Modeling and Multivariate Methods - SAS

298 Modeling Relationships With Gaussian Processes Chapter 11
The Gaussian Process Report
Functional interaction effects, computed in a similar way, are also listed in the Model Report table.
Summing the value for main effect and all interaction terms gives the Total Sensitivity, the amount of
influence a factor and all its two-way interactions have on the response variable.
Mu, Theta, and Sigma
The Gaussian correlation structure uses the product exponential correlation function with a power of 2 as
the estimated model. This comes with the assumptions that Y is Normally distributed with mean μ and
covariance matrix σ 2 R. The R matrix is composed of elements
r ij
–θ k
( x ik
– x jk
) 2
= exp
k
In the Model report, μ is the Normal distribution mean, σ 2 is the Normal Distribution parameter, and the
Theta column corresponds to the values of θ k in the definition of R.
These parameters are all fitted via maximum likelihood.
Note: If you see Nugget parameters set to avoid singular variance matrix, JMP has added a ridge
parameter to the variance matrix so that it is invertible.
The Cubic correlation structure also assumes that Y is Normally distributed with mean μ and covariance
matrix σ 2 R. The R matrix is composed of elements
r ij
=
∏
ρ( d ; θ k
)
k
d = x ik –x jk
where
ρ( d;
θ)
=
1 – 6( dθ) 2 + 6( d θ) 3 , d
1
≤ -----
2θ
21 ( – d θ) 3 1
, ----- < d
2θ
0,
1
-- < d
θ
1
≤ --
θ
For more information see Santer (2003). The theta parameter used in the cubic correlation is the reciprocal
of the parameter used in the literature. The reason is so that when a parameter (theta) has no effect on the
model, then it has a value of zero, instead of infinity.