Civil Engineering Reference
In-Depth Information
Table 4.5
Covariance
functions for kriging
metamodels.
exp
2
ʸ
2
h
2
Gaussian
g
(
h
)
=
−
1
3
ʸ
2
exp
√
5
√
5
|
h
|
ʸ
5
h
2
|
h
|
ʸ
Matérn
ʽ
=
5
/
2
g
(
h
)
=
+
+
−
1
exp
√
3
√
3
|
h
|
|
h
|
Matérn
ʽ
=
3
/
2
g
(
h
)
=
+
−
ʸ
ʸ
g
(
h
)
=
exp
−
|
h
|
ʸ
Exponential
exp
b
where 0
<b
|
h
|
ʸ
Power-exponential
g
(
h
)
=
−
≤
2
Figure
4.7
compares different covariance kernels for the same value of
ʸ
, also
known as characteristic length scales, which generally range over [0, 2]. The covari-
ance kernel parameters have a physical interpretation such that larger parameters
indicate that the response surface changes slowly along the respective variable, and
the surface changes quickly for smaller covariance parameters. Depending on the
value of
ʸ
, the kernels have different shapes. Consequently, the spatial influence
of a point at a distance
h
from another point is also different. There are also some
finer differences of these kernels as well. The Gaussian kernel is smooth and has
continuous derivatives at all orders. When the Matérn family of covariance kernels is
specified by a parameter
ʽ
1
/
2, where
k
is a non-negative integer, there is an
analytics expression. The Matérn kernels with
ʽ
=
k
+
5
/
2 are two popular
versions; the Matérn 3/2 kernel is once differentiable and the Matérn 5/2 kernel is
=
3
/
2 and
ʽ
=
Gaussian
Matern 3/2
Matern 5/2
0.01
0.1
0.5
1
2
0.01
0.1
0.5
1
2
0.01
0.1
0.5
1
2
0
2
4
6
8
10
0
2
4
6
8
10
0
2
4
6
8
10
h
h
h
Exponential
Power−exponential
0.01
0.1
0.5
1
2
0.01
0.1
0.5
1
2
0
2
4
6
8
10
0
2
4
6
8
10
h
h
Fig. 4.6
Examples of different covariance kernels used in kriging for a series of parameter values
for
ʸ
; the power-exponential function used
b
=
2. Each
curve
shows the spatial influence of a point
at a distance of
h
.
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