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Abstract Multivariate functional data present theoretical and practical complications that are not found in univariate functional data. One of these is a situation where the component functions of multivariate functional data are positive and are subject to mutual time warping. That is, the component processes exhibit a common shape but are subject to systematic phase variation across their domains in addition to subject‐specific time warping, where each subject has its own internal clock. This motivates a novel model for multivariate functional data that connect such mutual time warping to a latent‐deformation‐based framework by exploiting a novel time‐warping separability assumption. This separability assumption allows for meaningful interpretation and dimension reduction. The resulting latent deformation model is shown to be well suited to represent commonly encountered functional vector data. The proposed approach combines a random amplitude factor for each component with population‐based registration across the components of a multivariate functional data vector and includes a latent population function, which corresponds to a common underlying trajectory. We propose estimators for all components of the model, enabling implementation of the proposed data‐based representation for multivariate functional data and downstream analyses such as Fréchet regression. Rates of convergence are established when curves are fully observed or observed with measurement error. The usefulness of the model, interpretations, and practical aspects are illustrated in simulations and with application to multivariate human growth curves and multivariate environmental pollution data.
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Mean and covariance estimation for discretely observed high-dimensional functional data: Rates of convergence and division of observational regimes
Estimation of the mean and covariance parameters for functional data is a critical task, with local linear smoothing being a popular choice. In recent years, many scientific domains are producing multivariate functional data for which $$p$$, the number of curves per subject, is often much larger than the sample size $$n$$. In this setting of high-dimensional functional data, much of developed methodology relies on preliminary estimates of the unknown mean functions and the auto- and cross-covariance functions. This paper investigates the convergence rates of local linear estimators in terms of the maximal error across components and pairs of components for mean and covariance functions, respectively, in both $L^2$ and uniform metrics. The local linear estimators utilize a generic weighting scheme that can adjust for differing numbers of discrete observations $$N_{ij}$$ across curves $$j$$ and subjects $$i$$, where the $$N_{ij}$$ vary with $$n$$. Particular attention is given to the equal weight per observation (OBS) and equal weight per subject (SUBJ) weighting schemes. The theoretical results utilize novel applications of concentration inequalities for functional data and demonstrate that, similar to univariate functional data, the order of the $$N_{ij}$$ relative to $$p$$ and $$n$$ divides high-dimensional functional data into three regimes (sparse, dense, and ultra-dense), with the high-dimensional parametric convergence rate of $$\left\{\log(p)/n\right\}^{1/2}$$ being attainable in the latter two.
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- Award ID(s):
- 2310943
- PAR ID:
- 10547929
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Journal of Multivariate Analysis
- Volume:
- 204
- Issue:
- C
- ISSN:
- 0047-259X
- Page Range / eLocation ID:
- 105355
- Subject(s) / Keyword(s):
- Concentration inequalities High-dimensional data L^2 convergence Local linear smoothing Uniform Convergence
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.jmva.2024.105355
