26 Optimal Sensor Placement for Response Predictions Using Local and Global Methods 231 Parameter space Measured data space Prediction space m = 1 m = 2 Size of each green circle = posterior parameter uncertainty for particular data sample Average size = objective for parameter inference Size of each blue circle = posterior prediction uncertainty from particular parameter posterior Average size = objective for prediction inference Different designs explore different sub-domains of the measured data space. The red disturbances represent prediction error. Fig. 26.1 Flow of information from measured data to parameters and predictions Our choice of objective function Um is motivated by the fact that we wish to develop both parameter and prediction inference under the same framework. As already stated in the introduction, using the entropy (or any other related information-theoretic quantities such as Kullback-Leibler divergence or Mutual Information) has the drawback of requiring knowledge of the full posterior PDF for the quantity we wish to infer. For parameter inference from measured data (inverse problem), the posterior is given by Bayes’ rule (26.2). However, for prediction inference the posterior PDF of predictions g is required which is related to the posterior PDF of parameters through the known formula of transformation of variables fromθ tog(θ): p(g| d) =p(θ(g)| d) Jg−1 (26.4) where the posterior of the parameters p(θ(g)| d) in the right-hand side is written solely as a function of g using the inverse transformation g−1 fromg to θ, and Jg−1 is the Jacobian matrix of the inverse transformation. The transformation needs to be a unique mapping representing a one-to-one relation between g and θ. The inverse transformation g−1 fromg to θ and its Jacobian can be calculated only for very simple relations between g and θ. These factors severely limit the applicability of the method to very simple cases. For that reason we turn to the posterior covariance matrix to describe posterior uncertainty in a quantity, and specifically to its determinant as a scalar measure of uncertainty. Using the covariance matrix sidesteps the above mentioned issues because it does not require knowledge of the full functional form of the posterior PDF of the predicted quantity, but only the forward relationg =g(θ). Also, its calculation is relatively straightforward and can easily accommodate for both prediction and parameter inference. Therefore, we chose the objective function Um in (26.3) to be the determinant of the posterior covariance matrix of the prediction g, after measured data dm have been collected from design m: Um = g| dm (26.5) Next we demonstrate how the posterior covariance matrix can be evaluated using two different approaches, one global and one local.
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