# Math: acm

## mis.pyhasse.acm

ACM-Background and Demo

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.