Procedure: Predict simulator outputs using a BL emulator¶
Description and Background¶
A Bayes linear (BL) emulator is a stochastic representation of knowledge about the outputs of a simulator based on a second-order belief specification for an unknown function. The unknown function in this case is the simulator, viewed as a function that takes inputs and produces one or more outputs. One use for the emulator is to predict what the simulator would produce as output when run at one or several different points in the input space. This procedure describes how to derive such predictions in the case of a BL emulator such as is produced by ProcBuildCoreBL.
Inputs¶
- An adjusted Bayes linear emulator
- A single point \(x^\prime\) or a set of points \(x^\prime_1, x^\prime_2,\ldots,x^\prime_{n^\prime}\) at which predictions are required for the simulator output(s)
Outputs¶
- In the case of a single point, outputs are the adjusted expectation and variance at that point
- In the case of a set of points, outputs are the adjusted expectation vector and adjusted variance matrix for that set
Procedure¶
The adjusted Bayes linear emulator will supply the following necessary pieces of information:
- An adjusted expectation \(\text{E}_F[\beta]\) and variance \(\text{Var}_F[\beta]\) for the trend coefficients \(\beta\) given the model runs \(F\)
- An adjusted expectation \(\text{E}_F[w(x)]\) and variance \(\text{Var}_F[w(x)]\) for the residual process \(w(x)\), at any point \(x\), given the model runs \(F\)
- An adjusted covariance \(\text{Cov}_F[\beta,w(x)]\) between the trend coefficients and the residual process
The adjusted expectation and variance at the new point \(x'\) are obtained by application of ProcBLAdjust to the emulator as described below.
Predictive mean (vector)¶
Then our adjusted beliefs about the expected simulator output at a single further input configuration \(x'\) are given by:
In the case of a set of additional inputs \(X'\), where \(X'\) is the matrix with rows \(x^\prime_1, x^\prime_2,\ldots,x^\prime_{n^\prime}\), the adjusted expectation is:
where \(f(X)\) is the \(n^\prime\)-vector of simulator values with elements \((f(x^\prime_1), f(x^\prime_2),\ldots, f(x^\prime_{n^\prime}))\), \(H(X')\) is the \(n^\prime\times q\) matrix with rows \(h(x^\prime_1), h(x^\prime_2),\ldots, h(x^\prime_{n^\prime})\), and \(w(X)\) is the \(n^\prime\)-vector with elements \((w(x^\prime_1), w(x^\prime_2),\ldots, w(x^\prime_{n^\prime}))\).
Predictive variance (matrix)¶
Our adjusted variance of the simulator output at a single further input configuration \(x'\) is given by:
In the case of a set of additional inputs \(X'\), the adjusted variance is: