Generative Models for Functional Data using Phase and Amplitude Separation

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.

19 Pages, accepted for publication to Computational Statistics and Data Analysis (Dec 2012)

References in corpus (2)

• Kernel density estimation via diffusion
• Curve alignment by moments

Cited by in corpus (29)

• Functional Data Analysis of Amplitude and Phase Variation
• Statistical analysis of trajectories on Riemannian manifolds: Bird migration, hurricane tracking and video surveillance
• Amplitude and phase variation of point processes
• A Geometric Approach to Visualization of Variability in Functional Data
• Precise Matching of PL Curves in $R^N$ in the Square Root Velocity Framework
• Elastic depths for detecting shape anomalies in functional data
• Separating timing, movement conditions and individual differences in the analysis of human movement
• Functional PCA and Deep Neural Networks-based Bayesian Inverse Uncertainty Quantification with Transient Experimental Data
• Elastic Functional Principal Component Regression
• Permutation tests for the equality of covariance operators of functional data with applications to evolutionary biology
• Regression Models Using Shapes of Functions as Predictors
• Combined Analysis of Amplitude and Phase Variations in Functional Data
• Functional clustering via multivariate clustering
• Simultaneous inference for misaligned multivariate functional data
• Elastic Functional Changepoint Detection of Climate Impacts from Localized Sources
• Distance for Functional Data Clustering Based on Smoothing Parameter Commutation
• Multimodal Bayesian Registration of Noisy Functions using Hamiltonian Monte Carlo
• Estimation of Spatial Deformation for Nonstationary Processes via Variogram Alignment
• A Geometric Approach for Computing Tolerance Bounds for Elastic Functional Data
• Roughness regularization for functional data analysis with free knots spline estimation
• Registration for Incomplete Non-Gaussian Functional Data
• Explaining Neural Network Predictions for Functional Data Using Principal Component Analysis and Feature Importance
• Curvature and Torsion estimation of 3D functional data: A geometric approach to build the mean shape under the Frenet Serret framework
• Low-Rank Representation over the Manifold of Curves
• Phase-Amplitude Separation and Modeling of Spherical Trajectories
• A Shape-Based Functional Index for Objective Assessment of Pediatric Motor Function
• Functional Clustering of Discount Functions for Behavioral Investor Profiling
• Computer model calibration based on image warping metrics: an application for sea ice deformation
• Tractable Clustering of Data on the Curve Manifold