Definition of Term: Separable

An emulator’s correlation function specifies the correlation between the outputs of a simulator at two different points in its input space. The input space is almost always multi-dimensional, since the simulator will typically have more than one input. Suppose that there are \(p\) inputs, so that a point in the input space is a vector \(x\) of \(p\) elements \(x_1,x_2,\ldots,x_p\). Then a correlation function \(c(x,x^\prime)\) specifies the correlation between simulator outputs at input vectors \(x\) and \(x^\prime\). The inputs are said to be separable if the correlation function has the form of a product of one-dimensional correlation functions:

\[c(x,x^\prime) = \prod_{i=1}^p c_i(x_i,x_i^\prime).\]

Specifying the correlation between points that differ in more than one input dimension is potentially a very complex task, particularly because of the constraints involved in creating a valid correlation function. Separability is a property that greatly simplifies this task.

Separability is also used in the context of emulators with multiple outputs. In this case the term ‘separable’ is typically used to denote separability between the outputs and the inputs, i.e. that all the outputs have the same correlation function \(c(x,x')\) (which is often itself separable as defined above). The general covariance then takes the form:

\[\text{Cov}[f_u(x), f_{u'}(x')] = \sigma_{uu'}c(x,x')\]

where \(u\) and \(u'\) denote two different outputs and \(\sigma_{uu'}\) is a covariance between these two outputs.