Uncertainty Quantification Methodologies for Functional Data in Biomechanical Applications

Abstract

Massive data are feasibly collected or generated with the rapid development of sensing, high computing and computer simulation technologies. Among various types of data, functional data plays an important role in tracking system behaviors in various applications. However, functional data often shows complex data uncertainty caused by multiple factors such as experimental conditions, subject characteristics or computer simulation settings. To better understand the system behaviors for decision-making, new methodologies are expected to systematically quantify the uncertainty of functional data. Specifically, three major research issues are studied in the dissertation. First, the problem of constructing confidence bands (also known as corridors in biomechanical applications) of univariate functional signals is discussed. An effective method is developed for confidence bands generation that applies principal component analysis (PCA). Rather than using existing empirical models to account for the effects of subject variables on functional responses, linear regression models are further built to model the relationship between extracted PC features and subject variables, which makes the effects of subject variables interpretable. The advantage of the resultant confidence bands is reflected by the narrower bands than those generated by existing techniques while keeping a high coverage rate of sampled experimental functional data. Second, a generic method is developed to construct confidence bands for bivariate functional data. The effect of subject variables is quantified by non-parametric B-spline fitting and a polynomial regression model, which is capable of capturing non-linear dependencies between the subject variables and functional responses. Moreover, a Gaussian process model is developed to model the complicated covariance structure, which can fully consider between-subject and within-subject variability, auto-correlation between time points and cross-correlation between bivariate functional responses. Therefore, the constructed confidence bands can effectively capture the bivariate functional profile shape and functional variation patterns. As a byproduct, the developed model is effectively used for testing outliers of abnormal functional responses based on the property of the developed Gaussian process model. Third, a method to search for the optimal system design using an inexact computer simulation model with uncertainty quantification is developed. The uncertainty is quantified by specifying feasible regions instead of building a full probabilistic model, which makes the proposed method to be applicable when an emulator is not available. The use of feasible regions also narrows the potential simulation parameter set and reduces the computation load in generating simulation runs. An robust optimization problem is formulated and integrated with the model calibration. The proposed point and interval estimators of the optimal design are mathematically proved to have consistency and coverage properties.