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A regularization method for the parameter estimation problem in ordinary differential equations via discrete optimal control theory
Journal of Statistical Planning and Inference ( IF 0.8 ) Pub Date : 2021-01-01 , DOI: 10.1016/j.jspi.2020.04.007
Quentin Clairon

We present a parameter estimation method in Ordinary Differential Equation (ODE) models. Due to complex relationships between parameters and states the use of standard techniques such as nonlinear least squares can lead to the presence of poorly identifiable parameters. Moreover, ODEs are generally approximations of the true process and the influence of misspecification on inference is often neglected. Methods based on control theory have emerged to regularize the ill posed problem of parameter estimation in this context. However, they are computationally intensive and rely on a nonparametric state estimator known to be biased in the sparse sample case. In this paper, we construct criteria based on discrete control theory which are computationally efficient and bypass the presmoothing step of signal estimation while retaining the benefits of control theory for estimation. We describe how the estimation problem can be turned into a control one and present the numerical methods used to solve it. We show convergence of our estimator in the parametric and well-specified case. For small sample sizes, numerical experiments with models containing poorly identifiable parameters and with various sources of model misspecification demonstrate the acurracy of our method. We finally test our approach on a real data example.

中文翻译:

基于离散最优控制理论的常微分方程参数估计问题的正则化方法

我们在常微分方程 (ODE) 模型中提出了一种参数估计方法。由于参数和状态之间的复杂关系,使用诸如非线性最小二乘法之类的标准技术会导致存在难以识别的参数。此外,ODE 通常是真实过程的近似值,错误指定对推理的影响通常被忽略。在这种情况下,出现了基于控制理论的方法来规范参数估计的不适定问题。然而,它们是计算密集型的,并且依赖于已知在稀疏样本情况下有偏差的非参数状态估计器。在本文中,我们构建基于离散控制理论的标准,这些标准在计算上是有效的,并且绕过了信号估计的预平滑步骤,同时保留了控制理论对估计的好处。我们描述了如何将估计问题转化为控制问题,并介绍用于解决该问题的数值方法。我们展示了我们的估计量在参数化和明确指定的情况下的收敛性。对于小样本量,使用包含难以识别参数的模型和各种模型错误指定来源的数值实验证明了我们方法的准确性。我们最终在一个真实的数据示例上测试了我们的方法。我们展示了我们的估计量在参数化和明确指定的情况下的收敛性。对于小样本量,使用包含难以识别参数的模型和各种模型错误指定来源的数值实验证明了我们方法的准确性。我们最终在一个真实的数据示例上测试了我们的方法。我们展示了我们的估计量在参数化和明确指定的情况下的收敛性。对于小样本量,使用包含难以识别参数的模型和各种模型错误指定来源的数值实验证明了我们方法的准确性。我们最终在一个真实的数据示例上测试了我们的方法。
更新日期:2021-01-01
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