A Geometric Approach for Computing Tolerance Bounds for Elastic Functional Data

TL;DR

This paper introduces a geometric probabilistic model for constructing tolerance bounds for functional data with phase and amplitude variability, useful in process control and disease monitoring.

Contribution

It proposes two novel methods for tolerance bounds based on bootstrap and PCA within a geometric framework, addressing both amplitude and phase variability.

Findings

The bootstrap-based bounds effectively detect deviations in functional data.

PCA-based bounds provide a computationally efficient alternative.

The methods outperform existing approaches in simulated comparisons.

Abstract

We develop a method for constructing tolerance bounds for functional data with random warping variability. In particular, we define a generative, probabilistic model for the amplitude and phase components of such observations, which parsimoniously characterizes variability in the baseline data. Based on the proposed model, we define two different types of tolerance bounds that are able to measure both types of variability, and as a result, identify when the data has gone beyond the bounds of amplitude and/or phase. The first functional tolerance bounds are computed via a bootstrap procedure on the geometric space of amplitude and phase functions. The second functional tolerance bounds utilize functional Principal Component Analysis to construct a tolerance factor. This work is motivated by two main applications: process control and disease monitoring. The problem of statistical analysis and modeling of functional data in process control is important in determining when a production has moved beyond a baseline. Similarly, in biomedical applications, doctors use long, approximately periodic signals (such as the electrocardiogram) to diagnose and monitor diseases. In this context, it is desirable to identify abnormalities in these signals. We additionally consider a simulated example to assess our approach and compare it to two existing methods.