Math: pyhasse.core

Basic Calculations

Math explanations for basic of module for all pyhasse modules.

zeta-matrix

Order relations among objects can be described by different matrices, namely zeta-matrix and cover-matrix. There are other variants in literature possible,the two mentioned are most important.

The zeta matrix has entries 1, when the row determining object i is column defining object j. It is also possible to define the zeta matrix for object I object j. This is order theoretically irrelevant.

cover-matrix

Let us consider three objects i, j , k with I j k . Then the zeta matrix has the entries:

zeta(i,j) = 1, zeta(j,k) = 1,
and
zeta(i,k) = 1.

graphical representation would lead to a directed edge leading from i to j, from j to k and from i to k. A corresponding graph would be not clear enough for most purposes. As i j, j k implies by the transitivity axiom of partial order relations i k, the directed edge i - k can be eliminated as the remaining two lines i-j, j -k already imply i-k. The process of eliminating superfluous directed edges due to the axiom of transitivity is called a transitive reduction, and the corresponding matrix is called cover-matrix, denoted as cov. Taking the above mentioned case, the entries of the cover matrix would be

cov(i,j) = 1, cov(j,k) = 1, but cov(i,k) = 0.

Relational matrix

When data matrices with indicators qi and objects x,y,… are considered and the order relations are following the definition : x y, if qi(x) qi(y) for all indicators qi, then objects x,y may turn out as being equivalent with respect to the considered set of indicators. I.e. qi(x) = qi(y) for all I, written as x y Then it is convenient to consider x y as x y and y x . Then the entry of the relational matrix “relat” becomes : relat(x,y) = 1 and relat(y,x) = 1, which hurts the axiom of antisymmetry. Nevertheless for practical purposes it is convenient to describe in a first step the relations among objects by a relational matrix and reduce the relational matrix to a matrix describing only the representative elements of any equivalence class, which fulfills now the axioms of partial order and is a zeta matrix.

Distance matrix

Consider a set of indicators {q1,q2,q3}. Then it may be of interest which influence has any indicator on the structure of the Hasse diagram. Hereto the subsets {q1,q2}, {q,1,q3} and {q2, q3} can be analyzed. Any of these subsets, where just one indicator is omitted, can be compared with the original set {q1,q2,q3} by means of the squared Euclidian distance among the relational matrices. For example to see the influence of q1, the distance d(q1) is calculated. I.e. the squared Euclidian distance of the relational matrices due to {q1,q2,q3} and due to {q2, q3} is determined. Similarly the distances d(q2) and d(q3) can be determined. A large distance means that the corresponding indicator influences heavily the structure of the Hasse diagram. .