  Nowadays stochastic computer simulations with both numeral and distribution
inputs are widely used to mimic complex systems which contain a great deal of
uncertainty. This paper studies the design and analysis issues of such computer
experiments. First, we provide preliminary results concerning the Wasserstein
distance in probability measure spaces. To handle the product space of the
Euclidean space and the probability measure space, we prove that, through the
mapping from a point in the Euclidean space to the mass probability measure at
this point, the Euclidean space can be isomorphic to the subset of the
probability measure space, which consists of all the mass measures, with
respect to the Wasserstein distance. Therefore, the product space can be viewed
as a product probability measure space. We derive formulas of the Wasserstein
distance between two components of this product probability measure space.
Second, we use the above results to construct Wasserstein distance-based
space-filling criteria in the product space of the Euclidean space and the
probability measure space. A class of optimal Latin hypercube-type designs in
this product space are proposed. Third, we present a Wasserstein distance-based
Gaussian process model to analyze data from computer experiments with both
numeral and distribution inputs. Numerical examples and real applications to a
metro simulation are presented to show the effectiveness of our methods.
