Bayesian Information Fusion of Multmodality Nondestructive Measurements for Probabilistic Mechanical Property Estimation

Missing data occur when no data value is available for the variable in an observation. Missing data are fairly common in engineering due to incomplete record keeping, less rigorous testing, human or machine error, data censoring, etc. However, it can significantly influence the conclusions drawn from the data especially when only sparse data are available. Unfortunately, many current practices either throw away all subjects with incomplete data, or impute missing values with a population mean or some other fixed value, then proceed with the analysis. The first approach is bad because a potentially large amount of useful information are thrown away. The second is statistically incorrect, as it says people are certain about the values of the missing data when they are not actually observed. Therefore, a more rigorous methodology needs to be utilized to make the most of the available information for inference and reliability assessment while accounting for the effect of missing data.

In this research, Bayesian data augmentation method is adopted and implemented for prediction with missing data. The complete likelihood is defined as the product of the likelihoods of two factors: the likelihood of the complete data, and the inclusion matrix which indexes which potential data are observed. The appropriate likelihood for Bayesian inference is a marginal probability of the observed variables after integrating out the missing variables. The data augmentation process is conducted through Bayesian inference with missing data assuming the multivariate normal distribution. Gibbs sampling is used to draw posterior simulations of the joint distribution of unknown parameters and unobserved quantities. The missing elements of the data are sampled conditional on the observed elements. To account for the uncertainty due to missing data, the multiple imputation is carried out to create more than one set of replacements for the missing values in a dataset. Finally, the distribution of model parameters and variables with missing data can be obtained for reliability analysis.

Two examples are given to illustrate the engineering application of Bayesian inference with missing data. The first example is to predict the yield strength of aging pipeline by fusing the incomplete surface information. The input surface variables are yield strength obtained from surface indentation technique, chemical composition (P, Cr, Cu, Si), grain size, hardness, grade, and steel type. In this research, it is considered that not every variable is observed for each pipe. The predictive performance is compared among direct surface indentation technique, linear regression with complete data and Bayesian inference with missing data. The second example is to predict the fatigue life of corroded steel reinforcing bar from the input dataset including corrosion degree, depth and length of corrosion pit, stress range and nominal diameter. The dataset is incomplete with some of the observations missing. The predicted fatigue lives are compared with experiment and the results of linear regression with complete data. Both examples demonstrate that the Bayesian method can deal with missing data problem properly and show good predictive performance.