On Segmentation of Interaction Values - Developing Concepts in Applied Intelligence - page 29

3.2 Interaction Matrix

Each interestingness measure is studied the interaction relations with the q − 1

remaining interestingness measures. In our work, we only focus on the interac-
tion relations from each couple of interestingness measures m i ,m j so that the

interaction value ξ ij between two interestingness measures is peformed by the

formula [7][11]:

ξ ij = μ ( {m i ,m j } − μ ( {m i } − μ ( {m j }

) ) ) (3)

A remark is that the interaction value ξ ij between to interestingness measures m i
and m j has the symmetric property ξ ij = ξ ji . After calculating the interaction
values between each couple of interestingness measures, we have a matrix of
interaction values ξ (i.e., interaction matrix):

⎡
⎣

⎤
⎦

ξ 11 ξ 12 ... ξ 1 q
ξ 21 ξ 22 ... ξ 2 q
... ... ... ...
ξ q 1 ξ q 2 ... ξ qq

ξ =

3.3 Interaction Graph

For an intuitive observation of the interactive relations obtained from an inter-
action matrix, we have proposed a graph model (i.e., interaction graph). The
interaction graph is an extension of the correlation graph [5] [11].
In the interaction graph, G =( V, E )

- V : each vertex represents an interestingness measure.
- E : each edge is the interaction value between the two vertices

An interaction graph with q interestingness measures will have ( q− 1) 2 edges. It
is dicult to observe and to evaluate the interaction relations, especially the
strongly interaction relations. The interaction relation exists when the interac-
tion value is greater or equal a predetermined threshold τ , called τ -interaction.

3.4 Tool for Interaction Graph

The model of interaction graph is implemented in the INTERACTION module of
the ARQAT tool [6]. This module also provide the important means to calculate
the interaction values between the interestingness measures via the capacity
function Sugeno. The module performs in the following steps :

1. Determine the first initialized values of the capacity function for each in-
terestingness measure from the data set or the user's points of view by the
submodule CAPACITYINITIALISATION.
2. Calculate the values of the capacity function with the algorithm ”Singleton
Fuzzy Measure Ratio Standard” [10] by using the submodule CAPACITY-
COMPUTATION.