Optimality Criteria¶
Optimal designs begin with a pseudo-random set of model points (runs) that are capable of fitting the designed for model. The initial selection can usually be improved by replacing a subset of the points with better selections. Stat-Ease 360 software uses one of five criteria to decide which replacements are better and up to two exchange methods to decide how they are replaced.
I-optimal¶
The I-optimal algorithm chooses runs that minimize the integral of the prediction variance across the factor space
The I-optimal criteria is recommended to build response surface designs where the goal is to optimize the factor settings, requiring greater precision in the estimated model.
D-optimal¶
A D-optimal algorithm chooses runs that minimizes the determinant of the variance-covariance matrix. This has the effect of minimizing the volume of the joint confidence ellipsoid for the coefficients.
The D-optimal criteria is recommended to build factorial designs where the goal is to find factors important to the process. It is commonly used to create fractional general factorial experiments.
A-optimal¶
An A-optimal design minimizes the trace of the variance-covariance matrix. This has the effect of minimizing the average prediction variance of the polynomial model coefficients.
The A-optimal criteria is not recommended, but is available for research purposes.
Note
We do not recommend using either distance optimality criterion for physical experiments. It is too likely the proposed design will produce poor estimates for all the designed for model terms.
Modified Distance¶
The modified distance-based point selection algorithm selects model points to obtain a maximum spread throughout the design region while ensuring that adequate runs are chosen to fit the polynomial model (as described below).
It is a multiple criteria optimization. The first criteria is distance, and the second criteria is D-optimality to check that the model matrix is full rank (i.e. all coefficients in the chosen polynomial can be estimated). A Pareto front approach is used to evaluate the criteria. As the design is constructed, each step is evaluated for the distance and D-optimality criteria. If either of the criteria get worse, , it is not considered. If neither is decreased, a combined criteria is used to evaluate the exchange. The combined criteria averages the two criteria. Exchanges continue until a Pareto optimal solution is reached, where no point can be moved without decreasing one of the two criteria.
Modified distance-based point selection provides a set of points spread throughout the feasible design region. The D-optimality portion of the criterion helps to ensure the points chosen are able to estimate the terms in the designed for model.
Distance¶
The Distance-based point selection provides a set of points spread (approximately) evenly over the feasible design region. There is NO assurance the points chosen are able to estimate the terms in the designed for model. This is essentially a design built with nothing but lack-of-fit points.