cordexch - Coordinate exchange

Syntax

dCE = cordexch(nfactors,nruns) [dCE,X] = cordexch(nfactors,nruns) [dCE,X] = cordexch(nfactors,nruns,model) [dCE,X] = cordexch(...,param1,val1,param2,val2,...)

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 strings:

'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(...,param1,val1,param2,val2,...) specifies additional parameter/value pairs for the design. Valid parameters and their values are listed in the following table.

Parameter	Value
`'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.
`'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`.

Algorithm

Both cordexch and rowexch use iterative search algorithms. They operate by incrementally changing an initial design matrix X to increase D = |X^TX| 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.

Examples

Suppose you want a design to estimate the parameters in the following three-factor, seven-term interaction model:

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.

cordexch - Coordinate exchange

Syntax

Description

Algorithm

Examples

See Also

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