Multivariate functional principal component analysis: A normalization approach

Chiou, Jeng‐Min; Yang, Yu‐Shih; Chen, Yuting

doi:10.5705/ss.2013.305

Cited by 124 publications

(140 citation statements)

References 22 publications

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“…Ramsay and Silverman (2005, Chapter 10.3. in the context of hybrid data) or Chiou et al (2014). The associated weighted covariance operator Γ w is given by its elements (Γ w f ) (j) with f ∈ H and (Γ w f ) (j) (t j ) = C ·,j (·, t j ), f w , t j ∈ T j .…”

Section: Data Structure and Notationmentioning

confidence: 99%

“…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%

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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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Section: Data Structure and Notationmentioning

confidence: 99%

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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“…As noted in Lee and Jung (), one may alternatively use methods for MFPCA (Chiou, Yang, & Chen, ; Happ and Greven, ). These approaches are indeed more appropriate as they better reflect the characteristic nature of the data in terms of bivariate functions

s_{i} = false(w_{i}, v_{i} false) \in L^{2} false(scriptT false) \times L^{2} false(scriptT false) = : scriptH

, and therefore, we will use them in the following.…”

Section: Modes Of Joint Variation In Amplitude and Phasementioning

confidence: 99%

A general framework for multivariate functional principal component analysis of amplitude and phase variation

Happ

Scheipl

Gabriel

et al. 2019

Stat

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Functional data typically contain amplitude and phase variation. In many data situations, phase variation is treated as a nuisance effect and is removed during preprocessing, although it may contain valuable information. In this note, we focus on joint principal component analysis (PCA) of amplitude and phase variation. As the space of warping functions has a complex geometric structure, one key element of the analysis is transforming the warping functions to L2false(scriptTfalse). We present different transformation approaches and show how they fit into a general class of transformations. This allows us to compare their strengths and limitations. In the context of PCA, our results offer arguments in favour of the centred log‐ratio transformation. We further embed two existing approaches from the literature for joint PCA of amplitude and phase variation into the framework of multivariate functional PCA, where we study the properties of the estimators based on an appropriate metric. The approach is illustrated through an application from seismology.

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“…For the case where observations consist of samples of random trajectories that take values in

{double-struckR}^{p}

, the methodology of choice is often functional data analysis (FDA) (Ramsay and Silverman, ; Horvath and Kokoszka, ; Wang et al ., ), where methodology for one‐dimensional ( p =1) functional data is readily available. Models for functional data that consist of vector‐valued processes ( p >1) have been studied more recently (Zhou et al ., ; Berrendero et al ., ; Chiou et al ., ; Claeskens et al ., ; Verbeke et al ., ; Chiou et al ., ) as well as the case where at each time point one records a random function, i.e. function‐valued stochastic processes (Park and Staicu, ; Chen and Müller, ; Chen et al ., ).…”

Section: Introductionmentioning

confidence: 99%

Functional Models for Time-Varying Random Objects

Dubey

Müller

2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary Functional data analysis provides a popular toolbox of functional models for the analysis of samples of random functions that are real valued. In recent years, samples of time‐varying object data such as time‐varying networks that are not in a vector space have been increasingly collected. These data can be viewed as elements of a general metric space that lacks local or global linear structure and therefore common approaches that have been used with great success for the analysis of functional data, such as functional principal component analysis, cannot be applied. We propose metric covariance, a novel association measure for paired object data lying in a metric space (Ω,d) that we use to define a metric autocovariance function for a sample of random Ω‐valued curves, where Ω generally will not have a vector space or manifold structure. The proposed metric autocovariance function is non‐negative definite when the squared semimetric d2 is of negative type. Then the eigenfunctions of the linear operator with the autocovariance function as kernel can be used as building blocks for an object functional principal component analysis for Ω‐valued functional data, including time‐varying probability distributions, covariance matrices and time dynamic networks. Analogues of functional principal components for time‐varying objects are obtained by applying Fréchet means and projections of distance functions of the random object trajectories in the directions of the eigenfunctions, leading to real‐valued Fréchet scores. Using the notion of generalized Fréchet integrals, we construct object functional principal components that lie in the metric space Ω. We establish asymptotic consistency of the sample‐based estimators for the corresponding population targets under mild metric entropy conditions on Ω and continuity of the Ω‐valued random curves. These concepts are illustrated with samples of time‐varying probability distributions for human mortality, time‐varying covariance matrices derived from trading patterns and time‐varying networks that arise from New York taxi trips.

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Multivariate functional principal component analysis: A normalization approach

Cited by 124 publications

References 22 publications

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

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

A general framework for multivariate functional principal component analysis of amplitude and phase variation

Functional Models for Time-Varying Random Objects

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