Trimmed means for functional data

By Ricardo Fraiman, Graciela Muniz. 2001

360 citations recorded by the source corpus.

In practice, the use of functional data is often preferable to that of large finitedimensional vectors obtained by discrete approximations of functions. In this paper a new concept of data depth is introduced for functional data. The aim is to measure the centrality of a given curve within a group of curves. This concept is used to define ranks and trimmed means for functional data. Some theoretical and practical aspects are discussed and a simulation study is given. The results show a good performance of our method, in terms of efficiency and robustness, when compared with the mean. Finally, a real-data example based on the Nasdaq 100 index is discussed.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5a472e226c9c544af973eaa8056924b345397ae7

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In Applied Mathematics

In Statistics and Probability

Citations in the corpus

A survey of functional principal component analysis (2014, methodology): “Following this idea , Fraiman & Muniz ( 2001 ) proposed the trimmed means using the notion of functional depth .”
The spatial distribution in infinite dimensional spaces and related quantiles and depths (2014, background): “irical version is given by SD(c x)=1−kSb xk. Here, S x and Sb x are as deﬁned in Section 2. There are a few other notions of depth function for data in inﬁnite dimensional function spaces (see, e.g., [16, 25, 26] and [32]). However, as shown in [10], some of these depth functions exhibit degeneracy ”
Depth Statistics (2012, background): “Fraiman and Muniz (2001) calculate the univariate depths of the values of a function and integrate them over thewhole interval; this results in kind of ‘average’ depth.”
On data depth in infinite dimensional spaces (2014, methodology): “Depth-weighted L-type location estimators like trimmed means have been considered in Liu (1990), Donoho and Gasko (1992), Liu et al. (1999), Fraiman and Muniz (2001), Mosler (2002) and Zuo (2006).”
Statistical Computing in Functional Data Analysis: The R Package fda.usc (2012, background): “FM() that is based on the median (proposed by Fraiman and Muniz 2001), depth.”
Integrated depth for functional data: statistical properties and consistency (2016, methodology): “Integrated depth of Fraiman and Muniz [14] The consistency result of ([14], Thm.”
An anova test for functional data (2004, background): “For example, the analysis could be made in terms of median functions (or even " mode functions " ), although the required theory looks more involved, and so far less developed, that required for mean functions; see, however, Cadre (2001), Fraiman and Muniz (2001).”
On the Concept of Depth for Functional Data (2009, background): “Fraiman and Muniz (2001) defined and studied a concept of depth for functional observations based on the integrals of univariate depths.”
A Goodness-of-Fit Test for the Functional Linear Model with Scalar Response (2012, methodology): “Before applying the tests, four functional outliers corresponding to the 5% less depth curves according to the Fraiman and Muniz (2001) depth were removed.”
Outlier detection in functional data by depth measures, with application to identify abnormal NOx levels (2008, background): “The Fraiman and Muniz depth: Fraiman and Muniz (2001) were the first to introduce a functional data depth.”
Multivariate Functional Halfspace Depth (2014, background): “The β-trimmed mean and the β-trimmed variance (the mean and variance of all curves in the β-central region), see Fraiman and Muniz (2001), can be extended in a straightforward way too.”
Robust estimation and classification for functional data via projection-based depth notions (2007, methodology): “On the other hand, there is a need of depth notions valid for infinite-dimensional data in connection with the increasing demand of statistical tools for functional data analysis (FDA) where the available data are functions x = x(t) defined on some real interval (say [0,1] )s ee e.g., Ramsay and Sil”