1) The document provides a tutorial on using the Chain module in PyHasse to analyze partially ordered sets (posets) and find chains within the poset. 2) It explains how to use the module to view the Hasse diagram of a poset, find the number and characteristics of chains between objects, and calculate the similarity between chains. 3) The tutorial demonstrates these analyses on a dataset of object pollution profiles, finding 5 chains between objects 57 and 30 and calculating the Tanimoto similarity coefficients between each pair of chains.

1. Path to „Idagrotte“, Sächsische Schweiz
Tutorial: Chain
• Rainer Bruggemann
• Peter Koppatz
Rauschenstein,
Elbsandsteingebirge

2. As usual you have to select your set of interest.

3. A list of object sets pops up. The blue button at the left side: Selection of subsets.
Usually the Hasse diagram of the selected object set, i.e. „chain-pollution.csv“
Is of most interest: Select „General Info“

4. General Info provides information about the partially ordered set
(here „chain-pollution.csv“) which are not module specific.
Here we want to see the Hasse diagram.

5. An important information can be deduced by inspecting the Hasse diagram:
For example the sequence 30 < 6 < 22 < 14 < 57 shows that for some objects
the attribute values are simultaneously increasing (including equality).
Subsets of objects with no mutual incomparability are called chains.
Do we have more than one chain? Are chains different from each other?
For these two questions you will find answers in the menu „Module Specific“.

7. Selection: Chain statistics
Here you are informed about chains between the maximal and minimal elements of
the poset, having a height  a certain internally calculated average value.
For instance between 18 and 30 there is no chain with a height 4.556, whereas
between 57 and 30 five chains can be found having a sufficient large height.

8. Usually you want to find more conclusions examining pretty long chains.
1) Most often dissimilar chains are of interest, two chains are the dissimilar the less
coinciding objects they contain.
2) Sometimes one is interested in exceptions where the synchroneous increase of
attribute values leads to conflicting object profiles (see spyout).
The menu „Similarity as table“ calculates the Tanimoto-coefficient of pairs of chains
having the same height. Whereas the menu „Similarity as bar diagrams“ represents the
Tanimoto values as horizontal bars for each pair of chain.

9. The first part informs about potential candidates. For example of interest
is the object pair 57 and 30 because here 5 chains appear. Selecting the startvertex 57 and the
sinkvertex 30 the following results appear:

10. Note that startvertex, sinkvertex and maxch have to be carefully be selected.
After the selection, the button „Calc“ is to be pressed.
The probable common chain height is indicated in
„Number of levels = number of elements in a maximum chain“ which is 5 in the poset
selected (red ellipse).
The entries are the Tanimoto coefficients of each of the pairs of the five chains.
Chain 1 is similar with chain 2 (0.667), whereas chain 5 differs considerably from chain 1 (0.25).

11. „Similarity as bardiagram“
The presentation in a table as in the slide before contains redundant information,
because similarity is a symmetric property. Hence the presentation as a bar diagram
may be more appropriate.

12. The menu „chain selection“ aims at a information about any single chains the user
want to examine:
After selection of startvertex, sinkvertex and maxch (whereby the chain statistics could
give valuable information) one of the possible chains is to be selected.
We know, between 30 and 57 there are 5 chains, so the user has first to select the
Chain he is mostly interested. For example chain no 5.

13. Pressing the button „Show chain as Hasse diagram“ generates a Hasse diagram
where the special chain, selected by the user, is marked (next slide):

14. Chain 5 in the Hasse diagram, visualizing the poset of the dataset „chain-pollution.csv“.

15. By crossing with the mouse, a little window pops up,
Informing the user about equivalent elements and
about the profiles. In the chain 5 a simultaneous increase
from profile (0,0,0,0) of object 30 to (2,1,0,1) of object 57
can be observed.
A chain 30,6,45,14,57 would have the height 5 too and would
be very similar to chain no 5. Tanimoto-coefficient = 4/6 = 0.667.
Object 14 has the profile (1,1,0,1) , whereas object 38 has the
profile (2,1,0,0). Although object 14 does not contradict the
simultaneous increase of the attribute values from 30 to 57
it is in conflict with 38. Coincidence in the 2nd and third attribute,
but differences in the first and fourth attribute.

16. • See also:
• https://www.pyhasse.org,
• http://de.slideshare.net/pyhasse/tutorial-spyout
• http://chain.pyhasse.org (Demo)
• You want to contact us: The email address:
chain@pyhasse.org
• You want to call us?
– R.Bruggemann (+49) 30 6496676
– P.Koppatz (+49) 331 20029708