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Nonparametric Functional Graphical Modeling Through Functional Additive regression operator

JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION 2023年第543期118卷 1718-1732页

作者： Lee, Kuang-Yao Li, Lexin Li, Bing Zhao, Hongyu Temple Univ Philadelphia PA 19122 USA Univ Calif Berkeley Berkeley CA 94720 USA Penn State Univ University Pk PA 16802 USA Yale Univ New Haven CT USA

In this article, we develop a nonparametric graphical model for multivariate random functions. Most existing graphical models are restricted by the assumptions of multivariate Gaussian or copula Gaussian distributions, which also imply linear relations among the random variables or functions on different nodes. We relax those assumptions by building our graphical model based on a new statistical object-the functional additive regression operator. By carrying out regression and neighborhood selection at the operator level, our method can capture nonlinear relations without requiring any distributional assumptions. Moreover, the method is built up using only one-dimensional kernel, thus, avoids the curse of dimensionality from which a fully nonparametric approach often suffers, and enables us to work with large-scale networks. We derive error bounds for the estimated regression operator and establish graph estimation consistency, while allowing the number of functions to diverge at the exponential rate of the sample size. We demonstrate the efficacy of our method by both simulations and analysis of an electroencephalography dataset. Supplementary materials for this article are available online.

关键词： Additive conditional independence Brain connectivity analysis Graphical model Neuroimaging analysis regression operator Sparsistency

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Estimation in nonparametric functional-on-functional models with surrogate responses

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JOURNAL OF MULTIVARIATE ANALYSIS 2023年第1期198卷

作者： Boumahdi, Mounir Ouassou, Idir Rachdi, Mustapha Univ Cadi Ayyad Ecole Natl Sci Appl Marrakech Morocco Univ Grenoble Alpes Grenoble France

We construct an estimator for the regression operator of a functional response variable using surrogate data, given a functional random variable. The almost complete uniform convergence rate of the estimator is then established. Finally, to demonstrate the predictive utility and superiority of the estimator when dealing with incomplete data, we apply the methodology to both simulated data and meteorological data. & COPY;2023 Elsevier Inc. All rights reserved.

关键词： Almost complete convergence Entropy Functional data analysis Kernel estimator regression operator Semi-metric space Surrogate response

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k-Nearest neighbors local linear regression for functional and missing data at random

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STATISTICA NEERLANDICA 2021年第1期75卷 42-65页

作者： Rachdi, Mustapha Laksaci, Ali Kaid, Zoulikha Benchiha, Abbassia Al-Awadhi, Fahimah A. Univ Grenoble Alpes UFR SHS AGIM Team Lab AGEIS EA 7407 Bat Michel DuboisBP 47 F-38040 Grenoble 09 France King Khalid Univ Coll Sci Dept Math Abha Saudi Arabia Univ Djillali Liabes Sidi Bel Abbes Lab Stat & Proc Stochast Sidi Bel Abbes Algeria Kuwait Univ Fac Sci Dept Stat & Operat Res Safat Kuwait

We combine thek-Nearest Neighbors (kNN) method to the local linear estimation (LLE) approach to construct a new estimator (LLE-kNN) of the regression operator when the regressor is of functional type and the response variable is a scalar but observed with some missing at random (MAR) observations. The resulting estimator inherits many of the advantages of both approaches (kNN and LLE methods). This is confirmed by the established asymptotic results, in terms of the pointwise and uniform almost complete consistencies, and the precise convergence rates. In addition, a numerical study (i) on simulated data, then (ii) on a real dataset concerning the sugar quality using fluorescence data, were conducted. This practical study clearly shows the feasibility and the superiority of the LLE-kNN estimator compared to competitive estimators.

关键词： k-Nearest Neighbors method almost complete convergence functional data analysis local linear method missing data regression operator uniform consistency

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Local linear regression modelization when all variables are curves

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STATISTICS & PROBABILITY LETTERS 2017年 121卷 37-44页

作者： Demongeot, Jacques Naceri, Amina Laksaci, Ali Rachdi, Mustapha Univ Grenoble Alpes AGIM Team Lab AGEIS EA 7407 Fac Med Grenoble F-38700 La Tronche France Univ Djillali Liabes Sidi Bel Abbes Lab Stat & Proc Stochast BP 89 Sidi Bel Abbes 22000 Algeria Univ Grenoble Alpes UFR SHS AGIM Team Lab AGEIS EA 7407 BP 47 F-38040 Grenoble 09 France

A nonparametric local linear estimator of the regression function when both the response and the explanatory variables are of the functional kind is constructed. Then its rate of uniform almost-complete convergence is stated. This theoretical result will be a key tool for many further developments in nonparametric functional data analysis (FDA). (C) 2016 Elsevier B.V. All rights reserved.

关键词： Functional data analysis regression operator Local linear estimation Strong consistency Functional nonparametric statistics Functional response

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Relative-error prediction in nonparametric functional statistics: Theory and practice

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JOURNAL OF MULTIVARIATE ANALYSIS 2016年 146卷 261-268页

作者： Demongeot, Jacques Hamie, Ali Laksaci, Ali Rachdi, Mustapha Univ Grenoble 1 Fac Med Grenoble Univ Grenoble Alpes Lab AGIMFRE 3405CNRS F-38700 La Tronche France Univ Djillali Liabes Sidi Bel Abbes Lab Stat & Proc Stochast BP 89 Sidi Bel Abbes 22000 Algeria Univ Grenoble Alpes Lab AGIM FRE 3405 CNRSUPMFUFR SHS BP 47 F-38040 Grenoble 09 France

In this paper, an alternative kernel estimator of the regression operator of a scalar response variable Y given a random variable X taking values in a semi-metric space is considered. The constructed estimator is based on the minimization of the mean squared relative error. This technique is useful in analyzing data with positive responses, such as stock prices or life times. Least squares or least absolute deviation are among the most widely used criteria in statistical estimation for regression models. However, in many practical applications, especially in treating, for example, the stock price data, the size of the relative error rather than that of the error itself, is the central concern of the practitioners. This paper offers then an alternative to traditional estimation methods by considering the minimization of the least absolute relative error for operatorial regression models. We prove the strong and the uniform consistencies (with rates) of the constructed estimator. Moreover, the mean squared convergence rate is given and the asymptotic normality of the proposed estimator is proved. Finally, supportive evidence is shown by simulation studies and an application on some economic data was performed. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Mean square relative error Nonparametric estimation Functional data regression operator Positive responses Asymptotic normality Small ball property Stock price Economic data

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Variable selection via additive conditional independence

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JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY 2016年第5期78卷 1037-1055页

作者： Lee, Kuang-Yao Li, Bing Zhao, Hongyu Yale Univ New Haven CT USA Penn State Univ University Pk PA 16802 USA

We propose a non-parametric variable selection method which does not rely on any regression model or predictor distribution. The method is based on a new statistical relationship, called additive conditional independence, that has been introduced recently for graphical models. Unlike most existing variable selection methods, which target the mean of the response, the method proposed targets a set of attributes of the response, such as its mean, variance or entire distribution. In addition, the additive nature of this approach offers non-parametric flexibility without employing multi-dimensional kernels. As a result it retains high accuracy for high dimensional predictors. We establish estimation consistency, convergence rate and variable selection consistency of the method proposed. Through simulation comparisons we demonstrate that the method proposed performs better than existing methods when the predictor affects several attributes of the response, and it performs competently in the classical setting where the predictors affect the mean only. We apply the new method to a data set concerning how gene expression levels affect the weight of mice.

关键词： Additive covariance operator Heterogeneity Lasso regression operator Reproducing kernel Hilbert space Sparsity Variable selection consistency

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Modified kernel regression estimation with functional time series data

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STATISTICS & PROBABILITY LETTERS 2016年 114卷 78-85页

作者： Ling, Nengxiang Wang, Chao Ling, Jin Hefei Univ Technol Sch Math Hefei 230009 Peoples R China Hefei Univ Technol Sch Elect & Automat Engn Hefei 230009 PR Peoples R China

This paper derives the asymptotic distribution of a modified kernel regression estimator for strong mixing functional time series data. As a direct consequence, the approximate pointwise confidence interval of regress... 详细信息

关键词： regression operator Modified kernel estimator Functional time series data Asymptotic normality

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Nonparametric regression estimation for functional stationary ergodic data with missing at random

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JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2015年 162卷 75-87页

作者： Ling, Nengxiang Liang, Longlong Vieu, Philippe Hefei Univ Technol Sch Math Hefei 230009 Peoples R China Univ Toulouse 3 Inst Math F-31062 Toulouse France

In this paper, we investigate the asymptotic properties of the estimator for the regression function operator whenever the functional stationary ergodic data with missing at random (MAR) are considered. Concretely, we construct the kernel type estimator of the regression operator for functional stationary ergodic data with the responses MAR, and some asymptotic properties such as the convergence rate in probability as well as the asymptotic normality of the estimator are obtained under some mild conditions respectively. As an application, the asymptotic (1-zeta) confidence interval of the regression operator is also presented for 0 < zeta< 1. Finally, a simulation study is carried out to compare the finite sample performance based on mean square error between the classical functional regression in complete case and the functional regression with MAR. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Missing at random Functional data analysis Convergence in probability Asymptotic normality Ergodic processes regression operator

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Local smoothing regression with functional data

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COMPUTATIONAL STATISTICS 2007年第3期22卷 353-369页

作者： Benhenni, K. Ferraty, F. Rachdi, M. Vieu, P. Univ Grenoble LJK UMR CNRS 5224 UFR SHS F-38040 Grenoble 09 France Univ Toulouse 3 CNRS LSP UMR 5583 F-31062 Toulouse France

Kernel estimates of a regression operator are investigated when the explanatory variable is of functional type. The bandwidths are locally chosen by a data-driven method based on the minimization of a functional version of a cross-validated criterion. A short asymptotic theoretical support is provided and the main body of this paper is devoted to various finite sample size applications. In particular, it is shown through some simulations, that a local bandwidth choice enables to capture some underlying heterogeneous structures in the functional dataset. As a consequence, the estimation of the relationship between a functional variable and a scalar response, and hence the prediction, can be significantly improved by using local smoothing parameter selection rather than global one. This is also confirmed from a chemometrical real functional dataset. These improvements are much more important than in standard finite dimensional setting.

关键词： cross-validation functional data local versus global bandwidths regression operator

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