Response Surface Modeling for Design Space Exploration of Embedded Systems - Multi-objective Design Space Exploration of Multiprocessor SoC Architectures - page 89

The Kriging behavior (smoothness of the model) is controlled by a covariance
function, called the variogram function, which rules how varies the correlation be-
tween the response values in function of the distance between different points. A
function can be rougher or smoother, can exhibit large or small ranges of variation,
can be affected by a certain amount of noise, and all these features can be resumed
in a variogram model.
More precisely, the covariance function Cov( x 1 , x 2 ) only depends on the distance
between two points:

Cov( x 1 , x 2 ) = σ − γ ( x 1 − x 2 ) (4.18)

where γ ( h ) is the variogram function, and σ is the sill, i.e., the asymptotic value of
γ .

There are several variogram types that can be employed: Gaussian, Exponen-
tial, Matèrn, Rational Quadratic. Usually Gaussian is the first (default) choice: the
generated metamodel is infinitely differentiable.
Each variogram function is characterized by three different parameters: range,
sill, and noise.
The variogram range of the covariance function corresponds to a characteristic
scale of the problem. If the distance between two points is larger than the range the
corresponding outcomes should not influence each other (completely uncorrelated).
Range is inversely related to the number of oscillations of the function. Small ranges
mean sudden variations, while large ranges mean very regular trends, with very few
oscillations.
The variogram sill corresponds to the overall variability of the function. The gap
between the values of very distant points should be of the same scale of magnitude
of the sill.
Variogram noise can also be tuned to fit the expected standard error in the obser-
vations. Larger amount of noise will result in smoother responses, while zero noise
means exact interpolation.
Parameters determination can be based on previous knowledge on similar prob-
lems, or may be guessed by following two automatic fitting strategies: maximizing
the Likelihood of the model given the training dataset or maximizing the Leave-One-
Out (LOO) Predictive Probability. The Likelihood of the variogram is the probability
that a statistical distribution associated to the variogram parameters could generate
the given dataset. Likelihood is larger for good fitting models, but penalizes un-
necessarily complex models. Maximum likelihood models are the smoothest models
with best agreement to the dataset. The Leave-One-Out Predictive Probability gives a
measure of the goodness of the model also removing one point at a time in the dataset
and estimating the value at the removed site on the basis of the remaining designs.
Models predicting the smallest errors at “difficult” points are rewarded with high
LOO Predictive Probability. Maximum LOO Predictive Probability privileges good
fitting models which do not loose prediction performance by removing some design.
However, computation of the LOO Predictive Probability can be quite intensive,
more than the computation of the Likelihood.