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Functional Linear Regression Models. Application to High-throughput Plant Phenotyping Functional Data

Abstract : Functional data analysis (FDA) is a statistical branch that is increasingly being used in many applied scientific fields such as biological experimentation, finance, physics, etc. A reason for this is the use of new data collection technologies that increase the number of observations during a time interval. Functional datasets are realization samples of some random functions which are measurable functions defined on some probability space with values in an infinite dimensional functional space. There are many questions that FDA studies, among which functional linear regression is one of the most studied, both in applications and in methodological development. The objective of this thesis is the study of functional linear regression models when both the covariate X and the response Y are random functions and both of them are time-dependent. In particular we want to address the question of how the history of a random function X influences the current value of another random function Y at any given time t. In order to do this we are mainly interested in three models: the functional concurrent model (FCCM), the functional convolution model (FCVM) and the historical functional linear model. In particular for the FCVM and FCCM we have proposed estimators which are consistent, robust and which are faster to compute compared to others already proposed in the literature. Our estimation method in the FCCM extends the Ridge Regression method developed in the classical linear case to the functional data framework. We prove the probability convergence of this estimator, obtain a rate of convergence and develop an optimal selection procedure of the regularization parameter. The FCVM allows to study the influence of the history of X on Y in a simple way through the convolution. In this case we use the continuous Fourier transform operator to define an estimator of the functional coefficient. This operator transforms the convolution model into a FCCM associated in the frequency domain. The consistency and rate of convergence of the estimator are derived from the FCCM. The FCVM can be generalized to the historical functional linear model, which is itself a particular case of the fully functional linear model. Thanks to this we have used the Karhunen–Loève estimator of the historical kernel. The related question about the estimation of the covariance operator of the noise in the fully functional linear model is also treated. Finally we use all the aforementioned models to study the interaction between Vapour Pressure Deficit (VPD) and Leaf Elongation Rate (LER) curves. This kind of data is obtained with high-throughput plant phenotyping platform and is well suited to be studied with FDA methods.
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Submitted on : Monday, March 11, 2019 - 3:49:17 PM
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M Manrique. Functional Linear Regression Models. Application to High-throughput Plant Phenotyping Functional Data. Statistics [math.ST]. Université de Montpellier, 2016. English. ⟨tel-02063998⟩