- Source: Decision-theoretic rough sets
In the mathematical theory of decisions, decision-theoretic rough sets (DTRS) is a probabilistic extension of rough set classification. First created in 1990 by Dr. Yiyu Yao, the extension makes use of loss functions to derive
α
{\displaystyle \textstyle \alpha }
and
β
{\displaystyle \textstyle \beta }
region parameters. Like rough sets, the lower and upper approximations of a set are used.
Definitions
The following contains the basic principles of decision-theoretic rough sets.
= Conditional risk
=Using the Bayesian decision procedure, the decision-theoretic rough set (DTRS) approach allows for minimum-risk decision making based on observed evidence. Let
A
=
{
a
1
,
…
,
a
m
}
{\displaystyle \textstyle A=\{a_{1},\ldots ,a_{m}\}}
be a finite set of
m
{\displaystyle \textstyle m}
possible actions and let
Ω
=
{
w
1
,
…
,
w
s
}
{\displaystyle \textstyle \Omega =\{w_{1},\ldots ,w_{s}\}}
be a finite set of
s
{\displaystyle s}
states.
P
(
w
j
∣
[
x
]
)
{\displaystyle \textstyle P(w_{j}\mid [x])}
is
calculated as the conditional probability of an object
x
{\displaystyle \textstyle x}
being in state
w
j
{\displaystyle \textstyle w_{j}}
given the object description
[
x
]
{\displaystyle \textstyle [x]}
.
λ
(
a
i
∣
w
j
)
{\displaystyle \textstyle \lambda (a_{i}\mid w_{j})}
denotes the loss, or cost, for performing action
a
i
{\displaystyle \textstyle a_{i}}
when the state is
w
j
{\displaystyle \textstyle w_{j}}
.
The expected loss (conditional risk) associated with taking action
a
i
{\displaystyle \textstyle a_{i}}
is given
by:
R
(
a
i
∣
[
x
]
)
=
∑
j
=
1
s
λ
(
a
i
∣
w
j
)
P
(
w
j
∣
[
x
]
)
.
{\displaystyle R(a_{i}\mid [x])=\sum _{j=1}^{s}\lambda (a_{i}\mid w_{j})P(w_{j}\mid [x]).}
Object classification with the approximation operators can be fitted into the Bayesian decision framework. The
set of actions is given by
A
=
{
a
P
,
a
N
,
a
B
}
{\displaystyle \textstyle A=\{a_{P},a_{N},a_{B}\}}
, where
a
P
{\displaystyle \textstyle a_{P}}
,
a
N
{\displaystyle \textstyle a_{N}}
, and
a
B
{\displaystyle \textstyle a_{B}}
represent the three
actions in classifying an object into POS(
A
{\displaystyle \textstyle A}
), NEG(
A
{\displaystyle \textstyle A}
), and BND(
A
{\displaystyle \textstyle A}
) respectively. To indicate whether an
element is in
A
{\displaystyle \textstyle A}
or not in
A
{\displaystyle \textstyle A}
, the set of states is given by
Ω
=
{
A
,
A
c
}
{\displaystyle \textstyle \Omega =\{A,A^{c}\}}
. Let
λ
(
a
⋄
∣
A
)
{\displaystyle \textstyle \lambda (a_{\diamond }\mid A)}
denote the loss incurred by taking action
a
⋄
{\displaystyle \textstyle a_{\diamond }}
when an object belongs to
A
{\displaystyle \textstyle A}
, and let
λ
(
a
⋄
∣
A
c
)
{\displaystyle \textstyle \lambda (a_{\diamond }\mid A^{c})}
denote the loss incurred by take the same action when the object
belongs to
A
c
{\displaystyle \textstyle A^{c}}
.
= Loss functions
=Let
λ
P
P
{\displaystyle \textstyle \lambda _{PP}}
denote the loss function for classifying an object in
A
{\displaystyle \textstyle A}
into the POS region,
λ
B
P
{\displaystyle \textstyle \lambda _{BP}}
denote the loss function for classifying an object in
A
{\displaystyle \textstyle A}
into the BND region, and let
λ
N
P
{\displaystyle \textstyle \lambda _{NP}}
denote the loss function for classifying an object in
A
{\displaystyle \textstyle A}
into the NEG region. A loss function
λ
⋄
N
{\displaystyle \textstyle \lambda _{\diamond N}}
denotes the loss of classifying an object that does not belong to
A
{\displaystyle \textstyle A}
into the regions specified by
⋄
{\displaystyle \textstyle \diamond }
.
Taking individual can be associated with the expected loss
R
(
a
⋄
∣
[
x
]
)
{\displaystyle \textstyle R(a_{\diamond }\mid [x])}
actions and can be expressed as:
R
(
a
P
∣
[
x
]
)
=
λ
P
P
P
(
A
∣
[
x
]
)
+
λ
P
N
P
(
A
c
∣
[
x
]
)
,
{\displaystyle \textstyle R(a_{P}\mid [x])=\lambda _{PP}P(A\mid [x])+\lambda _{PN}P(A^{c}\mid [x]),}
R
(
a
N
∣
[
x
]
)
=
λ
N
P
P
(
A
∣
[
x
]
)
+
λ
N
N
P
(
A
c
∣
[
x
]
)
,
{\displaystyle \textstyle R(a_{N}\mid [x])=\lambda _{NP}P(A\mid [x])+\lambda _{NN}P(A^{c}\mid [x]),}
R
(
a
B
∣
[
x
]
)
=
λ
B
P
P
(
A
∣
[
x
]
)
+
λ
B
N
P
(
A
c
∣
[
x
]
)
,
{\displaystyle \textstyle R(a_{B}\mid [x])=\lambda _{BP}P(A\mid [x])+\lambda _{BN}P(A^{c}\mid [x]),}
where
λ
⋄
P
=
λ
(
a
⋄
∣
A
)
{\displaystyle \textstyle \lambda _{\diamond P}=\lambda (a_{\diamond }\mid A)}
,
λ
⋄
N
=
λ
(
a
⋄
∣
A
c
)
{\displaystyle \textstyle \lambda _{\diamond N}=\lambda (a_{\diamond }\mid A^{c})}
, and
⋄
=
P
{\displaystyle \textstyle \diamond =P}
,
N
{\displaystyle \textstyle N}
, or
B
{\displaystyle \textstyle B}
.
= Minimum-risk decision rules
=If we consider the loss functions
λ
P
P
≤
λ
B
P
<
λ
N
P
{\displaystyle \textstyle \lambda _{PP}\leq \lambda _{BP}<\lambda _{NP}}
and
λ
N
N
≤
λ
B
N
<
λ
P
N
{\displaystyle \textstyle \lambda _{NN}\leq \lambda _{BN}<\lambda _{PN}}
, the following decision rules are formulated (P, N, B):
P: If
P
(
A
∣
[
x
]
)
≥
γ
{\displaystyle \textstyle P(A\mid [x])\geq \gamma }
and
P
(
A
∣
[
x
]
)
≥
α
{\displaystyle \textstyle P(A\mid [x])\geq \alpha }
, decide POS(
A
{\displaystyle \textstyle A}
);
N: If
P
(
A
∣
[
x
]
)
≤
β
{\displaystyle \textstyle P(A\mid [x])\leq \beta }
and
P
(
A
∣
[
x
]
)
≤
γ
{\displaystyle \textstyle P(A\mid [x])\leq \gamma }
, decide NEG(
A
{\displaystyle \textstyle A}
);
B: If
β
≤
P
(
A
∣
[
x
]
)
≤
α
{\displaystyle \textstyle \beta \leq P(A\mid [x])\leq \alpha }
, decide BND(
A
{\displaystyle \textstyle A}
);
where,
α
=
λ
P
N
−
λ
B
N
(
λ
B
P
−
λ
B
N
)
−
(
λ
P
P
−
λ
P
N
)
,
{\displaystyle \alpha ={\frac {\lambda _{PN}-\lambda _{BN}}{(\lambda _{BP}-\lambda _{BN})-(\lambda _{PP}-\lambda _{PN})}},}
γ
=
λ
P
N
−
λ
N
N
(
λ
N
P
−
λ
N
N
)
−
(
λ
P
P
−
λ
P
N
)
,
{\displaystyle \gamma ={\frac {\lambda _{PN}-\lambda _{NN}}{(\lambda _{NP}-\lambda _{NN})-(\lambda _{PP}-\lambda _{PN})}},}
β
=
λ
B
N
−
λ
N
N
(
λ
N
P
−
λ
N
N
)
−
(
λ
B
P
−
λ
B
N
)
.
{\displaystyle \beta ={\frac {\lambda _{BN}-\lambda _{NN}}{(\lambda _{NP}-\lambda _{NN})-(\lambda _{BP}-\lambda _{BN})}}.}
The
α
{\displaystyle \textstyle \alpha }
,
β
{\displaystyle \textstyle \beta }
, and
γ
{\displaystyle \textstyle \gamma }
values define the three different regions, giving us an associated risk for classifying an object. When
α
>
β
{\displaystyle \textstyle \alpha >\beta }
, we get
α
>
γ
>
β
{\displaystyle \textstyle \alpha >\gamma >\beta }
and can simplify (P, N, B) into (P1, N1, B1):
P1: If
P
(
A
∣
[
x
]
)
≥
α
{\displaystyle \textstyle P(A\mid [x])\geq \alpha }
, decide POS(
A
{\displaystyle \textstyle A}
);
N1: If
P
(
A
∣
[
x
]
)
≤
β
{\displaystyle \textstyle P(A\mid [x])\leq \beta }
, decide NEG(
A
{\displaystyle \textstyle A}
);
B1: If
β
<
P
(
A
∣
[
x
]
)
<
α
{\displaystyle \textstyle \beta
, decide BND(
A
{\displaystyle \textstyle A}
).
When
α
=
β
=
γ
{\displaystyle \textstyle \alpha =\beta =\gamma }
, we can simplify the rules (P-B) into (P2-B2), which divide the regions based solely on
α
{\displaystyle \textstyle \alpha }
:
P2: If
P
(
A
∣
[
x
]
)
>
α
{\displaystyle \textstyle P(A\mid [x])>\alpha }
, decide POS(
A
{\displaystyle \textstyle A}
);
N2: If
P
(
A
∣
[
x
]
)
<
α
{\displaystyle \textstyle P(A\mid [x])<\alpha }
, decide NEG(
A
{\displaystyle \textstyle A}
);
B2: If
P
(
A
∣
[
x
]
)
=
α
{\displaystyle \textstyle P(A\mid [x])=\alpha }
, decide BND(
A
{\displaystyle \textstyle A}
).
Data mining, feature selection, information retrieval, and classifications are just some of the applications in which the DTRS approach has been successfully used.
See also
Rough sets
Granular computing
Fuzzy set theory
References
External links
The International Rough Set Society
The Decision-theoretic Rough Set Portal
Kata Kunci Pencarian:
- Penggalian data
- Decision-theoretic rough sets
- Rough set
- Game-theoretic rough sets
- Set theory
- Dominance-based rough set approach
- Fuzzy set
- Rationality
- Mathematical logic
- Integer factorization
- Garbage can model
