From Deterministic to Probabilistic Rough Sets
Wojciech Ziarko
University of Regina
Computer Science
Saskatchewan, Canada
ziarko@sasktel.net

May 14, 2024

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Overview
1

Original Rough SetsClassification and Decision Tables

2

Variable Precision Rough Sets
Single Parameter Variable Precision Rough Sets
Two Parameter Variable Precision Rough Sets

3

Probabilistic Rough Sets
Bayesian Rough Sets
Probabilistic Decision Tables
Probabilistic Rules
Evaluation of Decision Tables

4

Optimization of Decision Tables
Attribute Reduction and Significance

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Pawlak Rough Sets: Approximation Space

Introduced by late Professor Zdzislaw Pawlak
Collection of objects or observations of interest: universe U, assumed
finite, but can be generalized to infinite
Classification knowledge, an ability to define categories (notnecessarily data-based), represented by an equivalence indiscernibility
relation: R ⊆ U × U
Elementary sets collection, assumed finite: R ∗ = {E1 , E2 , ..., En }Approximation space: (U, R)

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Pawlak Rough Sets: Undefinable Sets

It is not possible, in general, to form precise discriminating definition,in terms of the available classification knowledge, of an arbitrary set
X ⊆U
Some sets (sometimes called concepts) can never be defined, or
learned, with a given classification knowledge
At the best, only approximate definitions can be created, or learned
Sets for which discriminating definitions do not exist are called
undefinable, or rough

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Pawlak Rough Sets: Rough Approximations of Sets
Lower approximation:
R(X ) = ∪{E ∈ R ∗ : E ⊆ X }Upper approximation:
R = ∪{E ∈ R ∗ : E ∩ X 6= ∅}Disjoint approximation regions:
Positive region:
POS(X ) = ∪{E ∈ R ∗ : E ⊆ X }Negative region:
NEG (X ) = ∪{E ∈ R ∗ : E ∩ X = ∅}Boundary region:
BND(X ) = ∪{E ∈ R ∗ : E ∩ X 6= ∅ ∧ E * X }Wojciech Ziarko (UofR)

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Pawlak Rough Sets: Rough Approximations of Sets

If BND(X ) = ∅ then X is definable
Elementary sets and approximation are definable and disjoint
A rough set X is approximately defined by specifying definitions of its
approximation regions POS(X ), NEG (X ) and BND(X )This can be done in a tabular form by creating a rough decision table
The approximation space is defined via classification of objects,
collected in a classification table, based on identity of values of their
attributes

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Classification Table: Representation of Instances
Obj
e1
e2
e3
e4
e5
e6
e7
e8
e9
e10
e11
e12
e13

HRate
High
High
High
High
High
NormalNormal
Low
Low
Low
Low
High
Normal

BP
High
High
High
NormalNormalNormalNormal
High
High
High
High
NormalNormal

Temp
High
High
NormalHigh
High
NormalNormalHigh
High
NormalNormalHigh
NormalTreatm
1
1
2
1
1
3
3
1
1
2
2
2
1

Response
Positive
Negative
Negative
Positive
Positive
Positive
Negative
Negative
Negative
Positive
Negative
Negative
Positive

Table: Classification Table of medical records
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Rough Decision Table: Representation of Classes

HRate
High
High
High
NormalLow
Low
High
Normal

BP
High
High
NormalNormalHigh
High
NormalNormalTemp
High
NormalHigh
NormalHigh
NormalHigh
NormalTreatment
1
2
1
3
1
2
2
1

Appr Region
BND
NEG
POS
BND
NEG
BND
NEG
POS

Table: Rough Decision Table representation of the rough set:
Response=Positive

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Acquisition of Rough Decision Tables

From data, based on analysis and pre-processing of existing data:
data mining, most common
Based on prior human expert knowledge: expert specifies the classesof rough decision table
Through learning from individual observations using pre-selectedtraining data: objects, cases, instances
The decision tables acquired from data are likely to be incomplete,
due to the nature of learning from data

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Analysis and Processing of Rough Decision Tables

Analysis of dependencies occurring in the decision table: functional,
partial functional
Reduction - elimination of redundant or unrelated parts of thedecision, such as:
Elimination of redundant columns (attribute reducts)
Elimination of redundant values (value reducts)

Significance analysis of individual attributes
Formation of minimal length, that is, most generalized predictive rules

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10 / 44Typical Issues with Applications to Real-World Data

Imperfections of practical application data
Presence of measurement noise
Lack of consistency
Extensive boundary regions
Inter-data relationships are often probabilistic in nature, rather thandeterministic
Difficulty in creating any deterministic predictive rules or models fromdata

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11 / 44Variable Precision Approach to Rough Sets

An attempt to create ”softer” rough sets, more applicable to realworld problems and imperfect data
To utilize frequency distribution info in data when creating decisiontables and rules (probabilistic knowledge)
To allow for use of subjective probabilities obtained from humanexperts
To enhance the scope of applications of rough set theory fundamental
ideas

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12 / 44Single Parameter Variable Precision Model of Rough Sets

Misclassification degree of set X with respect to Y :c(X , Y ) = 1 −card(X ∩ Y )card(X ∩ −Y )== P(¬Y |X )card(X )card(X )if card(X ) > 0 and c(X , Y ) = 0 if card(X ) = 0
The partial majority inclusion of the set X within Y :Y ⊇β X if and only if c(X , Y ) ≤ β with 0 ≤ β < 0.5
Example:Let U = {x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 }, X = {x1 , x2 , x3 , x4 } andY = {x1 , x2 , x3 , x8 }. Then ¬Y = {x4 , x5 , x6 , x7 } giving Y ⊇0.25 X

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Set Approximations
Lower approximation[[
R β (X ) = {E ∈ R ∗ : c(E , X ) ≤ β} = {E ∈ R ∗ : X ⊇β E }Upper approximation
R β (X ) =[{E ∈ R ∗ : c(E , X ) < 1 − β}Negative region
NEGRβ (X ) =[{E ∈ R ∗ : c(E , X ) ≥ 1 − β}Boundary region
BNRβ (X ) =Wojciech Ziarko (UofR)[{E ∈ R ∗ : β < c(E , X ) < 1 − β}From Deterministic to Probabilistic RS

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Relationship to Pawlak Rough Sets

R 0 (X ) = R(X )R 0 (X ) = R(X )BNR0 (X ) = BNR(X )NEGR0 (X ) = NEGR(X )Wojciech Ziarko (UofR)

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15 / 44Variable Precision Rough Sets: Asymetric Bounds
Two parameters used to control approximation regions
Lower and upper limit 0 ≤ l < u ≤ 1
Lower approximation[
R l (X ) = {E ∈ R ∗ : c(E , X ) ≤ l}Upper approximation
R u (X ) =[{E ∈ R ∗ : c(E , X ) < u}Negative region
NEGRu (X ) =[{E ∈ R ∗ : c(E , X ) ≥ u}

Boundary region
BNRl,u (X ) =[{E ∈ R ∗ : l < c(E , X ) < u}When l = 0 and u = 1, the above definitions reduce to the original roughsets approximation regions (check)
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Example: A Classification Table of Finishing Mill Data
Situation
e1
e2
e3
e4
e5
e6
e7
e8
e9
e10
e11
e12
e13
e14
e15
e16
e17
e18
e19
e20
e21
e22

Width
Wide
Wide
Wide
Wide
Wide
Wide
Narrow
Narrow
Narrow
Narrow
Narrow
Narrow
Wide
Wide
Wide
Wide
Wide
Narrow
Narrow
Narrow
Narrow
Narrow

Gauge
Heavy
Heavy
MediumMedium
Thin
Thin
Heavy
Heavy
MediumMedium
Thin
Thin
Heavy
Heavy
Medium
Thin
Thin
Heavy
Heavy
MediumThin
Thin

Result
Good
Bad
Bad
Good
Bad
Bad
Bad
Good
Good
Good
Good
Good
Good
Good
Good
Good
Good
Good
Good
Bad
Bad
Bad

Condition attributes: C={Width, Gauge}
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Approximation Space Based on Width and Gauge
Condition Classes Forming Approximation Space:(Width := Wide)AND(Gauge := Heavy ) with C1 = {1, 2, 13, 14},
Width := Wide)AND(Gauge := Medium) with C2 = {3, 4, 15},
(Width := Wide)AND(Gauge := Thin) with C3 = {5, 6, 16, 17},
(Width := Narrow )AND(Gauge := Heavy ) with C4 = {7, 8, 18, 19},
(Width := Narrow )AND(Gauge := Medium) with C5 = {9, 10, 20},
(Width := Narrow )AND(Gauge := Thin) with C6 = {11, 12, 21, 22}.
Decision Classes:(Result := Good) with
D1 = {1, 4, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}(Result := Bad) with D2 = {2, 3, 5, 6, 7, 20, 21, 22}.
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Approximations of (Result := Bad)C (D2 ) = ∅ then no deterministic decision rule exists for (Result := Bad)c(C1 , D2 ) = 0.75, c(C2 , D2 ) = 0.67, c(C3 , D2 ) = 0.5
c(C4 , D2 ) = 0.75, c(C5 , D2 ) = 0.67, c(C6 , D2 ) = 0.5[[
C 0.6 (D2 ) = {Ci ∈ C ∗ : c(Ci , D2 ) ≤ 0.6} = {C3 , C6 }The above means that non-deterministic rule exists for (Result := Bad)[[
BNC0.6,0.7 (D2 ) = {Ci ∈ C ∗ : 0.6 < c(Ci , D2 ) < 0.7} = {C2 , C5 }[[{Ci ∈ C ∗ : c(Ci , D2 ) < 0.7} = {C2 , C3 , C5 , C6 }[[NEGC0.7 (D2 ) = {Ci ∈ C ∗ : c(Ci , D2 ) ≥ 0.7} = {C1 , C4 }C 0.7 (D2 ) =Wojciech Ziarko (UofR)

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Probabilistic Rough Sets
Rough set X prior probability , representing the probability ofoccurrence of X , in the absence of any other information:P(X ) =

card(X )card(U)(1)

with the assumption 0 < P(X ) < 1
Elementary set E prior probability:
P(E ) =

card(E )card(U)(2)

with the assumption 0 < P(E ) < 1
Conditional occurrence probability of the rough set X withinelementary set E :card(X ∩ E )P(X |E ) =,(3)
card(E )if U if infinite, the set measure theory can be used to substitute for
cardinalities
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Probabilistic Approximation Regions 1
Based on the degree deviation of conditional probabilities from priorprobability rather than presence or absence of a set inclusion
Approximation regions characterize areas with significantly increased,significantly decreased, or approximately unchanged rough set X
occurrence probability in relation to prior probability
The approximation regions are defined in terms of two precisioncontrol parameters:
The upper limit u and the lower limit l, with the constraint:
0 ≤ l < P(X ) < u ≤ 1(4)

The upper and lower limit parameters give precise meaning of theterms significantly decreased, or significantly increased rough set X
occurrence probability
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Probabilistic Approximation Regions 2
Positive region:
POSu (X ) = ∪{E : 0 < P(X ) < u ≤ P(X |E ) ≤ 1}.

(5)

Negative region:
NEGl (X ) = ∪{E : 0 ≤ P(X |E ) ≤ l < P(X ) < 1}.

(6)

Boundary region
BNDl,u (X ) = ∪{E : l < P(X |E ) < u}.

(7)

When l = 0 and u = 1, the above definitions reduce to the originalrough sets approximation regions

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Probabilistic Rough Sets in the Limit: Bayesian Rough Sets
The lower and upper limit parameters are constrained by
0 ≤ l < P(X ) < u ≤ 1(8)When both l and u approach P(X ), l → P(X ) and P(X ) ← u then:
POSu (X ) → POS ∗ (X ) = ∪{E : 0 < P(X ) < P(X |E ) ≤ 1}

(9)

NEGu (X ) → NEG ∗ (X ) = ∪{E : 0 ≤ P(X |E ) < P(X ) < 1}

(10)

BNDl,u (X ) → BND ∗ (X ) = ∪{E : 0 < P(X |E ) = P(X ) < 1}. (11)The limit approximation regions are called Absolute Approximation
Regions
They provide the basis of Bayesian Rough Set model

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Probabilistic Approximations: Example

A rough set X is probabilistically defined by specifying definitions of
its approximation regions elementary sets and their probabilistic
relations to the rough set
This can be done in a tabular form by creating a probabilistic decisiontable

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Classification Table
Obj
e1
e2
e3
e4
e5
e6
e7
e8
e9
e10
e11
e12
e13

HRate
High
High
High
High
High
NormalNormal
Low
Low
Low
Low
High
Normal

BP
High
High
High
NormalNormalNormalNormal
High
High
High
High
NormalNormal

Temp
High
High
NormalHigh
High
NormalNormalHigh
High
NormalNormalHigh
NormalTreatm
1
1
2
1
1
3
3
1
1
2
2
2
1

Response
Positive
Negative
Negative
Positive
Positive
Positive
Negative
Negative
Negative
Positive
Negative
Negative
Positive

Table: Classification Table of medical records
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Probabilistic Decision Tables

HRate
High
High
High
NormalLow
Low
High
Normal

BP
High
High
NormalNormalHigh
High
NormalNormalTemp
High
NormalHigh
NormalHigh
NormalHigh
NormalTreatment
1
2
1
3
1
2
2
1

Region
BND
NEG
POS
BND
NEG
BND
NEG
POS

P(Ei )
0.05200.13540.15620.15620.1406
0.10930.1562
0.0941

P(X |Ei )
0.78
0.02
0.99
0.36
0.11
0.41
0.27
0.85

Table: Probabilistic decision table with l = 0.3, u = 0.8, P(X ) = 0.4366 of
Response := Positive

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Probabilistic Rules From Decision Tables

The Probabilistic rule rX |Y is an expression: des(Y ) → s(X )Y is a definable set with a defining description des(Y )The description des(Y ) is a conjunction of attribute-value pairs
X is a rough set referenced by s(X )Three kinds of rules:
Positive rule: Y ⊆ POSu (X )Negative rule: Y ⊆ NEGl (X )Boundary rule: Y ⊆ BNDl,u (X )Rules corresponding to elementary sets E are elementary rules
In practice, minimal length rules are of most interest

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Minimal Probabilistic Rules
HRate
High
High
High
NormalLow
Low
High
Normal

BP
High
High
NormalNormalHigh
High
NormalNormalTemp
High
NormalHigh
NormalHigh
NormalHigh
NormalTreatment
1
2
1
3
1
2
2
1

Region
BND
NEG
POS
BND
NEG
BND
NEG
POS

P(Ei )
0.05200.13540.15620.15620.1406
0.10930.1562
0.0941

P(X |Ei )
0.78
0.02
0.99
0.36
0.11
0.41
0.27
0.85

Correspond to value reducts of rough set theory
Are maximally general (strong data support)
Example: BP = Normal ∧ Treatment = 1 → PositiveResponse

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Rule Evaluative Parameters

The rule rX |Y certainty parameter defined as the conditional
probability cert(rX |Y ) = P(X |Y )The rule rX |Y generality (support,strength ) parameter defined as theprobability gen(rX |Y ) = P(Y )The rule certainty gain parameter gain(rX |Y ) = |P(X |Y ) − P(X )|For all positive rules: u ≤ cert(rX |Y )For all negative rules: cert(rX |Y ) ≤ l

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Rule Evaluative Measures
The rules and the evaluative measures can be computed from theprobabilistic decision table
Rule certainty:P
cert(rX |Y ) =E ⊆Y P(E )P(X |E )P

E ⊆Y P(E )}Rule generality:gen(rX |Y ) =X

P(E )}E ⊆Y

Certainty gain:
gain(rX |Y ) =Wojciech Ziarko (UofR)|PE ⊆Y P(E )P(X |E ) − P(X )PE ⊆Y P(E )From Deterministic to Probabilistic RSP

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Evaluation of Rules: Example
HRate
High
High
High
NormalLow
Low
High
Normal

BP
High
High
NormalNormalHigh
High
NormalNormalTemp
High
NormalHigh
NormalHigh
NormalHigh
NormalTreatment
1
2
1
3
1
2
2
1

Region
BND
NEG
POS
BND
NEG
BND
NEG
POS

P(Ei )
0.05200.13540.15620.15620.1406
0.10930.1562
0.0941

P(X |Ei )
0.78
0.02
0.99
0.36
0.11
0.41
0.27
0.85

P(X ) = 0.4366
BP = Normal ∧ Treatment = 1 → PositiveResponse
Generality = 0.2503, Certainty = 0.9374, CertaintyGain = 0.5008

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Evaluation of Decision Tables: Dependency Measures
The focus is on partial functional and probabilistic dependencies
They capture the quality of approximation of the rough set X interms of the elementary sets of the approximation space
Parametric partial functional measure, γl,u (X |C ), is a variable
precision generalization of Pawlak’s partial functional dependency
measure:γl,u (X |C ) = P(POSu (X |C ) ∪ NEGl (X |C ))where C is a set of condition attributes.
It represents the relative degree of accuracy of approximation of arough set X
It can be computed directly from probabilistic decision table by:
γl,u (X |C ) =

X

P(E )

E ⊆POSu (X |C )∪NEGl (X |C )Wojciech Ziarko (UofR)

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Partial Functional Dependencies in Decision Tables

Parametric partial functional measure, γl,u (X |C ) becomes partial
functional dependency measure γ(X |C ) when l = 0 and u = 1:γ(X |C ) = γ0,1 (X |C ) =

X

P(E )

E ⊆POS1 (X |C )∪NEG0 (X |C )Approximation regions POS,BND and NEG are definable and disjoint
The dependency between condition attributes C and the attributeRegion is functional

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33 / 44Dependency Monotonicity

Monotonicity is a property of a dependency which ensures that thedependency will not decrease with the addition of an attribute
It allows for efficient computation (linear in the number of attributes)
of a minimal subset of attributes preserving the dependency, called
attribute reduct

Theorem (Monotonicity of Partial Functional Dependency)
Let B ⊂ C be a subset of condition attributes on U and let ”a” be any
condition attribute. Then the following relation holds:
γ(X |B) ≤ γ(X |B ∪ {a})Wojciech Ziarko (UofR)

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34 / 44Parametric Probabilistic Dependencies
The probabilistic dependency represents a degree of probabilistic
dependency between classification formed by condition attributes andthe classification {X , ¬X }The parametric probabilistic dependency, λ − dependency , is anormalized expected maximum degree of deviation of the probabilityof the rough set P(X |E ), or P(¬X |E ), from the prior probability of
the rough set P(X ), or from the prior probability of the rough set
P(¬X ), respectively:
X
P(E )|P(X |E ) − P(X )|λl,u (X |C ) =E ⊆POSu (X |C )∪NEGl (X |C )P(POSu (X |C ))P(¬X ) + P(NEGl (X |C ))P(X )The higher deviation reflects stronger probabilistic connection
between elementary set E and the rough set X
If all elementary sets E are probabilistically independent from roughset X , that is, if P(X ∩ E ) = P(X )P(E ), then λl,u (X |C ) = 0
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Probabilistic Dependencies in Bayesian Model
To evaluate and optimize probabilistic decision tables derived in theframework of Bayesian Rough Set model, the non-parametricprobabilistic dependency (probabilistic dependency), λ − Dependency ,can be used:X
P(E )|P(X |E ) − P(X )|λ(X |C ) =E ∈U/C(12)

2P(X )(1 − P(X ))It represents the average degree of probabilistic dependency between
elementary sets corresponding to combinations of attribute values andrough set X
If all elementary sets E are probabilistically independent from roughset X , then λ(X |C ) = 0
It can be computed directly from the probabilistic decision table
It is monotonic
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Mnotonicity of Probabilistic Dependency

Theorem (Monotonicity of Probabilistic Dependency)
Let B ⊆ C be a subset of condition attributes on U and let ”a” be any
condition attribute. Then the following relation holds:
λ(X |B) ≤ λ(X |B ∪ {a})(13)

It allows for efficient procedure for decision table optimization andanalysis of significance of attributes

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Probabilistic Decision Table Reduction 1

Minimal subset of attributes RED ⊆ C preserving γ(Region|C ) = 1 iscalled functional reduct:
γ(Region|RED) = 1(14)

and for any attribute a ∈ RED:γ(Region|RED − {a}) < 1(15)

After reduction, some elementary sets may combine requiring
re-computation of P(E ) and P(X |E )Wojciech Ziarko (UofR)

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Probabilistic Decision Table Reduction Example

HRate
High
High
High
NormalLow
Low
High
Normal

BP
High
High
NormalNormalHigh
High
NormalNormalTemp
High
NormalHigh
NormalHigh
NormalHigh
NormalTreatment
1
2
1
3
1
2
2
1

Region
BND
NEG
POS
BND
NEG
BND
NEG
POS

P(Ei )
0.05200.13540.15620.15620.1406
0.10930.1562
0.0941

P(X |Ei )
0.78
0.02
0.99
0.36
0.11
0.41
0.27
0.85

The functional reduct is (HRate, BP, Treatments)
The attribute Temp can be eliminatedWojciech Ziarko (UofR)

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Reduced Probabilistic Decision Table Example

HRate
High
High
High
NormalLow
Low
High
Normal

BP
High
High
NormalNormalHigh
High
NormalNormal

Wojciech Ziarko (UofR)

Treatment
1
2
1
3
1
2
2
1

Region
BND
NEG
POS
BND
NEG
BND
NEG
POS

From Deterministic to Probabilistic RS

P(Ei )
0.05200.13540.15620.15620.1406
0.10930.1562
0.0941

P(X |Ei )
0.78
0.02
0.99
0.36
0.11
0.41
0.27
0.85

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Probabilistic Reduct

Minimal subset of attributes preserving either the non-parametricγ − dependency or λ − dependency ), called reduct, or probabilistic
reduct, respectively, RED):λ(X |RED) = λ(X |C )(16)

and for any attribute a ∈ RED:λ(X |RED − {a}) < λ(X |RED)(17)

Similarly for the partial functional dependencies

Wojciech Ziarko (UofR)

From Deterministic to Probabilistic RS

May 14, 2024

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Significance and Core of Attributes
Determining the contribution of individual reduct RED attributes tothe dependency in question via their significance analysis
Can be done by evaluating the relative decrease of dependency due toremoval of an attribute a from the reduct RED:
sigRED (a) =λ(X |RED) − λ(X |RED − {a})>0λ(X |RED)(18)

Finding the subset of most essential attributes with respect to theprobabilistic dependency, the ones contained in ALL reducts called
core attributes
Any core attribute {a} satisfies the following inequality:λ(X |C ) > λ(X |C − {a}).

(19)Similarly for the non-parametric partial functional dependency
Wojciech Ziarko (UofR)

From Deterministic to Probabilistic RS

May 14, 2024

42 / 44Issues and Directions
The problem of converting precise numeric values into general
qualitative values
Value Range-based discretization is not good - possible formation of
spurious new combinations
Development of methods for avoiding overfitting and for reliable
estimates of probabilities
Reduction of combinatorial complexity: minimizing the number of
attributes and their values
Use of classification hierarchy
Dealing with incomplete observations
Developing applications: finance, medicine, pharmacy, control,
pattern classification

Wojciech Ziarko (UofR)

From Deterministic to Probabilistic RS

May 14, 2024

43 / 44

Thank You!

Wojciech Ziarko (UofR)

From Deterministic to Probabilistic RS

May 14, 2024

44 / 44