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

Happ, Clara; Greven, Sonja

doi:10.1080/01621459.2016.1273115

Cited by 276 publications

(358 citation statements)

References 26 publications

Supporting

Mentioning

358

Contrasting

Order By: Relevance

“…Yet, in this paper, the PCA is applied on a spline functional model of temperature and salinity profiles instead of applying it directly to the profiles themselves (hence the term functional). The extension of univariate functional PCA to multivariate functional data is of high practical relevance to reveal joint variation of the different variables (Happ and Greven 2015). A multivariate functional PCA provides an effective way to understand the structure of a multivariate system such as the 3D temperaturesalinity structure of the Southern Ocean.…”

Section: Introductionmentioning

confidence: 99%

A Linear Decomposition of the Southern Ocean Thermohaline Structure

Pauthenet

Roquet

Madec

et al. 2017

Journal of Physical Oceanography

View full text Add to dashboard Cite

The thermohaline structure of the Southern Ocean is deeply influenced by the presence of the Antarctic Circumpolar Current (ACC), where water masses of the World Ocean are advected, transformed, and redistributed to the other basins. It remains a challenge to describe and visualize the complex 3D pattern of this circulation and its associated tracer distribution. Here, a simple framework is presented to analyze the Southern Ocean thermohaline structure. A functional principal component analysis (PCA) is applied to temperature u and salinity S profiles to determine the main spatial patterns of their variations. Using the Southern Ocean State Estimate (SOSE), this study determines the vertical modes describing the Southern Ocean thermohaline structure between 5 and 2000 m. The first two modes explain 92% of the combined u-S variance, thus providing a surprisingly good approximation of the thermohaline properties in the Southern Ocean. The first mode (72% of total variance) accurately describes the north-south property gradients. The second mode (20%) mostly describes salinity at 500 m in the region of Antarctic Intermediate Water formation. These two modes present circumpolar patterns that can be closely related with standard frontal definitions. By projecting any given hydrographic profile onto the SOSE-based modes, it is possible to determine its position relative to the fronts. The projection is successfully applied on the hydrographic profiles of the WOCE SR3 section. The Southern Ocean thermohaline decomposition provides an objective way to define water mass boundaries and their spatial variability and has useful application for comparing model output with observations.

show abstract

Section: Introductionmentioning

confidence: 99%

A Linear Decomposition of the Southern Ocean Thermohaline Structure

Pauthenet

Roquet

Madec

et al. 2017

Journal of Physical Oceanography

View full text Add to dashboard Cite

show abstract

“…Smooth principal component images can be calculated using a regularized tensor product decomposition based on a rank‐one approximation (CANDECOMP/PARAFRAC) . In principle, the multivariate functional principal component analysis (MFPCA) approach for (multivariate) functional data can also be used to calculate eigenimages, interpreting the images as multivariate functional data with a single element on a 2D or 3D domain.…”

Section: Overview Of Methods For Scalar‐on‐image Regressionmentioning

confidence: 99%

“…For the calculation of the eigenimages in PCR2D , we use the implementation of the rank‐one–based approach in the MFPCA package, which, at present, works only for 2D images. The reconstruction of the coefficient image

\overset{β}{true}

using the estimated eigenimages and the regression coefficients can easily be done using the expandBasisFunction method in MFPCA.…”

Section: Overview Of Methods For Scalar‐on‐image Regressionmentioning

confidence: 99%

The impact of model assumptions in scalar‐on‐image regression

Happ

Greven

Schmid

2018

Statistics in Medicine

Self Cite

View full text Add to dashboard Cite

Complex statistical models such as scalar-on-image regression often require strong assumptions to overcome the issue of nonidentifiability. While in theory, it is well understood that model assumptions can strongly influence the results, this seems to be underappreciated, or played down, in practice. This article gives a systematic overview of the main approaches for scalar-on-image regression with a special focus on their assumptions. We categorize the assumptions and develop measures to quantify the degree to which they are met. The impact of model assumptions and the practical usage of the proposed measures are illustrated in a simulation study and in an application to neuroimaging data. The results show that different assumptions indeed lead to quite different estimates with similar predictive ability, raising the question of their interpretability. We give recommendations for making modeling and interpretation decisions in practice based on the new measures and simulations using hypothetic coefficient images and the observed data.

show abstract

“…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%

“…The manuscript proceeds as follows: In Section , we review several transformations from

normalΓ false(scriptT false)

or the space of probability density functions to

L^{2} false(scriptT false)

proposed in the literature and give reasons why the clr approach should be preferred to other transformations when defining PCA for warping functions. Section embeds the existing methods for joint analysis of amplitude and phase variation into the framework of multivariate functional principal component analysis (MFPCA; Happ and Greven, ) and gives new insights into the properties of the joint principal components. The theoretical results are illustrated in Section by means of data from a from a multiphysics computational seismology experiment.…”

Section: Introductionmentioning

confidence: 99%

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

Happ

Scheipl

Gabriel

et al. 2019

Stat

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

Cited by 276 publications

References 26 publications

A Linear Decomposition of the Southern Ocean Thermohaline Structure

A Linear Decomposition of the Southern Ocean Thermohaline Structure

The impact of model assumptions in scalar‐on‐image regression

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

Contact Info

Product

Resources

About