Dependencies in Structures of Decision Tables

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Dependencies in Structures of Decision Tables

• Wojciech Ziarko
• University of Regina
• Saskatchewan, Canada

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Contents

• Pawlaks 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

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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

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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
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Classical Pawlaks Rough Set
Negative region X ? E?
Elementary set E
U
Boundary region X ? E ? ? , E? X
set X
Positive region E ? X
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Approximation Regions

• Based on the lower and upper approximations
• of ,U can be divided into three
  disjoint
• definable regions

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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)

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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

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Undiscretized Data
Complex multidimensional functions on features
can be used to create final discrete
attribute-value representation
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Discretized Representation
D
C

• peak Peak of the Wave
• size Area of Peak
• m1 Steroid Oral therapy
• m2 Double Filtration Plasmapheresis

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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
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Probabilistic Background of Rough Sets

• U - outcome space the set of possible outcomes
• s(U) s-algebra of measurable subsets of U
• Event an element of s(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)
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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 lt P(X) lt 1

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

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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

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Probabilistic Interpretation for Pawlaks
Approximations
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Pawlaks Approximation Measures in Probabilistic
Terms
Let FX1,, 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

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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

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C
D
Example Classification Table
Atoms
Elementary sets
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Variable Precision RS Model

• An extension of the classical RS (Pawlaks) 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

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Variable Precision RS Model
Negative region
Elementary set E
U
Boundary region l lt P(XE) lt u
set X
Positive region
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VPRSM Approximations

• Positive Region (u-lower approximation)
• Negative Region
• Boundary Region
• Upper Approximation

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Probabilistic Decision Tables
where t is a tuple in C(U)
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C
D
Example Classification Table
Atoms
Elementary sets
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(No Transcript)
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?-Dependency Between Attributes in the VPRSM
Generalization of partial functional dependency
measure ?
Represents the size of positive and negative
regions of X
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?-Dependency Between Attributes Preliminaries

• The degree of influence the occurrence of an
  elementary set E has on the likelihood of X
  occurrence.

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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
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Properties of Gain Functions
- summary deviation from independence
- analogous to Bayes equation
Basis for generalized measure of attribute
dependency
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?-Dependency Between Attributes
Measure of dependency between attributes
Applicable to both classification tables and
probabilistic decision tables
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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

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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

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U
DT Hierarchy Formation
POS
U
BND
U
NEG
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Hierarchical Condition Partition
U
UBND
Based on nested structure of condition attributes
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Decision Partition
U
Based on values of the decision attribute
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? -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

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?-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

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Faces Example
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Hierarchy of DTs Based on Faces
Layer 1
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Layer 2
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Layer 3
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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