Extended Random Sets for Knowledge Discovery in Information Systems

1
Extended Random Sets for Knowledge Discovery in
Information Systems

• Yuefeng Li
• School of Software Engineering and Data
  Communications
• Queensland University of Technology
• Brisbane 4001, Australia
• Email y2.li_at_qut.edu.au

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Data Mining

• Data mining, which is also referred to as
  knowledge discovery in database is a process of
  nontrivial extraction of implicit, previously
  unknown and potentially useful information
  (patterns) from data in databases
• Typical approaches
• Data classification
• Data clustering
• Dining association rules

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Association Rules

• The objective of mining association rules
• is to discover all rules that have support and
  confidence greater than the user-specified
  minimum support and minimum confidence
• The form of a rule is
• A1 ? A2 ??Am ? B1 ? B2 ??Bm ,
• where Ai and Bj are sets of attributes values
  from the relevant datasets in a database.

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Association Rules cont.

• A ? B is an interesting rule iff P(BA) P(B) is
  greater than a suitable constant.
• Criteria
• Frequency of occurrence is a well-accepted
  criterion.
• The rules should reflect real world phenomena,
  that is, data mining is to find real world
  patterns.
• It is desirable to use some mathematical models
  to interpret association rules in order to obtain
  useful patterns.

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Databases to Decision Tables

Table 1. A decision table
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An Example cont.

• Attributes driver, vehicle type, weather,
  road, time, accident
• Condition attributes weather, road
• Decision attributes time, accident
• Decision rule, e.g., if the weather is foggy and
  road is icy then the accident occurred at night
  in 140 cases.

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Formalization Rough Set

• S (U, A) -- an information system
• Where, U, a database, a set of records.
• A, a set of attributes and
• There is a function for every attribute a?A such
  that a U ? Va, where Va is the set of all values
  of a. We call Va the domain of a.

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Formalization Rough Set cont.

• B-granule
• Let B be a subset of A. B determines a binary
  relation I(B) on U such that (x, y) ? I(B) if and
  only if a(x) a(y) for every a?B, where a(x)
  denotes the value of attribute a for element x?U.
  
• I(B) is an equivalent relation, it determines a
  family of all equivalent classes of I(B)
• The partition determined by B, is denoted by U/B.
  
• The classes in U/B are referred to B-granules.
• The class which contains x is called B-granule
  induced by x, and is denoted by B(x).

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Formalization Rough Set cont.

• (U, C, D) is called a decision table of (U, A),
  iff
• C?D?A, where C, condition attributes, and D,
  decision attributes, are disjoint sets of A.
• C(x) and D(x) indicate the condition granule and
  the decision granule induced by x, respectively.
• L is a language defined using attributes of A, an
  atomic formula is given by a v, where a?A and
  v?Va.
• Formulas can be also formed by logical negation,
  conjunction and disjunction.
• A formula is called a basic formula in this paper
  if it is an atomic formula or is formed only by
  conjunction.

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Formalization Rough Set cont.

• In table 1, if C weather, road, and D
  time, accident. Then we have
• U/C 1,7, 2,5, 3,6, 4 c1, c2, c3,
  c4 the set of condition granules
• U/D 1, 2,3,7, 4, 5,6 d1, d2, d3,
  d4 the set of decision granules
• (U,C,D) is a decision table of (U,A), where U is
  a database which includes 1000 records.

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Pawlaks Interpretation

• Assumption - Each fact in the decision table is a
  subset of U in which all elements have the same
  values for all attributes
• Every class f determines a rule f(C )? f(D).
• The strength of the decision rule f(C )? f(D) is
  defined as C(f)?D(f) / U and
• The certainty factor of the decision rule is
  defined as C(f)?D(f) / C(f).

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Pawlaks Interpretation cont.
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Pawlaks Interpretation cont.
Table 2. Strengths and certainty factors of
decision rules
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Extended Random Sets

• The relationships between the premises and the
  conclusions of decision rules.
• c1 ? (d1, 80/100), (d2, 20/100)
• c2 ? (d2, 140/160), (d4, 20/160)
• c3 ? (d2 , 40/240), (d4, 200/240)
• c4 ? (d3, 500/500)

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Extended Random Sets cont.

• We use a mapping to formalize the relationship

and
for all ci?U/C.
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Extended Random Sets cont.

• Use the frequency in the decision table for
  support degree of each condition granule. We
  have

for every condition granule ci, where, Nx is the
number of analogous cases of fact x. By
normalizing, we can get a probability function P
on U/C such that
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Extended Random Sets cont.

• We call the pair (?, P) an extended random set.
• for a given condition granule ci, we assume

we can obtain the following decision rules
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Extended Random Sets cont.

• We define the strengths of the decision rules are

And, the corresponding certainty factors are
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Extended Random Sets cont.

• A decision rule

is an interesting rule if
is greater than a suitable constant.
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Extended Random Sets cont.

• Where,

We can prove that pr is a probability function on
(U/D).
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Extended Random Sets cont.

• Example of an extended random set

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Extended Random Sets cont.
Table 3. Probability function on the set of
decision granules
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Extended Random Sets cont.
Table 4. Interesting rules
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Interpretation of Extended Random Sets

• A very interesting phenomena form Table 3
• Only some descriptions on the set of decision
  granules are meaningful for a given information
  system if we use or to combine decision
  granules.
• E.g.,
• d1 or d2 -- (accident yes)
• d1 or d3 -- ?
• The concept of meaningful
• A description X on the set of decision granules
  of decision table (U, C, D) is meaningful if
  there is a decision table (U, E, F), such that
  E?C, and X?F.

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Interpretation of Extended Random Sets cont.

• The derived random set (?, P) from the extended
  random set (?, P)

It can determines a Dempster-Shafer mass function
m on ? such that
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Interpretation of Extended Random Sets cont.
Table 5. Uncertain measures on the set of
decision granules
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Algorithm 1 from Pawlaks Method

• let UN 0
• for (i 1 to n ) // n is the number of classes
• UN UN Ni
• for (i 1 to n)
• strength(i) Ni/UN CN Ni
• for (j 1 to n)
• if ((j?i) and (fj(C) fi(C)))
• CN CN Nj
• certainty_factor(i) Ni/CN .

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Algorithm 2 from extended random sets

• let UN 0, U/C ?
• for (i 1 to n)
• UN UN Ni
• for (i 1 to n do ) // create the data structure
• if (fi(C)? U/C)
• insert((fi(D), Ni)) to ?(fi(C))
• else
• add(fi(C)) into U/C, and set ?(fi(C))?
• for (i 1 to U/C)
• P(ci) (1/UN ) ? ()
• for (i 1 to U/C) // normalization
• temp 0
• for (j 1 to ?(ci))
• temp temp sndi,j
• for (j 1 to ?(ci))
• sndi,j sndi,j/temp
• for (i 1 to U/C) // calculate rule strengths
  
• for (j 1 to ?(ci))
• strength(ci?fsti,j) P(ci) ? sndi,j

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Algorithm Analysis

• Algorithm 1
• time complexity is O(n2), where n is the number
  of classes in the decision table.
• Algorithm 2
• the time complexity is O(n?U/C)
• U/C n, Algorithm 2 is better than Algorithm
  1 for the time complexity.

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Summary

• The advantages of our approach can be summarized
  as follows
• It provides a new algorithm to calculate decision
  rules, which is faster than Pawlaks algorithm
• In addition to the well-accepted criterion
  frequencies, the extended random sets are
  easily to include other criteria when determining
  association rules
• The extended random sets can provide more than
  one measures for dealing with uncertainties in
  the association rules. This is a significant
  distinguished characteristic from other methods.