adjacency_matrix

1-0 matrix defining which categorical variable levels are related.

Specification

• Alias: None

• Arguments: INTEGERLIST

Description

The adjacency_matrix keyword is associated with discrete_design_set variables that are specified to be categorical. Each such variable is associated with one \(k \times k\) symmetric matrix, where \(k\) is the number of values (or levels) of the variable. Entry \(i,j\) of a matrix should be 1 if level \(i\) and level \(j\) are related by some subjective criteria or if \(i=j\) ; it should be 0 otherwise. The matrices for all variables of the same type ( string, real, or integer) are entered sequentially as a list of integers as shown in the examples below.

Default Behavior

The adjacency_matrix keyword is only relevant for discrete_design_set real and discrete_design_set integer variables if one or more of them have been specified to be categorical. It is always relevant for discrete_design_set string variables. If the user does not define an adjacency matrix, the default is method dependent. Currently, the only method that makes use of the adjacency matrix is mesh_adaptive_search, which uses a tri-diagonal adjacency matrix by default.

Expected Output

The expected output is method dependent.

Usage Tips

If an adjacency matrix is defined for one type of (categorical) discrete_design_set variable, if must be defined for all variables of that type, even for those not defined to be categorical. Those for the non-categorical set variables will be ignored.

Examples

The following example shows a variables specification where some real and some integer discrete_design_set variables are categorical.

variables
  continuous_design = 3
    initial_point  -1.0    1.5   2.0
    lower_bounds  -10.0  -10.0 -10.0
    upper_bounds   10.0   10.0  10.0
    descriptors    'x1'   'x2'  'x3'
  discrete_design_range = 2
    initial_point  2     2
    lower_bounds   1     1
    upper_bounds   4     9
    descriptors   'y1'  'y2'
  discrete_design_set
    real = 2
      elements_per_variable = 4 5
      elements  =  1.2 2.3 3.4 4.5 1.2 3.3 4.4 5.5 7.7
      descriptors  'y3'            'y4'
      categorical  'no'            'yes'
      adjacency_matrix 1 1 0 0  # Begin entry of 4x4 matrix for y3
                       1 1 1 0
                       0 1 1 1
                       0 0 1 1
                       1 0 1 0 1 # Begin entry of 5x5 matrix for y4
                       0 1 0 1 0
                       1 0 1 0 1
                       0 1 0 1 0
                       1 0 1 0 1
    integer = 2
      elements_per_variable = 2 3
      elements  =  4  7  8  9  12
      descriptors  'z1'  'z2'
      categorical  'yes' 'yes'

Note that for the real case, the user wants to define an adjacency matrix for the categorical variable, so adjacency matrices for both variables must be specified. The matrix for the first one will be ignored. Note that no adjacency matrix is specified for either integer categorical variable. The default will be used in both cases. Currently the only method taking advantage of adjacency matrices is mesh_adaptive_search, which uses a tri-diagonal adjacency matrix by default. Thus, the matrices used would be

z1: 1 1
    1 1
z2: 1 1 0
    1 1 1
    0 1 1

The following example shows a variables specification for string variables. Note that string variables are always considered to be categorical. If an adjacency matrix is not specified, a method-dependent default matrix will be used.

variables,
  continuous_design = 2
    initial_point  0.5  0.5
    lower_bounds   0.   0.
    upper_bounds   1.   1.
    descriptors =  'x'  'y'
  discrete_design_set string = 1
    elements = 'aniso1' 'aniso2' 'iso1' 'iso2' 'iso3'
    descriptors = 'ancomp'
    adjacency_matrix 1 1 0 0 0
                     1 1 0 0 0
                     0 0 1 1 1
                     0 0 1 1 1
                     0 0 1 1 1