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Thresholding projection estimators in functional linear models
"... We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule allows to get consistency under broad assumptions as well as ..."
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We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule allows to get consistency under broad assumptions as well as minimax rates of convergence under additional regularity hypotheses. We also consider the particular case of Sobolev spaces generated by the trigonometric basis which permits to get easily estimators of the derivatives of the regression function and prove they are minimax. Rates of convergence are given for some particular cases.
Single and multiple index functional regression models with nonparametric link Annals of Statistics 39
 Probability Theory: Independence, Interchangeability, Martingales (3rd ed
, 2011
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Asymptotic equivalence of functional linear regression and a white noise inverse problem
 Ann. Statist
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Continuously additive models for nonlinear functional regression
 Biometrika
, 2012
"... We introduce continuously additive models, which can be motivated as extensions of additive regression models with vector predictors to the case of infinitedimensional predictors. This approach provides a class of flexible functional nonlinear regression models, where random predictor curves are c ..."
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We introduce continuously additive models, which can be motivated as extensions of additive regression models with vector predictors to the case of infinitedimensional predictors. This approach provides a class of flexible functional nonlinear regression models, where random predictor curves are coupled with scalar responses. In continuously additive modeling, integrals taken over a smooth surface along graphs of predictor functions relate the predictors to the responses in a nonlinear fashion. We use tensor product basis expansions to fit the smooth regression surface that characterizes the model. In a theoretical investigation, we show that the predictions obtained from fitting continuously additive models are consistent and asymptotically normal. We also consider extensions to generalized responses. The proposed approach outperforms existing functional regression models in simulations and data illustrations.
Local linear regression for functional data, submitted
, 2008
"... We study a non linear regression model with functional data as inputs and scalar response. We propose a pointwise estimate of the regression function that maps a Hilbert space onto the real line by a local linear method. We provide the asymptotic mean square error. Computations involve a linear inve ..."
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Cited by 4 (2 self)
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We study a non linear regression model with functional data as inputs and scalar response. We propose a pointwise estimate of the regression function that maps a Hilbert space onto the real line by a local linear method. We provide the asymptotic mean square error. Computations involve a linear inverse problem as well as a representation of the small ball probability of the data and are based on recent advances in this area. The rate of convergence of our estimate outperforms those already obtained in the literature on this model.
Using basis expansions for estimating functional PLS regression. Applications with chemometric data
, 2010
"... There are many chemometric applications, such as spectroscopy, where the objective is to explain a scalar response from a functional variable (the spectrum) whose observations are functions of wavelengths rather than vectors. In this paper, PLS regression is considered for estimating the linear mode ..."
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There are many chemometric applications, such as spectroscopy, where the objective is to explain a scalar response from a functional variable (the spectrum) whose observations are functions of wavelengths rather than vectors. In this paper, PLS regression is considered for estimating the linear model when the predictor is a functional random variable. Due to the infinite dimension of the space to which the predictor observations belong, they are usually approximated by curves/functions within a finite dimensional space spanned by a basis of functions. We show that PLS regression with a functional predictor is equivalent to finite multivariate PLS regression using expansion basis coefficients as the predictor, in the sense that, at each step of the PLS iteration, the same prediction is obtained. In addition, from the linear model estimated using the basis coefficients, we derive the expression of the PLS estimate of the regression coefficient function from the model with a functional predictor. The results provided by this functional PLS approach are compared with those given by functional PCR and discrete PLS and PCR using different sets of simulated and spectrometric data.
Minimax adaptive tests for the Functional Linear model
"... Abstract:. We introduce two novel procedures to test the nullity of the slope function in the functional linear model with real output. The test statistics combine multiple testing ideas and random projections of the input data through functional Principal Component Analysis. Interestingly, the proc ..."
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Abstract:. We introduce two novel procedures to test the nullity of the slope function in the functional linear model with real output. The test statistics combine multiple testing ideas and random projections of the input data through functional Principal Component Analysis. Interestingly, the procedures are completely datadriven and do not require any prior knowledge on the smoothness of the slope nor on the smoothness of the covariate functions. The levels and powers against local alternatives are assessed in a nonasymptotic setting. This allows us to prove that these procedures are minimax adaptive (up to an unavoidable log log n multiplicative term) to the unknown regularity of the slope. As a side result, the minimax separation distances of the slope are derived for a large range of regularity classes. A numerical study illustrates these theoretical results.
Continuous timevarying kriging for spatial prediction of functional data: An environmental application
"... Spatially correlated functional data is present in a wide range of environmental disciplines and, in this context, efficient prediction of curves is a key issue. We present an approach for spatial prediction based on the functional linear pointwise model adapted to the case of spatially correlated ..."
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Spatially correlated functional data is present in a wide range of environmental disciplines and, in this context, efficient prediction of curves is a key issue. We present an approach for spatial prediction based on the functional linear pointwise model adapted to the case of spatially correlated curves. First, a smoothing process is applied to the curves by expanding the curves and the functional parameters in terms of a set of Fourier basis functions. The number of basis functions is chosen by crossvalidation. Then, the spatial prediction of a curve is obtained as a pointwise linear combination of the smoothed data. The prediction problem is solved by estimating a linear model of coregionalization to set the spatial dependence among the fitted coefficients. We extend an optimization criterion used in multivariable geostatistics to the functional context. The method is illustrated by smoothing and predicting temperature curves measured at 35 Canadian weather stations. ∗Corresponding author.