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Hae-Jin ChoiSchool of Mechanical Engineering,
Chung-Ang University
7. Kriging – Gaussian Process Model(Ch.11. 5 Experiments with Computer Models)
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Introduction to Metamodel— Metamodel is ‘model of a model’— Computer simulation model (such as FEM) is a model of a physical
system.— Metamodeling is modeling the computer simulation model with a
mathematical construct. (i.e., model of a simulation model)— Why? Metamodel
— Computer simulation tends to require a large amount of computing resources (i.e., computationally expensive)
— Design optimization, which is an iterative process to find an optimum, cannot be performed with computationally expensive simulation model
— A metamodel reduces the computing expenses significantly, since it is an mathematical equation.
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Metamodeling of computer experiments— Two types of computer simulation models
— Stochastic simulation model : output (response) is random variable— Deterministic simulation model : output (response) is deterministic by
mathematic behind the simulation. (i.e., NO CHANGE in INPUT then NO CHANGE in OUTPUT)
— For the optimization with Stochastic simulation model, traditional DOE techniques and RSM methodology may be applied.
— For the optimization with Deterministic simulation model, those are not available since it does not include randomness in response.
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Metamodeling of computer experiments— Design of computer experiments: Space-filling methods
— Latin Hypercube Design— Sphere-Packing Design— Uniform Design— Maximum Entropy Design
— Modeling of computer experiments— Kriging – Gaussian Process Model
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Kriging – a metamodeling method— Kriging (Gaussian Process Model) is a representative metamodeling
technique originated from Geostatistics. (D.G. Krige,South African geologist)
Original elevation data and sample points: 300 randomly placed points where elevation data was sampled.
Predicted elevation with 300 randomly placed points where elevation data was sampled.
(Source: http://casoilresource.lawr.ucdavis.edu/drupal/node/442) 5
Kriging Theory— Assumption:
— Response function is composed of a regression model and stochastic process.
1 2
1 2
[ ( ), ( ),..., ( )] : ( 1) vector of regression functions
[ , ,..., ] : ( 1) vector of unknown coefficients
Tp
Tp
where f f f p
pb b b
= ´
= ´
f(x) x x x
β
Kriging model
Z(s1)
Z(s2)
Z(s3)
Z(s4)Regression model
s1 s2 s3 s4
( ) ( ) Y Z= +Tx f(x) β x
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Kriging Theory – continued..— In many case, is assumed to be a constant (simple or
ordinary kriging), or it may be first or second order polynomial.— Z(x) is assumed to be a Gaussian process with zero mean, E[Z(x)]=0,
and covariance, Cov [Z(xi), Z(xj)].— While globally approximates the design space, Z(x) creates
“localized” deviations so that the kriging model interpolate nsampled data.
— The covariance matrix of the samples is
Tf(x) β
Tf(x) β
2
s s
[ ( ), ( )] ([ ( , )])
is an (n n ) correlation matix with ones along the diagonal
[ ( , )] is correlation function
i j i j
i j
Cov Z Z R s swhere
R s s
s=
´
s s R
R
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Kriging Theory – correlation function— Correlation function, , in the correlation matrix, R, is user
defined function and a variety of correlation function exist.— Gaussian correlation function (the mostly preferred correlation
function) is
( , )i jR s s
2
1( , ) exp
: unknown correlation parameters : the number of design variables and : components of the samples, and
k k
k k
ni j i j
kk
k
i j th i j
R
where
nk
q
q
=
é ù= - -ê ú
ê úë ûås s s s
s s s s
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Kriging Theory – correlation function— Various correlation functions
( )e, k kk i jijwher d = -s s
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Kriging Theory – correlation function
— Various correlation functions for0 ≤ ≤ 2
— Dash, full, and dashed-dotted line for
kijd
0.2, 1, 5kq = kijd k
ijd
kijd
kijd
0.5
0.2
1
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Kriging for Prediction
— Predicted estimates at the untried point x is
Here,
ˆ ˆˆ( )
ˆ
Ywhere
=
=
T T -1
T -1 -1 T -1
x f(x) β + r(x) R (Y -Fβ)
β (F R F) F R Y
1 2
2
2
: correlation matrix[ , ,..., ]
[ ( , ), ( , ),..., ( , )][ , ,..., ]
s
s
ns
nT T T T
n
Ts s s
R R Ry y y
=
=
=
1
1
RF f(s ) f(s ) f(s )r(x) x s x s x sY
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Kriging for Prediction
— Predicted estimates at the untried point x is
ˆ ˆˆ( )Y = T T -1x f(x) β + r(x) R (Y -Fβ)
Z(s1)
Z(s2)
Z(s3)
Z(s4)
s1 s2 s3 s4x12
Kriging Theory – correlation parameter
— How to determine the correlation parameters, ?— Maximizing Maximum Likelihood Estimator (MLE) to find the
optimum correlation parameters.
2
2
ˆln lnmaximize
2
ˆ ˆˆ (estimated process variance) /
s z
Tz s
n
where
n
s
s
é ù+-ê úë û
=
θ
-1
R
(Y -Fβ) R (Y -Fβ)
1 2[ , ,..., ]nq q q=θ
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Kriging – different types of Kriging— Simple kriging assumes a known constant trend: .— Ordinary kriging assumes an unknown constant trend: .— Universal kriging assumes a general polynomial trend model, such as
linear trend model .
ˆ ˆˆ( )Y = T T -1x f(x) β + r(x) R (Y -Fβ)
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Kriging Example
Ordinary Kriging Example
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Kriging Example
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Kriging Example
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Kriging Example
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Kriging Example
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Kriging Example
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Kriging Example
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Latin Hypercube Design— A space-filling sampling technique with random permutation
(McKay, Conover, and Beckman,1979)— X1=[2,5,1,4,3,6], X2=[4,5,6,3,2,1]
Random field with two variables (uniform distribution)
Random field with two variables (normal distribution)
x1
x2
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Maximum Entropy Design— Popular space-filling sampling technique (Shewry and Wynn 1987)— Entropy : a measure of the amount of information contained in the
distribution of a data set— The maximum entropy is achieve by maximizing determinant
of()
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References— Kriging
— Sacks J., W. J. Welch, T. J. Mitchell, and H.P. Wynn, (1989) “Design and Analysis of Computer Experiments,” Statistical Science, Vol. 4, No. 4, pp. 409-435, Simpson, T., T.M, Mauery, J. J. Korte, and F. Mistree, “Kriging Models for Global Approximation in Simulation-Based Multidisciplinary Design Optimization,” AIAA Journal, Vol. 39, No. 12, 2001
— Latin Hypercube Design— McKay, M. D., Beckman, R. J. and Conover, W. J. (1979), “A comparison of
three methods for selecting values of input variables in the analysis of output from a computer code,” TechnometricsVol. 21, pp. 239-245.
— http://prod.sandia.gov/techlib/access-control.cgi/1998/980210.pdf— Maximum Entropy Design
— Shewry, M. C. and Wynn, H. P. (1987) “Maximum entropy sampling,” J. Appl. Statist., Vol. 14, pp. 165-170.
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