256
Y. He et al.
cov
⎛
⎜
⎝
cov
Y
(1)
, Y
(1)
· · · cov
Y
(1)
, Y
(n)
. . .
. . .
. . .
cov
Y
(n)
, Y
(1)
· · · cov
Y
(n)
, Y
(n)
⎞
⎟
⎠
(20.3)
These correlations depend on the absolute distance between the sample points
x
(i)
j − x
(l)
j
and the parameters θ and p.
The likelihood function is defined as Eq. (20.4):
L
Y
(1)
, · · · , Y
(n)
|μ, σ
=
1
2πσ 2
n
2
exp
−
(Y
(i)
− μ)
2
2σ 2
(20.4)
which can be expressed in terms of the sample data as Eq. (20.5)
L =
1
2πσ 2
n
2
|Ψ |
1/2
exp
−
(y − 1μ)
T
Ψ
−1
(y − 1μ)
2σ 2
(20.5)
where Ψ is the correlation matrix between the random variables. Then we take the
natural logarithm to simplify the likelihood function as Eq. (20.6)
ln(L) = −
n
2
ln(2π ) −
n
2
ln
σ
2
− ln|Ψ | −
(y − 1μ)
T
Ψ
−1
(y − 1μ)
2σ 2
(20.6)
And we can obtain the maximum likelihood estimates (MLE) for μ (Eq. 20.7)
and σ
2 (Eq. 20.8):
ˆ
μ =
1
T
Ψ
−1 y
1 T Ψ −1 1
(20.7)
ˆ
σ
2
=
(y − 1μ)
T
Ψ
−1
(y − 1μ)
n
(20.8)
Then the concentrated ln-likelihood function [14] can be deduced as Eq. (20.9)
ln(L) ≈ −
n
2
ln
ˆ
σ
2
−
1
2
ln|Ψ |
(20.9)
The value of this function depends on the unknown parameters θ and p. Therefore
we are supposed to find values for these parameters which maximize Eqs. (20.3),
(20.9) to build the best Kriging model with high accuracy. According to previous
study [15], we can fix the parameter at p = 2, which shows the best smooth correlation
with a continuous gradient,
To test the error of surrogate model, such metrics as maximum relative error
(e max ), average relative error
e avg
and R-square (R
2 ) are employed [16]. And the
results of error analysis are shown in Table 20.3.
Y. He et al.
cov
⎛
⎜
⎝
cov
Y
(1)
, Y
(1)
· · · cov
Y
(1)
, Y
(n)
. . .
. . .
. . .
cov
Y
(n)
, Y
(1)
· · · cov
Y
(n)
, Y
(n)
⎞
⎟
⎠
(20.3)
These correlations depend on the absolute distance between the sample points
x
(i)
j − x
(l)
j
and the parameters θ and p.
The likelihood function is defined as Eq. (20.4):
L
Y
(1)
, · · · , Y
(n)
|μ, σ
=
1
2πσ 2
n
2
exp
−
(Y
(i)
− μ)
2
2σ 2
(20.4)
which can be expressed in terms of the sample data as Eq. (20.5)
L =
1
2πσ 2
n
2
|Ψ |
1/2
exp
−
(y − 1μ)
T
Ψ
−1
(y − 1μ)
2σ 2
(20.5)
where Ψ is the correlation matrix between the random variables. Then we take the
natural logarithm to simplify the likelihood function as Eq. (20.6)
ln(L) = −
n
2
ln(2π ) −
n
2
ln
σ
2
− ln|Ψ | −
(y − 1μ)
T
Ψ
−1
(y − 1μ)
2σ 2
(20.6)
And we can obtain the maximum likelihood estimates (MLE) for μ (Eq. 20.7)
and σ
2 (Eq. 20.8):
ˆ
μ =
1
T
Ψ
−1 y
1 T Ψ −1 1
(20.7)
ˆ
σ
2
=
(y − 1μ)
T
Ψ
−1
(y − 1μ)
n
(20.8)
Then the concentrated ln-likelihood function [14] can be deduced as Eq. (20.9)
ln(L) ≈ −
n
2
ln
ˆ
σ
2
−
1
2
ln|Ψ |
(20.9)
The value of this function depends on the unknown parameters θ and p. Therefore
we are supposed to find values for these parameters which maximize Eqs. (20.3),
(20.9) to build the best Kriging model with high accuracy. According to previous
study [15], we can fix the parameter at p = 2, which shows the best smooth correlation
with a continuous gradient,
To test the error of surrogate model, such metrics as maximum relative error
(e max ), average relative error
e avg
and R-square (R
2 ) are employed [16]. And the
results of error analysis are shown in Table 20.3.