Principal components for multivariate functional data

Berrendero, José R.; Justel, Ana; Svarc, Marcela

doi:10.1016/j.csda.2011.03.011

Cited by 115 publications

(68 citation statements)

References 31 publications

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“…The wide variety of disciplines where FDA is applied is also shown through the Special Issue on Statistics for Functional Data (González-Manteiga and Vieu, 2007) published in this journal in 2007. Some examples of these fields of application are: climatology, chemicals, geophysics and oceanology, economics, remote sensing, demographics (Delicado, 2011), materials science (Berrendero et al, 2011), biostatistics or genetics (López-Pintado and Romo, 2011;Li and Chiou, 2011). A mixture of practical and theoretical aspects is found in Ferraty and Romain (2011).…”

Section: Introductionmentioning

confidence: 99%

Functional data analysis in shape analysis

Epifanio

Ventura‐Campos

2011

Computational Statistics & Data Analysis

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Mid-level processes on images often return outputs in functional form. In this context the use of functional data analysis (FDA) in image analysis is considered. In particular, attention is focussed on shape analysis, where the use of FDA in the functional approach (contour functions) shows its superiority over other approaches, such as the landmark based approach or the set theory approach, on two different problems (principal component analysis and discriminant analysis) in a well-known database of bone outlines. Furthermore, a problem that has hardly ever been considered in the literature is dealt with: multivariate functional discrimination. A discriminant function based on independent component analysis for indicating where the differences between groups are and what their level of discrimination is, is proposed. The classification results obtained with the methodology are very promising. Finally, an analysis of hippocampal differences in Alzheimer's disease is carried out.

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Section: Introductionmentioning

confidence: 99%

Functional data analysis in shape analysis

Epifanio

Ventura‐Campos

2011

Computational Statistics & Data Analysis

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“…Often the focus is on functions with a univariate domain, such as time series or spectra. The function values may be multivariate, such as temperatures measured at 3, 9 and 12 cm below ground (Berrendero et al, 2011) or human ECG data measured at 8 different places on the body (Pigoli and Sangalli, 2012). In this paper we will also consider functions whose domain is multivariate.…”

Section: Introductionmentioning

confidence: 99%

A Measure of Directional Outlyingness With Applications to Image Data and Video

Rousseeuw

Raymaekers

Hubert

2018

Journal of Computational and Graphical Statistics

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Functional data covers a wide range of data types. They all have in common that the observed objects are functions of of a univariate argument (e.g. time or wavelength) or a multivariate argument (say, a spatial position). These functions take on values which can in turn be univariate (such as the absorbance level) or multivariate (such as the red/green/blue color levels of an image). In practice it is important to be able to detect outliers in such data. For this purpose we introduce a new measure of outlyingness that we compute at each gridpoint of the functions' domain. The proposed directional outlyingness (DO) measure accounts for skewness in the data and only requires O(n) computation time per direction. We derive the influence function of the DO and compute a cutoff for outlier detection. The resulting heatmap and functional outlier map reflect local and global outlyingness of a function. To illustrate the performance of the method on real data it is applied to spectra, MRI images, and video surveillance data.

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“…Existing approaches for multivariate functional principal component analysis (MFPCA) are restricted to functions observed on the same finite, one-dimensional interval (Ramsay and Silverman, 2005;Jacques and Preda, 2014;Chiou et al, 2014;Berrendero et al, 2011). Except for Berrendero et al (2011), they all aim at a multivariate functional Karhunen-Loève representation of the data.…”

Section: Introductionmentioning

confidence: 99%

“…Except for Berrendero et al (2011), they all aim at a multivariate functional Karhunen-Loève representation of the data. For data measured e.g.…”

Section: Introductionmentioning

confidence: 99%

Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains

Happ

Greven

2018

Journal of the American Statistical Association

279

322

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Existing approaches for multivariate functional principal component analysis are restricted to data on the same one-dimensional interval. The presented approach focuses on multivariate functional data on different domains that may differ in dimension, e.g. functions and images. The theoretical basis for multivariate functional principal component analysis is given in terms of a Karhunen-Loève Theorem. For the practically relevant case of a finite Karhunen-Loève representation, a relationship between univariate and multivariate functional principal component analysis is established. This offers an estimation strategy to calculate multivariate functional principal components and scores based on their univariate counterparts. For the resulting estimators, asymptotic results are derived. The approach can be extended to finite univariate expansions in general, not necessarily orthonormal bases. It is also applicable for sparse functional data or data with measurement error. A flexible R implementation is available on CRAN. The new method is shown to be competitive to existing approaches for data observed on a common one-dimensional domain. The motivating application is a neuroimaging study, where the goal is to explore how longitudinal trajectories of a neuropsychological test score covary with FDG-PET brain scans at baseline. Supplementary material, including detailed proofs, additional simulation results and software is available online.

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Principal components for multivariate functional data

Cited by 115 publications

References 31 publications

Functional data analysis in shape analysis

Functional data analysis in shape analysis

A Measure of Directional Outlyingness With Applications to Image Data and Video

Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains

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