Methodology - Design of Experiments for Reinforcement Learning - page 83

geological surfaces, where the goal is to exactly interpolate a 3-dimensional surface;
also see the seminal works by Matheron ( 1963 ) and Cressie ( 1993 ). Sacks et al.
( 1989 ) developed this modeling approach specifically for deterministic computer
models, which was then extended to random or stochastic models (van Beers and
Kleijnen 2003 ). The kriging formulations herein are described in a number of works,
including Qu and Fu ( 2013 ), Chen and Kim ( 2014 ), Ankenman et al. ( 2010 ), Xie
et al. ( 2010 ), and Roustant et al. ( 2012a ), and the reader is directed to these sources
for additional reading, as well as Rasmussen and Williams ( 2006 ) for an authoritative
review of Gaussian process models.
Let y be a response value to a d -dimensional parameter space x

= ( x 1 , ... , x d ) ∈

n × d denote a set of n (row-wise) parameter vectors x , also called
the experimental design in this context, where the i th design point is denoted by x ( i ) .
Similarly, let y

d . Let X

D ↂ R ∈ R

= y ( x (1) ), ... , y ( x ( n ) ) denote the corresponding vector of response
values for the n design points, where y ( x ( i ) ) is the response value of the i th design
point.

4.2.2 Deterministic Kriging

Deterministic kriging assumes that the response at a point x has no intrinsic variation
(Ankenman et al. 2010 ) and represents the response surface by:

f ( x ) β

Y ( x ) = + M ( x ) (4.6)

where f ( x ) is a vector of trend functions and M ( x ) is a centered square-integrable
process that accounts for spatial correlation or dependence. Simple kriging uses a
constant trend value such that f ( x )

= μ ( x ), whereas universal kriging uses trend

j = 1 ʲ j f j ( x ), where f j is a fixed basis function
(e.g., first order, second order, etc.) and ʲ j is an unknown trend coefficient.
The predicted mean response at design point x and its variance for simple kriging,
respectively, are:

p

functions that take the form of μ ( x ) =

Y ( x )

) − M

= + · −

μ ( x ) M ( x , ( y μ ) (4.7)

Va r

Y ( x )

) − M M ( x ,

= − · ·

( x , x ) M ( x , ) (4.8)

and for universal kriging are:

Y ( x )

f ( x ) β F β )

) − M

= + · −

M ( x , ( y (4.9)

Va r

Y ( x )

) − M M ( x ,

= − · ·

( x , x ) M ( x , ) (4.10)

+ f ( x ) − F F − M F − 1 f ( x ) − F ) − M ) − M

M ( x , · M ( x , ·