Source: Ann. Statist.
Volume 38, Number 1
Ordinary differential equations (ODEs) are commonly used to model dynamic behavior of a system. Because many parameters are unknown and have to be estimated from the observed data, there is growing interest in statistics to develop efficient estimation procedures for these parameters. Among the proposed methods in the literature, the generalized profiling estimation method developed by Ramsay and colleagues is particularly promising for its computational efficiency and good performance. In this approach, the ODE solution is approximated with a linear combination of basis functions. The coefficients of the basis functions are estimated by a penalized smoothing procedure with an ODE-defined penalty. However, the statistical properties of this procedure are not known. In this paper, we first give an upper bound on the uniform norm of the difference between the true solutions and their approximations. Then we use this bound to prove the consistency and asymptotic normality of this estimation procedure. We show that the asymptotic covariance matrix is the same as that of the maximum likelihood estimation. Therefore, this procedure is asymptotically efficient. For a fixed sample and fixed basis functions, we study the limiting behavior of the approximation when the smoothing parameter tends to infinity. We propose an algorithm to choose the smoothing parameters and a method to compute the deviation of the spline approximation from solution without solving the ODEs.
 Antosiewicz, H. A. (1962). An inequality for approximate solutions of ordinary differential equations. Math. Z. 78 44–52.
Mathematical Reviews (MathSciNet): MR153910
 Arora, N. and Biegler, L. T. (2004). A trust region SQP algorithm for equality constrained parameter estimation with simple parametric bounds. Comput. Optim. Appl. 28 51–86.
 Bock, H. G. (1983). Recent advances in parameter identification techniques for ODE. In Numerical Treatment of Inverse Problems in Differential and Integral Equations 95–121. Birkhäuser, Basel.
Mathematical Reviews (MathSciNet): MR714563
 Breto, C., He, D., Ionides, E. L. and King, A. A. (2009). Time series analysis via mechanistic models. Ann. Appl. Stat. 3 319–348.
 Cao, J. and Zhao, H. (2008). Estimating dynamic models for gene regulation networks. Bioinformatics 24 1619–1624.
 Chicone, C. (1999). Ordinary Differential Equations with Applications, 1st ed. Springer, New York.
 Coddington, E. A. and Levinson, N. (1955). Theory of Ordinary Differential Equations. McGraw-Hill, New York.
Mathematical Reviews (MathSciNet): MR69338
 DeBoor, C. (2001). A Practical Guide to Splines. Springer, New York.
 Gardner, T. S., di Bernardo, D., Lorenz, D. and Collins, J. J. (2003). Inferring genetic networks and identifying compound mode of action via expression profiling. Science 301 102–105.
 Hall, C. A. and Meyer, W. W. (1976). Optimal error bounds for cubic spline interpolation. J. Approx. Theory 16 105–122.
Mathematical Reviews (MathSciNet): MR397247
 Hartman, P. (1964). Ordinary Differential Equations. Wiley, New York.
Mathematical Reviews (MathSciNet): MR171038
 Huang, J. (2007). Discussion on the paper by Ramsay, Hooker, Campbell and Cao. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 783.
 Ionides, E. L. (2007). Discussion on the paper by Ramsay, Hooker, Campbell and Cao. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 783–784.
 Ionides, E. L., Breto, C. and King, A. A. (2006). Inference for nonlinear dynamical systems. Proc. Nat. Acad. Sci. India Sect. A 103 18438–18443.
 Jeffrey, A. and Xia, X. (2005). Identifiability of HIV/AIDS model. In Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention (W. Y. Tan and H. Wu, eds.). World Scientific, Singapore.
 Lele, S. (2007). Discussion on the paper by Ramsay, Hooker, Campbell and Cao. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 787.
 Liu, J. and West, M. (2001). Combined parameter and state estimation in simulation based filtering. In Sequential Monte Carlo Methods in Practice (A. Doucet, J. Freitas and K. Gordon, eds.). Springer, New York.
 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 65 292–300.
 Olhede, S. (2007). Discussion on the paper by Ramsay, Hooker, Campbell and Cao. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 772–779.
 Poyton, A. A., Varziri, M. S., McAuley, K. B., McLellan, P. J. and Ramsay, J. O. (2006). Parameter estimation in continuous dynamic models using principal differential analysis. Comput. Chem. Eng. 30 698–708.
 Ramsay, J. O., Hooker, G., Campbell, D. and Cao, J. (2007). Parameter estimation for differential equations: A generalized smoothing approach (with discussions). J. R. Stat. Soc. Ser. B Stat. Methodol. 69 741–796.
 Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis. Springer, New York.
 Schumaker, L. (2007). Spline Functions: Basic Theory. Cambridge Univ. Press, Cambridge.
 Tjoa, I. B. and Biegler, L. (1991). Simultaneous solution and optimization strategies for parameter estimation of differential-algebraic equation systems. Ind. Eng. Chem. Res. 30 376–385.
 van der Vaart, A. (2000). Asymptotic Statistics. Cambridge Univ. Press, Cambridge.
 van der Vaart, A. and Wellner, J. (2000). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.
 Varah, J. M. (1982). A spline least squares method for numerical parameter estimation in differential equations. SIAM J. Sci. Comput. 3 28–46.
Mathematical Reviews (MathSciNet): MR651865
 Xia, X. (2003). Estimation of HIV/AIDS parameters. Automatica 39 1983–1988.
 Xia, X. and Moog, C. (2003). Identifiability of nonlinear systems with applications to HIV/AIDS models. IEEE Trans. Automat. Control 48 330–336.