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Parametric incomplete data models defined by ordinary differential equations (ODEs) are widely used in biostatistics to describe biological processes accurately. Their parameters are estimated on approximate models, whose regression functions are evaluated by a numerical integration method. Accurate and efficient estimations of these parameters are critical issues. This paper proposes parameter estimation methods involving either a stochastic approximation EM algorithm (SAEM) in the maximum likelihood estimation, or a Gibbs sampler in the Bayesian approach. Both algorithms involve the simulation of non-observed data with conditional distributions using Hastings–Metropolis (H–M) algorithms. A modified H–M algorithm, including an original local linearization scheme to solve the ODEs, is proposed to reduce the computational time significantly. The convergence on the approximate model of all these algorithms is proved. The errors induced by the numerical solving method on the conditional distribution, the likelihood and the posterior distribution are bounded. The Bayesian and maximum likelihood estimation methods are illustrated on a simulated pharmacokinetic nonlinear mixed-effects model defined by an ODE. Simulation results illustrate the ability of these algorithms to provide accurate estimates. 相似文献
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Sévérien Nkurunziza 《Statistical Papers》2009,50(3):527-551
This paper deals with a testing problem for each of the interaction parameters of the Lotka–Volterra ordinary differential
equations system~(ODE). In short, when the rates of birth and death are fixed, we would like to test if each interaction parameter
is higher or lower than a fixed reference rate. We choose a statistical model where the actual population sizes are modelled
as random perturbations of the solutions to this ODE. By assuming that the random perturbations follow correlated Ornstein–Uhlenbeck
processes, we propose the uniformly most powerful test concerning each interaction parameter of the ODE and, we establish
the asymptotic properties of the test. Further, we illustrate the suggested test on the Canadian mink–muskrat data set.
This research has received the financial support from Natural Sciences and Engineering Research Council of Canada and Institut
des Sciences Mathématiques. 相似文献
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王爱峰 《淮海工学院学报(社会科学版)》2010,8(12):75-76
结合几年来在常微分方程课堂教学的一点体会,从优化课堂教学设计,尊重学生的主体地位,发掘微分方程中的思想方法、趣味性以及在教学中渗透数学建模的思想等方面探讨了在课堂教学中提高学生数学素养的途径。 相似文献
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Lie级数是常微分方程的以初始条件为系数的幂级数解,它适于微分方程的性态的研究,并易于说明其对初始条件敏感的混沌性,在此利用Mathematica繁育地此问题予以实现,给出产生ODEs的近似解析解的方法。 相似文献
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