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Space-filling design

In recent years, space-filling designs are used in computer experiments as they are thought to be more appropriate for deterministic computer models as opposed to traditional design of experiments.

Contents

1. Background
2. Types
3. References

Background

Space-filling designs are a type of design of experiments, that spread the design points evenly or uniformly throughout the region of experimentation. This is a desirable feature if the experimenter doesn't know the form of the model that is required, and believes that interesting phenomena are likely to be found in different regions of the experimental space. In space-filling designs, traditional design of experiments approaches such as replication, randomization, and blocking are irrelevant. For a deterministic computer model this is desirable, because a single run of the computer model at a design point provides all the information about the response at that point and, since an actual physical experiment is not conducted, there is no need for blocking against external sources of variation.

Types

Some of the most widely used space-filling designs are

1. Latin Hypercube Designs: In a Latin Hypercube designs, each factor has as many levels as there are runs in the design. The levels are spaced evenly from the lower bound to the upper bound of the factor. The Latin Hypercube method chooses points to maximize the minimum distance between design points, but with a constraint. The constraint maintains the even spacing between factor levels.
2. Sphere Packing Designs: The Sphere-Packing design method maximizes the minimum distance between pairs of design points. The effect of this maximization is to spread the points out as much as possible inside the design region.
3. Uniform Designs: The Uniform design minimizes the discrepancy between the design points (empirical uniform distribution) and a theoretical uniform distribution.
4. Maximum Entropy Designs: The Maximum Entropy design is an alternative to the Latin Hypercube design for computer experiments. The Maximum Entropy design optimizes a measure of the amount of information contained in an experiment. They maximize the Shannon information of an experiment, assuming that the data come from a normal (m, s² R) distribution, where $R_{i,j}=\exp {\biggl (}-\textstyle \sum _{k=1}^{N}\displaystyle \theta _{k}(x_{i,k}-x_{j,k})^{2}{\biggr )}$ is the correlation of response values at two different design points, x_i and x_j. Computationally, these designs maximize |R|, the determinant of the correlation matrix of the sample. If x_i and x_j are far apart, then R_ij approaches zero. If x_i and x_j are close together, then R_ij is near one.
5. Minimum Potential Designs: These design spread points out inside a sphere. This design can be imagined with points as electrons with springs attached to every other point. The coulomb force pushes the points apart, but the springs pull them together. The design is the spacing of points that minimizes the potential energy of the system. Minimum Potential designs have spherical symmetry and uniform spacing, and are nearly orthogonal.

References