Math: pyhasse.acm

ACM-Background

The defining equation within Hassediagram technique (HDT) and its software realization by PyHasse is:

Let be objects of an object set X, and be the indicators of a multi-indicator system (mis)

The objects , are called comparable, if equation (1) holds:

: if and only if for all (1)

is the value of object , with respect to attribute . If (1) does not hold, and are »incomparable«, denoted as

The incomparability of with caused by a specific attribute pair is denoted by

(2)

The comparability of to is consequently defined by

(3)

The module ACM analyzes the incomparabilities which may appear among the objects of a user-selected subset .

1. Rendering the ac (antichain) matrix:
   Let s be the row determined by the object pair , and t the column determined by the pair , . Then an entry of the ac-matrix is defined as follows:

   (4)

2. Rowsums of ac-matrix (eventually restricted to the pairs of objects resulting from a user-selected subset X’), called

   (5)

   A large value of rss indicates that many attribute pairs are causing an incomparability of with .

3. Columnsums of ac-matrix (eventually restricted to the pairs of objects resulting from a user-selected subset , called .

   (6)

   with n the number of selected objects i.e.:

   

   A large value of css indicates that the selected attribute pair (corresponding to the column t of ac) is causing incomparabilities for many object pairs taken from X'.

4. let w be either or , then it is of interest to find: max(w) and the corresponding list of attribute pairs (w = rs) or of object pairs (w = cs). These maximal values and associated lists inform about crucial pairs of objects and attributes, resp.