Coordinate exchange
dCE = cordexch(nfactors,nruns)
[dCE,X] = cordexch(nfactors,nruns)
[dCE,X] = cordexch(nfactors,nruns,'model
')
[dCE,X] = cordexch(...,'name
',value
)
dCE = cordexch(nfactors,nruns)
uses
a coordinateexchange algorithm to generate a Doptimal
design dCE
with nruns
runs (the
rows of dCE
) for a linear additive model with nfactors
factors
(the columns of dCE
). The model includes a constant
term.
[dCE,X] = cordexch(nfactors,nruns)
also
returns the associated design matrix X
, whose columns
are the model terms evaluated at each treatment (row) of dCE
.
[dCE,X] = cordexch(nfactors,nruns,'
uses
the linear regression model specified in model
')model
. model
is
one of the following:
'linear'
— Constant and
linear terms. This is the default.
'interaction'
— Constant,
linear, and interaction terms
'quadratic'
— Constant,
linear, interaction, and squared terms
'purequadratic'
— Constant,
linear, and squared terms
The order of the columns of X
for a full
quadratic model with n terms is:
The constant term
The linear terms in order 1, 2, ..., n
The interaction terms in order (1, 2), (1, 3), ..., (1, n), (2, 3), ..., (n – 1, n)
The squared terms in order 1, 2, ..., n
Other models use a subset of these terms, in the same order.
Alternatively, model
can be a matrix
specifying polynomial terms of arbitrary order. In this case, model
should
have one column for each factor and one row for each term in the model.
The entries in any row of model
are powers
for the factors in the columns. For example, if a model has factors X1
, X2
,
and X3
, then a row [0 1 2]
in model
specifies
the term (X1.^0).*(X2.^1).*(X3.^2)
. A row of all
zeros in model
specifies a constant term,
which can be omitted.
[dCE,X] = cordexch(...,'
specifies
one or more optional name/value pairs for the design. Valid parameters
and their values are listed in the following table. Specify name
',value
)name
inside
single quotes.
name  Value 

bounds  Lower and upper bounds for each factor, specified as
a 
categorical  Indices of categorical predictors. 
display  Either 
excludefun  Handle to a function that excludes undesirable runs.
If the function is f, it must support the syntax b = f(S),
where S is a matrix of treatments with 
init  Initial design as a 
levels  Vector of number of levels for each factor. Not used
when 
maxiter  Maximum number of iterations. The default is 
tries  Number of times to try to generate a design from a new
starting point. The algorithm uses random points for each try, except
possibly the first. The default is 
options  A structure that specifies whether to run in parallel,
and specifies the random stream or streams. Create the

Suppose you want a design to estimate the parameters in the following threefactor, seventerm interaction model:
$$y={\beta}_{0}+{\beta}_{1}x{}_{1}+{\beta}_{2}x{}_{2}+{\beta}_{3}x{}_{3}+{\beta}_{12}x{}_{1}x{}_{2}+{\beta}_{13}x{}_{1}x{}_{3}+{\beta}_{23}x{}_{2}x{}_{3}+\epsilon $$
Use cordexch
to generate a Doptimal
design with seven runs:
nfactors = 3; nruns = 7; [dCE,X] = cordexch(nfactors,nruns,'interaction','tries',10) dCE = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Columns of the design matrix X
are the model
terms evaluated at each row of the design dCE
.
The terms appear in order from left to right: constant term, linear
terms (1, 2, 3), interaction terms (12, 13, 23). Use X
to
fit the model, as described in Linear Regression, to response data measured at the design
points in dCE
.
Both cordexch
and rowexch
use
iterative search algorithms. They operate by incrementally changing
an initial design matrix X to increase D =
X^{T}X
at each step. In both algorithms, there is randomness built into the
selection of the initial design and into the choice of the incremental
changes. As a result, both algorithms may return locally, but not
globally, Doptimal designs. Run each algorithm
multiple times and select the best result for your final design. Both
functions have a 'tries'
parameter that automates
this repetition and comparison.
Unlike the rowexchange algorithm used by rowexch
, cordexch
does
not use a candidate set. (Or rather, the candidate set is the entire
design space.) At each step, the coordinateexchange algorithm exchanges
a single element of X with a new element evaluated
at a neighboring point in design space. The absence of a candidate
set reduces demands on memory, but the smaller scale of the search
means that the coordinateexchange algorithm is more likely to become
trapped in a local minimum.