26 Optimal Sensor Placement for Response Predictions Using Local and Global Methods 233 Expression (26.13) is a local approximation because it is derived from a Taylor series expansion of the log-posterior around the “optimal” value θ = θ∗ of the model parameters, which minimizes the fit with the data. However, since the data are not available during this stage, a nominal value of θ is selected based on engineering judgement. The Gaussian approximation is exact for the case of linear models. Then the posterior covariance matrix of predictions g is given by: g| dm = [∇θ g(θ)] θ | dm [∇θ g(θ)] T | θ=θ∗ (26.14) where ∇θ g(θ) is the Ng ×Nθ matrix of sensitivities of the predicted quantity with respect to model parameters (Jacobian matrix). Again the above expression is accurate for predicted quantities that are linear with respect to model parameters. Note that for both methods (global and local) the case of optimizing the design for parameter inference can be recovered as a special case by setting g(θ) =θ. 26.5 Application: Simple Linear Model We demonstrate and verify the two approaches using a simple linear model. For this problem the required integrals can be evaluated analytically due to the simple form of the likelihood function, and therefore the analytical form of the objective function is found. Next we approximate the objective function with the sampling method and compare with the analytical solution as the number of samples is increased. Finally we also calculate the objective function using the sensitivity method. Due to the linearity of the problem we anticipate the sensitivity method to be exact. We consider the simple case of the single-parameter linear model for the measured quantity: dx(θ) =θx (26.15) where θ is the uncertain parameter and x is the measured location specifying the experimental design. The prediction error equation is then: dx =θx +e (26.16) where the error terme is assumed to follow a zero-mean univariate Gaussian distributionN(0,σ 2 e ). Three cases are examined for the standard deviation of the error σe: 1. Constant: σe =σ e 2. Proportional to location: σe =σ ex 3. Inversely proportional to location: σe =σ e/x The likelihood function then takes the known form: p( dx|θ,x) = 1 ,2πσ2 e exp − 1 2 ( dx −θx) 2 σ2 e (26.17) We also assume a Gaussian prior for θ, p(θ) =N(mp,σ 2 p). Since we have only one parameter the predicted quantity is the parameter itself (g =g(θ) =θ) and therefore the expected utility function is simply the expected posterior variance of θ. So our objective is to find the optimal measure location x∗ such that we learn the most about θ. The integrals of Eqs. (26.10)–(26.12) simplify to: E[θ 2| dx]= 1 p( dx) +∞ −∞ θ 2 p( dx|θ)p(θ)dθ (26.18) E[θ| dx]= 1 p( dx) +∞ −∞ θ p( dx|θ)p(θ)dθ (26.19) p( dx) = +∞ −∞ p( dx|θ)p(θ)dθ (26.20)
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