Defining relation

The aliasing of all the factors of a design can be summarized by a defining relation. The defining relation of a design contains all the words used to generate the added factors of the design. Therefore, it uniquely defines any regular design.

For example, consider the 32-run design with one four-level factor \(A\), and two added two-level factors 27 and 30. Since the design involves 32 runs, there are 5 basic factors (\(a\) to \(e\)), so that the added factors are labeled \(f\) and \(g\). The column numbers, 27 and 30, correspond to the generators \(abde\) and \(bcde\), respectively. After relabeling the pseudo-factors, the generators become \(A_3de\) and \(A_2cde\). Therefore, the defining relation of the design can be written as $\( (A_3def, \; A_2cdeg) \)$

In the fatld package, we can create the defining relation of the design using the fatld.relation.Relation class that create a Relation object, which holds a defining relation.

>>> from fatld.relation import Relation, num2gen
>>> added_factors = [27, 30]
>>> letters = ['f', 'g']
>>> words = [num2gen(x) + letters[i] for i,x in enumerate(added_factors)]
>>> words
['abdef', 'bcdeg']
>>> # We specify `m` so that the words can be relabeled
>>> r = Relation(words, m=1)
>>> r
['A3def', 'A2cdeg']

Tip

If you are using a Design object, created using fatld.design.Design, you can obtain the defining relation of the design using the fatld.design.Design.defining_relation() method, instead of creating it by hand.

Expanding

Two or more words multiplied together can generate another word. This means that a defining relation with \(k\) words can be expanded to \(2^k-1\) words in total, by multliplying all the words together. All the aliasing between the factors of a design can be infered from such an expanded relation.

>>> full_relation = r.expand(relabel=True)
>>> full_relation
['A3def', 'A2cdeg', 'A1cfg']

When the full expanded relation is too long you can summarise it using the word length pattern. This pattern counts the number of words of each type and each length.

>>> [word_length(w) for w in full_relation]
[4, 5, 4]
>>> [word_type(w) for w in full_relation]
[1, 1, 1]
>>> r.word_length_pattern()
[[0, 0], [0, 2], [0, 1]]