This talk was presented as part of the panel discussion on Day 2 of the MUMS Opening Workshop.

1. Facts about some statistical models in
calibrating imperfect mathematical models
Mengyang Gu
Department of Applied Mathematics and Statistics
Johns Hopkins University

2. The calibration problem
A mathematical model fM(x, u), where x ∈ X is a
p-dimensional vector of the observable inputs and u is a
q-dimensional vector of the unobservable parameters. The
experimental/field data are denoted as
yO = yO(x1), ..., yO(xn)
T
.
When the mathematical model is imperfect, it is usual to
model
yO
(x) = fM
(x, u) + b(x) + ,
with being random errors. The reality is
yR(x) = fM(x, u) + b(x).
How to model the bias function b(·)?

3. Outline
1. Gaussian stochastic process (GaSP) calibration
2. L2 calibration
3. Scaled Gaussian stochastic process (S-GaSP) calibration

4. Outline
1. Gaussian stochastic process (GaSP) calibration
2. L2 calibration
3. Scaled Gaussian stochastic process (S-GaSP) calibration

5. Gaussian stochastic process calibration
Assume the trend and intercept are properly modeled in the
mathematical model.
Kennedy and O’Hagan (2001) modeled b(·) via a
stationary Gaussian stochastic process (GaSP), meaning
that any marginal distribution
(b(x1), ..., b(xn))T
∼ MN(0, σ2
R),
where Ri,j = K(xi, xj), with K(·, ·) being a kernel function.
This statistical model for the bias function is followed by
many works in calibration (e.g. Bayarri et al. (2007b,a);
Higdon et al. (2008); Liu et al. (2009); Paulo et al. (2012);
Gu (2018)).
The predictive accuracy is improved by combining the
mathematical model and discrepancy function.

6. Two types of inconsistency in Gaussian stochastic
process calibration
(Random bias function.) When the bias is generated
from the Gaussian stochastic process, some usual
estimators for the calibration parameters u are inconsistent
when the sample size goes to infinity. ("Adding
spatially-correlated errors can mess up the fixed effect you
love" Reich et al. (2006); Hodges and Reich (2010);
Hughes and Haran (2013)).
(Deterministic bias function.) When the bias is a fixed
deterministic function in some functional space (e.g.
Sobolev space), the estimator of u does not minimize
some frequently used norms between the reality and the
mathematical model, e.g. the L2 norm:
||yR(·) − fM(·, ˆu)||L2(X) (Arendt et al. (2012b,a); Tuo and
Wu (2015, 2016); Wong et al. (2017); Plumlee (2016)).

7. Inconsistency when data is from the Gaussian
stochastic process
Example 1
Assume fM(x, u) = u and the experimental data is noise-free,
i.e. yO(x) = fM(x, u) + b(x) and b(·) ∼ GaSP(0, σ2K(·, ·)), with
K(xi, xj) = exp(−|xi − xj|/γ), the exponential correlation
function. n observations are obtained, equally spaced at
xi ∈ [0, 1]. Assume both σ2 and γ are known.
Lemma 2
Assume σ2 > 0 and γ > 0 are both finite. When n → ∞, the
maximum likelihood estimator ˆuMLE = (1T
n R−11n)−11T
n R−1yO in
Example 1 has the following limiting distribution
ˆuMLE ∼ N u,
2σ2γ
2γ + 1
.
The variance of the estimator does not go to zero when the
sample size increases to infinity.

8. 20 40 60 80 100
0.000.050.100.150.200.25
n
MSE
GaSP
S−GaSP
20 40 60 80 100
0.000.050.100.150.200.25
n
MSE
GaSP
S−GaSP
Figure 1: Mean squared error (MSE) of MLE of u in Example 1 when
the data is from a zero-mean GaSP (red triangles) and from a
zero-mean S-GaSP (blue dots) for different number of observations.
105
simulations are implemented for each point. The left panel is for
γ = 0.1 and the right panel is for γ = 0.02, both assuming σ2
= 1.

9. The equivalence to the kernel ridge regression in
GaSP calibration
After marginalize out b, the marginal distribution of yO
follows
[yO
| u, σ2
0, λ] ∼ MN(fM
u , σ2
0((nλ)−1
R + In)). (1)
where fM
u := (fM
(x1, u), ..., fM
(xn, u))T
. Denote L(u) the likelihood of
u in (1) and the regularization parameter λ := σ2
0/(nσ2
).
Lemma 3
The maximum likelihood estimator ˆuλ,n := argmaxu∈U L(u) and
predictive mean estimator ˆbλ,n(·) := E[b(·) | yO
, ˆuλ,n, λ] can be
expressed as the estimator of the kernel ridge regression (KRR)
(ˆuλ,n, ˆbλ,n(·)) = argmin
b(·)∈H, u∈U
λ,n(u, b),
λ,n(u, b) =
1
n
n
i=1
(yO
(xi ) − fM
(xi , u) − b(xi ))2
+ λ||b||2
H . (2)
where || · ||H is the native norm or the reproducing kernel Hilbert
space norm. The || · ||H is quite different than the L2 norm || · ||L2(X).

10. Outline
1. Gaussian stochastic process (GaSP) calibration
2. L2 calibration
3. Scaled Gaussian stochastic process (S-GaSP) calibration

11. L2 calibration
The L2 approach in Tuo and Wu (2015, 2016). First they use a
GaSP to estimate yR
(·) based on yO
and then they estimate uL2
by
ˆuL2
= argmin
u∈U x∈X
(ˆyR
(x) − fM
(x, u))2
dx

12. L2 calibration
The L2 approach in Tuo and Wu (2015, 2016). First they use a
GaSP to estimate yR
(·) based on yO
and then they estimate uL2
by
ˆuL2
= argmin
u∈U x∈X
(ˆyR
(x) − fM
(x, u))2
dx
The L2 approach does not use the mathematical model to
predict the reality. However, the mathematical model is often
developed by experts, meaning that it often contains information
about the reality.

13. Criteria
Denote L2 loss ||f(·)||2
L2(X) := x∈X f2(x)dx for any squared
integrable function f. We focus on two types of predictions.
i. The L2 loss between the reality and the estimator of the
reality L2(ˆyR(·, ˆu)) = ||yR(·) − ˆyR(·, ˆu)||2
L2(X).

14. Criteria
Denote L2 loss ||f(·)||2
L2(X) := x∈X f2(x)dx for any squared
integrable function f. We focus on two types of predictions.
i. The L2 loss between the reality and the estimator of the
reality L2(ˆyR(·, ˆu)) = ||yR(·) − ˆyR(·, ˆu)||2
L2(X).
ii. The L2 loss between the reality and calibrated mathematical
model L2(ˆu) = ||yR(·) − fM(·, ˆu)||2
L2(X) = ||bˆu(·)||2
L2(X),
where ˆu is the estimator of the calibration parameter.
Is it possible to satisfy both criteria when the sample size is
finite and infinite?

15. Outline
1. Gaussian stochastic process (GaSP) calibration
2. L2 calibration
3. Scaled Gaussian stochastic process (S-GaSP) calibration

16. The scaled Gaussian stochastic process
Consider the following process
yO
(x) = fM
(x, u) + bz(x) + ,
bz(x) = b(x) | ξ∈X
b2
(ξ)dξ = Z ,
b(·) ∼ GaSP(0, σ2
K(·, ·)),
Z ∼ pZ (·), ∼ N(0, σ2
0).
(3)
The bz(·) is called the scaled Gaussian stochastic process (S-GaSP).
Given Z = z, the S-GaSP becomes a GaSP constrained at the space
related to the L2 norm of the discrepancy function x∈X
b2
(x)dx = z.
Conditional on all the parameters, the default choice of pZ (·) is
pZ (z) =
gZ (z) pb (Z = z)
∞
0
gZ (t) pb (Z = t) dt
, (4)
with
gZ (z) =
λz
2σ2
exp −
λzz
2σ2
. (5)

17. Fact about the S-GaSP
Starting from a GaSP with any reasonable kernel, the S-GaSP is
a GaSP with a transformed kernel. For a GaSP with zero mean
and covariance function σ2
K(·, ·), one has
K(xi , xj ) =
∞
k=1
ρk φk (xi )φk (xj ).
where ρk and φk (·) are the ordered eigenvalues and
orthonormal eigenfunctions, respectively.
Lemma 4 (Mercer theorem for S-GaSP)
Any marginal distribution of the S-GaSP defined in (3) is a
multivariate normal distribution
[bz(x1), ..., bz(xn) | σ2
Rz] ∼ MN(0, σ2
Rz)
where the (i, j) entry of Rz is Kz(xi , xj ) as follows
Kz(xi , xj ) =
∞
k=1
ρk
1 + λzρk
φk (xi )φk (xj ). (6)

18. The equivalence to the penalized kernel ridge
regression in S-GaSP
After marginalizing out bz in (3),
[yO
| u, σ2
0, λ, λz] ∼ MN(fM
u , σ2
0((nλ)−1
Rz + In)). (7)
Denote Lz(u) the likelihood for u in (7).
Lemma 5
The maximum likelihood estimator ˆuλ,λz ,n := argmaxu Lz(u) and
predictive mean ˆbλ,λz ,n(·) := E[bz(·) | yO
, ˆuλ,λz ,n, λ, λz] are the same
as the estimator of the penalized kernel ridge regression (KRR) as
follows
(ˆuλ,λz ,n, ˆbλ,λz ,n(·)) = argmin
b(·)∈H, u∈u
λ,λz ,n(u, b),
λ,λz ,n(u, b) =
1
n
n
i=1
(yO
(xi ) − fM
(xi , u) − b(xi ))2
+ λ||b||2
Hz
. (8)
where ||b||2
Hz
= ||b||2
H + λz||b||2
L2(X).

19. The nonparametric regression setting
Let us first consider the nonparametric regression model,
y(xi ) = f(xi ) + i , i ∼ N(0, σ2
0), (9)
where f is assumed to follow a zero mean S-GaSP prior with the
default choice of pZ (·) and gZ (·) in Equation (4) and (5). For
simplicity, we assume xi
i.i.d.
∼ Unif([0, 1]p
).

20. The nonparametric regression setting
Let us first consider the nonparametric regression model,
y(xi ) = f(xi ) + i , i ∼ N(0, σ2
0), (9)
where f is assumed to follow a zero mean S-GaSP prior with the
default choice of pZ (·) and gZ (·) in Equation (4) and (5). For
simplicity, we assume xi
i.i.d.
∼ Unif([0, 1]p
).
(Function space of the reality). We assume the underlying
truth f0(·) := Ey [y(·)] reside in p dimensional Sobolev space with
order m > p/2 as follows
Wm
2 (X) = {f(·) =
∞
k=1
fk φ(·) ∈ L2(X) :
∞
k=1
k2m/p
f2
k < ∞},

21. The nonparametric regression setting
Let us first consider the nonparametric regression model,
y(xi ) = f(xi ) + i , i ∼ N(0, σ2
0), (9)
where f is assumed to follow a zero mean S-GaSP prior with the
default choice of pZ (·) and gZ (·) in Equation (4) and (5). For
simplicity, we assume xi
i.i.d.
∼ Unif([0, 1]p
).
(Function space of the reality). We assume the underlying
truth f0(·) := Ey [y(·)] reside in p dimensional Sobolev space with
order m > p/2 as follows
Wm
2 (X) = {f(·) =
∞
k=1
fk φ(·) ∈ L2(X) :
∞
k=1
k2m/p
f2
k < ∞},
(Choice of kernel). Denote {(ρj , φj )}∞
j=1 the eigenvalues and
eigenfunctions of the reproducing kernel K(·, ·). For all j, assume
cρj−2m/p
≤ ρj ≤ Cρj−2m/p
(10)
for some constant cρ, Cρ > 0. For all j and x ∈ X, we assume
the eigenfunctions are bounded. As an example, the widely
used Matérn kernel satisfies this assumption.

22. Convergence rate of the S-GaSP
Theorem 1
Assume the eigenvalues of K(·, ·) satisfy (10) and the
eigenfunctions are bounded. Assume f0 ∈ Wm
2 (X) and denote
β = (2m−p)2
2m(2m+p) . For the nonparametric regression model (9), for
sufficiently large n and any α > 2, with probability at least
1 − exp −α−2
3 (1 − 2 exp(−nβ))
||ˆfλ,λz ,n −f0||L2(X) ≤ 2
√
2(||f0||L2(X) + ||f0||H) + CK ασ0 n
− m
2m+p ,
(11)
by choosing λ = n−2m/(2m+p) and λz = λ−1/2, where CK is a
constant that only depends on the kernel K(·, ·).

23. Convergence rate of the S-GaSP in calibration
Define the estimator for the reality in the S-GaSP by the penalized
KRR for any x ∈ X
yR
λ,λz ,n(x, ˆu) := f(x, ˆuλ,λz ,n) + ˆbλ,λz ,n(x).
Corollary 6
Assume yR
(·) − fM
(·, u) ∈ Wm
2 (X) for any u ∈ U and
supu∈U yR
(·) − fM
(·, u) H < ∞. Let the eigenvalues of K(·, ·) satisfy
(10). For sufficiently large n and any α > 2 and Cβ ∈ (0, 1), with
probability at least 1 − exp{−(α − 2)/3} − exp(−nCβ β
),
yR
(·) − ˆyR
λ,λz ,n(·, ˆu) L2(X) ≤ 2
√
2 sup
u∈U
yR
(·) − fM
(·, u) L2(X)
+ sup
u∈U
yR
(·) − fM
(·, u) H + CK σ0
√
α n− m
2m+p
by choosing λ = n−2m/(2m+p)
and λz = λ−1/2
, where CK is a constant
depending on the kernel K(·, ·) and β = (2m − p)2
/(2m(2m + p)).

24. Convergence rate of the S-GaSP in calibration
Define the estimator for the reality in the S-GaSP by the penalized
KRR for any x ∈ X
yR
λ,λz ,n(x, ˆu) := f(x, ˆuλ,λz ,n) + ˆbλ,λz ,n(x).
Corollary 6
Assume yR
(·) − fM
(·, u) ∈ Wm
2 (X) for any u ∈ U and
supu∈U yR
(·) − fM
(·, u) H < ∞. Let the eigenvalues of K(·, ·) satisfy
(10). For sufficiently large n and any α > 2 and Cβ ∈ (0, 1), with
probability at least 1 − exp{−(α − 2)/3} − exp(−nCβ β
),
yR
(·) − ˆyR
λ,λz ,n(·, ˆu) L2(X) ≤ 2
√
2 sup
u∈U
yR
(·) − fM
(·, u) L2(X)
+ sup
u∈U
yR
(·) − fM
(·, u) H + CK σ0
√
α n− m
2m+p
by choosing λ = n−2m/(2m+p)
and λz = λ−1/2
, where CK is a constant
depending on the kernel K(·, ·) and β = (2m − p)2
/(2m(2m + p)).
First criterion: L2(ˆyR
(·, ˆu)) = ||yR
(·) − ˆyR
(·, ˆu)||2
L2(X).

25. Convergence to the L2 minimizer in S-GaSP
Denote the L2 minimizer in calibration that minimizes the reality
and mathematical models as follows
uL2
= argminu∈U
x∈X
(yR
(x) − fM
(x, u))2
dx. (12)
Theorem 2
Under some more regular conditions, the estimator of the
penalized kernel ridge regression by the S-GaSP calibration
model satisfies
ˆuλ,λz ,n = uL2
+ Op(n
− m
2m+p ),
by choosing λ = O(n
− 2m
2m+p ) and λz = O(λ−1/2).

26. Convergence to the L2 minimizer in S-GaSP
Denote the L2 minimizer in calibration that minimizes the reality
and mathematical models as follows
uL2
= argminu∈U
x∈X
(yR
(x) − fM
(x, u))2
dx. (12)
Theorem 2
Under some more regular conditions, the estimator of the
penalized kernel ridge regression by the S-GaSP calibration
model satisfies
ˆuλ,λz ,n = uL2
+ Op(n
− m
2m+p ),
by choosing λ = O(n
− 2m
2m+p ) and λz = O(λ−1/2).
Second criterion: L2(ˆu) = ||yR(·) − fM(·, ˆu)||2
L2(X).

27. Convergence to the L2 minimizer in S-GaSP
Denote the L2 minimizer in calibration that minimizes the reality
and mathematical models as follows
uL2
= argminu∈U
x∈X
(yR
(x) − fM
(x, u))2
dx. (12)
Theorem 2
Under some more regular conditions, the estimator of the
penalized kernel ridge regression by the S-GaSP calibration
model satisfies
ˆuλ,λz ,n = uL2
+ Op(n
− m
2m+p ),
by choosing λ = O(n
− 2m
2m+p ) and λz = O(λ−1/2).
Second criterion: L2(ˆu) = ||yR(·) − fM(·, ˆu)||2
L2(X).
The GaSP calibration does not have this property.

28. Example 2
Let yO
(x) = yR
(x) + , where yR
(x) = 2
∞
j=1
j−3
cos(π(j − 0.5)x) sin(j)
and ∼ N(0, 0.052
) is an independent Gaussian noise. Let the
mathematical model be a mean parameter, i.e. fM
(x) = u. The goal
is to predict yR
(x) at x ∈ [0, 1] and estimate u. This function is in the
Sobolev space with order m = 3.
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q q
q
q q
q q q q q q q q q q q q q q q q q q q q q q q q q
5 6 7 8 9 10
0.0000.0040.0080.012
log(n)
AvgRMSEfM+δ
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q q
q
q q
q q q q q q q q q q q q q q q q q q q q q q q q q
q
q
GaSP
S−GaSP
q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q
5 6 7 8 9 10−8−6−4−20
log(n)
log(RMSEθ)
q
q q q
q q
q
q q
q
q q q q
q q
q
q q
q q q q
q q
q q q q q q
q q q q q q q q q q q q q q q q q q q
q
q
GaSP
S−GaSP
Figure 2: Calibration and prediction by the GaSP and discretized S-GaSP
calibration models for Example 2. In the left panel, the black curve is the
theoretical upper bound from Corollary 6 (up to a constant). The blue and red
circles overlap in the left panel. In the right panel, the black curve is the
theoretical upper bound from Theorem 2. λ = n−2m/(2m+p)
× 10−4
with
m = 3, p = 1 and λz = λ−1/2
are assumed.

29. MLE for Example 2
q
q
q
q
q
q q
q
q
q
q
q
q q
q q
q q
q q q q
q q
q q q q q q q q q q q q q q q q q q q q q q q q q q
5 6 7 8 9 10
0.0000.0040.0080.012
log(n)
AvgRMSEfM+δ
q
q
q
q
q
q
q
q
q
q
q
q
q q
q q
q q
q q q q
q q
q q q q q q q q q q q q q q q q q q q q q q q q q qq
q
GaSP
S−GaSP
q q q q q q q
q
q
q q q q q
q q q q
q
q q q q
q
q q q q q
q
q q q
q
q q q
q
q
q q q q
q q q q q q q
5 6 7 8 9 10
−8−6−4−20
log(n)
log(RMSEθ)
q
q q q
q q
q
q q
q
q q q
q
q q
q
q
q
q
q q q
q q q
q q q
q q
q q q
q q q q
q q q q q q q q q q q q
q
q
GaSP
S−GaSP
Figure 3: Calibration and prediction for Example 2 when the
parameters (θ, σ2
0, γ, λ) are estimated by MLE. In the left panel, the
black curve represents the theoretical upper bound by Corollary 6 (up
to a constant). In the right panel, the black curve represents the
theoretical upper bound in Theorem 2 (up to a constant). λz = 1/
√
λ
is assumed for the S-GaSP calibration.

30. Orthogonality
Corollary 7
Under some regular conditions, the penalized KRR estimator for the
calibration parameters in the S-GaSP calibration in (8) satisfies
ˆbλ,λz ,n(·),
∂fM
(·, ˆuλ,λz ,n)
∂uj H
+ λz
ˆbλ,λz ,n(·),
∂fM
(·, ˆuλ,λz ,n)
∂uj L2(X)
= 0;
Further assuming the mathematical model is differentiable at ˆuλ,n, the
KRR estimator of the calibration parameters in the GaSP calibration
in (2) satisfies
ˆbλ,n(·),
∂fM
(·, ˆuλ,n)
∂uj H
= 0,
for any uj , j = 1, ..., q.

31. The discretized scaled Gaussian stochastic process
One can select NC distinct points to discretize the input space
[0, 1]p to replace ξ∈X b(ξ)2dξ by 1
NC
NC
i=1 b(xC
i )2 in the
S-GaSP model in (3). More specifically, we let the discretized
points be the observed variable inputs, i.e. xC
i = xi, for
i = 1, ..., NC and NC = n. The discretized S-GaSP is then
defined as
yO
(x) = fM
(x, u) + bzd
(x) + ,
bzd
(x) = b(x) |
1
n
n
i=1
b(xi)2
= Zd
b(·) ∼ GaSP(0, σ2
K(·, ·)),
Zd ∼ pZd
(·), ∼ N(0, σ2
0).
(13)
Still we assume the default choice of pZd
(·) and gZd
(·) defined
in (4) and (5), respectively.

32. Theorem 3 (Predictive distribution of the discretized
S-GaSP)
Assume the bzd
(·) in (13) with pZd
(·) and gZd
(·) defined in (4) and (5),
respectively. The predictive distribution of the field data at any x ∈ X
by the discretized S-GaSP model in (13) is as follows
yO
(x) | yO
, u, σ2
0, λ, λz ∼ MN(ˆµzd
(x), σ2
0((nλ)−1
K∗
zd
(x, x) + 1)),
where
ˆµzd
(x) = fM
(x, u) +
rT
(x)
1 + λλz
R +
nλ
1 + λλz
In
−1
yO
− fM
u ,
K∗
zd
(x, x) = K(x, x) − rT
(x) In − R +
nλ
1 + λλz
In
−1
n
(1 + λλz)λz
˜R−1
r(x)
for any x ∈ X where r(x) = (K(x, x1), ..., K(x, xn))T
and ˜R = R + n
λz
In
with the (i, j) entry of R being K(xi , xj ) and λ = σ2
0/(nσ2
).
A more interesting result is that the predictive mean and variance
from discretized S-GaSP is exactly the same as the GaSP, when the
data is noise-free.

33. Some of our works in calibration and identifiability
Gu, M., Xie, F. and Wang, L. (2018). A theoretical framework of the
scaled Gaussian stochastic process. arXiv:1807.03829.
Gu, M. and Wang, L. (2017). Scaled Gaussian stochastic process for
computer model calibration and prediction. arXiv:1707.08215.
Gu, M. (2018) “RobustCalibration" available at CRAN, an R package
for robust calibration of imperfect mathematical model. R package
version 0.5.1.
Gu, M. (2018). Jointly robust prior for emulation, variable selection
and calibration. arXiv:1804.09329.
Gu, M. and Shen, W. (2018) Generalized probabilistic principal
component analysis (GPPCA) for correlated data.

34. Thanks!

35. Kyle R Anderson and Michael P Poland. Bayesian estimation of
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39. Application: ground deformation and Kilauea Volcano
(just had the biggest eruption in 100 years...)
Figure 4: Two satellite interferograms used in Anderson and Poland
(2016, 2017).
A geophysical/mathematical model fM
(x, u) is used to model the
ground displacement.
The calibration parameters are the spatial location of the magma
chamber, depth of the chamber, magma storage rate, host rock
properties, etc.
“All models are wrong".

40. Calibrating the geophysical model for Kilauea Volcano
Table 1: Input variables and calibration parameters of the geophysical
model for Kilauea Volcano in 2011 to 2012.
Input variable (x) Name Description
x1 Latitude Spatial coordinate
x2 Longitude Spatial coordinate
Parameter (u) Name Description
u1 ∈ [−2000, 3000] Chamber east (m) Spatial coordinate for the chamber
u2 ∈ [−2000, 5000] Chamber north (m) Spatial coordinate for the chamber
u3 ∈ [500, 6000] Chamber depth (m) Depth of the chamber
u4 ∈ [0, 0.15] Res. vol. change rate (m3/s) Volume change rate of the reservoir
u5 ∈ [0.25, 0.33] Poisson’s ratio Host rock property
We compare GaSP and S-GaSP for calibrating the geophysical model
in Anderson and Poland (2016) – the displacement of the ground’s
surface caused by addition of magma to a spherical reservoir.

41. Posterior distribution of the calibration parameters
−2000 −1000 0 1000 2000 3000
0.00000.00050.00100.00150.0020
θ1
Density
GaSP
S−GaSP
−2000 0 1000 3000 5000
0.00000.00050.00100.00150.0020
θ2
Density
GaSP
S−GaSP
1000 2000 3000 4000 5000 6000
0.00000.00040.00080.0012
θ3
Density
GaSP
S−GaSP
0.00 0.05 0.10 0.15
010203040
θ4
Density
GaSP
S−GaSP
0.26 0.28 0.30 0.32
68101214
θ5
Density
GaSP
S−GaSP
Figure 5: Marginal posterior density of u by the GaSP (red curves)
and S-GaSP (blue curves).
The marginal posterior of u3, the chamber depth, is centered at
the large value ≈ 5000 in the GaSP calibration.

42. Prediction for the first interferogram
−4000 −2000 0 2000 4000
−200020006000
Computer model by GaSP
−0.02
0.00
0.02
0.04
−4000 −2000 0 2000 4000
−200020006000
Computer model by S−GaSP
−0.02
0.00
0.02
0.04
−4000 −2000 0 2000 4000
−200020006000
Computer model and discrepancy by GaSP
−0.02
0.00
0.02
0.04
−4000 −2000 0 2000 4000
−200020006000
Computer model and discrepancy by S−GaSP
−0.02
0.00
0.02
0.04
Figure 6: Prediction by different models of the first interferogram.
The predictive mean squared error by the GaSP calibration and
S-GaSP calibration is 2.70 × 10−5
and 2.40 × 10−5
using the
calibrated computer model, respectively.

43. Prediction for the second interferogram
−4000 −2000 0 2000 4000
−200020006000
Computer model by GaSP
−0.02
0.00
0.02
0.04
−4000 −2000 0 2000 4000
−200020006000
Computer model by S−GaSP
−0.02
0.00
0.02
0.04
−4000 −2000 0 2000 4000
−200020006000
Computer model and discrepancy by GaSP
−0.02
0.00
0.02
0.04
−4000 −2000 0 2000 4000
−200020006000
Computer model and discrepancy by S−GaSP
−0.02
0.00
0.02
0.04
Figure 7: Prediction by different models of the second interferogram.
The predictive mean squared error by the GaSP calibration and
S-GaSP calibration is 1.43 × 10−4
and 1.21 × 10−4
using the
calibrated computer model, respectively.