From Reachability Matrix to Hierarchical Matrix: Difference between revisions
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==== Process ==== | ==== Process ==== | ||
# Two subsets of the element set are constructed for each element e<sub>i</sub>: | # Two subsets of the element set are constructed for each element e<sub>i</sub>: | ||
# | #* The reachability set R<sub>i</sub> which involves all of those elements that are reachable from element e<sub>i</sub> | ||
# | #* An antecedent set A<sub>i</sub> which involves all of those elements that reach element e<sub>i</sub>. <br>In other words for each element e<sub>i</sub> the set R<sub>i</sub> can be determined as the set of elements whose columns have an entry of 1 in row i of the Reachability Matrix. | ||
In other words for each element e<sub>i</sub> the set R<sub>i</sub> can be determined as the set of elements whose columns have an entry of 1 in row i of the Reachability Matrix. | # After constructing the sets Ri and Ai their intersection Ri Ai is found next, that is all of those elements which are common to both Ri and Ai. Those elements ei for which Ri Ai = Ri are not reachable from any of the remaining element and thus constitute top/first level elements of the hierarchy (Table I). | ||
# During the next step these top level elements – the rows and columns corresponding to them – are striking from the Reachability Matrix and the procedure is repeated. The resulting elements are also top level elements of the hierarchy; hence they constitute second/top level elements (Table II). By repeating the above process – until there are no more elements to take part in – it is feasible to determine all the levels of the element system. In Tables I – IV you can see step-by-step the calculation of reachability and antecedent sets based upon the Reachability Matrix of the above example. | |||
Latest revision as of 14:15, 7 January 2022
The Hierarchical Reachability Matrix is constructed in order to partition the element set into levels.
Process
- Two subsets of the element set are constructed for each element ei:
- The reachability set Ri which involves all of those elements that are reachable from element ei
- An antecedent set Ai which involves all of those elements that reach element ei.
In other words for each element ei the set Ri can be determined as the set of elements whose columns have an entry of 1 in row i of the Reachability Matrix.
- After constructing the sets Ri and Ai their intersection Ri Ai is found next, that is all of those elements which are common to both Ri and Ai. Those elements ei for which Ri Ai = Ri are not reachable from any of the remaining element and thus constitute top/first level elements of the hierarchy (Table I).
- During the next step these top level elements – the rows and columns corresponding to them – are striking from the Reachability Matrix and the procedure is repeated. The resulting elements are also top level elements of the hierarchy; hence they constitute second/top level elements (Table II). By repeating the above process – until there are no more elements to take part in – it is feasible to determine all the levels of the element system. In Tables I – IV you can see step-by-step the calculation of reachability and antecedent sets based upon the Reachability Matrix of the above example.