Generative Models for Functional Data using Phase and Amplitude Separation

TL;DR

This paper introduces a novel approach for modeling functional data by separating phase and amplitude variability, leading to more accurate generative models and improved classification performance across various applications.

Contribution

It proposes a new method that separates phase and amplitude in functional data using elastic shape analysis, then models these components jointly with probabilistic techniques.

Findings

Models outperform traditional methods ignoring phase variability.

Effective in classifying SONAR signals, signatures, and body movements.

Demonstrates superiority through simulations and real data applications.

Abstract

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.