Skip to Main content Skip to Navigation
Theses

Statistical study of functional principal component analysis in uni and multivariate frameworks

Abstract : This work is the concatenation of two parts, having for point how to relate to the analysis of functional data and in particular to be interested in the questions related to the high dimension in this context. The first part concerns functional principal component analysis in the univariate case. Our approach aims to give non-asymptotic results for different projection estimators of the eigenelements of a covariance operator. We first define an estimator based on a projection operator. This operator can be seen as a raw data reconstruction step in the context of functional data analysis. We show that the naive estimator, which computes the eigen elements without regularization after the projection step, is optimal in the minimax sense for a good basis choice. To this end, we set both a lower and an upper bound on the root mean square error of eigenelement reconstruction. We also prove general results for general Lipschitz and Daubechies bases which do not achieve minimax rates. In the case of Daubechies, thresholding is necessary to reach its optimal level. This part is concluded by numerical simulations which confirm the acuity of the approach and an application to genomic data. The second part concerns the generalization of the model to the multivariate functional case. As in the first part, our approach aims to give non-asymptotic results for the estimation of the first principal component of a multivariate random process. We first define the covariance function and the covariance operator in the multivariate case. We then define a projection operator. This operator can be seen as a reconstruction step from raw data in the context of functional data analysis. Next, we show that eigen-elements can be expressed as the solution of an optimization problem, and we introduce the LASSO variant of this optimization problem and the associated plugin estimator. Finally, we evaluate the precision of the estimator. We establish a minimax lower bound on the mean square error of reconstruction of the proper element, which proves that the procedure has an optimal variance in the minimax sense
Document type :
Theses
Complete list of metadata

https://hal.archives-ouvertes.fr/tel-03704410
Contributor : ABES STAR :  Contact
Submitted on : Wednesday, July 20, 2022 - 12:30:35 AM
Last modification on : Wednesday, July 20, 2022 - 3:46:30 AM

File

2022UPSLD006.pdf
Version validated by the jury (STAR)

Identifiers

  • HAL Id : tel-03704410, version 2

Collections

Citation

Ryad Belhakem. Statistical study of functional principal component analysis in uni and multivariate frameworks. Functional Analysis [math.FA]. Université Paris sciences et lettres, 2022. English. ⟨NNT : 2022UPSLD006⟩. ⟨tel-03704410v2⟩

Share

Metrics

Record views

79

Files downloads

36