1. Martin Ralbovský Jan Rauch KIZI FIS VŠE

2. Contents Motivation & introduction Graphs of quantifiers Classes of quantifiers, tables of critical frequencies Graphs of tables of critical frequencies

3. Motivation Association measures = quantifiers are crucial for quality association mining They have been extensively studied The formulas are hard to comprehend Sometimes interesting results

4. Four-fold contingency table Mψ ψ abr cds kln

5. Considered quantifiers 1 Founded implication Lower critical implication Upper critical implication

6. Considered quantifiers 2 Above average dependence Fishers quantifier Simple deviation

7. Considered quantifiers 3 Founded equivalence Pairing

8. Contents Motivation & introduction Graphs of quantifiers Classes of quantifiers, tables of critical frequencies Graphs of tables of critical frequencies

9. Initial remarks We used the Maple software: possibility to graph 2 dimensions usage of animation parameter to graph 3 dimensions We wanted to see the graphs and compare them, to know more about quantifiers from the graphs.

10. Graph and animation examples Founded implication graph

11. Findings after graphing Most graphs difficult to interpret and - founded & critical implications One interesting result obtained – comparison of founded equivalence and pairing quantifiers

12. Founded equivalence & pairing – known facts Founded equivalence equivalence > implication founded equivalence > founded implication finds equivalent occurrence of ψ and (in terms of positive/negative examples) [Kupka] Pairing quantifier new quantifier [Kupka] pairing of tuple of examined items

13. Graphs of FE, Pairing Founded equivalencePairing

14. Learning from graphs Characterizing properties: Founded equivalence – the bigger a+d, the better Pairing – the more a=d, the better When shouldnt be used: Founded equivalencePairing 1032000 099903 How to help Look at the contingency tables Proper base settings could help Combining the quantifiers

15. Contents Motivation & introduction Graphs of quantifiers Classes of quantifiers, tables of critical frequencies Graphs of tables of critical frequencies by Jan Rauch

16. Contents Motivation & introduction Graphs of quantifiers Classes of quantifiers, tables of critical frequencies Graphs of tables of critical frequencies

17. Comparing implicational quantifiers Founded implication – confidence, basic measure for association mining, simple to comprehend Lower and upper critical equivalence – statistical binomial test, hard to comprehend, computationally demanding If and when can be critical implications replaced by founded implication? What is relation between them?

18. Using table of maximal bs Table of maximal b is another definition for the quantifier It reduces the dimension (b and p), can be used to compare the implicational quantifiers For founded implication table of maximal b can be written as a function For critical implications, we cannot separate the variables

19. Tables of maximal bs for implicational quantifiers

20. Learning from graphs Lower critical < founded < upper critical Founded implication graph – linear curve Critical implications graphs – ??? We examined slopes of graphs: 101003005007009001000 Lower critical impl.0.10.160.20.210.2150.22 Upper critical impl.0.70.360.3060.2940.2850.2820.281

21. Learning from graphs II Seems that critical implications graphs are symmetric with respect to founded implication graph (slope 2.5) Our working hypothesis: For all natural a: lower critical < founded < upper critical

22. Creating tables of minimal |b-c| Constructing tables of minimal |b-c| for symmetrical quantifier with F property Algorithm: For given N, finding quadruples, contingency tables (a,b,c,d) for which the quantifier is valid For given a, d searching for maximal |b-c| when the quantifier is still valid Matrix indexed with a, d created Graphing the matrix

23. Construction

24. Fishers quantifier vs. simple deviation

25. For smaller n, the quantifiers are different For higher n, the quantifiers tend to be the same Why???

26. Above average quantifier

27. Above average quantifier - analysis In region with high a and low d with respect to n, the quantifier is not valid In this region the two fractions tend to have the same value, therefore it is hard to fulfill the inequality Again, not implicational behavior