I. Introduction
Ordinary differential equations (ODE) are widely used to describe dynamic processes in physics, engineering, chemistry, biology, etc. These ODEs usually involve some unknown parameters. How to estimate ODE parameters is an old but difficult problem. In principle, we can estimate these unknown parameters by some classical parametric estimators, such as least squares estimators or maximum likelihood estimators (MLE). When ODEs have analytical solutions, this is essentially a non-linear regression problem ([1]). However, most nonlinear ODE systems do not have analytical solutions and some methods have been proposed. One method is using nonlinear least squares (NLS). NLS approach is computationally intensive since it requires numerically solving the ODEs repeatedly to update the parameters and initial conditions. NLS only gives point estimates of parameters, and when interval estimation is needed, a great deal more computation may be required. In addition, the optimization algorithm needs to be considered carefully to avoid the local minima, see [2], [3], [4]. Another approach commonly used up to now is a two-stage method, also called collocation methods, where the first step fits the observed data by nonlinear least squares using piecewise polynomial functions or spline functions without considering satisfying the ODEs, and the second step estimates the parameters by linear least squares solution of the differential equations sampled at a set of points, see [5], [6], [7], [8], [9]. Ramsay and Silverman [10] and Poyton et al. [11] modified two-stage method by iterating the two steps with principal differential analysis, which converged quickly to the estimates of both the solution and the parameters and had substantially improved bias and precision. Ramsay et al. [12] proposed a new collocation method called generalized profiling procedure. In this approach, the ODE solution is approximated by a linear combination of basis functions. However, the coefficients of the basis functions are estimated by a penalized smoothing procedure with an ODE-defined penalty. The smoothing parameter controls the trade off between fitting the data with the basis functions and fidelity of the basis functions to the ODEs. Cao and Ramsay [13] used this method to estimate ODE parameters in a gene regulation network and a goodness-of-fit test was developed. Chen and Wu [14] proposed a two-stage local polynomial method for estimating the time varying parameters. However all of the existing two-stage estimation method didn't consider the property of the ODE, such as the periodic property, which may help to ensure a satisfactory estimation of the state variable and their derivates.