Statistical Inference Of Nonlinear Ordinary Differential Equations

Nonlinear ordinary differential equation (ODE) has been widely used in variousfields, for example medicine, biology, finance, engineering and so on. Parameters ofODE are usually unknown, it is necessary to estimate parameters through observationsfirst, which need to find the solution (numerical solution or analytic solution). But forusual nonlinear ODE, seeking its analytic solution is very difficult, and the numericalsolution always with large computation. Thus traditional methods led to a large errorand low efficiency result. Besides, due to the effects from objective factors, differentobservations have different effects on parameter estimation. Therefore identifyingobservations which affect strongly is also very critical. But the calculations of commonmethod is too large, which makes statistical diagnose become a difficult work. Becauseof the same reason, common data transformation in statistical inference is also difficultto achieve.In order to overcome these drawbacks of statistical inference of ODE, parameterestimation, statistical diagnose, data transformation and other issues are discussedbasing on the two-step estimation method of ODE in this paper. Firstly, we reviewtwo-step parameter estimation method. At the same time, a reasonable parameter valueof Prey-predator model will be estimated through this way. Then, this paper proposesstatistical interference method basing on two-step method. In this regard, case deletionmodel and perturbation model are considered. In the end, this paper introduces datatransformation idea, discusses data transformation of ODE according to the two-stepmethod. In this part, we consider state variable transformation and time transformation.The paper makes corresponding simulations and real data experiments for statisticalinference methods above. All the results show that the statistical diagnostic method caneffectively detect anomalies in data, and appropriate data transformation can improvefitting. This achieves our desired effect.