Optimal design

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This article is about the topic in the design of experiments. For the topic in optimal control theory, see shape optimization.

In the design of experiments, optimal designs are experimental designs that are generated based on a particular optimality criterion and are generally 'optimal' only for a specified statistical model. As a result, they generally do not satisfy the desirable properties (such as independence among the estimators) of standard classical designs (such as a factorial experiment or a fractional factorial design).

The reasons for using optimal designs instead of standard classical designs generally fall into two categories:

standard factorial or fractional factorial designs require too many runs for the amount of resources or time allowed for the experiment
the design space is constrained (the process space contains factor settings that are not feasible or are impossible to run).

Optimal designs are always an option regardless of the terms included in the model (for example, first order, first order plus some interactions, full quadratic, cubic, etc.) and are applicable for many experimental purposes such as sampling and response surface methodology.

The experimenter must specify a model for the design before the algorithm can generate the specific treatment combinations. Given the total number of treatment runs for an experiment and a specified model, the algorithm chooses the optimal set of design runs from a candidate set of possible design treatment runs. This candidate set of treatment runs consists of all possible combinations of various factor levels that one wishes to consider in the experiment. If the size of the candidate set is too large to allow all possible designs to be searched, the algorithm generally uses a stepping and exchanging process to select the set of treatment runs.

1 Optimality criteria
2 Drawbacks
3 External links
4 References

[edit] Optimality criteria

There are several optimality criteria used to select the points for a design.

[edit] A-optimality

One criterion is A-optimality, which seeks to minimize the trace of the inverse of the information matrix. This criterion results in minimizing the average variance of the estimates of the regression coefficients.

[edit] D-optimality

A popular criterion is D-optimality, which seeks to maximize |X'X|, the determinant of the information matrix X'X of the design. This criterion results in maximizing the differential Shannon information content of the parameter estimates.

[edit] E-optimality

A lesser known design is E-optimality, which maximizes the minimum eigenvalue of the information matrix.

[edit] G-optimality

Several designs are concerned with prediction variance. Among these, a popular criterion is G-optimality, which seeks to minimize the maximum entry in the diagonal of the hat matrix X'(X'X)^-1X. This has the effect of minimizing the maximum variance of the predicted values.

[edit] I-optimality

A second criterion on prediction variance is I-optimality, which seeks to minimize the mean squared prediction error.

[edit] V-optimality

A third criterion on prediction variance is V-optimality, which seeks to minimize the average prediction variance.

[edit] Drawbacks

Since the optimality criterion of most optimal designs is based on some function of the information matrix, the 'optimality' of a given design is model dependent. That is, the experimenter must specify a model for the design and the final number of design points desired before the 'optimal' design' can be generated. The design generated is 'optimal' only for that model.