Defining probability density for a distribution of random functions

By Aurore Delaigle, Peter Hall. 2010

166 citations recorded by the source corpus.

The notion of probability density for a random function is not as straightforward as in finite-dimensional cases. While a probability density function generally does not exist for functional data, we show that it is possible to develop the notion of density when functional data are considered in the space determined by the eigenfunctions of principal component analysis. This leads to a transparent and meaningful surrogate for density defined in terms of the average value of the logarithms of the densities of the distributions of principal components for a given dimension. This density approximation is estimable readily from data. It accurately represents, in a monotone way, key features of small-ball approximations to density. Our results on estimators of the densities of principal component scores are also of independent interest; they reveal interesting shape differences that have not previously been considered. The statistical implications of these results and properties are identified and discussed, and practical ramifications are illustrated in numerical work.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/091d5c4a091973ed02d1e1c439e538a0dd38249a

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Citations in the corpus

Peter Hall, functional data analysis and random objects (2016, methodology): “In Delaigle and Hall (2010), complex arguments are used to show that the finite approximation approximation approach in Gasser, Hall and Presnell (1998) can be refined by replacing the fixed basis functions ψ j by eigenfunctions φ j of the underlying processes X , more precisely by estimated…”
Functional data clustering: a survey (2013, background): “In the clustering context, the lack of a deﬁnition for the probability density of a functional random variable (Delaigle and Hall 2010) makes the probabilistic model-based methods not directly applicable.”
Componentwise classification and clustering of functional data (2012, methodology): “In the rainfall data, which are available at http://dss.ucar.edu/datasets/ds482.1, we considered N = 190 rainfall curves from N0 = 43 northern and N1 = 147 southern Australian weather stations, used by Delaigle and Hall (2010).”
Achieving near perfect classification for functional data (2012, methodology): “In the second example we consider the N = 191 rainfall curves from N0 = 43 northern (Π0) and N1 = 148 southern (Π1) Australian weather stations, used by Delaigle and Hall (2010) and available at http://dss.”
Nonparametric modelling for functional data: selected survey and tracks for future (2018, background): “as Radon–Nikodym derivative) of a density is discussed in [3,119].”
Model-based clustering for multivariate functional data (2013, methodology): “Approximation of the density for multivariate functional data As the notion of probability density is not well defined for functional data (Delaigle and Hall, 2010), we can use an approximation of the density based on the KarhunenLoeve expansion (2).”
Functional Data Analysis (2016, background): “This “curse” is quantiﬁable in terms of unfavorable small ball probabilities in function space (Delaigle & Hall 2010).”