sens.minimax.control {ICAOD}

Returns Control Parameters for Verifying General Equivalence Theorem For Minimax Optimal Designs

Description

This function returns a list of control parameters that are used to find the “answering set” for minimax and standardized maximin designs. The answering set is required to obtain the sensitivity (derivative) function in order to verify the optimality of a given design.

Usage

sens.minimax.control(n_seg = 6, merge_tol = 0.005)

Arguments

n_seg

For a given design, the number of starting points in the local search to find all the local maxima of the minimax criterion over the parameter space is equal to (n_seg + 1)^p. Defaults to 6. Please increase its value when the parameter space is large. It is also applicable for standardized maximin designs. See 'Details' of sens.minimax.control.

merge_tol

Merging tolerance. It is used to specify the elements of the answering set by choosing only the local maxima (found by the local search) that are nearer to the global maximum. See 'Details' of sens.minimax.control. Defaults to 0.005. We advise to not change its default value because it has been successfully tested on many optimal design problems.

Details

Given a design, an “answering set” is a subset of all the local optima of the optimality criterion over the parameter space. Answering set is used to obtain the sensitivity function of a minimax or standardized maximin criterion. Therefore, an invalid answering set may result in a false sensitivity plot and ELB. Unfortunately, there is no theoretical rule on how to choose the number of elements of the answering set; and they have to be found by trial and error. Given a design, the answering set for a minimax criterion is obtained as follows:

• Step 1: Find all the local maxima of the optimality criterion (minimax) over the parameter space. For this purpose, the parameter space is divided into (n_seg + 1)^p segments, where p is the number of unknown model parameters. Then, each boundary point of the resulted segments (intervals) is assigned to the argument par of the function optim in order to start a local search using the "L-BFGS-B" method.

• Step 2: Pick the ones nearest to the global minimum subject to a merging tolerance merge_tol (default 0.005).

Obviously, the answering set is a subset of all the local maxima over the parameter space (or local minima in case of standardized maximin criteria) Therefore, it is very important to be able to find all the local maxima to create the true answering set with no missing elements. Otherwise, even when the design is optimal, the sensitivity (derivative) plot may not reveal its optimality.

Note that the minimax criterion (or standardized maximin criterion) is a multimodel function especially near the optimal design and this makes the job of finding all the locall maxima (minima) over the parameter space very complicated.

Value

A list of control parameters for verifying the general equivalence theorem for minimax and standardized maximin optimal designs.

Examples

sens.minimax.control()
sens.minimax.control(n_seg = 4)