Dynamical systems describe the changes in processes that arise naturally from their underlying physical principles, such as the laws of motion or the conservation of mass, energy or momentum. These models facilitate a causal explanation for the drivers and impediments of the processes. But do they describe the behaviour of the observed data? And how can we quantify the models' parameters that cannot be measured directly? This paper addresses these two questions by providing a methodology for estimating the solution; and the parameters of linear dynamical systems from incomplete and noisy observations of the processes. The proposed procedure builds on the parameter cascading approach, where a linear combination of basis functions approximates the implicitly defined solution of the dynamical system. The systems' parameters are then estimated so that this approximating solution adheres to the data. By taking advantage of the linearity of the system, we have simplified the parameter cascading estimation procedure, and by developing a new iterative scheme, we achieve fast and stable computation. We illustrate our approach by obtaining a linear differential equation that represents real data from biomechanics. Comparing our approach with popular methods for estimating the parameters of linear dynamical systems, namely, the non-linear least-squares approach, simulated annealing, parameter cascading and smooth functional tempering reveals a considerable reduction in computation and an improved bias and sampling variance.

中文翻译：

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线性动力系统的快速稳定参数估​​计

动力系统描述了由其基本物理原理自然产生的过程变化，例如运动定律或质量、能量或动量守恒。这些模型有助于对过程的驱动因素和障碍进行因果解释。但是它们是否描述了观察到的数据的行为？我们如何量化无法直接测量的模型参数？本文通过提供一种估计解决方案的方法来解决这两个问题；以及来自过程的不完整和嘈杂观察的线性动力系统的参数。所提出的程序建立在参数级联方法的基础上，其中基函数的线性组合近似于动态系统的隐式定义的解决方案。系统的 然后估计参数，以便该近似解符合数据。利用系统的线性，我们简化了参数级联估计过程，并通过开发新的迭代方案，实现了快速稳定的计算。我们通过获得代表生物力学真实数据的线性微分方程来说明我们的方法。将我们的方法与用于估计线性动力系统参数的流行方法（即非线性最小二乘法、模拟退火、参数级联和平滑函数调和）进行比较，发现计算量显着减少，偏差和采样方差有所改善。我们简化了参数级联估计过程，并通过开发新的迭代方案，实现了快速稳定的计算。我们通过获得代表生物力学真实数据的线性微分方程来说明我们的方法。将我们的方法与用于估计线性动力系统参数的流行方法（即非线性最小二乘法、模拟退火、参数级联和平滑函数调和）进行比较，发现计算量显着减少，偏差和采样方差有所改善。我们简化了参数级联估计过程，并通过开发新的迭代方案，实现了快速稳定的计算。我们通过获得代表生物力学真实数据的线性微分方程来说明我们的方法。将我们的方法与用于估计线性动力系统参数的流行方法（即非线性最小二乘法、模拟退火、参数级联和平滑函数调和）进行比较，发现计算量显着减少，偏差和采样方差有所改善。