发表机构
Université de Sherbrooke（舍布鲁克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文综述了函数型数据分析在参数化曲线上的理论与应用，通过希尔伯特空间建模、平滑惩罚及函数型主成分分析，突出其捕捉微分几何特征的几何优势。
AI 中文摘要
本文提供了函数型数据分析（FDA）在 $\mathbb{R}^p$ 中参数化曲线上的全面数学与实践综述。我们观察到，依赖于连续参数的曲线自然存在于许多领域，例如儿科中监测儿童发育、评估气象现象、分析金融投资组合、跟踪神经功能以及绘制地理污染水平。我们考虑一个由实可分希尔伯特空间的笛卡尔积给出的形式理论框架，以在数学上对这些曲线进行建模。为了弥合带有误差的离散实验测量与光滑连续函数之间的差距，本文详细阐述了数据平滑与拟合的关键阶段，并通过普通最小二乘或惩罚最小二乘准则设定其数学公式；对于后者，利用索伯列夫空间框架，引入平滑参数 ${\lambda}$ 和微分算子，通过惩罚过度的曲线粗糙度来防止不规则的几何行为。此外，我们将通常的主成分分析扩展为其在 $\mathbb{R}^3$ 中的函数型对应物，使用变分法和欧拉-拉格朗日乘子定理来寻找协方差算子的特征函数和特征值，以展示空间方差的最优分解；最后，我们强调FDA相对于传统多元数据分析的几何优势，突出其捕捉速度、加速度、曲率和挠率等关键微分特征的独特能力。
英文摘要
In this paper, one provides a comprehensive mathematical and practical overview of Functional Data Analysis (FDA) specifically applied to parametrized curves in Rp. One observes that curves depending on continuously parameter are naturally present across many fields, such as, for example, monitoring child development in pediatrics, assessing meteorological phenomena, analyzing financial portfolios, tracking neurological functions, and mapping geographic pollution levels. One considers a formal theoretical framework given by a Cartesian product of real separable Hilbert spaces, to model these curves mathematically. To bridge the gap between discrete experimental measurements subject to errors, and smooth continuous functions, the paper details the essential phase of data smoothing and fitting and sets its mathematical formulation via of the Ordinary or Penalized Least Squares criteria which, for the later, using the Sobolev spaces framework, incorporates a smoothing parameter $λ$ and differential operators to prevent erratic geometric behaviors by penalizing excessive curve roughness. Furthermore, one extends the usual Principal Component Analysis to its functional counterpart in $\mathbb{R}^3$, using the calculus of variations and the Euler-Lagrange multiplier theorem in order to find eigenfunctions and eigenvalues of the covariance operator to exhibit the optimal decomposition of the spatial variance and finally, one highlights the geometric advantages of FDA over traditional multivariate data analysis, emphasizing its unique capacity to capture critical differential features as velocity, acceleration, curvature, and torsion.
