OptimalDesign (version 1.0.2.1)

od_SYM: Symmetrization of an approximate design

Description

Attempts to "symmetrize" an approximate design w by minimizing its norm while keeping its information matrix.

Usage

od_SYM(Fx, w, b1=NULL, A1=NULL, b2=NULL, A2=NULL, b3=NULL, A3=NULL, w0=NULL,
      crit="D", h=NULL, echo=TRUE)

Value

A list with the following components:

call

The call of the function

w.sym

The resulting "symmetrized" approximate design

Arguments

Fx

the n times m (where m>=2, m<=n) matrix containing all candidate regressors (as rows), i.e., n is the number of candidate design points, and m is the number of parameters

w

a non-negative vector of length n representing the design

b1,A1, b2,A2, b3,A3

the real vectors and matrices that define the constraints on permissible designs w as follows: A1 %*% w <= b1, A2 %*% w >= b2, A3 %*% w == b3. Each of the arguments can be NULL, but at least one of b1, b2, b3 must be non-NULL. If some bi is non-NULL and Ai is NULL, then Ai is set to be matrix(1, nrow =1, ncol = n).

w0

a non-negative vector of length n representing the design to be augmented (i.e., the function adds the constraint w >= w0 for permissible designs w). This argument can also be NULL; in that case, w0 is set to the vector of zeros.

crit

the optimality criterion. Possible values are "D", "A", "I", "C", "c".

h

a non-zero vector of length m corresponding to the coefficients of the linear parameter combination of interest. If crit is not "C" nor "c" then h is ignored. If crit is "C" or "c" and h=NULL then h is assumed to be c(0,...,0,1).

echo

Print the call of the function?

Author

Radoslav Harman, Lenka Filova

Details

For some models, the optimum approximate design is not unique (although the optimum information matrix usually is unique). This function uses one optimal approximate design to produce an optimal approximate design with a minimum Euclidean norm, which is unique and usually more "symmetric".

References

Harman R, Filova L, Richtarik P (2019). A randomized exchange algorithm for computing optimal approximate designs of experiments. Journal of the American Statistical Association, 1-30. (Subsection 5.1)

Examples

# Compute a D-optimal approximate design using the randomized method REX.
# Visualize both the design obtained by REX and its symmetrized version.

form.q <- ~x1 + x2 + x3 + I(x1^2) + I(x2^2) + I(x3^2) + I(x1*x2) + I(x1*x3) + I(x2*x3)
Fx <- Fx_cube(form.q, n.levels = c(5, 5, 5))
w.app <- od_REX(Fx)$w.best
od_plot(Fx, w.app, X=Fx[, 2:3])
w.app.sym <- od_SYM(Fx, w.app, b3 = 1)$w.sym
od_plot(Fx, w.app.sym, X=Fx[, 2:3])