Computational considerations in functional principal component analysis

Ocaña, Francisco A.; Aguilera, Ana M.; Escabias, Manuel

doi:10.1007/s00180-007-0051-2

Cited by 34 publications

(18 citation statements)

References 11 publications

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“…Theoretical and asymptotic properties of FPCA for Hilbert-valued random functions were studied in [50][51][52][53][54]. In the case of a basis expansion for each functional variable (see Equation ( 1)), each functional PCA is equivalent to the multivariate PCA of the matrix AΨ 1/2 , with A = (a ij ) being the n × p matrix of basis coefficients and Ψ being the p × p matrix of inner products between basis functions, Ψ = (Ψ ij ) =< φ i , φ j >, i, j = 1, ..., p. The vector of basis coefficients of the the l−th PC weight function f l (t) is given by b l = Ψ −1/2 v l , where v l is the l−th eigenvector of the sample covariance matrix of AΨ 1/2 (see [55] for a detailed study).…”

Section: Functional Principal Component Regressionmentioning

confidence: 99%

COVID-19 Data Imputation by Multiple Function-on-Function Principal Component Regression

et al. 2021

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The aim of this paper is the imputation of missing data of COVID-19 hospitalized and intensive care curves in several Spanish regions. Taking into account that the curves of cases, deceases and recovered people are completely observed, a function-on-function regression model is proposed to estimate the missing values of the functional responses associated with hospitalized and intensive care curves. The estimation of the functional coefficient model in terms of principal components’ regression with the completely observed data provides a prediction equation for the imputation of the unobserved data for the response. An application with data from the first wave of COVID-19 in Spain is developed after properly homogenizing, registering and smoothing the data in a common interval so that the observed curves become comparable. Finally, Canonical Correlation Analysis is performed on the functional principal components to interpret the relationship between hospital occupancy rate and illness response variables.

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Section: Functional Principal Component Regressionmentioning

confidence: 99%

COVID-19 Data Imputation by Multiple Function-on-Function Principal Component Regression

et al. 2021

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“…For computational aspects of FPCA, see Yao and Lee who suggested an iterative procedure for estimating functional PCs, using penalized spline regression for updating the mean function within the iteration process, and Ocana et al . who focused on an algorithm for computing estimates of FPCA, that is based on classic multivariate PCA settings. Contributions to various aspects of FPCA include those of Boente and Fraiman who discussed kernel‐based smoothed functional PCs, Cardot who extended FPCA to nonparametric functional mixed effect models, called conditional FPCA, and Mas who considered local FPCA in which local covariance operators are used.…”

Section: Functional Principal Component Analysismentioning

confidence: 99%

Functional analysis of glaucoma data

Hosseini‐Nasab

2013

Statistics in Medicine

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We refer glaucoma to a category of eye disorders often associated with a dangerous buildup of intraocular pressure (IOP), which can damage the eyes' optic nerve that transmits visual information to the brain. Because IOP changes over time, it is a function of time, and it is an advantage that we analyze the phenomenon using functional data analysis. In this paper, we treat the data related to the IOP of 35 patients with right eye glaucoma, collected in Rasul-e-Akram Hospital at Tehran, Iran, over the years 2007–2011. We shall explore the structure of the data in search of the features that describe them, and find the characteristics that give a comprehensible presentation of the structure of the variability in the data.We extract patterns of variation in the data by using a generalization of the smoothed functional principal component analysis to obtain the main factors causing glaucoma and then determine their importance. We also explore the correlation patterns between the IOP of right and left eyes, and then model the left eye IOP of the glaucoma patients at each time on the basis of their right eye IOP in a previous interval of time.We can use the model to predict the values of the former variable by using the latter one in a previous time interval.

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“…Functional PCA and functional PLS regression were introduced as natural extensions of their multivariate counterparts to solve the problems of high dimension and multicollinearity associated with the scalar-on-function linear model (Deville, 1974;Dauxois et al, 1982;Ocaña et al, 1999Ocaña et al, , 2007Preda and Saporta, 2005;Aguilera et al, 2016). Both methodologies were compared on different simulated datasets concluding that they have similar forecasting performance, but the estimated parameter function provided by functional PLS regression is more accurate and needs fewer components (Reiss and Ogden, 2007;Aguilera et al, 2010;Delaigle and Hall, 2012b,c).…”

Section: Introductionmentioning

confidence: 99%

Multi-class classification of biomechanical data: A functional LDA approach based on multi-class penalized functional PLS

Aguilera-Morillo

Aguilera

2020

Statistical Modelling

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A functional linear discriminant analysis approach to classify a set of kinematic data (human movement curves of individuals performing different physical activities) is performed. Kinematic data, usually collected in linear acceleration or angular rotation format, can be identified with functions in a continuous domain (time, percentage of gait cycle, etc.). Since kinematic curves are measured in the same sample of individuals performing different activities, they are a clear example of functional data with repeated measures. On the other hand, the sample curves are observed with noise. Then, a roughness penalty might be necessary in order to provide a smooth estimation of the discriminant functions, which would make them more interpretable. Moreover, because of the infinite dimension of functional data, a reduction dimension technique should be considered. To solve these problems, we propose a multi-class approach for penalized functional partial least squares (FPLS) regression. Then linear discriminant analysis (LDA) will be performed on the estimated FPLS components. This methodology is motivated by two case studies. The first study considers the linear acceleration recorded every two seconds in 30 subjects, related to three different activities (walking, climbing stairs and down stairs). The second study works with the triaxial angular rotation, for each joint, in 51 children when they completed a cycle walking under three conditions (walking, carrying a backpack and pulling a trolley). A simulation study is also developed for comparing the performance of the proposed functional LDA with respect to the corresponding multivariate and non-penalized approaches.

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Computational considerations in functional principal component analysis

Abstract: Functional data analysis, Hilbert spaces, Principal components, Covariance estimation, Orthogonal projection,

Cited by 34 publications

References 11 publications

COVID-19 Data Imputation by Multiple Function-on-Function Principal Component Regression

COVID-19 Data Imputation by Multiple Function-on-Function Principal Component Regression

Functional analysis of glaucoma data

Multi-class classification of biomechanical data: A functional LDA approach based on multi-class penalized functional PLS

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