# cordexch

Coordinate exchange

## Syntax

dCE = cordexch(nfactors,nruns)
[dCE,X] = cordexch(nfactors,nruns)
[dCE,X] = cordexch(nfactors,nruns,'model')
[dCE,X] = cordexch(...,'name',value)

## Description

dCE = cordexch(nfactors,nruns) uses a coordinate-exchange algorithm to generate a D-optimal 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,'model') uses the linear regression model specified in 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:

1. The constant term

2. The linear terms in order 1, 2, ..., n

3. The interaction terms in order (1, 2), (1, 3), ..., (1, n), (2, 3), ..., (n – 1, n)

4. 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(...,'name',value) specifies one or more optional name/value pairs for the design. Valid parameters and their values are listed in the following table. Specify name inside single quotes.

nameValue
bounds

Lower and upper bounds for each factor, specified as a 2-by-nfactors matrix. Alternatively, this value can be a cell array containing nfactors elements, each element specifying the vector of allowable values for the corresponding factor.

categorical

Indices of categorical predictors.

display

Either 'on' or 'off' to control display of the iteration counter. The default is 'on'.

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 nfactors columns and b is a vector of Boolean values with the same number of rows as S. b(i) is true if the method should exclude ith row S.

init

Initial design as a nruns-by-nfactors matrix. The default is a randomly selected set of points.

levels

Vector of number of levels for each factor. Not used when bounds is specified as a cell array.

maxiter

Maximum number of iterations. The default is 10.

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 1.

options

A structure that specifies whether to run in parallel, and specifies the random stream or streams. This option requires Parallel Computing Toolbox™.

Create the options structure with statset. Structure fields:

• UseParallel — Set to true to compute in parallel. Default is false.

• UseSubstreams — Set to true to compute in parallel in a reproducible fashion. Default is false. To compute reproducibly, set Streams to a type allowing substreams: 'mlfg6331_64' or 'mrg32k3a'.

• Streams — A RandStream object or cell array of such objects. If you do not specify Streams, cordexch uses the default stream or streams. If you choose to specify Streams, use a single object except in the case

• UseParallel is true

• UseSubstreams is false

In that case, use a cell array the same size as the Parallel pool.

## Examples

Suppose you want a design to estimate the parameters in the following three-factor, seven-term 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 D-optimal 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.

## Algorithms

Both cordexch and rowexch use iterative search algorithms. They operate by incrementally changing an initial design matrix X to increase D = |XTX| 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, D-optimal 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 row-exchange 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 coordinate-exchange 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 coordinate-exchange algorithm is more likely to become trapped in a local minimum.

## Version History

Introduced before R2006a