I have a research problem where I want to build an emulator (surrogate/metamodel) for a stochastic computer model for efficient uncertainty & sensitivity analysis. The go-to here is to run the computer model a handful of times and then perform a GP regression to predict the output of the computer model at untried inputs. My computer model is an offshore wind farm simulator.
What's interesting about my computer model is that a lot of the interesting inputs are represented as counts. One such input is the number of spare parts that I have available to fix the wind farm when things break. The number of spares is a non-negative integer, but potentially quite large so possibly in the hundreds. Another discrete input is the number of repair boats I have; this will be a fairly small non-negative integer, in the region of $0 - 20$. There are also lots of continuous inputs, e.g. failure rates of components.
My go-to covariance function (when everything is continuous) is squared exponential $C(x,x') = \sigma^2 \exp \{ -\sum \theta_i d_i^2 \} + \lambda^2 \delta_{x, x'} $ where $d_i$ are the (Euclidean) distances between each element of $x$ and $x'$. Squared exponential is nice because of the properties it has about differentiability of the function we infer. However, I'm not sure if a function with count inputs should automatically possess such properties. However, I understand a function defined on $\mathbb{N}$ could have a generalisation on $\mathbb{R}$(e.g. factorial $\to$ gamma function). This might be enough to justify using a covariance function like squared exponential, Matérn, ..., the usual candidates.
My question is, can I just use any old covariance function for functions with count data inputs or are some covariance functions better suited to the problem? Or will a simple trick like using a different distance metric? (e.g. Manhattan distance might make more sense for count data).
