Uses the BLKL algorithm to construct designs for estimating response surfaces.

### Options

`PRINT` = string token |
Printed output required (`monitoring` ); default `*` i.e. no printing |
---|---|

`TERMS` = formula |
Model to be fitted when the design is used; no default i.e. this option must be specified |

`CONSTANT` = string token |
How to treat the constant in the model (`estimate, omit` ); default `esti` |

`FACTORIAL` = scalar |
Limit for expansion of terms in the model; default 2 |

`NUNITS` = scalar |
Number of units (or trials) in the design |

`NDELETION` = scalar |
Number of design points to consider for deletion; default takes `NUNITS` /4, or 4 is this is larger |

`NINCLUSION` = scalar |
Number of design points to consider for inclusion; default takes `NUNITS` /4, or 4 is this is larger |

`NRUNS` = scalar |
Number of times to run the algorithm; default 100 |

`ADJUSTMENTSTEP` = scalar |
Maximum amount by which to perturb the design points in the adjustment algorithm; default `*` i.e. no adjustments are tried |

`NBLOCKS` = scalar |
Number of blocks; default 1 i.e. design not blocked |

`BLOCKFACTOR` = factor |
Saves the block factor (if any) for the design |

`BLOCKSIZE` = scalar or variate |
Number of units in each block of the design |

`PREVIOUSBLOCKS` = factor |
Supplies values of the blocking factor for any previous experiments that are to be included in the analysis of the results of the design |

`MIXTURE` = variates |
Lists any variates that are part of a mixture (their values must be greater than zero and sum to one) |

`SEED` = scalar |
Seed for random numbers used to construct the initial design; default 124195 |

`DETERMINANT` = scalar |
Saves the determinant of the information matrix for the best design |

`MEANGRID` = scalar |
Saves the mean value of the standardized variance of predictions obtained from the design over a grid of x-values |

`MAXGRID` = scalar |
Saves the maximum value of the standardized variance of predictions obtained from the design over a grid of x-values |

`NGRIDPOINTS` = scalar |
Number of grid points in each x-direction to use for `MEANGRID` and `MAXGRID` ; default 5 |

### Parameters

`X` = variates |
Lists the variates to be investigated in the design; these need not be supplied if none of the other parameters are required |
---|---|

`X2` = variates |
Lists identifiers to be used to represent squares of the x-variates in the model |

`X3` = variates |
Lists identifiers to be used to represent cubes of the x-variates in the model |

`SUPPORTPOINTS` = variates |
Support points for each x-variate in the design; if these are not (all) specified, they are formed automatically |

`PREVIOUSVALUES` = variates |
Supplies values of the x-variates for any previous experiments that are to be included in the analysis of the results of the design |

### Description

`AFRESPONSESURFACE`

uses the BLKL algorithm of Atkinson & Donev (1992) to construct a design to estimate parameters of a response-surface model. The algorithm searches for a D-optimal design: that is, a design that will provide a maximum value for the determinant of the information matrix of the model parameters. The model is specified using the `TERMS`

option, with the `CONSTANT`

option indicating whether or not it is to contain the constant term (or intercept). The `FACTORIAL`

sets a limit on the number of variates in each model term; by default this is 2.

The `NUNITS`

option specifies the number of units in the design. If there is to be a blocking factor in the design, the `NBLOCKS`

option specifies its number of levels, and the `BLOCKFACTOR`

option saves its values. The `BLOCKSIZE`

option specifies the number of units to be contained in each block of the design, in a scalar (if they are all the same) or a variate. If the block sizes are fixed, the specified sizes must sum to the number of units. However, if you specify sizes that sum to a value greater that the required number of units, the algorithm will search for the optimum block sizes.

When the model is to contain squares or cubes of x-variables, you will need to specify identifiers to represent these using the parameters of the directive. (When using regression directives such as `FIT`

to fit the model, you can use the `POL`

function but this is not recognised by `AFRESPONSESURFACE`

.) The x-variates in the model must then all be listed by the `X`

parameter. The corresponding squares are listed by the `X2`

parameter, and the cubes by the `X3`

parameter.

After specifying the `X`

parameter, you can also use the `SUPPORTPOINTS`

parameter to specify the x-values of the points to be considered when constructing the design; if this is not specified, these support points are formed automatically. Note that the variates are all assumed to be scaled to have values between -1 and 1. However, the criterion for D-optimality is unaffected by linear transformations of the X-variables. So you can rescale afterwards in any way you like. `AFRESPONSESURFACE`

allows for a set of mixture variates, whose values must all be positive and which must sum to 1. The variates in the mixture are specified using the `MIXTURE`

option.

The `PREVIOUSVALUES`

parameter can be used to supply values of the x-variates for any previous experiments that are to be included in the analysis of the results of the new experiment, or to specify points that must be included in the design. The `PREVIOUSBLOCKS`

option should then indicate the blocks to which these previous observations belonged.

The BLKL algorithm starts by forming an initial design by making a random selection of points from the set of support points. The `SEED`

option defines the seed for the random numbers used to make the selection (default 124195). The algorithm then uses an exchanges algorithm to improve the design. At each exchange, the *K* points with the lowest variance of prediction amongst the points of design are considered for replacement by the *L* points with the highest variance of prediction amongst the candidate points for inclusion in the design. The algorithm makes the best one of these exchanges, continuing until there are none that increase the determinant. The values for *K* and *L* are specified by the `NDELETION`

and `NINCLUSION`

options respectively. The best values depend on the design parameters, including the number of model parameters and the number of residual degrees of freedom. If they are unset, `AFRESPONSESURFACE`

sets them to the number of units divided by 4, or 4 if this larger. The `NRUNS`

option can be set to request that the algorithm is run several times, with different starting designs; the default is 100. The design parameters are saved only for the best design found, but you can set option `PRINT=monitoring`

to print information about each attempt.

There is also be a final adjustment algorithm which can be used except when the design contains mixtures. This examines the design points one at a time to see whether the design can be improved by moving it a small amount along any x-axis. If an increase is possible, the point providing the greatest increase is moved. The process is then repeated until no improvment is possible. This phase is selected by setting the `ADJUSTMENTSTEP`

option to the maximum amount (e.g. 0.2) by which the point may be moved on any axis.

The `DETERMINANT`

option allows you to saves the determinant of the information matrix for the best design. An alternative way of evaluating the design is to examine the standardized variance of the predictions that would be obtained from the design at other points, not in the design. The `MEANGRID`

option can save the mean value of the standardized variance of prediction over a grid of x-values, and the `MAXGRID`

option can save the maximum value. Number of grid points in each x-direction is specified by the `NGRIDPOINTSMETHOD`

option (default 5).

Options: `PRINT`

, `TERMS`

, `CONSTANT`

, `FACTORIAL`

, `NUNITS`

, `NDELETION`

, `NINCLUSION`

, `NRUNS`

, `ADJUSTMENTSTEP`

, `NBLOCKS`

, `BLOCKFACTOR`

, `BLOCKSIZE`

, `PREVIOUSBLOCKS`

, `MIXTURE`

, `SEED`

, `DETERMINANT`

, `MEANGRID`

, `MAXGRID`

, `NGRIDPOINTS`

.

Parameters: `X`

, `X2`

, `X3`

, `SUPPORTPOINTS`

, `PREVIOUSVALUES`

.

### Method

The algorithm is described in Sections 15.6 and 15.7 of Atkinson & Donev (1992). The source code of the algorithm was provided by Alex Donev. This, and his assistance generally with this Genstat implementation, is gratefully acknowledged.

### Action with `RESTRICT`

Restrictions on the `X`

, `X2`

, `X3`

, `SUPPORTPOINTS`

or `PREVIOUSVALUES`

parameters are ignored.

### Reference

Atkinson, A.C. & Donev, A.N. (1992) *Optimum Experimental Designs*. Oxford University Press, Oxford.

### See also

Procedures: `AFNONLINEAR`

, `AGBOXBEHNKEN`

, `AGCENTRALCOMPOSITE`

, `AGFACTORIAL`

, `AGMAINEFFECT`

, `RQUADRATIC`

.

Commands for: Design of experiments, Regression analysis.

### Example

" Examples 2:4.9.14a-c " AFRESPONSESURFACE [NDELETION=10; NINCLUSION=40; NRUNS=1000; NUNITS=17;\ TERMS=X1 * X2 * X3 * X4 + X1_2 + X2_2 + X3_2 + X4_2;\ CONSTANT=estimate; DETERMINANT=Det; MAXGRID=Dmax; MEANGRID=Dave]\ X=X1,X2,X3,X4; X2=X1_2,X2_2,X3_2,X4_2 PRINT X1,X2,X3,X4; FIELD=8; DECIMALS=3 & Det,Dmax,Dave VARIATE [NVALUES=15] S1,S2,S3 READ [PRINT=data,errors] S1,S2,S3 -0.5774 -0.5774 -0.5774 0.5774 -0.5774 -0.5774 -0.5774 0.5774 -0.5774 0.5774 0.5774 -0.5774 -0.5774 -0.5774 0.5774 0.5774 -0.5774 0.5774 -0.5774 0.5774 0.5774 0.5774 0.5774 0.5774 1.0000 0.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 : VARIATE [NVALUES=6] P1,P2,P3 READ [PRINT=data,errors] P1,P2,P3 1.0000 0.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 -1.0000 : AFRESPONSESURFACE [NUNITS=16; NDELETION=3; NINCLUSION=3;\ TERMS=X1 * X2 * X3 + X1_2 + X2_2 + X3_2; CONSTANT=estimate;\ DETERMINANT=Det; MAXGRID=Dmax; MEANGRID=Dave] \ X=X1,X2,X3; X2=X1_2,X2_2,X3_2; SUPPORTPOINTS=S1,S2,S3 SORT [INDEX=X1,X2,X3] X1,X2,X3 PRINT X1,X2,X3; FIELD=8; DECIMALS=3 & Det,Dmax,Dave AFRESPONSESURFACE [NUNITS=12; NDELETION=3; NINCLUSION=3; NRUNS=500; \ TERMS=X1 + X2 + X3 + X4 + X1.X2 + X1.X3 + X2.X3 + X4_2; CONSTANT=omit; MIXTURE=X1,X2,X3;\ DETERMINANT=Det; MAXGRID=Dmax; MEANGRID=Dave]\ X=X1,X2,X3,X4; X2=*,*,*,X4_2 PRINT X1,X2,X3,X4; FIELD=8; DECIMALS=3 & Det,Dmax,Dave