D-optimal design with fixed covariates
dCV = dcovary(nfactors,fixed)
[dCV,X] = dcovary(nfactors,fixed)
[dCV,X] = dcovary(nfactors,fixed,
[dCV,X] = daugment(...,
dCV = dcovary(nfactors,fixed) uses
a coordinate-exchange algorithm to generate a D-optimal
design for a linear additive model with
subject to the constraint that the model include the fixed covariate
fixed. The number of runs in the design
is the number of rows in
fixed. The design
initial columns for treatments of the model terms.
[dCV,X] = dcovary(nfactors,fixed) also
returns the design matrix
X associated with the
[dCV,X] = dcovary(nfactors,fixed, uses
the linear regression model specified in
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.
model can be a matrix
specifying polynomial terms of arbitrary order. In this case,
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
X3, then a row
[0 1 2] in
(X1.^0).*(X2.^1).*(X3.^2). A row of all
model specifies a constant term,
which can be omitted.
[dCV,X] = daugment(..., specifies
additional parameter/value pairs for the design. Valid parameters
and their values are listed in the following table.
Lower and upper bounds for each factor, specified as
Indices of categorical predictors.
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
Initial design as an
Vector of number of levels for each factor.
Maximum number of iterations. The default is
The value is a structure that contains options specifying
whether to compute multiple tries in parallel, and specifying how
to use random numbers when generating the starting points for the
tries. Create the options structure with
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
Return Design for Linear Additive Model with Two Factors
The following example uses the
dummyvar function to block an eight-run experiment into 4 blocks of size 2 for estimating a linear additive model with two factors:
fixed = dummyvar([1 1 2 2 3 3 4 4]); dCV2 = dcovary(2,fixed(:,1:3),'linear')
dCV2 = 8×5 -1 1 1 0 0 1 -1 1 0 0 -1 -1 0 1 0 1 1 0 1 0 1 1 0 0 1 -1 -1 0 0 1 -1 1 0 0 0 1 -1 0 0 0
The first two columns of
dCV2 contain the settings for the two factors; the last three columns are the dummy variable encodings for the four blocks.
Return Design and Design Matrix for Linear Additive Model
Suppose you want a design to estimate the parameters in a three-factor linear additive model, with eight runs that necessarily occur at different times. If the process experiences temporal linear drift, you may want to include the run time as a variable in the model. Produce the design as follows:
time = linspace(-1,1,8)'; [dCV1,X] = dcovary(3,time,'linear')
dCV1 = 8×4 -1.0000 1.0000 1.0000 -1.0000 1.0000 -1.0000 -1.0000 -0.7143 -1.0000 -1.0000 -1.0000 -0.4286 1.0000 -1.0000 1.0000 -0.1429 1.0000 1.0000 -1.0000 0.1429 -1.0000 1.0000 -1.0000 0.4286 1.0000 1.0000 1.0000 0.7143 -1.0000 -1.0000 1.0000 1.0000
X = 8×5 1.0000 -1.0000 1.0000 1.0000 -1.0000 1.0000 1.0000 -1.0000 -1.0000 -0.7143 1.0000 -1.0000 -1.0000 -1.0000 -0.4286 1.0000 1.0000 -1.0000 1.0000 -0.1429 1.0000 1.0000 1.0000 -1.0000 0.1429 1.0000 -1.0000 1.0000 -1.0000 0.4286 1.0000 1.0000 1.0000 1.0000 0.7143 1.0000 -1.0000 -1.0000 1.0000 1.0000
The column vector
time is a fixed factor, normalized to values between ±1. The number of rows in the fixed factor specifies the number of runs in the design. The resulting design
dCV gives factor settings for the three controlled model factors at each time.
Automatic Parallel Support
Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.
To run in parallel, specify the
Options name-value argument in the call to
this function and set the
UseParallel field of the
options structure to
For more information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).
Introduced before R2006a