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...
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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.
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 e...
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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.
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 ...
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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.
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...
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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.
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 base...
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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.
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 independe...
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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.
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...
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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 regression operator is presented. (C) 2016 Elsevier B.V. All rights reserved.
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...
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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.
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 versi...
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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.
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