On Some Theoretical Aspects of Functional Data Analysis of Parametrized Curves in ℝp
Abstract
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 ℝ3, 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.
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BibTeX
@misc{colin2026some,
title = {{On Some Theoretical Aspects of Functional Data Analysis of Parametrized Curves in $\mathbbR^p$}},
author = {Bernard Colin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28689},
eprint = {2609.28689},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28689},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On Some Theoretical Aspects of Functional Data Analysis of Parametrized Curves in ℝ^p
AU - Bernard Colin
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28689
UR - https://arxiv.org/abs/2609.28689
AB - 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 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.
ER -