About NIS-Apriori
PART1 :
Apriori based rule generation
1. Background I: How do we generate rules from table data sets?
Rough sets by Pawlak (1991), Apriori algorithm by Agrawal (1994)
2. Background II: How do we cope with information incompleteness in table data sets?
Incomplete information databases by Lipski, NISs by Orlowska (1984)
3. What are the rules with information incompleteness?
NIS-Apriori based rule generation (2008 - ), certain rules, possible rules
(NIS) age sex Tom {25,26,27} {male}
Mary {23} {female}
(DIS) age sex Tom 25 male Mary 23 female
(DIS) age sex Tom 26 male Mary 23 female
(DIS) age sex Tom 27 male Mary 23 female
All possible tables from NIS, derived DISs
NIS
(Non-deterministic Information System) DIS
(Deterministic
Information System
)(NIS) age sex Tom {25,26,27} {male}
Mary {23} {female}
[age,23]=>[sex,female], A rule in each derived DISs
This rule certainly holds in NIS.
Characteristic implication!!
Certain rule!!
(DIS) age sex Tom 25 male Mary 23 female
(DIS) age sex Tom 26 male Mary 23 female
(DIS) age sex Tom 27 male Mary 23 female
[age,23]=>[sex,female] support ≧ 0.3, accuracy ≧ 0.7
support=0.5, accuracy=1.0
support=0.5, accuracy=1.0
support=0.5, accuracy=1.0
(NIS) age sex Tom {25,26,27} {male}
Mary {23} {female}
[age,27]=>[sex,male]
A rule in one of derived DISs This rule may hold in NIS Characteristic implication!!
Possible rule!!
(DIS) age sex Tom 25 male Mary 23 female
(DIS) age sex Tom 26 male Mary 23 female
(DIS) age sex Tom 27 male Mary 23 female
[age,27]=>[sex,male] support ≧ 0.3, accuracy ≧ 0.7
support=0, accuracy=0
support=0, accuracy=0
support=0.5, accuracy=1.0
Certain rules and possible rules Ⅰ
NIS
DIS1 … Actual DIS … DISn
τ is a possible rule in NIS, if τ is a rule at least one derived DIS
τ is a certain rule in NIS, if τ is a rule in each of derived DIS
(definition by possible world semantics)
Not rules possible
rules certain
rules Set of all implications
2021/6/11 7
NIS
DIS1 … Real DIS … DISn
PROBLEM
The definition is natural, however how do we deal with?
The number of derived DISs increases exponentially, and the number is more than 10 power 100 in the
mammographic data set in UCI machine learning repository.
It is hard to handle each DIS sequentially.
Certain rules and possible rules Ⅱ
I gave a solution to this problem, and realized NIS-
Apriori !!
Φ
Ψ 1 … Ψ k … Ψ n
11
(maxsupp,maxacc)
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(minsupp,minacc)
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● ●
●
●
●
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(support(τ),accuracy(τ)) by
Ψmax(support(τ),accuracy(τ)) by
Ψminsupport(τ) accuracy(τ)
Each point (support( τ ),accuracy( τ ) ) by Ψk is located in the rectangle area.
(support(τ),accuracy(τ)) by
ΨkProved Property for each implication τ
For each implication τ
inf and sup for calculating
(minsupp,minacc) and (maxsupp,maxacc)
P Q R S T
1 3 {1,3} 3 2 3
2 2 {2,3} {1,3} {1,3} 2
3 {1,2} 2 {1,2} 3 1
4 1 3 3 {2,3} {1,2,3}
5 3 1 {1,2} 3 3
[P,1] [P,2] [P,3] ::::: [T,2] [T,3]
inf {4} {2} {1,5} ::::: {2} {1,5}
sup {3,4} {2,3} {1,5} ::::: {2,4} {1,4,5}
Calculation by inf and sup Ⅰ
OUTACC: A set of objects with the same condition and the different decision
INACC: A set of objects with the same condition
and the same decision
Calculation by inf and sup Ⅱ
• The calculation of (minsupp,minacc),
(maxsupp,maxacc) does not depend on the
number of derived DISs.
Apriori algorithm adjusted to table
data set
Apriori NIS-Apriori certain rule
• accuracy minacc
• support minsupp possible rule
• accuracy maxacc
• support maxsupp
Theoretical unique properties of NIS-Apriori
• Certain rules and possible rules are strictly defined by possible world semantics.
(Rules from tables may not be defined strictly.)
• NIS-Apriori includes Apriori as a special case.
(NIS-Apriori handles DISs, NISs, missing values ‘?’.)
• NIS-Apriori is sound and complete for rules, i.e., any rule can certainly be obtained.
(Few algorithms for tables are sound and complete.)
• The exponential order complexity of NIS-Apriori is reduced to linear order complexity of the Apriori algorithm.
(Due to this solution, NIS-Apriori works effectively.)
Papers for theoretical properties
• H. Sakai, M. Nakata, J. Watada: NIS-Apriori-based rule generation with three-way decisions and its application system in SQL, Information
Sciences, Vol. 507, pp.755-771 (2020)
• H. Sakai, M. Nakata, M.: Rough set-based rule generation and Apriori based rule generation from table data sets: A survey and a combination, CAAI Transactions on Intelligence Technology, Vol.4 No.4, pp. 214-222 (2019)
• H. Sakai, M. Wu, M. Nakata: Apriori-Based Rule Generation in Incomplete Information Databases and Non-Deterministic Information Systems,
Fundamenta Informaticae, Vol. 130, No.3, pp.343-376 (2014)
• H.Sakai, R.Ishibashi, M.Nakata: Rules and Apriori Algorithm in Non-
deterministic Information Systems, Transactions on Rough Sets, Springer- Verlag, Vol.9, pp.328-350 (2008)
