Weak convergence of discretely observed functional data with applications

Nagy, Stanislav; Gijbels, Irène; Hlubinka, Daniel

doi:10.1016/j.jmva.2015.06.006

Cited by 10 publications

(11 citation statements)

References 19 publications

(25 reference statements)

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“…With (A6) we are able to formulate the consistency result also for depths based on noisy observations. This theorem substantially improves on the related consistency results for discretely observed noiseless random functions in C ([0, 1]) provided in Nagy et al (2016a).…”

Section: Depth Functionals Insupporting

confidence: 76%

“…The choice σ 2 ≡ 0 covers the case when the random functions are observed discretely without noise. This setup was considered by Nagy et al (2016a) in the space C ([0, 1]), and in the present contribution we extend these results substantially to discontinuous functions contaminated with measurement errors, using very different proof techniques.…”

Section: Weak Convergence For Noisy Functional Datamentioning

confidence: 65%

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Data depth for measurable noisy random functions

Nagy¹,

Ferraty

2019

Journal of Multivariate Analysis

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In the literature on data depth applicable to random functions it is usually assumed that the trajectories of all the random curves are continuous, known at each point of the domain, and observed exactly. These assumptions turn out to be unrealistic in practice, as the functions are often observed only at a finite grid of time points, and in the presence of measurement errors. In this work, we provide the necessary theoretical background enabling the extension of the statistical methodology based on data depth to measurable (not necessarily continuous) random functions observed within the latter framework. It is shown that even if the random functions are discontinuous, observed discretely, and contaminated with additive noise, many common depth functionals maintain the fine consistency properties valid in the ideal case of completely observed noiseless functions. For the integrated depth for functions, we provide uniform rates of convergence over the space of integrable functions.

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Section: Depth Functionals Insupporting

confidence: 76%

Section: Weak Convergence For Noisy Functional Datamentioning

confidence: 65%

Data depth for measurable noisy random functions

Nagy¹,

Ferraty

2019

Journal of Multivariate Analysis

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“…Nagy et al (2016) used piecewise linear interpolation as described in Lemma 2 to obtain continuous versions of densely observed curves. Therein, it is shown that basing statistical inference on a sample of curves X 1 , .…”

Section: Law Of Large Numbers For Discretely Observed Random Functionsmentioning

confidence: 99%

Law of large numbers for discretely observed random functions

Nagy

Gijbels

2017

Journal of the Korean Statistical Society

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Abstract. A strong law of large numbers for continuous random functions, and associated tensor product surfaces is established in the setup of discretely observed functional data. The result is shown in the framework of uniform convergence of functions, and stated without imposing any distributional assumptions. It is demonstrated that, under mild conditions, laws of large numbers for continuously observed functional data imply the corresponding laws under the discrete observational design of functions. Applications to the problem of estimation of expectation functions and covariance surfaces for discretely observed functional data are discussed.

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“…If L k is selected from M k , then, as in Corollary 1, the Bayes risk is achieved with respect to the empirical distributions. Theorem 1(ii) of Nagy & Gijbels & Hlubinka (2015) says that under the above sampling scheme it holds P [G 0,k → G 0 weakly] = 1 and P [G 1,k → G 1 weakly] = 1.…”

Section: Functional Optimalitymentioning

confidence: 99%

Fast DD-classification of functional data

Mosler

Mozharovskyi²

2015

Stat Papers

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A fast nonparametric procedure for classifying functional data is introduced. It consists of a two-step transformation of the original data plus a classifier operating on a low-dimensional space. The functional data are first mapped into a finite-dimensional location-slope space and then transformed by a multivariate depth function into the DD-plot, which is a subset of the unit square. This transformation yields a new notion of depth for functional data. Three alternative depth functions are employed for this, as well as two rules for the final classification in [0, 1] 2 . The resulting classifier has to be cross-validated over a small range of parameters only, which is restricted by a Vapnik-Chervonenkis bound. The entire methodology does not involve smoothing techniques, is completely nonparametric and allows to achieve Bayes optimality under standard distributional settings. It is robust, efficiently computable, and has been implemented in an R environment. Applicability of the new approach is demonstrated by simulations as well as by a benchmark study.

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Weak convergence of discretely observed functional data with applications

Cited by 10 publications

References 19 publications

Data depth for measurable noisy random functions

Data depth for measurable noisy random functions

Law of large numbers for discretely observed random functions

Fast DD-classification of functional data

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