Robust nonparametric estimation for functional data

Crambes, Christophe; Delsol, Laurent; Laksaci, Ali

doi:10.1080/10485250802331524

Cited by 37 publications

(15 citation statements)

References 16 publications

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“…They can appear as outlying measurements at a single or several time points, or as an outlying shape of an entire function. Current approaches to deal with outliers and contamination, and visual exploration of functional data more generally, include exploratory box plots (Hyndman & Shang 2010, Sun & Genton 2011 and robust versions of FPCA (Crambes et al 2008, Gervini 2008, Bali et al 2011, Kraus & Panaretos 2012, Boente & Salibián-Barrera 2014. Owing to the practical importance of this topic, more research on outlier detection and robust FDA approaches is needed.…”

Section: Functional Principal Component Analysismentioning

confidence: 99%

Functional Data Analysis

Wang

Chiou

Müller

2016

Annu. Rev. Stat. Appl.

750

450

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With the advance of modern technology, more and more data are being recorded continuously during a time interval or intermittently at several discrete time points. These are both examples of functional data, which has become a commonly encountered type of data. Functional data analysis (FDA) encompasses the statistical methodology for such data. Broadly interpreted, FDA deals with the analysis and theory of data that are in the form of functions. This paper provides an overview of FDA, starting with simple statistical notions such as mean and covariance functions, then covering some core techniques, the most popular of which is functional principal component analysis (FPCA). FPCA is an important dimension reduction tool, and in sparse data situations it can be used to impute functional data that are sparsely observed. Other dimension reduction approaches are also discussed. In addition, we review another core technique, functional linear regression, as well as clustering and classification of functional data. Beyond linear and single-or multiple-index methods, we touch upon a few nonlinear approaches that are promising for certain applications. They include additive and other nonlinear functional regression models and models that feature time warping, manifold learning, and empirical differential equations. The paper concludes with a brief discussion of future directions.Annu. Rev. Stat. Appl. 2016.3:257-295. Downloaded from www.annualreviews.org Access provided by Academia Sinica -Life Science Library on 06/04/16. For personal use only. Section 2, and several approaches for dimension reduction in functional regression are discussed in Section 3. Clustering and classification of functional data are useful and important tools with wide-ranging applications in FDA. Methods include extensions of classical k-means and hierarchical clustering, Bayesian and model approaches to clustering, and classification via functional regression and functional discriminant analysis. These topics are explored in Section 4. The classical methods for FDA have been predominantly linear, such as functional principal components (FPCs) or the FLM. As more and more functional data are generated, it has emerged that many such data have inherent nonlinear features that make linear methods less effective. Sections 5 reviews some nonlinear approaches to FDA, including time warping, nonlinear manifold modeling, and nonlinear differential equations to model the underlying empirical dynamics.A well-known and well-studied nonlinear effect is time warping, where in addition to the common amplitude variation, one also considers time variation. This creates a basic nonidentifiability problem, and Section 5.1 provides a discussion of these foundational issues. A more general approach to modeling nonlinearity in functional data is to assume that the functional data lie on www.annualreviews.org • Review of Functional Data Analysis 259

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Section: Functional Principal Component Analysismentioning

confidence: 99%

Functional Data Analysis

Wang

Chiou

Müller

2016

Annu. Rev. Stat. Appl.

750

450

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“…These results are extended to the dependent case by Attouch, Laksaci, and Ould Saïd (submitted for publication). Crambes, Delsol, and Laksaci (2008) stated the convergence in the L q norm in both cases (i.i.d. and strong mixing).…”

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confidence: 98%

Asymptotic normality of a robust estimator of the regression function for functional time series data

Attouch

Laksaci

Saïd

2010

Journal of the Korean Statistical Society

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a b s t r a c tWe propose a family of robust nonparametric estimators for a regression function based on the kernel method. We establish the asymptotic normality of the estimator under the concentration property on small balls probability measure of the functional explanatory variable when the observations exhibit some kind of dependence. This approach can be applied in time series analysis to make prediction and build confidence bands. We illustrate our methodology on the US electricity consumption data.

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“…The nonparametric M ‐estimation for functional data has been theoretically studied by Crambes, Delsol, and Laksaci () and J. Chen and Zhang () for strongly mixing data and extended to general ergodic data by Xiong and Lin (). Similarly to many results cited in Section 2.1, these results were obtained under the assumption that ψ (·) in (14) is a known function, which does not depend on unknown parameters such as scale.…”

Section: Other Nonparametric Regression Modelsmentioning

confidence: 99%

Robust nonparametric regression: A review

Čı́žek

Sadikoglu

2019

WIREs Computational Stats

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Nonparametric regression methods provide an alternative approach to parametric estimation that requires only weak identification assumptions and thus minimizes the risk of model misspecification. In this article, we survey some nonparametric regression techniques, with an emphasis on kernel-based estimation, that are additionally robust to atypical and outlying observations. While the main focus lies on robust regression estimation, robust bandwidth selection and conditional scale estimation are discussed as well. Robust estimation in popular nonparametric models such as additive and varying-coefficient models is summarized too. The performance of the main methods is demonstrated on a real dataset. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Robust Methods Statistical and Graphical Methods of Data Analysis > Nonparametric Methods K E Y W O R D S nonparametric regression, outliers, robust estimation 1 | INTRODUCTIONNonparametric regression models have received substantial attention in the theoretical and applied statistics literature over the last few decades. The underlying reason for their growing popularity and importance is that they do not rely on any particular form of the regression function characterizing the relationship between the dependent and explanatory variables and that they are thus robust to model misspecification. On the other hand, many classical nonparametric estimators are not robust in the sense that they are sensitive to atypical and outlying observations in the data. To address this, many authors have explored robust nonparametric regression estimation. As many nonparametric estimators of the regression function are local versions of the estimators of the location-scale model or the linear regression model, many initially proposed robust nonparametric regression estimators were inspired by the developments and construction of robust estimators in those two simple models, which we briefly recall in Section 1.1. Later, we introduce the main concepts of nonparametric regression in Section 1.2 and discuss the robust nonparametric methods from Section 2 on. | Robust estimation of the location and regression modelsConsider the simple univariate location-scale model Y = μ + σε, where Y is a continuously distributed univariate random variable, μ and σ are the location and scale parameters, respectively, and ϵ is an error term. For a random sample

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Robust nonparametric estimation for functional data

Cited by 37 publications

References 16 publications

Functional Data Analysis

Functional Data Analysis

Asymptotic normality of a robust estimator of the regression function for functional time series data

Robust nonparametric regression: A review

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