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. 2021 Oct 30;40(24):5152-5173.
doi: 10.1002/sim.9117. Epub 2021 Jun 23.

Principal component analysis of hybrid functional and vector data

Jeong Hoon Jang  1
Affiliations

Principal component analysis of hybrid functional and vector data

Jeong Hoon Jang. Stat Med. .

Abstract

We propose a practical principal component analysis (PCA) framework that provides a nonparametric means of simultaneously reducing the dimensions of and modeling functional and vector (multivariate) data. We first introduce a Hilbert space that combines functional and vector objects as a single hybrid object. The framework, termed a PCA of hybrid functional and vector data (HFV-PCA), is then based on the eigen-decomposition of a covariance operator that captures simultaneous variations of functional and vector data in the new space. This approach leads to interpretable principal components that have the same structure as each observation and a single set of scores that serves well as a low-dimensional proxy for hybrid functional and vector data. To support practical application of HFV-PCA, the explicit relationship between the hybrid PC decomposition and the functional and vector PC decompositions is established, leading to a simple and robust estimation scheme where components of HFV-PCA are calculated using the components estimated from the existing functional and classical PCA methods. This estimation strategy allows flexible incorporation of sparse and irregular functional data as well as multivariate functional data. We derive the consistency results and asymptotic convergence rates for the proposed estimators. We demonstrate the efficacy of the method through simulations and analysis of renal imaging data.

Keywords: dimension reduction; functional data analysis; multiple data modalities; multivariate data analysis; multivariate functional data; principal component analysis.

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Conflict of interest statement

CONFLICT OF INTEREST

The authors declare no potential conflict of interests.

Figures

FIGURE 1
FIGURE 1
The top panel presents the baseline (left) and post-furosemide (right) renogram curves of 2 kidneys that are diagnosed as “non-obstructed” (dashed lines) and “obstructed” (solid lines). The bottom panel shows baseline and post-furosemide renogram curves of 253 kidneys. [Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 2
FIGURE 2
Estimated 1st hybrid PC ξ^1 of the Emory renal study data. The top panel represents the functional part of the estimated 1st hybrid PC: ψ^1(1) (left; baseline) and ψ^1(2) (right; post-furosemide. The middle panel plots the mean functions (solid line) and the effects of adding (+) and subtracting (−) a suitable multiple of ψ^1(1) (left) and ψ^1(2) (right). The bottom panel is the barplot for the vector part of the estimated 1st hybrid PC θ1. [Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 3
FIGURE 3
Estimated 2nd hybrid PC ξ2 of the Emory renal study data. The top panel represents the functional part of the estimated 2nd hybrid PC: Ψ2(1) (left; baseline) and Ψ2(2) (right; post-furosemide. The middle panel plots the mean functions (solid line) and the effects of adding (+) and subtracting (−) a suitable multiple of Ψ2(1) (left) and Ψ2(2) (right). The bottom panel is the barplot of the vector part of the estimated 2nd hybrid PC θ2. [Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 4
FIGURE 4
Scatterplot of first and second hybrid PC scores of 253 kidneys. [Colour figure can be viewed at wileyonlinelibrary.com]
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References

    1. Besse P, Ramsay JO. Principal component analysis of sampled functions. Psychometrika. 1986;51:285–311.
    1. Castro P, Lawton W, Sylvestre E. Principal modes of variation for processes with continuous sample curves. Technometrics. 1986;28:329–337.
    1. Rice JA, Silverman BW. Estimating the mean and covariance structure nonparametrically when the data are curves. J R Stat Soc Ser B Methodol. 1991;53:233–243.
    1. Silverman BW. Smoothed functional principal component analysis by choice of norm. Ann Stat. 1996;24:1–24.
    1. Shi M, Weiss RE, Taylor JMG. An analysis of paediatric CD4 counts for acquired immune deficiency syndrome using flexible random curves. J R Stat Soc Ser C Appl Stat. 1996;45:151–163.

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