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Journal of the American Statistical Association Volume 104, 2009 - Issue 487
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Applications and Case Studies

Predicting Vehicle Crashworthiness: Validation of Computer Models for Functional and Hierarchical Data

M. J. Bayarri M. J. Bayarri is Professor, Department of Statistics and Operations Research, University of Valencia, Valencia, Spain . James O. Berger is the Arts and Sciences Professor of Statistics, Department of Statistical Science, Duke University, Durham, NC 27708 . Marc C. Kennedy is Senior Risk Analyst, Central Science Laboratory, Sand Hutton, York YO41 1LZ, U.K. . Athanasios Kottas is Associate Professor, Department of Applied Mathematics and Statistics, University of California, Santa Cruz, CA 95064 . Rui Paulo is Assistant Professor, Department of Mathematics, ISEG, Technical University of Lisbon, Portugal . Jerry Sacks is Director Emeritus and Senior Fellow, National Institute of Statistical Sciences, Research Triangle Park, NC 27709 . John A. Cafeo, Chin-Hsu Lin, and Jian Tu are with Research and Development, General Motors, Warren, MI 48090. This research was supported in part by grants from General Motors and the National Science Foundation (grant DMS-0073952) to the National Institute of Statistical Sciences. The authors thank the editor, David Banks, an associate editor, and two referees for their valuable comments and suggestions.
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James O. Berger M. J. Bayarri is Professor, Department of Statistics and Operations Research, University of Valencia, Valencia, Spain . James O. Berger is the Arts and Sciences Professor of Statistics, Department of Statistical Science, Duke University, Durham, NC 27708 . Marc C. Kennedy is Senior Risk Analyst, Central Science Laboratory, Sand Hutton, York YO41 1LZ, U.K. . Athanasios Kottas is Associate Professor, Department of Applied Mathematics and Statistics, University of California, Santa Cruz, CA 95064 . Rui Paulo is Assistant Professor, Department of Mathematics, ISEG, Technical University of Lisbon, Portugal . Jerry Sacks is Director Emeritus and Senior Fellow, National Institute of Statistical Sciences, Research Triangle Park, NC 27709 . John A. Cafeo, Chin-Hsu Lin, and Jian Tu are with Research and Development, General Motors, Warren, MI 48090. This research was supported in part by grants from General Motors and the National Science Foundation (grant DMS-0073952) to the National Institute of Statistical Sciences. The authors thank the editor, David Banks, an associate editor, and two referees for their valuable comments and suggestions.
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Marc C. Kennedy M. J. Bayarri is Professor, Department of Statistics and Operations Research, University of Valencia, Valencia, Spain . James O. Berger is the Arts and Sciences Professor of Statistics, Department of Statistical Science, Duke University, Durham, NC 27708 . Marc C. Kennedy is Senior Risk Analyst, Central Science Laboratory, Sand Hutton, York YO41 1LZ, U.K. . Athanasios Kottas is Associate Professor, Department of Applied Mathematics and Statistics, University of California, Santa Cruz, CA 95064 . Rui Paulo is Assistant Professor, Department of Mathematics, ISEG, Technical University of Lisbon, Portugal . Jerry Sacks is Director Emeritus and Senior Fellow, National Institute of Statistical Sciences, Research Triangle Park, NC 27709 . John A. Cafeo, Chin-Hsu Lin, and Jian Tu are with Research and Development, General Motors, Warren, MI 48090. This research was supported in part by grants from General Motors and the National Science Foundation (grant DMS-0073952) to the National Institute of Statistical Sciences. The authors thank the editor, David Banks, an associate editor, and two referees for their valuable comments and suggestions.
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Athanasios Kottas M. J. Bayarri is Professor, Department of Statistics and Operations Research, University of Valencia, Valencia, Spain . James O. Berger is the Arts and Sciences Professor of Statistics, Department of Statistical Science, Duke University, Durham, NC 27708 . Marc C. Kennedy is Senior Risk Analyst, Central Science Laboratory, Sand Hutton, York YO41 1LZ, U.K. . Athanasios Kottas is Associate Professor, Department of Applied Mathematics and Statistics, University of California, Santa Cruz, CA 95064 . Rui Paulo is Assistant Professor, Department of Mathematics, ISEG, Technical University of Lisbon, Portugal . Jerry Sacks is Director Emeritus and Senior Fellow, National Institute of Statistical Sciences, Research Triangle Park, NC 27709 . John A. Cafeo, Chin-Hsu Lin, and Jian Tu are with Research and Development, General Motors, Warren, MI 48090. This research was supported in part by grants from General Motors and the National Science Foundation (grant DMS-0073952) to the National Institute of Statistical Sciences. The authors thank the editor, David Banks, an associate editor, and two referees for their valuable comments and suggestions.
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Rui Paulo M. J. Bayarri is Professor, Department of Statistics and Operations Research, University of Valencia, Valencia, Spain . James O. Berger is the Arts and Sciences Professor of Statistics, Department of Statistical Science, Duke University, Durham, NC 27708 . Marc C. Kennedy is Senior Risk Analyst, Central Science Laboratory, Sand Hutton, York YO41 1LZ, U.K. . Athanasios Kottas is Associate Professor, Department of Applied Mathematics and Statistics, University of California, Santa Cruz, CA 95064 . Rui Paulo is Assistant Professor, Department of Mathematics, ISEG, Technical University of Lisbon, Portugal . Jerry Sacks is Director Emeritus and Senior Fellow, National Institute of Statistical Sciences, Research Triangle Park, NC 27709 . John A. Cafeo, Chin-Hsu Lin, and Jian Tu are with Research and Development, General Motors, Warren, MI 48090. This research was supported in part by grants from General Motors and the National Science Foundation (grant DMS-0073952) to the National Institute of Statistical Sciences. The authors thank the editor, David Banks, an associate editor, and two referees for their valuable comments and suggestions.
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Jerry Sacks M. J. Bayarri is Professor, Department of Statistics and Operations Research, University of Valencia, Valencia, Spain . James O. Berger is the Arts and Sciences Professor of Statistics, Department of Statistical Science, Duke University, Durham, NC 27708 . Marc C. Kennedy is Senior Risk Analyst, Central Science Laboratory, Sand Hutton, York YO41 1LZ, U.K. . Athanasios Kottas is Associate Professor, Department of Applied Mathematics and Statistics, University of California, Santa Cruz, CA 95064 . Rui Paulo is Assistant Professor, Department of Mathematics, ISEG, Technical University of Lisbon, Portugal . Jerry Sacks is Director Emeritus and Senior Fellow, National Institute of Statistical Sciences, Research Triangle Park, NC 27709 . John A. Cafeo, Chin-Hsu Lin, and Jian Tu are with Research and Development, General Motors, Warren, MI 48090. This research was supported in part by grants from General Motors and the National Science Foundation (grant DMS-0073952) to the National Institute of Statistical Sciences. The authors thank the editor, David Banks, an associate editor, and two referees for their valuable comments and suggestions.
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John A. Cafeo M. J. Bayarri is Professor, Department of Statistics and Operations Research, University of Valencia, Valencia, Spain . James O. Berger is the Arts and Sciences Professor of Statistics, Department of Statistical Science, Duke University, Durham, NC 27708 . Marc C. Kennedy is Senior Risk Analyst, Central Science Laboratory, Sand Hutton, York YO41 1LZ, U.K. . Athanasios Kottas is Associate Professor, Department of Applied Mathematics and Statistics, University of California, Santa Cruz, CA 95064 . Rui Paulo is Assistant Professor, Department of Mathematics, ISEG, Technical University of Lisbon, Portugal . Jerry Sacks is Director Emeritus and Senior Fellow, National Institute of Statistical Sciences, Research Triangle Park, NC 27709 . John A. Cafeo, Chin-Hsu Lin, and Jian Tu are with Research and Development, General Motors, Warren, MI 48090. This research was supported in part by grants from General Motors and the National Science Foundation (grant DMS-0073952) to the National Institute of Statistical Sciences. The authors thank the editor, David Banks, an associate editor, and two referees for their valuable comments and suggestions.
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Chin-Hsu Lin M. J. Bayarri is Professor, Department of Statistics and Operations Research, University of Valencia, Valencia, Spain . James O. Berger is the Arts and Sciences Professor of Statistics, Department of Statistical Science, Duke University, Durham, NC 27708 . Marc C. Kennedy is Senior Risk Analyst, Central Science Laboratory, Sand Hutton, York YO41 1LZ, U.K. . Athanasios Kottas is Associate Professor, Department of Applied Mathematics and Statistics, University of California, Santa Cruz, CA 95064 . Rui Paulo is Assistant Professor, Department of Mathematics, ISEG, Technical University of Lisbon, Portugal . Jerry Sacks is Director Emeritus and Senior Fellow, National Institute of Statistical Sciences, Research Triangle Park, NC 27709 . John A. Cafeo, Chin-Hsu Lin, and Jian Tu are with Research and Development, General Motors, Warren, MI 48090. This research was supported in part by grants from General Motors and the National Science Foundation (grant DMS-0073952) to the National Institute of Statistical Sciences. The authors thank the editor, David Banks, an associate editor, and two referees for their valuable comments and suggestions.
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Jian Tu M. J. Bayarri is Professor, Department of Statistics and Operations Research, University of Valencia, Valencia, Spain . James O. Berger is the Arts and Sciences Professor of Statistics, Department of Statistical Science, Duke University, Durham, NC 27708 . Marc C. Kennedy is Senior Risk Analyst, Central Science Laboratory, Sand Hutton, York YO41 1LZ, U.K. . Athanasios Kottas is Associate Professor, Department of Applied Mathematics and Statistics, University of California, Santa Cruz, CA 95064 . Rui Paulo is Assistant Professor, Department of Mathematics, ISEG, Technical University of Lisbon, Portugal . Jerry Sacks is Director Emeritus and Senior Fellow, National Institute of Statistical Sciences, Research Triangle Park, NC 27709 . John A. Cafeo, Chin-Hsu Lin, and Jian Tu are with Research and Development, General Motors, Warren, MI 48090. This research was supported in part by grants from General Motors and the National Science Foundation (grant DMS-0073952) to the National Institute of Statistical Sciences. The authors thank the editor, David Banks, an associate editor, and two referees for their valuable comments and suggestions.
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Pages 929-943 | Received 01 Dec 2006, Published online: 01 Jan 2012
 
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Abstract

The CRASH computer model simulates the effect of a vehicle colliding against different barrier types. If it accurately represents real vehicle crashworthiness, the computer model can be of great value in various aspects of vehicle design, such as the setting of timing of air bag releases. The goal of this study is to address the problem of validating the computer model for such design goals, based on utilizing computer model runs and experimental data from real crashes. This task is complicated by the fact that (i) the output of this model consists of smooth functional data, and (ii) certain types of collision have very limited data. We address problem (i) by extending existing Gaussian process-based methodology developed for models that produce real-valued output, and resort to Bayesian hierarchical modeling to attack problem (ii). Additionally, we show how to formally test if the computer model reproduces reality. Supplemental materials for the article are available online.

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