The catalogue of types of geometric neighbors over four variables
Jiří Vomlel and Milan Studený
This is a catalogue of types of
differential imsets (for standard imsets that are geometric neighbors)
over 4 variables. The catalogue is meant as an extension of the paper
M. Studený, J. Vomlel: On open
questions in the geometric approach to structural learning Bayesian
nets, submitted to International Journal of Approximate Reasoning,
special issue from WUPES 09.
The motivation and deeper theoretical background is explained in that
paper, including rough description of the catalogue in Section 5.2.
Recall that the items in our catalogue are types (= permutation
equivalence classes) of differential imsets over 4 variables for those
standard imsets that are geometric neighbors. Altogether we have 319 of
such types, that is, 319
items in the catalogue.
Each type is described (in the catalogue) by one representative (= a
member of the respective permutation equivalence class). It is a
differential imset w=u-v,
where $u,v$ are standard imsets over N={ a,b,c,d} that are geometric
neighbors. The items are classified using three
criteria/characteristics:
- the degree difference,
which takes values 0,1,2, and 3,
- the squared Euclidean
length (of w),
whose possible values are even numbers between 4 and 22,
- the number of non-zero
values (in w),
falling within limits 4 and 12.
Each item in the catalogue is ascribed an identification number, which
is an integer between 1 and 319.
The catalogue has a form of a pdf file, which is divided into chapters,
sections, subsections and sub-subsections.
- Chapters correspond to the degree difference,
- sections (within each chapter) correspond to the squared
Euclidean length,
- subsections (within each section) correspond to the number of
non-zero values,
- sub-subsections within each subsections correspond items of the
catalogue (= types) that fall within that particular subsection.
The numbering of chapters, sections and subsections is given by the
value of the respective characteristic.
One can walk through the catalogue with the help of bookmarks
(available by advanced versions of Adobe Reader). The top level of
those bookmarks correspond to chapters (= the degree difference). If
one clicks on the chapter icon, the Adobe Reader moves to the beginning
of the chapter. One can also click on the plus sign +
(attached left to the chapter icon) to roll out the icons for sections
available within that chapter. (These correspond to the squared
Euclidean length.) Again, if one clicks on the section icon the Adobe
reader moves to its beginning. Conversely, the
icons for sections within a particular chapter can be back hidden if
one clicks on the minus sign -, which replaced the former plus sign
(attached left to the chapter icon). The same principle holds for
revealing (and hiding) subsections within sections and items within
subsections.
Thus, the leaves of that virtual tree are items in the catalogue. Each
such item contains the following information:
- Hasse diagram (over N={
a,b,c,d}) of the differential imset w (that was chosen to represent
that particular type)
- record of (the same) w
in terms of basic imsets, that is, in terms of delta notation,
- one of the shortest possible combinations of elementary imsets
(with +/- 1 coefficients) giving w
(see Section 5.1 of the paper for an explanation),
- the list of all pairs of standard imsets [u,v] that are geometric neighbors
and yield w as their
differential imset w=u-v,
- the essential graphs for those (pairs of) standard imsets.
The catalogue is available as a PDF file
here.