Optimal Sensor Placement for Response Predictions Using Local and Global Methods

Abstract

A Bayesian framework for model-based optimal sensor placement for response predictions is presented. Our interest lies in determining the parameters of the model in order to make predictions about a particular response quantity of interest. This problem is not adequately explored since the majority of currently available literature is focused on parameter inference, rather than prediction inference. The model parameters are inferred by collecting experimental data which depends on the chosen sensor locations. The parameter values are uncertain and their uncertainty is described by a prior probability density function. The measured quantity, or data, is a quantity that can be predicted by the model which depends on both parameters and sensor locations. A prediction error equation is used to describe the discrepancy between the model-predicted measured quantity and the actual data collected from the experiment. The sensor locations are optimized with respect to prediction inference, while the case of parameter inference is derived as a special case under a more general framework. The posterior covariance matrix is used as a measure of uncertainty in the predictions. Two approaches are developed for its calculation, one global and one local. The local approach is based on sensitivities at a fixed value of the parameters, while the global approach uses Monte Carlo sampling and explores the full range of uncertainty in the parameters. A simple numerical example is presented in order to illustrate and verify the two approaches.

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