1,859 research outputs found
Local functional principal component analysis
Covariance operators of random functions are crucial tools to study the way
random elements concentrate over their support. The principal component
analysis of a random function X is well-known from a theoretical viewpoint and
extensively used in practical situations. In this work we focus on local
covariance operators. They provide some pieces of information about the
distribution of X around a fixed point of the space x₀. A description of
the asymptotic behaviour of the theoretical and empirical counterparts is
carried out. Asymptotic developments are given under assumptions on the
location of x₀ and on the distributions of projections of the data on the
eigenspaces of the (non-local) covariance operator
Multilevel functional principal component analysis
The Sleep Heart Health Study (SHHS) is a comprehensive landmark study of
sleep and its impacts on health outcomes. A primary metric of the SHHS is the
in-home polysomnogram, which includes two electroencephalographic (EEG)
channels for each subject, at two visits. The volume and importance of this
data presents enormous challenges for analysis. To address these challenges, we
introduce multilevel functional principal component analysis (MFPCA), a novel
statistical methodology designed to extract core intra- and inter-subject
geometric components of multilevel functional data. Though motivated by the
SHHS, the proposed methodology is generally applicable, with potential
relevance to many modern scientific studies of hierarchical or longitudinal
functional outcomes. Notably, using MFPCA, we identify and quantify
associations between EEG activity during sleep and adverse cardiovascular
outcomes.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS206 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Functional principal component analysis of spatially correlated data
This paper focuses on the analysis of spatially correlated functional data. We propose a parametric model for spatial correlation and the between-curve correlation is modeled by correlating functional principal component scores of the functional data. Additionally, in the sparse observation framework, we propose a novel approach of spatial principal analysis by conditional expectation to explicitly estimate spatial correlations and reconstruct individual curves. Assuming spatial stationarity, empirical spatial correlations are calculated as the ratio of eigenvalues of the smoothed covariance surface Cov (Xi(s),Xi(t))(Xi(s),Xi(t)) and cross-covariance surface Cov (Xi(s),Xj(t))(Xi(s),Xj(t)) at locations indexed by i and j. Then a anisotropy Matérn spatial correlation model is fitted to empirical correlations. Finally, principal component scores are estimated to reconstruct the sparsely observed curves. This framework can naturally accommodate arbitrary covariance structures, but there is an enormous reduction in computation if one can assume the separability of temporal and spatial components. We demonstrate the consistency of our estimates and propose hypothesis tests to examine the separability as well as the isotropy effect of spatial correlation. Using simulation studies, we show that these methods have some clear advantages over existing methods of curve reconstruction and estimation of model parameters
Structured Functional Principal Component Analysis
Motivated by modern observational studies, we introduce a class of functional
models that expands nested and crossed designs. These models account for the
natural inheritance of correlation structure from sampling design in studies
where the fundamental sampling unit is a function or image. Inference is based
on functional quadratics and their relationship with the underlying covariance
structure of the latent processes. A computationally fast and scalable
estimation procedure is developed for ultra-high dimensional data. Methods are
illustrated in three examples: high-frequency accelerometer data for daily
activity, pitch linguistic data for phonetic analysis, and EEG data for
studying electrical brain activity during sleep
MULTILEVEL SPARSE FUNCTIONAL PRINCIPAL COMPONENT ANALYSIS
The basic observational unit in this paper is a function. Data are assumed to have a natural hierarchy of basic units. A simple example is when functions are recorded at multiple visits for the same subject. Di et al. (2009) proposed Multilevel Functional Principal Component Analysis (MFPCA) for this type of data structure when functions are densely sampled. Here we consider the case when functions are sparsely sampled and may contain as few as 2 or 3 observations per function. As with MFPCA, we exploit the multilevel structure of covariance operators and data reduction induced by the use of principal component bases. However, we address inherent methodological differences in the sparse sampling context to: 1) estimate the covariance operators; 2) estimate the functional scores and predict the underlying curves. We show that in the sparse context 1) is harder and propose an algorithm to circumvent the problem. Surprisingly, we show that 2) is easier via new BLUP calculations. Using simulations and real data analysis we show that the ability of our method to reconstruct underlying curves with few observations is stunning. This approach is illustrated by an application to the Sleep Heart Health Study, which contains two electroencephalographic (EEG) series at two visits for each subject
Functional Principal Component Analysis for Non-stationary Dynamic Time Series
Motivated by a highly dynamic hydrological high-frequency time series,
we propose time-varying Functional Principal Component Analysis (FPCA)
as a novel approach for the analysis of non-stationary Functional Time Series
(FTS) in the frequency domain. Traditional FPCA does not take into account
(i) the temporal dependence between the functional observations and (ii) the
changes in the covariance/variability structure over time, which could result in
inadequate dimension reduction. The novel time-varying FPCA proposed adapts
to the changes in the auto-covariance structure and varies smoothly over frequency
and time to allow investigation of whether and how the variability structure
in an FTS changes over time. Based on the (smooth) time-varying dynamic
FPCs, a bootstrap inference procedure is proposed to detect significant changes
in the covariance structure over time. Although this time-varying dynamic FPCA
can be applied to any dynamic FTS, it has been applied here to study the daily
processes of partial pressure of CO2 in a small river catchment in Scotland
Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains
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-Lo\`eve Theorem. For the practically relevant case of a finite
Karhunen-Lo\`eve 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.Comment: Revised Version. R-Code for the online appendix is available in the
.zip file associated with this article in subdirectory "/Software". The
software associated with this article is available on CRAN (packages funData
and MFPCA
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