Minimum Potential Energy Designs

Copyright © 2005, SAS Institute Inc. All rights reserved. 1

Statistical Discovery.TM From SAS.

Minimum Potential Energy Designs

Bradley Jones &

Christopher Gotwalt

SAS Institute Inc.



Abstract• Introducing a new class of space filling designs based

on a physical analogy of design points as protons connected by springs.

• Properties• Spherical symmetry

• Nearly orthogonal

• Uniform Spacing

• Easy to compute with unconstrained optimization code

• Outline• Show how to generate these designs

• Discuss their properties

• Give examples with different numbers of factors & sample sizes



Illustration of Core Idea



Objective function

12

1 1

1d

d

n n

iji j i ij

where dij is the distance between the ith and jth points.

Goal: Find the design that minimizes the above function



Spherical Symmetry – Uniform Spacing



Near Orthogonal

12 Factor 24 Run Design



Estimation Efficiency for Low Order Polynomial Models

Factors Runs D-Efficiency

2 6-8 100%

3 11 98.5%

4 15 98.6

5 21 99.3

D-Efficiency for full quadratic model.Four and five factor designs have an added center point.



Two Factor Designs



4 points



5 points



6 points



7 points



8 points



9 points



10 points



11 points



12 points



13 points



14 points



15 points



16 Points



17 Points



18 Points



19 Points



20 Points



50 points



96 Points



200 Points



Three Factors

x

y

z

X1X2X3

1.0000-0.00020.0003

-0.00021.0000

-0.0005

0.0003-0.00051.0000

X1 X2 X3

Correlations



Four Factors

X1X2X3X4

1.0000-0.00020.0006

-0.0002

-0.00021.0000

-0.00010.0000

0.0006-0.00011.0000

-0.0001

-0.00020.0000

-0.00011.0000

X1 X2 X3 X4

Correlations



Conclusions

• Benefits• Spherical symmetry

• Nearly orthogonal

• Uniform Spacing

• Available in commercial software

• Negative• Not “space filling” in higher dimensions (except in low

dimensional projections.



Contact Information

[email protected]

Documents

Minimum Potential Energy Designs