space-filling designs for computer...
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Space-Filling Designs for Computer Experiments
V. Roshan Joseph
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Supported by ARO W911NF-14-1-0024
Computer model
System
Mathematicalmodel
Computermodel
• Computer model is a numerical implementation of the mathematical model.
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Computer experiments
Computer model outputsinputs ..
.
.
• Expensive black-box code• Deterministic outputs• Complex relationships
Experiment
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Applications
• Approximation • Sensitivity analysis• Calibration• Optimization• Uncertainty quantification
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Uncertainty QuantificationUncertainty
sources
Input Output
Propagation of uncertainty
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Gul, E., Joseph, V. R., Yan, H., and Melkote, S. N. (2015). “Uncertainty Quantification in Machining Simulations Using In Situ Emulator,” Under Review.
Review of Space-Filling Designs
• No need to worry about– Replication– Randomization– Blocking
• What type of designs?– Fractional factorial designs, orthogonal
arrays,…? No, use space-filling designs!
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Space-Filling Designs
• Definition: designs that fill the space!
• What is the meaning of filling the space?– Maximin distance– Minimax distance– Uniform
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Maximin distance design
• Johnson, Moore, and Ylvisaker (1991)
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Minimax distance design
• Johnson, Moore, and Ylvisaker (1991)
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Consider the simple case of four-run design in two factors (n=4, p=2).
Maximin distance design. Minimax distance design.
Examples of Maximin and Minimax Designs
Latin hypercube design
• McKay, Conover, Beckman (1979)
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Latin hypercube design
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Maximin Latin hypercube design
• Morris and Mitchell (1995): Maximindistance design within the class of Latin hypercube designs.
where D is an LHD.
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Uniform design
• Fang (1980)
• Good for approximating integrals by sample averages.
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Maximum Projection Designs
Joseph, V. R., Gul, E., and Ba, S. (2015). “Maximum Projection Designs for Computer Experiments,” Biometrika, 102, 371-380.
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MmLHD• MmLHD (20,2)
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MmLHD• A two-dimensional projection of MmLHD (20,10)
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MmLHD
• Ensures good space-filling in p dimensions and uniform one-dimensional projections, but their projections in 2,…,p-1 dimensions can be poor.
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Improvements to MmLHD
• Draguljic, Santner, Dean (2012)
• Criterion is computationally expensive.
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Uniform Design
• Hickernell (1998) proposed CL2 criterion that ensures projections to all subspaces.
• But is uniformity important in computer experiments?
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Generalized LHD
• Dette and Pepelyshev (2010): placing more points in the boundaries than around the center can minimize the prediction errors from GP modeling.
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MaxPro criterion• Weighted Euclidean distance:
• Modify the Morris-Mitchell criterion to
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Bayesian criterion
• We don’t know about 𝜃𝜃 before the experiment!
• Prior:
• Then, the criterion becomes
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MaxPro criterion
• MaxPro criterion:
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LHD property
•• MaxPro design must have n distinct levels
for each factor.• LHD requirement is automatically enforced
in the criterion!
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Design construction algorithm
• Minimizing MaxPro objective function is not easy!– np number of variables– many local minima
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R Package: MaxProJMP 12 (under FFF)
ExamplesMaxProLHD(20,2) and MaxPro(20,2)
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Numerical comparisons• n=100, p=10• Designs:
– MaxPro– MaxProLHD– MmLHD– UD– GLHD
• Criteria:– Maximin– miniMax– CL2
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Minimum distance (larger-the-better)
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GP-based criteria
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An optimality result
• Prior:
• MaxPro design is good in terms of Entropy, condition number, prediction variance, etc.
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Applications
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Solid End Milling Process• Simulation on computer model of Production Module
software
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UQ for Solid End Milling
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MaxPro LHD
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After Inverse probability transform
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In-Situ Emulator
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𝑓𝑓 𝒖𝒖, 𝑡𝑡 = 𝑒𝑒𝑒𝑒𝑒𝑒{log 𝑓𝑓 𝒖𝒖𝟎𝟎, 𝑡𝑡 + �̂�𝛿𝑡𝑡 (𝒖𝒖)}
�̂�𝛿𝑡𝑡(u) = 𝒓𝒓 𝒖𝒖 ′𝑹𝑹−1(𝒘𝒘𝑡𝑡 − 𝒘𝒘0,𝑡𝑡)
𝑅𝑅𝑡𝑡(𝒖𝒖𝑖𝑖 − 𝒖𝒖𝑖𝑖) = 𝑅𝑅𝑡𝑡,𝒖𝒖 𝒖𝒖𝑖𝑖 − 𝒖𝒖𝑗𝑗 − 𝑅𝑅𝑡𝑡,𝒖𝒖 𝒖𝒖𝑖𝑖 − 𝒖𝒖0 − 𝑅𝑅𝑡𝑡,𝒖𝒖 𝒖𝒖𝑗𝑗 − 𝒖𝒖0 + 1,
𝑅𝑅𝑡𝑡,𝒖𝒖 𝒖𝒖𝑖𝑖 − 𝒖𝒖𝑗𝑗 = exp −∑𝑘𝑘=1𝑝𝑝 𝜃𝜃𝑡𝑡,𝑘𝑘 𝑢𝑢𝑖𝑖𝑘𝑘 − 𝑢𝑢𝑗𝑗𝑘𝑘
2.
where
Output with 95% confidence region
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Computational time
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UD (n=2000) In-Situ EmulatorCurve cutting 1 day 8 mins5-axis cutting 9 days 1 hour
References• V. Roshan Joseph (2016). “Space-Filling Designs for Computer
Experiments: A Review,” (with discussions and rejoinder), Quality Engineering, 28, 28-44.
• V. Roshan Joseph, Gul, E., and Ba, S. (2015). “Maximum Projection Designs for Computer Experiments,” Biometrika, 102, 371-380.
• Gul, E., V. Roshan Joseph, Yan, H., and Melkote, S. N. “Uncertainty Quantification in Machining Simulations Using In Situ Emulator,” Under review.
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