Electronic Journal of Statistics

Analysis of nonlinear modes of variation for functional data

Rima Izem and J.S. Marron

Full-text: Open access

Abstract

A set of curves or images of similar shape is an increasingly common functional data set collected in the sciences. Principal Component Analysis (PCA) is the most widely used technique to decompose variation in functional data. However, the linear modes of variation found by PCA are not always interpretable by the experimenters. In addition, the modes of variation of interest to the experimenter are not always linear. We present in this paper a new analysis of variance for Functional Data. Our method was motivated by decomposing the variation in the data into predetermined and interpretable directions (i.e. modes) of interest. Since some of these modes could be nonlinear, we develop a new defined ratio of sums of squares which takes into account the curvature of the space of variation. We discuss, in the general case, consistency of our estimates of variation, using mathematical tools from differential geometry and shape statistics. We successfully applied our method to a motivating example of biological data. This decomposition allows biologists to compare the prevalence of different genetic tradeoffs in a population and to quantify the effect of selection on evolution.

Article information

Source
Electron. J. Statist., Volume 1 (2007), 641-676.

Dates
First available in Project Euclid: 17 December 2007

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1197908969

Digital Object Identifier
doi:10.1214/07-EJS080

Mathematical Reviews number (MathSciNet)
MR2369029

Zentralblatt MATH identifier
1320.62175

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Functional data analysis nonlinear modes of variation analysis of variance Fréchet mean Fréchet variance variation in manifolds

Citation

Izem, Rima; Marron, J.S. Analysis of nonlinear modes of variation for functional data. Electron. J. Statist. 1 (2007), 641--676. doi:10.1214/07-EJS080. https://projecteuclid.org/euclid.ejs/1197908969


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