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rowexch

Row exchange

Syntax

dRE = rowexch(nfactors,nruns)
[dRE,X] = rowexch(nfactors,nruns)
[dRE,X] = rowexch(nfactors,nruns,model)
[dRE,X] = rowexch(...,param1,val1,param2,val2,...)

Description

dRE = rowexch(nfactors,nruns) uses a row-exchange algorithm to generate a D-optimal design dRE with nruns runs (the rows of dRE) for a linear additive model with nfactors factors (the columns of dRE). The model includes a constant term.

[dRE,X] = rowexch(nfactors,nruns) also returns the associated design matrix X, whose columns are the model terms evaluated at each treatment (row) of dRE.

[dRE,X] = rowexch(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:

  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.

[dRE,X] = rowexch(...,param1,val1,param2,val2,...) specifies additional parameter/value pairs for the design. Valid parameters and their values are listed in the following table.

ParameterValue
'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 ith row S should be excluded.

'init'

Initial design as an 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.

options

A structure that specifies whether to run in parallel, and specifies the random stream or streams. Create the options structure with statset. Option 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, rowexch uses the default stream or streams. If you choose to specify Streams, use a single object except in the case

    • You have an open Parallel pool

    • UseParallel is true

    • UseSubstreams is false

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

'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.

Examples

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

Use rowexch to generate a D-optimal design with seven runs:

nfactors = 3;
nruns = 7;
[dRE,X] = rowexch(nfactors,nruns,'interaction','tries',10)
dRE =
    -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 dRE. 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 dRE.

More About

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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.

At each step, the row-exchange algorithm exchanges an entire row of X with a row from a design matrix C evaluated at a candidate set of feasible treatments. The rowexch function automatically generates a C appropriate for a specified model, operating in two steps by calling the candgen and candexch functions in sequence. Provide your own C by calling candexch directly. In either case, if C is large, its static presence in memory can affect computation.

See Also

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