A geometric approach for computing tolerance bounds for elastic functional data

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.

KEYWORDS:

• Compositional noise
• functional data analysis
• functional tolerance bounds
• functional Principal Component Analysis

Acknowledgments

This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. The authors would like to thank Dr. Marc Welliver at Sandia National Laboratories for his technical support during this work. They would also like to acknowledge the Associate Editor and Reviewer for providing constructive comments that have significantly improved the content of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was in part supported by the National Technical Nuclear Forensics Center (NTNFC) of the U.S. Department of Homeland Security (DHS). Sebastian Kurtek's work was partially supported by NSF grants DMS-1613054, CCF-1740761 and CCF-1839252, and by NIH grant R37 CA214955.