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.

本文简要介绍了功能数据分析领域中的一个重要难题 —— 相位变异问题，并总结了几种现有的相位分离方法，阐述了它们的差异以及对应的优化算法。

振幅和相位变化的函数数据分析

Functional Data Analysis of Amplitude and Phase Variation

Constructing generative models for functional observations is an important
task in statistical functional analysis. In general, functional data contains
both phase (or x or horizontal) and amplitude (or y or vertical) variability.
Tradi- tional methods often ignore the phase variability and focus solely on
the amplitude variation, using cross-sectional techniques such as fPCA for
dimensional reduction and data modeling. Ignoring phase variability leads to a
loss of structure in the data and inefficiency in data models. This paper
presents an approach that relies on separating the phase (x-axis) and amplitude
(y-axis), then modeling these components using joint distributions. This
separation, in turn, is performed using a technique called elastic shape
analysis of curves that involves a new mathematical representation of
functional data. Then, using individual fPCAs, one each for phase and amplitude
components, while respecting the nonlinear geometry of the phase representation
space; impose joint probability models on principal coefficients of these
components. These ideas are demonstrated using random sampling, for models
estimated from simulated and real datasets, and show their superiority over
models that ignore phase-amplitude separation. Furthermore, the generative
models are applied to classification of functional data and achieve high
performance in applications involv- ing SONAR signals of underwater objects,
handwritten signatures, and periodic body movements recorded by smart phones.

本文提出了一种新的方法，利用弹性形状曲线分析对功能性数据中的相位和振幅进行分离建模，并通过联合分布和主要系数进行建模，此方法优于传统方法。通过随机抽样和实际数据集构建的模型证明了新方法的有效性。此外，所提出的生成模型应用于功能数据分类方面表现出了优异表现，包括声纳信号、手写签名和智能手机记录的周期性身体运动。