A geometric approach for computing tolerance bounds for elastic functional data.

Area of Science:

• Statistics
• Functional Data Analysis
• Time Series Analysis

Background:

• Functional data analysis is crucial for understanding complex signals in fields like process control and biomedical monitoring.
• Existing methods may not adequately capture random warping variability in functional data.
• Identifying deviations from baseline data is essential for quality control and disease diagnosis.

Purpose of the Study:

• To develop a novel method for constructing tolerance bounds for functional data with random warping variability.
• To define a generative, probabilistic model for amplitude and phase components of functional observations.
• To create tolerance bounds capable of identifying deviations in both amplitude and phase.

Main Methods:

• A generative, probabilistic model for amplitude and phase components was defined.
• Two types of functional tolerance bounds were developed: one using bootstrap on the geometric space of amplitude and phase functions, and another using functional Principal Component Analysis (fPCA).
• A simulated example was used to assess the proposed approach and compare it with existing methods.

Main Results:

• The proposed method successfully constructs tolerance bounds for functional data with random warping.
• The developed bounds can identify when data exceeds established amplitude and/or phase limits.
• The approach demonstrated effectiveness in simulated scenarios, outperforming or matching existing methods.

Conclusions:

• The new method provides a robust framework for tolerance bound construction in functional data with warping.
• This approach has significant implications for process control, enabling early detection of production anomalies.
• The method is also valuable for biomedical applications, aiding in the identification of abnormal physiological signals for disease monitoring.