Statistical Diagnosis Of Ordinary Differential Equations With Applications To HIV Dynamics

The parameters of ordinary differential equation(ODE)are usually unknown in practical application,which need to be estimated based on the factual observation data.Because of the influence of objective factor,different observations have different effects on parameter estimation,so it is necessary to make a statistical diagnosis for the equation.But the calculation of traditional parameter estimation methods is large,which makes statistical diagnose become a difficult work.In order to overcome this drawback,in this paper we consider statistical diagnosis based on the spline basis function involved in the two –step estimation.Firstly,this paper analysis the statistical diagnosis method of ODE based on the two-step estimation,which contains case deletion model and perturbation model,and diagnostic statistic is constructed in this paper,and this approach is applied to the statistical diagnosis of HIV model,diagnostic statistics of HIV model are constructed under different conditions.By the simulated experiment it find:(1)the statistical diagnosis approach based on the two-step estimation can detect anomalies in data;(2)compared to the approaches of other literatures,this approach can detect points of tiny perturbation availably;(3)this approach can weaken the influence of the boundary points.Secondly,this paper put forward the statistical diagnosis method of ODE based on the full perturbation,and this approach is applied to the statistical diagnosis of HIV model.Compared to the other statistical diagnosis approaches,this approach adds full perturbation to fitting curve rather than adds perturbation to data directly,this fact greatly simplifies the procedures of impact analysis and makes the procedures more intuitive.By analyzing simulated experiment it find:(1)the statistical diagnosis approach of ODE based on the full perturbation can detect influential area of the data effectively,and detect the influential point;(2)this approach reduces the error of fitting procedures by spline basis function and makes diagnosis more accurate.The proposed conclusion can be generalized to other statistical diagnosis of ODE.Finally,the similar ideas are applied to the statistical diagnosis of high order ODE in this paper.