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Generative Models for Functional Data using Phase and Amplitude Separation
J. Derek Tucker, Wei Wu, Anuj Srivastava
TL;DR
The paper addresses generative modeling of functional data when variability arises from both amplitude and phase, which traditional approaches often ignore. It separates these components using elastic shape analysis and models their fPCA representations jointly. The resulting models generate samples that better preserve observed structure and support classification across several functional-data applications.
Problem
Many functional-data methods ignore phase variability by assuming observations are temporally aligned, potentially losing patterns and producing inefficient models.
Method
The framework uses elastic shape analysis to separate aligned amplitudes from warping phases, then applies separate fPCAs and joint Gaussian or nonparametric models to their coefficients.
Results
The proposed models generate samples visually similar to simulated and Berkeley growth data and support model-based classification on several functional datasets.
Takeaways & Limitations
Phase-amplitude separation provides a joint metric-based framework for generative modeling and classification of functional data with phase variability.
Takeaways & Limitations
The framework assumes absolutely continuous functions and boundary-preserving diffeomorphic warping functions; the SONAR application is additionally complicated by compositional and additive noise.
Abstract
from arXiv · showhide
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.
1. Introduction
Functional data modeling must account for both amplitude variability and phase variability, because ignoring temporal misalignment can lose essential structure. The paper proposes elastic phase-amplitude separation, nonlinear phase modeling, and joint probabilistic models for generative sampling and classification.
- Motivation: Functional observations vary through both aligned-function amplitude and warping-function phase, but many methods implicitly assume temporal alignment.Amplitude is y-axis variability after alignment; phase is x-axis variability represented by warping functions.
- Need for Phase-Amplitude Separation: Ignoring phase variability can produce inefficient models that fail to capture patterns present in the data.In simulated data, direct fPCA modeling generated curves with multiple peaks instead of preserving the original unimodal structure.
- Modeling Results: Random samples from the proposed models visually resemble simulated and Berkeley growth data more closely than samples from direct fPCA Gaussian models.The non-separated model produced excessive variation and curves with three peaks or poor resemblance to the original data.
- Phase-Amplitude Separation: Elastic shape analysis uses a square-root slope function representation to address the nonlinear and infinite-dimensional geometry of warping functions.The paper represents a warping function γ by ψ = √γ̇ and applies horizontal fPCA after transformation.
- Past Literature on Phase-Amplitude Separation: The elastic method bases alignment on a proper distance with symmetry, positive definiteness, and triangle inequality, unlike the cited existing statistical methods.It also performs component separation and fPCA jointly under the same metric rather than in distinct metric-based steps.
- Proposed Framework: The framework aligns functions, estimates separate phase and amplitude summaries, performs separate fPCAs, and models their principal coefficients jointly.The proposed models use joint Gaussian or nonparametric distributions on the two fPCA representations.
- Applications: The models are also applied to likelihood-based classification of simulated, signature, smartphone movement, and SONAR functional data.The SONAR data are noisy because of compositional and additive noise, increasing within-class variability and reducing separation across classes.
2. Phase and Amplitude Separation Using Elastic Analysis
Elastic shape analysis separates functional phase and amplitude variability by representing functions with SRSFs and optimizing alignment over nonlinear warping functions. The resulting amplitude distances are warping-invariant, while phase is modeled through the estimated warping functions.
- Elastic alignment: Elastic alignment uses nonlinear time warping to separate phase and amplitude components and derive metrics for comparing and classifying functions.The framework is designed to automate alignment while retaining distinct x- and y-variability.
- Mathematical representation: The SRSF representation converts functions into square-integrable objects and makes alignment symmetric, positive definite, and compatible with a proper distance.Its isometry property supports registration in SRSF space before mapping aligned functions back to function space.
- Amplitude distance: The amplitude y-distance is invariant to random warpings, so it measures amplitude differences independently of phase variation.This distance is defined using aligned SRSFs and supports mean-function construction.
- Karcher alignment: The Karcher-mean procedure produces a preferred mean SRSF, aligned SRSFs, and optimal warping functions for the observed data.The preferred mean is selected so the mean warping is the identity element.
- Results: In simulated data, aligned functions have sharper peaks, tighter alignment, and thinner bands around the mean after the generated warping effects are removed.The remaining variation is attributed to the original amplitude-generating functions.
- Results: The elastic method is selected over MBM and MSE alignment methods based on superior performance and theoretical advantages on simulated unimodal and real SONAR data.The comparison reports amplitude and phase variances for the different algorithms.
3. Analysis and Modeling of Components
The component-analysis framework models phase on the nonlinear geometry of warping functions and amplitude through fPCA of aligned SRSFs. Principal coefficients from both components can then support joint generative models of functional observations.
- Phase variability: Phase variability is represented by warping functions, whose nonlinear and infinite-dimensional space is transformed to SRSF space and reduced using fPCA.The square-root transformation maps warping functions to a unit Hilbert sphere.
- Phase variability: The Karcher mean of warping functions and tangent-space shooting vectors provide a vector-space basis for phase principal-component analysis.The tangent space is taken at the Karcher mean on the SRSF sphere.
- Phase variability: The phase covariance matrix is decomposed by SVD to obtain principal directions and observed coefficients for the warping functions.Principal directions are estimated from the tangent-space covariance matrix.
- Amplitude variability: Vertical fPCA analyzes aligned SRSFs in L2 while treating each function’s initial value separately because SRSFs do not retain that value.Principal directions can be integrated back into function space.
- Amplitude variability: For the simulated data, the first three vertical singular values are 0.0481, 0.0307, and 0.0055, with negligible remaining variability.The first component captures second-peak height, the second captures first-peak height, and the third has negligible variability.
- Generative models: Generative modeling combines dominant amplitude and phase fPCA coefficients, with joint Gaussian and non-parametric models considered for those coefficients.The reconstructed random function combines an amplitude function with a reconstructed warping function.
4. Modeling Results
Random sampling from phase–amplitude models produced samples visually similar to the original simulated and Berkeley growth data. For Simulated Data 1, the separated model was more consistent with the original data than a Gaussian model imposed directly on f.
- Estimated phase and amplitude models generated random warping functions, amplitude functions, and their compositions for simulated and Berkeley growth data.Each dataset used 35 randomly generated amplitude functions and 35 domain-warping functions before composition.
- The Simulated Data 1 samples were very similar to the original dataset, indicating that the proposed model captured its variability under visual inspection.
- Figure 7 separates random samples into warping functions, amplitude functions, their compositions, and direct Gaussian samples of f.
- The separated model was more consistent with the original Simulated Data 1 data than the fPCA-based Gaussian model imposed directly on f.The non-separated model produced samples with three peaks, higher variation, or weak resemblance to the original data.
- Figure 8 presents analogous phase, amplitude, and composition samples for Simulated Data 2.
- Figure 9 presents analogous phase, amplitude, and composition samples for the Berkeley growth data.
5. Classification Using Phase and Amplitude Models
The paper classifies functional observations by modeling amplitude and phase separately or jointly after alignment and dimensional reduction. Across signatures, iPhone actions, and SONAR signals, these models generally outperform standard L2 classification, with joint models reaching 62% on iPhone data and 54% on SONAR data.
- Classification models: Classification uses training-data principal subspaces for aligned amplitude functions and phase shooting vectors, with Gaussian or kernel-density models fitted to their coefficients.Test samples are projected into each class subspace and assigned according to the largest likelihood.
- Classification models: Joint classification combines amplitude and phase rules under an independence assumption.
- Signature data: 93% and 75% are the best amplitude-only classification rates for signature subjects U1 and U13, respectively, outperforming phase-only and standard L2 rules.Because phase-only classification performs poorly, combining it with amplitude-only classification lowers overall performance.
- iPhone action data: 62% is the best classification performance for the iPhone action data, achieved by combining phase and amplitude models and exceeding standard L2.Both phase-only and amplitude-only rules drastically outperform standard L2.
- Model comparison: Kernel-density models produce only minor iPhone improvements and minimal change for subject U13, suggesting Gaussian assumptions are sufficient in those cases.For signature subject U1, kernel density improves phase-only classification while reducing other methods’ performance, suggesting non-Gaussian warping behavior.
- SONAR data: 54% is the best classification performance for SONAR data, achieved by combining phase and amplitude models and exceeding standard L2.The reported classification rates have low standard deviation, indicating good generalization.
6. Conclusions
The paper proposes joint modeling of phase and amplitude variability through an elastic, metric-based framework, improving functional-data classification across signatures, motion, and SONAR applications. Figure 13 presents aligned and smoothed SONAR functions for nine classes.
- The proposed framework separates phase and amplitude using an elastic distance, then models each component with fPCA and coefficient distributions.The coefficient models may be multivariate Gaussian or nonparametric.
- Separate phase-amplitude models improve classification performance for handwritten signatures, iPhone motion data, and SONAR signals.
- Figure 13 displays aligned and smoothed SONAR functions across 9 classes.