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A Geometric Approach for Computing Tolerance Bounds for Elastic Functional Data

In this paper, we develop a method for constructing tolerance bounds for functional data with random warping variability. In particular, we present a general technique that is able to appropriately model both amplitude and phase variabilities. In particular, it is desirable to define a generative, probabilistic model for such observations. The model is expected to properly and parsimoniously characterize the nature and 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 many 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 simple simulated example to assess our approach and compare to two existing methods.


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1 Introduction

2 Combined Phase-Amplitude fPCA

3 Functional Tolerance Bounds

4 Simulation Results

5 Applications to Real Data

6 Discussion and Future Work



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