Uncertainty quantification for the distribution-to-warping function regression method used in distribution reconstruction of missing structural health monitoring data

Abstract

Data loss is a common problem of structural health monitoring and adversely affects many structural health monitoring applications. Tremendous progress in missing structural health monitoring data imputation has been made in recent years, forming an important part of sensor validation. Most of the imputed data are based on estimates obtained by data-driven statistical or machine learning models; quantifying their estimation uncertainties is significant in terms of being able to perform accuracy assessments and providing more insights into the imputed data. However, this procedure has been surprisingly neglected in the structural health monitoring community. This article focuses on uncertainty quantification for the distribution-to-warping function regression method (an indirect distribution-to-distribution regression method) used in reconstructing distributions of missing data. The distribution-to-warping function regression method belongs to the framework of functional data analysis as both the predictor and response are continuous functions. The challenge of performing uncertainty quantification for the distribution-to-warping function regression method comes not only from the functional nature of warping functions but also from their inherent constraints. To this end, a functional transformation is employed to transform warping functions into a vector space, and the confidence estimation for the regression operator is conducted in the vector space based on functional principal component analysis and bootstrapping. Then, the confidence region of the conditional expectation of missing distribution (caused by data loss) can be further estimated and visualized. In addition, a calibration processing procedure is also considered to obtain improved estimates of the confidence intervals with a better coverage accuracy under the desired probability. Simulation studies are conducted to validate and illustrate the proposed method, and then, it is applied to field strain monitoring data.