1. Dependencies in Structures of Decision Tables
Wojciech Ziarko
University of Regina
Saskatchewan, Canada

2. Contents Pawlak’s rough sets Attribute-based classifications
Probabilities and rough sets
VPRS model
Probabilistic decision tables
Dependencies between sets
Gain function
-dependencies between attributes
-dependencies between attributes
Hierarchies of decision tables
Dependencies between partitions in DT hierarchies
Faces example

3. Approximation Space (U,R)
U – universe of objects of interest , can be infinite
target set of interest
equivalence relation, U/R is finite
elementary sets
are atoms, the set of atoms is finite

4. Approximate Definitions
If a set can be expressed as a union
of some elementary classes of R, we say that the X is
R-definable otherwise, we say that the X is undefinable,
i.e. it is impossible to describe X precisely using
knowledge R.
In this case, X can be represented by a pair of lower and upper approximations:

5. Classical Pawlak’s Rough Set
Negative region: X  E=
Elementary set E U
Boundary region:
X  E   , E X
set X
Positive region: E  X

6. Approximation Regions
Based on the lower and upper approximations
of ,U can be divided into three disjoint
definable regions:

7. Attribute-Based Classifications
The observations about objects are typically expressed via finite-valued functions called attributes:
The attribute-based classifications may not produce classification of the universe U (for example, when the attribute values are affected by random noise)
This means attributes are not always functions on U (they could be better modeled by approximate functions)

8. Attributes and Classifications
The attributes fall into two disjoint categories: condition attributes C and decision attributes D
Each subset of attributes defines a mapping:
The subset B of condition attributes generates partition U/B of U into B-elementary classes
The corresponding equivalence relation is called B-indiscernibility relation

9. Undiscretized Data
Month
Max. Temp.
Min. Temp.
Max. Rel. Hum.
Min. Rel. Hum.
Cloudiness
Precipitation 1
-17.1
-25.8 75 65 2
-3.4
-18.6 96 73 3
-16.7
-23.9 53 4
4.6
-8.9 88 38 5
14.4
-6.3 70 18 6 7
26.6
5.2 85 33 8
22.3
8.5
100 43 9 19 94 47 10
13.6
3.2 74 35 11
8.9
-4.9 87 34 12
-5.8
-18.5 93 81
Complex multidimensional functions on features can be used to
create final discrete attribute-value representation

10. Discretized RepresentationG
Peak
Size
Therapy
Prognosis G1
Low
Large m1
Bad G2
Good G3
High G4 G5
Small m2 G6 G7 G8
high
small
bad
peak: Peak of the Wave
size: Area of Peak
m1: Steroid Oral therapy
m2: Double Filtration Plasmapheresis

11. Attributes and Classifications
-elementary sets atoms
C-elementary sets elementary sets
D-elementary sets decision categories
We assume that the set of all atoms is finite
Each B-elementary set is a union of some atoms

12. Probabilistic Background of Rough Sets
U - outcome space: the set of possible outcomes
σ(U) – σ-algebra of measurable subsets of U
Event – an element of σ(U), a subset of U
Assumption 1: all outcomes are equally likely .
Assumption 2: event X occurs if an outcome e belongs to X.
Assumption 3: the prior probability of every event exists, and
Probability estimators (other estimators are possible):

13. Probabilistic Approximation Space (U, R, P)
U – universe of objects of interest
target set of interest
equivalence relation, U/R is finite
elementary sets
atoms, the set of atoms is finite
P(G) probability function on atoms and X
0 < P(X) < 1

14. Probabilistic Approximation Space
Atoms G
Elementary sets E U
Set X
Atoms are assigned probabilities P(G)

15. Probabilities of Interest
Each atom is assigned joint probability P(G)
The probability P(E) of an elementary set
Prior probability P(X) of the decision category
This is the probability of X in the absence of any attribute value-based information, the reference probability

16. Conditional Probabilities and Elementary Sets
To represent the degree of confidence in the occurrence of decision category X, based on the knowledge that elementary set E occurred, the conditional probabilities are used:
The conditional probabilities can be expressed in terms of joint probabilities:

17. Probabilistic Interpretation for Pawlak’s Approximations

18. Pawlak’s Approximation Measures in Probabilistic Terms
Let F={X1,…, Xn} be a partition of U corresponding to U/D,
in the approximation space (U, U/C)
Accuracy measure of approximation of F by U/C
-dependency measure between C and D

19. Classification Table
The classification table represents complete classification and probabilistic information about the universe U
It is a collection of tuples representing individual atoms and their joint probabilities

20. Prob. Attr. a Attr. b Attr. c d 0.15 1 2 0.20 0.13 0.02 0.01 0.08 0.30
Example Classification Table C D
Prob.
Attr. a
Attr. b
Attr. c d
0.15 1 2
0.20
0.13
0.02
0.01
0.08
0.30 3
Atoms
Elementary
sets

21. Variable Precision RS Model
An extension of the classical RS (Pawlak’s) model
Other related extensions are VC-DRSA (Greco, Mattarazo, Slowinski), decision theoretic approach (Yao)
The classical approach is to define the positive and negative regions of a set X based on total inclusion, or exclusion with X, respectively;
There is no uncertainty in these regions
In the VPRSM the positive and negative regions are defined in terms of controlled certainty improvement (gain) with respect to the set X

22. Variable Precision RS Model
Negative region:
Elementary set E U
Boundary region:
l < P(X|E) < u
set X
Positive region:

23. VPRSM Approximations Positive Region (u-lower approximation)
Negative Region
Boundary Region
Upper Approximation

24. Probabilistic Decision Tables
where t is a tuple in C(U)

25. Prob. Attr. a Attr. b Attr. c d 0.15 1 2 0.20 0.13 0.02 0.01 0.08 0.30
Example Classification Table C D
Prob.
Attr. a
Attr. b
Attr. c d
0.15 1 2
0.20
0.13
0.02
0.01
0.08
0.30 3
Atoms
Elementary
sets

26. P(E)
Attr. a
Attr. b
Attr. c
P(X|E)
Region
0.23 1 2
POS
0.33
0.61
BND
0.11
0.27
0.01
0.32
0.06
NEG

27. -Dependency Between Attributes in the VPRSM
Generalization of partial functional dependency measure 
Represents the size of positive and negative regions of X:

28. -Dependency Between Attributes: Preliminaries
The degree of influence the occurrence of an elementary set E has on the likelihood of X occurrence.

29. Expected Gain Functions
Expected change of occurrence certainty of a given decision
category X due to occurrence of any elementary set:
Average expected change of occurrence certainty of any
decision category X due to occurrence of any elementary set:

30. Properties of Gain Functions
- summary deviation from independence
- analogous to Bayes equation
Basis for generalized measure of attribute dependency

31. -Dependency Between Attributes
Measure of dependency between attributes
Applicable to both classification tables and
probabilistic decision tables

32. Hierarchies of Decision Tables
Decision tables learned from data suffer from both the low accuracy and incompleteness
Increasing the number of attributes or increasing their precision leads to exponential growth of the tables
An approach to deal with these problems is forming decision table hierarchies

33. Hierarchies of Decision Tables
The hierarchy is formed by treating the boundary area as a sub-approximation space
The sub-approximation space is independent from “parent” approximation space, normally defined in terms attributes different from the ones used by the parent
The hierarchy is constructed recursively, subject to dependency, attribute and elementary set support constraints.
The resulting hierarchical approximation space is not definable in terms of condition attributes over U

34. DT Hierarchy FormationU
DT Hierarchy Formation
POS U’
BND
U’’
NEG

35. Hierarchical “Condition” PartitionU
U’=BND
Based on nested structure of condition attributes

36. Based on values of the decision attribute
“Decision” Partition U
Based on values of the decision attribute

37.  -Dependency Between Partitions in the Hierarchy of Decision Tables
Let (X, X) be the partition corresponding to the decision attribute
Let R be the hierarchical partition of U and R’ be the hierarchical partition of boundary area of X,
The dependency can be computed recursively by:

38. -Dependency Between Partitions in the Hierarchy of Decision Tables
Let (X, X) be the partition corresponding to the decision attribute
Let R be the hierarchical partition of U and R’ be the hierarchical partition of boundary area of X,
The dependency can be computed recursively by:

39. “Faces”
Example

40. Hierarchy of DT’s
Based on “Faces”
Layer 1

41. Layer 2

42. Layer 3

43. Conclusions
The original rough set approach is mainly applicable to problems in which the probability distributions in the boundary area do not matter
When the distributions are of interest, the extensions such as VPRSM, Bayesian etc. are applicable
The contradiction between DT learnability vs. its completeness and accuracy is a serious practical problem
The DT hierarchy construction provides only partial remedy
Softer techniques are needed for attribute value representation, to better handle noisy data – incorporation of fuzzy set ideas