In functional data analysis, replicate observations of a smooth functional process and its derivatives offer a unique opportunity to flexibly estimate continuous-time ordinary differential equation models. Ramsay (1996) first proposed to estimate a linear ordinary differential equation from functional data in a technique called Principal Differential Analysis, by formulating a functional regression in which the highest-order derivative of a function is modelled as a time-varying linear combination of its lower-order derivatives. Principal Differential Analysis was introduced as a technique for data reduction and representation, using solutions of the estimated differential equation as a basis to represent the functional data. In this work, we re-formulate PDA as a generative statistical model in which functional observations arise as solutions of a deterministic ODE that is forced by a smooth random error process. This viewpoint defines a flexible class of functional models based on differential equations and leads to an improved understanding and characterisation of the sources of variability in Principal Differential Analysis. It does, however, result in parameter estimates that can be heavily biased under the standard estimation approach of PDA. Therefore, we introduce an iterative bias-reduction algorithm that can be applied to improve parameter estimates. We also examine the utility of our approach when the form of the deterministic part of the differential equation is unknown and possibly non-linear, where Principal Differential Analysis is treated as an approximate model based on time-varying linearisation. We demonstrate our approach on simulated data from linear and non-linear differential equations and on real data from human movement biomechanics. Supplementary R code for this manuscript is available at \url{https://github.com/edwardgunning/UnderstandingOfPDAManuscript}.

翻译：在函数数据分析中，对光滑函数过程及其导数的重复观测为灵活估计连续时间常微分方程模型提供了独特机遇。Ramsay（1996）首次提出通过函数回归框架——将函数的最高阶导数建模为其低阶导数的时变线性组合——从函数数据中估计线性常微分方程，该技术被称为主微分分析。主微分分析最初是作为数据降维与表示技术提出的，其使用估计微分方程的解作为基函数来表示函数数据。在本研究中，我们将PDA重新构建为生成式统计模型，其中函数观测值被视作由光滑随机误差过程驱动的确定性ODE的解。这一视角定义了基于微分方程的灵活函数模型类，并深化了对主微分分析中变异来源的理解与刻画。然而，在PDA标准估计方法下，该模型可能导致参数估计存在严重偏差。为此，我们提出了一种可应用于改进参数估计的迭代偏差缩减算法。我们还探讨了当微分方程确定性部分形式未知且可能为非线性的情形下本方法的实用性，此时主微分分析被视为基于时变线性化的近似模型。我们通过线性与非线性微分方程的模拟数据以及人体运动生物力学的真实数据验证了所提方法。本文的补充R代码可在 \url{https://github.com/edwardgunning/UnderstandingOfPDAManuscript} 获取。