Parameter Estimation of Autoregressive-Exogenous and Autoregressive Models Subject to Missing Data Using Expectation Maximization

Abstract

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Missing observations may present several problems for statistical analyses on datasets if they are not accounted for. This paper concerns a model-based missing data analysis procedure to estimate the parameters of regression models fit to datasets with missing observations. Both autoregressive-exogenous (ARX) and autoregressive (AR) models are considered. These models are both used to simulate datasets, and are fit to existing structural vibration data, after which observations are removed. A missing data analysis is performed using maximum-likelihood estimation, the expectation maximization (EM) algorithm, and the Kalman filter to fill in missing observations and regression parameters, and compare them to estimates for the complete datasets. Regression parameters from these fits to structural vibration data can thereby be used as damage-sensitive features. Favorable conditions for accurate parameter estimation are found to include lower percentages of missing data, parameters of similar magnitude with one another, and selected model orders similar to those true to the dataset. Favorable conditions for dataset reconstruction are found to include random and periodic missing data patterns, lower percentages of missing data, and proper model order selection. The algorithm is particularly robust to varied noise levels.

Keywords

• regression analysis
• vibration
• damage assessment
• probability
• estimation
• structural dynamics