Statistical Science

Functional Data Analysis of Amplitude and Phase Variation

J. S. Marron, James O. Ramsay, Laura M. Sangalli, and Anuj Srivastava

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The abundance of functional observations in scientific endeavors has led to a significant development in tools for functional data analysis (FDA). This kind of data comes with several challenges: infinite-dimensionality of function spaces, observation noise, and so on. However, there is another interesting phenomena that creates problems in FDA. The functional data often comes with lateral displacements/deformations in curves, a phenomenon which is different from the height or amplitude variability and is termed phase variation. The presence of phase variability artificially often inflates data variance, blurs underlying data structures, and distorts principal components. While the separation and/or removal of phase from amplitude data is desirable, this is a difficult problem. In particular, a commonly used alignment procedure, based on minimizing the $\mathbb{L}^{2}$ norm between functions, does not provide satisfactory results. In this paper we motivate the importance of dealing with the phase variability and summarize several current ideas for separating phase and amplitude components. These approaches differ in the following: (1) the definition and mathematical representation of phase variability, (2) the objective functions that are used in functional data alignment, and (3) the algorithmic tools for solving estimation/optimization problems. We use simple examples to illustrate various approaches and to provide useful contrast between them.

Article information

Statist. Sci., Volume 30, Number 4 (2015), 468-484.

First available in Project Euclid: 9 December 2015

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Zentralblatt MATH identifier

Functional data analysis registration warping alignment elastic metric dynamic time warping Fisher–Rao metric


Marron, J. S.; Ramsay, James O.; Sangalli, Laura M.; Srivastava, Anuj. Functional Data Analysis of Amplitude and Phase Variation. Statist. Sci. 30 (2015), no. 4, 468--484. doi:10.1214/15-STS524.

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