Martin RalbovskýJan Rauch

KIZI FIS VŠE

ContentsMotivation & introductionGraphs of quantifiersClasses of quantifiers, tables of critical

frequenciesGraphs of tables of critical frequencies

MotivationAssociation measures = quantifiers are

crucial for quality association miningThey have been extensively studiedThe formulas are hard to comprehendSometimes interesting results

Four-fold contingency tableM ψ ψ

a b r

c d s

k l n

Considered quantifiers 1Founded implication

Lower critical implication

Upper critical implication

Considered quantifiers 2

Above average dependence

Fisher’s quantifier

Simple deviation

Considered quantifiers 3

Founded equivalence

Pairing

ContentsMotivation & introductionGraphs of quantifiersClasses of quantifiers, tables of critical

frequenciesGraphs of tables of critical frequencies

Initial remarksWe used the Maple software:

possibility to graph 2 dimensionsusage of animation parameter to graph 3 dimensions

We wanted to see the graphs and compare them, to know more about quantifiers from the graphs.

Graph and animation examplesFounded implication graph

Findings after graphingMost graphs difficult to interpret

and - founded & critical implications

One interesting result obtained – comparison of founded equivalenceand pairing quantifiers

Founded equivalence & pairing – known facts Founded equivalenceequivalence “>” implication

founded equivalence “>” founded implication finds equivalent occurrence of ψ and (in

terms of positive/negative examples) [Kupka]Pairing quantifiernew quantifier [Kupka]pairing of tuple of examined items

Graphs of FE, PairingFounded equivalence Pairing

Learning from graphsCharacterizing properties:

Founded equivalence – “the bigger a+d, the better”Pairing – “the more a=d, the better”

When shouldn’t be used:Founded equivalence Pairing1 0 3 20000 999 0 3

How to helpLook at the contingency tables Proper base settings could helpCombining the quantifiers

ContentsMotivation & introductionGraphs of quantifiersClasses of quantifiers, tables of critical

frequenciesGraphs of tables of critical frequencies

by Jan Rauch

ContentsMotivation & introductionGraphs of quantifiersClasses of quantifiers, tables of critical

frequenciesGraphs of tables of critical frequencies

Comparing implicational quantifiersFounded 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?

Using table of maximal b’sTable of maximal b is another definition for

the quantifierIt reduces the dimension (b and p), can be

used to compare the implicational quantifiersFor founded implication table of maximal b

can be written as a function

For critical implications, we cannot separate the variables

Tables of maximal b’s for implicational quantifiers

Learning from graphsLower critical “<“ founded “<“ upper criticalFounded implication graph – linear curveCritical implications graphs – ??? We examined slopes of graphs:

10 100 300 500 700 900 1000

Lower critical impl. 0.1

0.16 0.2 0.21 0.215 0.22 0.22

Upper critical impl. 0.7

0.36 0.306

0.294 0.285 0.282 0.281

Learning from graphs IISeems 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

Creating tables of minimal |b-c|Constructing tables of minimal |b-c| for

symmetrical quantifier with F propertyAlgorithm: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 createdGraphing the matrix

Construction

Fisher’s quantifier vs. simple deviation

Fisher’s quantifier vs. simple deviationFor smaller n, the quantifiers are differentFor higher n, the quantifiers tend to be the

sameWhy???

Above average quantifier

Above average quantifier - analysisIn 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