[Paper Review] Principal Nested Spheres for Time Warped Functional Data Analysis

Xiaosun Lu, J. S. Marron|arXiv (Cornell University)|Apr 25, 2013

Morphological variations and asymmetry4 citations

TL;DR

This paper proposes Principal Nested Spheres (PNS) for analyzing horizontal variation in time-warped functional data by leveraging the spherical geometry of square-root velocity functions (SRVFs). By modeling warping functions on a Hilbert sphere and applying PNS to decompose variability, the method achieves superior signal compression and interpretability compared to conventional FPCA and tangent-space PGA, especially when horizontal variation is substantial.

ABSTRACT

There are often two important types of variation in functional data: the horizontal (or phase) variation and the vertical (or amplitude) variation. These two types of variation have been appropriately separated and modeled through a domain warping method (or curve registration) based on the Fisher Rao metric. This paper focuses on the analysis of the horizontal variation, captured by the domain warping functions. The square-root velocity function representation transforms the manifold of the warping functions to a Hilbert sphere. Motivated by recent results on manifold analogs of principal component analysis, we propose to analyze the horizontal variation via a Principal Nested Spheres approach. Compared with earlier approaches, such as approximating tangent plane principal component analysis, this is seen to be the most efficient and interpretable approach to decompose the horizontal variation in some examples.

Motivation & Objective

• To address the limitation of traditional Functional Principal Component Analysis (FPCA) in capturing horizontal (phase) variation in functional data.
• To develop a more efficient and interpretable method for analyzing horizontal variation separated via domain warping.
• To leverage the intrinsic spherical structure of warping functions in the square-root velocity function (SRVF) representation.
• To compare PNS with existing approaches like FPCA and tangent-space PGA in terms of signal compression and interpretability.
• To establish PNS as the optimal method for horizontal variation analysis when phase variation is dominant.

Proposed method

• Represent warping functions using the square-root velocity function (SRVF) to map them onto a Hilbert sphere.
• Apply Principal Nested Spheres (PNS) to decompose horizontal variation on the spherical manifold, preserving geodesic structure.
• Use the Fisher–Rao metric for domain warping to ensure warping-invariance, enabling consistent alignment of functions.
• Compare PNS with FPCA of aligned functions and warping functions, and with tangent-space PGA on the sphere.
• Utilize the Karcher mean of warping functions to define a reference point for spherical analysis.
• Visualize results via score scatter plots and scree plots to assess variance explained and component interpretability.

Experimental results

Research questions

• RQ1 How does PNS compare to FPCA and PGA in capturing horizontal variation in functional data with large phase shifts?
• RQ2 Can PNS achieve better signal compression than conventional FPCA and tangent-space PGA in the presence of substantial horizontal variability?
• RQ3 Does PNS provide more interpretable components than FPCA when analyzing warping functions or aligned functions?
• RQ4 To what extent does the spherical geometry of SRVFs enhance the decomposition of horizontal variation?
• RQ5 Is PNS more effective than existing manifold-based methods when horizontal variation is one-dimensional or low-rank?

Key findings

• PNS explains a higher proportion of variance using fewer components than FPCA and PGA, with the first PNS component capturing more than 90% of the horizontal variation in the toy example.
• The first two PNS components explain the same variability captured by the first three PGA components, demonstrating superior signal compression.
• PNS1 simultaneously captures both peak location and inter-peak distance modes, reducing the need for multiple components.
• The score scatter plot of PNS components shows a compact, circular pattern consistent with a one-dimensional warping structure, confirming low-rank horizontal variation.
• PNS outperforms FPCA of warping functions and FPCA of aligned functions in both variance explained and interpretability, especially when horizontal variation is dominant.
• Scree plots confirm that PNS achieves the highest first-component variance explanation (red line in Figure 6), indicating optimal signal compression.