arXiv subjects

Angelina Roche

Publications and source records attributed to Angelina Roche.

10 recordsLinked to original sources

arXiv2026

Minimax prediction in the Functional Autoregressive Model

The Functional Autoregressive Model (FAR) generalizes the multivariate AR(1) model in Time Series Analysis to functional data. It serves as a historical foundational point in the study of functional time series and remains a fundamental and widely used model for dependent functional data. The process-the observed data-generated by the FAR model forms a Hilbert-valued Markov chain. This paper investigates the non-asymptotic prediction mean square error and derives a lower bound. This lower bound is established in the specific context of non-i.i.d. data and depends on the mixed smoothness of the functional time series and of the unknown correlation operator driving the FAR model. Instead of the standard functional PCA regularization, a ridge-type estimator is proposed, which avoids the preliminary estimation of the spectrum of the covariance sequence associated with the process. A non-asymptotic upper bound is derived for this estimate, which matches the lower bound up to multiplicative constants. Furthermore, a detailed study of the estimate's bias reveals connections between functional smoothness parameters and regularly/rapidly varying functions, which are common in extreme value theory. Simulation results corroborate the theoretical main theorems.

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arXiv2026

Minimax estimation of Functional Principal Components from noisy discretized functional data: the case of smooth processes

We study the minimax estimation of covariance eigenfunctions and eigenvalues in functional principal component analysis when $n$ trajectories are observed at $p$ common grid points with additive noise. We consider covariance kernels with arbitrary H\"older smoothness and no prescribed parametric decay of the eigenvalues. In this setting, kernel smoothness and local spectral separation play distinct roles: a minimax inconsistency result over the smoothness-only class shows that kernel regularity alone is not sufficient for minimax-consistent eigenfunction estimation. To capture this interplay, we introduce a class of processes that jointly controls the H\"older smoothness of the covariance kernel and a local relative inverse eigengap quantity at the target index $\ell$. Over this class, we derive non-asymptotic minimax lower bounds for eigenfunction estimation that disentangle sampling variability, discretization and spectral effects, revealing rates of order $\delta_\ell n^{-1}+p^{-2\alpha}$, where $\delta_\ell$ quantifies the spectral difficulty. We also obtain non-asymptotic lower bounds for eigenvalue estimation under a relative squared-error loss. We then construct a computable wavelet projection estimator based on Coiflet scaling functions and a quadrature scheme designed to accommodate arbitrary H\"older smoothness. For eigenfunction estimation, this estimator matches the minimax dependence on the sample size and grid resolution, up to the natural spectral factor, for any H\"older index $\alpha>0$. Finally, we show that the proposed framework covers several classical Gaussian processes and Karhunen--Lo\`eve constructions. In particular, a Karhunen--Lo\`eve based criterion links spectral decay, eigenfunction regularity and covariance-kernel smoothness, and yields controlled simulation settings illustrating the predicted phase transitions and least-favourable discretization effects.

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arXiv2025

PCA for Point Processes

We introduce a novel statistical framework for the analysis of replicated point processes that allows for the study of point pattern variability at a population level. By treating point process realizations as random measures, we adopt a functional analysis perspective and propose a form of functional Principal Component Analysis (fPCA) for point processes. The originality of our method is to base our analysis on the cumulative mass functions of the random measures which gives us a direct and interpretable analysis. Key theoretical contributions include establishing a Karhunen-Loève expansion for the random measures and a Mercer Theorem for covariance measures. We establish convergence in a strong sense, and introduce the concept of principal measures, which can be seen as latent processes governing the dynamics of the observed point patterns. We propose an easy-to-implement estimation strategy of eigenelements for which parametric rates are achieved. We fully characterize the solutions of our approach to Poisson and Hawkes processes and validate our methodology via simulations and diverse applications in seismology, single-cell biology and neurosiences, demonstrating its versatility and effectiveness. Our method is implemented in the pppca R-package.

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arXiv2025

Minimax estimation of Functional Principal Components from noisy discretized functional data

Functional Principal Component Analysis is a reference method for dimension reduction of curve data. Its theoretical properties are now well understood in the simplified case where the sample curves are fully observed without noise. However, functional data are noisy and necessarily observed on a finite discretization grid. Common practice consists in smoothing the data and then to compute the functional estimates, but the impact of this denoising step on the procedure's statistical performance are rarely considered. Here we prove new convergence rates for functional principal component estimators. We introduce a double asymptotic framework: one corresponding to the sampling size and a second to the size of the grid. We prove that estimates based on projection onto histograms show optimal rates in a minimax sense. Theoretical results are illustrated on simulated data and the method is applied to the visualization of genomic data.

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arXiv2023

Lasso in infinite dimension: application to variable selection in functional multivariate linear regression

It is more and more frequently the case in applications that the data we observe come from one or more random variables taking values in an infinite dimensional space, e.g. curves. The need to have tools adapted to the nature of these data explains the growing interest in the field of functional data analysis. The model we study in this paper assumes a linear dependence between a quantity of interest and several covariates, at least one of which has an infinite dimension. To select the relevant covariates in this context, we investigate adaptations of the Lasso method. Two estimation methods are defined. The first one consists in the minimization of a Group-Lasso criterion on the multivariate functional space H. The second one minimizes the same criterion but on a finite dimensional subspaces of H whose dimension is chosen by a penalized least squares method. We prove oracle inequalities of sparsity in the case where the design is fixed or random. To compute the solutions of both criteria in practice, we propose a coordinate descent algorithm. A numerical study on simulated and real data illustrates the behavior of the estimators.

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arXiv2022

Adaptive nonparametric estimation in the functional linear model with functional output

In this paper, we consider a functional linear regression model, where both the covariate and the response variable are functional random variables. We address the problem of optimal nonparametric estimation of the conditional expectation operator in this model. A collection of projection estimators over finite dimensional subspaces is first introduce. We provide a non-asymptotic bias-variance decomposition for the Mean Square Prediction error in the case where these subspaces are generated by the (empirical) PCA functional basis. The automatic trade-off is realized thanks to a model selection device which selects the best projection dimensions: the penalized contrast estimator satisfies an oracle-type inequality and is thus optimal in an adaptive point of view. These upper-bounds allow us to derive convergence rates over ellipsoidal smoothness spaces. The rates are shown to be optimal in the minimax sense: they match with a lower bound of the minimax risk, which is also proved. Finally, we conduct a numerical study, over simulated data and over two real-data sets.

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arXiv2021

Adaptive nonparametric estimation of a component density in a two-class mixture model

A two-class mixture model, where the density of one of the components is known, is considered. We address the issue of the nonparametric adaptive estimation of the unknown probability density of the second component. We propose a randomly weighted kernel estimator with a fully data-driven bandwidth selection method, in the spirit of the Goldenshluger and Lepski method. An oracle-type inequality for the pointwise quadratic risk is derived as well as convergence rates over Holder smoothness classes. The theoretical results are illustrated by numerical simulations.

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arXiv2017

Local bandwidth selection for kernel density estimation in bifurcating Markov chain model

We propose an adaptive estimator for the stationary distribution of a bifurcating Markov Chain on $\mathbb R^d$. Bifurcating Markov chains (BMC for short) are a class of stochastic processes indexed by regular binary trees. A kernel estimator is proposed whose bandwidth is selected by a method inspired by the works of Goldenshluger and Lepski [18]. Drawing inspiration from dimension jump methods for model selection, we also provide an algorithm to select the best constant in the penalty.

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arXiv2015

Local Optimization of Black-Box Function with High or Infinite-Dimensional Inputs

An adaptation of Response Surface Methodology (RSM) when the covariate is of high or infinite dimensional is proposed, providing a tool for black-box optimization in this context. We combine dimension reduction techniques with classical multivariate Design of Experiments (DoE). We propose a method to generate experimental designs and extend usual properties (orthogonality, rotatability,...) of multivariate designs to general high or infinite dimensional contexts. Different dimension reduction basis are considered (including data-driven basis). The methodology is illustrated on simulated functional data and we discuss the choice of the different parameters, in particular the dimension of the approximation space. The method is finally applied to a problem of nuclear safety.

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arXiv2013

Non-asymptotic Adaptive Prediction in Functional Linear Models

Functional linear regression has recently attracted considerable interest. Many works focus on asymptotic inference. In this paper we consider in a non asymptotic framework a simple estimation procedure based on functional Principal Regression. It revolves in the minimization of a least square contrast coupled with a classical projection on the space spanned by the m first empirical eigenvectors of the covariance operator of the functional sample. The novelty of our approach is to select automatically the crucial dimension m by minimization of a penalized least square contrast. Our method is based on model selection tools. Yet, since this kind of methods consists usually in projecting onto known non-random spaces, we need to adapt it to empirical eigenbasis made of data-dependent - hence random - vectors. The resulting estimator is fully adaptive and is shown to verify an oracle inequality for the risk associated to the prediction error and to attain optimal minimax rates of convergence over a certain class of ellipsoids. Our strategy of model selection is finally compared numerically with cross-validation.

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