In physics-based engineering modeling, the two primary sources of model uncertainty, which account for the differences between computer models and physical experiments, are parameter uncertainty and model discrepancy. Distinguishing the effects of the two sources of uncertainty can be challenging. For situations in which identifiability cannot be achieved using only a single response, we propose to improve identifiability by using multiple responses that share a mutual dependence on a common set of calibration parameters. To that end, we extend the single response modular Bayesian approach for calculating posterior distributions of the calibration parameters and the discrepancy function to multiple responses. Using an engineering example, we demonstrate that including multiple responses can improve identifiability (as measured by posterior standard deviations) by an amount that ranges from minimal to substantial, depending on the characteristics of the specific responses that are combined.

References

References
1.
Kennedy
,
M. C.
, and
O’Hagan
,
A.
, 2001, “
Bayesian Calibration of Computer Models
,”
J. R. Stat. Soc. Ser. B (Stat. Methodol.)
,
63
(
3
), pp.
425
464
.
2.
Arendt
,
P.
,
Apley
,
D.
, and
Chen
,
W.
, 2012, “
Quantification of Model Uncertainty: Calibration, Model Discrepancy, and Identifiability
,”
J. Mech. Des.
,
134
(
10
), p.
100908
.
3.
Qian
,
P.
,
Wu
,
H.
, and
Wu
,
C.
, 2008, “
Gaussian Process Models for Computer Experiments With Qualitative and Quantitative Factors
,”
Technometrics
,
50
(
3
), pp.
383
396
.
4.
McMillan
,
N.
,
Sacks
,
J.
,
Welch
,
W.
, and
Gao
,
F.
, 1999, “
Analysis of Protein Activity Data by Gaussian Stochastic Process Models
,”
J. Biopharm. Stat.
,
9
(
1
), pp.
145
160
.
5.
Cressie
,
N.
, 1993,
Statistics for Spatial Data
,
Wiley
,
New York
.
6.
Ver Hoef
,
J.
, and
Cressie
,
N.
, 1993, “
Multivariable Spatial Prediction
,”
Math. Geol.
,
25
(
2
), pp.
219
240
.
7.
Conti
,
S.
,
Gosling
,
J. P.
,
Oakley
,
J. E.
, and
O’Hagan
,
A.
, 2009, “
Gaussian Process Emulation of Dynamic Computer Codes
,”
Biometrika
,
96
(
3
), pp.
663
676
.
8.
Conti
,
S.
, and
O’Hagan
,
A.
, 2010, “
Bayesian Emulation of Complex Multi-Output and Dynamic Computer Models
,”
J. Stat. Plann. Inference
,
140
(
3
), pp.
640
651
.
9.
McFarland
,
J.
,
Mahadevan
,
S.
,
Romero
,
V.
, and
Swiler
,
L.
, 2008, “
Calibration and Uncertainty Analysis for Computer Simulations With Multivariate Output
,”
AIAA J.
,
46
(
5
), pp.
1253
1265
.
10.
Bayarri
,
M. J.
,
Berger
,
J. O.
,
Cafeo
,
J.
,
Garcia-Donato
,
G.
,
Liu
,
F.
,
Palomo
,
J.
,
Parthasarathy
,
R. J.
,
Paulo
,
R.
,
Sacks
,
J.
, and
Walsh
,
D.
, 2007, “
Computer Model Validation With Functional Output
,”
Ann. Stat.
,
35
(
5
), pp.
1874
1906
.
11.
Williams
,
B.
,
Higdon
,
D.
,
Gattiker
,
J.
,
Moore
,
L. M.
,
McKay
,
M. D.
, and
Keller-McNulty
,
S.
, 2006, “
Combining Experimental Data and Computer Simulations, With an Application to Flyer Plate Experiments
,”
Bayesian Anal.
,
1
(
4
), pp.
765
792
.
12.
Drignei
,
D.
, 2009, “
A Kriging Approach to the Analysis of Climate Model Experiments
,” J. Agric., Biol.,
Environ. Stat.
,
14
(
1
), pp.
99
114
.
13.
Apley
,
D.
,
Liu
,
J.
, and
Chen
,
W.
, 2006, “
Understanding the Effects of Model Uncertainty in Robust Design With Computer Experiments
,”
J. Mech. Des.
,
128
(
4
), pp.
945
958
.
14.
Sacks
,
J.
,
Welch
,
W. J.
,
Mitchell
,
T. J.
, and
Wynn
,
H. P.
, 1989, “
Design and Analysis of Computer Experiments
,”
Stat. Sci.
,
4
(
4
), pp.
409
423
.
15.
Jin
,
R.
, 2004, “
Enhancements of Metamodeling Techniques in Engineering Design
,” Ph.D. thesis, University of Illinois at Chicago, Chicago, Illinois.
16.
Kennedy
,
M. C.
,
Anderson
,
C. W.
,
Conti
,
S.
, and
O’Hagan
,
A.
, 2006, “
Case Studies in Gaussian Process Modelling of Computer Codes
,”
Reliab. Eng. Syst. Saf.
,
91
(10–11)
, pp.
1301
1309
.
17.
Rasmussen
,
C. E.
, 1996, “
Evaluation of Gaussian Processes and Other Methods for Non-Linear Regression
,” Ph.D. thesis, University of Toronto, Toronto, ON, CA.
18.
Lancaster
,
T.
, 2004,
An Introduction to Modern Bayesian Econometrics
,
Blackwell Publishing
,
Malden, MA
.
19.
Johnson
,
R.
, and
Wichern
,
D.
, 2007,
Applied Multivariate Statistical Analysis
,
Prentice-Hall, Upper Saddle River
,
NJ
.
20.
Kennedy
,
M. C.
, and
O’Hagan
,
A.
, 2000, “
Supplementary Details on Bayesian Calibration of Computer Models
,” University of Sheffield, Sheffield, UK, http://www.isds.duke.edu/∼fei/samsi/Oct_09/01Sup-KenOHa.pdfhttp://www.isds.duke.edu/∼fei/samsi/Oct_09/01Sup-KenOHa.pdf, last accessed on June 27, 2012
21.
Billingsley
,
P.
, 1995,
Probability and Measure
,
John Wiley & Sons, Inc.
,
New York, NY
.
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