Discussion: Nugget effects in uncertainty and sensitivity analyses¶
Description and Background¶
The possibility of including a nugget term in the correlation function for a Gaussian process emulator is discussed in the alternatives page on emulator prior correlation function (AltCorrelationFunction). In the procedure pages for uncertainty analysis (ProcUAGP) and variance based sensitivity analysis (ProcVarSAGP) using a GP emulator, concerned respectively with uncertainty analysis and variance based sensitivity analysis for the core problem, closed form expresssions are given for various integrals for some cases where a nugget term may be included. This page provides a technical discussion of the reason for some apparent anomalies in those formulae.
Discussion¶
The closed form expressions are derived for the generalised Gaussian correlation function with nugget (see AltCorrelationFunction)
where \(I_{x=x'}\) equals 1 if \(x=x'\) but is otherwise zero, and \(\nu\) represents the nugget term. They also assume a normal distribution for the uncertain inputs.
In both ProcUAGP and ProcVarSAGP a number of integrals are computed depending on \(c(x,x')\), and we might expect the resulting integrals in each case to have two parts, one multiplied by \(\nu\) and the other multiplied by \((1-\nu)\). In most cases, however, only the second term is found.
The reason for this is the fact that the nugget only arises when the two arguments of the correlation function are equal. The integrals all integrate with respect to the distribution of the uncertain inputs \(x\), and since this is a normal distribution it gives zero probability to \(x\) equalling any particular value. We therefore find that in almost all of these integrals the nugget only appears in the integrand for a set of probability zero, and so it does not appear in the result of evaluating that integral. The sole exception is the integral denoted by \(U_p\), where the nugget does appear leading to a result \(U_p=\nu+(1-\nu)=1\).
The heuristic explanation for this is that a nugget term is a white noise addition to the smooth simulator output induced by the generalised Gaussian correlation term. If we average white noise over even a short interval the result is necessarily zero because in effect we are averaging an infinite number of independent terms.
The practical implication is that the nugget reduces the amount that we learn from the training data, and also reduces the amount that we could learn in sensitivity analysis calculations by learning the values of a subset of inputs.