identification, model selection, and multimodel inference., Biometrics 65 (1), 292-300. · Zbl 1159.62079 · doi:10.1111/j.1541-0420.2008.01059.x
[34] Miao, H., Xia, X., Perelson, A. S., and Wu, H. (2011). On identifiability of nonlinear ode models and applications in viral dynamics., SIAM Review 53 (1), 3-39. · Zbl 1215.34015 · doi:10.1137/090757009
[35] Nagumo, J., Arimoto, S., and Yoshizawa, S. (1962). An active pulse transmission line simulating nerve axon., Proceedings of the IRE 50 (10), 2061-2070.
[36] Nowak, M. and May, R. M. (2000)., Virus Dynamics: Mathematical Principles of Immunology and Virology . Oxford University Press on Demand. · Zbl 1101.92028
[37] Qi, X. and Zhao, H. (2010). Asymptotic efficiency and finite-sample properties of the generalized profiling estimation of parameters in ordinary differential equations., The Annals of Statistics 38 (1), 435-481. · Zbl 1181.62156 · doi:10.1214/09-AOS724 · arxiv:0903.3400
[38] Ramsay, J. O., Hooker, G., Campbell, D., and Cao, J. (2007). Parameter estimation for differential equations: A generalized smoothing approach., Journal of the Royal Statistical Society: Series B (Statistical Methodology) 69 (5), 741-796. · doi:10.1111/j.1467-9868.2007.00610.x
[39] Tank, D., Regehr, W., and Delaney, K. (1995). A quantitative analysis of presynaptic calcium dynamics that contribute to short-term enhancement., The Journal of neuroscience 15 (12), 7940-7952.
[40] Tjoa, I. B. and Biegler, L. T. (1991). Simultaneous solution and optimization strategies for parameter estimation of differential-algebraic equation systems., Industrial & Engineering Chemistry Research 30 (2), 376- 385.
[41] Tsybakov, A. B. (2009)., Introduction to Nonparametric Estimation . Springer. · Zbl 1176.62032
[42] Vajda, S., Valko, P., and Yermakova, A. (1986). A direct-indirect procedure for estimation of kinetic parameters., Computers & Chemical Engineering 10 (1), 49-58.
[43] Varah, J. (1982). A spline least squares method for numerical parameter estimation in differential equations., SIAM Journal on Scientific and Statistical Computing 3 (1), 28-46. · Zbl 0481.65050 · doi:10.1137/0903003
[44] Voit, E. O. (2000)., Computational Analysis of Biochemical Systems: A Practical Guide for Biochemists and Molecular Biologists . Cambridge University Press.
[45] Voit, E. O. and Almeida, J. (2004). Decoupling dynamical systems for pathway identification from metabolic profiles., Bioinformatics 20 (11), 1670-1681.
[46] Vujačić, I., Dattner, I., González, J., and Wit, E. (2014). Time-course window estimator for ordinary differential equations linear in the parameters., Statistics and Computing , doi: 10.1007/s11222-014-9486-9. · Zbl 1331.62109
[47] Wu, H., Zhu, H., Miao, H., and Perelson, A. S. (2008). Parameter identifiability and estimation of hiv/aids dynamic models., Bulletin of Mathematical Biology 70 (3), 785-799. · Zbl 1146.92021 · doi:10.1007/s11538-007-9279-9
[48] Xia, X. and Moog, C. (2003). Identifiability of nonlinear systems with application to hiv/aids models., IEEE Transactions on Automatic Control 48 (2), 330-336. · Zbl 1364.93838 · doi:10.1109/TAC.2002.808494
[49] Xue, H., Miao, H., and Wu, H. (2010). Sieve estimation of constant and time-varying coefficients in nonlinear ordinary differential equation models by considering both numerical error and measurement error., The Annals of Statistics 38 (4), 2351-2387. · Zbl 1203.62049 · doi:10.1214/09-AOS784 · arxiv:1010.4162
[50] Xun, X., Cao, J., Mallick, B., Maity, A., and Carroll, R. J. (2013). Parameter estimation of partial differential equation models., Journal of the American Statistical Association 108 (503), 1009-1020. · Zbl 06224983 · doi:10.1080/01621459.2013.794730
[51] Yermakova, A., Vajda, S., and Valko, P. (1982). Direct integral method via spline-approximation for estimating rate constants., Applied Catalysis 2 (3), 139-154.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.