Forms a reduced similarity matrix referring to the `GROUPS`

instead of the original units (synonym of `HREDUCE`

).

### Options

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

`METHOD` = string token |
Method used to form the reduced similarity matrix (`first, last, mean, minimum, maximum, zigzag` ); default `firs` |

### Parameters

`SIMILARITY` = symmetric matrices |
Input similarity matrix |
---|---|

`REDUCEDSIMILARITY` = symmetric matrices |
Output (reduced) similarity matrix |

`GROUPS` = factors |
Factor defining the groups |

`PERMUTATION` = variates |
Permutation order of units (for `METHOD` = `firs` , `last` or `zigz` ) |

### Description

This directive was renamed `HREDUCE`

in Release 14, but the original name of `REDUCE`

is currently still retained as a synonym. However, it may be removed in a future release.

Sometimes you may want to regard an *n*-by-*n* similarity matrix *S* as being partitioned into *b*-by-*b* rectangular blocks. You might then want to form a reduced matrix of similarities, between the different blocks instead of between the individual units. To do this you have to arrange for each of the *b*^{2} blocks of the full matrix to be replaced by a single value. Each diagonal block must be replaced by unity. The `METHOD`

option specifies how to replace the off-diagonal blocks, for example the maximum, minimum or mean similarity within the block. The *zigzag* method (Rayner 1966) is relevant in particular when the data consist of *b* soil samples for each of which information is recorded on several soil horizons, possibly different in the different samples. The method recognizes that certain horizons might be absent from some soil samples; this leads to finding successive optimal matches, conditional on the constraint that one horizon cannot match a horizon that has already been assigned to a higher level; after finding these optima, an average is taken for each horizon.

The `SIMILARITY`

parameter specifies the similarity matrix for the full set of *n* observations; this must be present and have values. The `REDUCEDSIMILARITY`

parameter specifies an identifier for the reduced similarity matrix, of order *b*; this will be declared implicitly if you have not declared it already. The factor that defines the classification of the units into groups must be specified by the `GROUPS`

parameter. The units can be in any order, so that for example the units of the first group need not be all together nor given first. The labels of the factor label the reduced similarity matrix.

The `PERMUTATION`

parameter, if present, must specify a variate. It defines the ordering of samples within each group, and so must be specified for methods `first`

, `last`

and `zigzag`

. Within each group, the unit with the lowest value of the permutation variate is taken to be the first sample, and so on. Genstat will, if necessary, use a default permutation of one up to the number of rows of the similarity matrix.

If you set option `PRINT=similarities`

, the values of the reduced symmetric matrix are printed, as percentages.

Options: `PRINT`

, `METHOD`

.

Parameters: `SIMILARITY`

, `REDUCEDSIMILARITY`

, `GROUPS`

, `PERMUTATION`

.

### Reference

Rayner, J.H. (1966). Classification of soils by numerical methods. *Journal of Soil Science*, 17, 79-92.

### See also

Directive: `HREDUCE`

.

Commands for: Multivariate and cluster analysis.