A.Introduction-RoughSets), a data set is
represented as a table called an information system
$\mathcal{A} = (U, A)$, where $U$ is a non-empty
set of finite objects as the universe of discourse (note:
it refers to all instances/experiments/rows in datasets)
and $A$ is a non-empty finite set of attributes, such
that $a : U \to V_{a}$ for every $a \in A$. The
set $V_{a}$ is the set of values that attribute
$a$ may take. Information systems that involve a
decision attribute, containing classes or decision values
of each objects, are called decision systems (or said as
decision tables). More formally, it is a pair
$\mathcal{A} = (U, A \cup {d})$, where $d
\notin A$ is the decision attribute. The elements of
$A$ are called conditional attributes. However,
different from RST, the FRST has several ways to express
indiscernibility.Fuzzy indiscernibility relation (FIR) is used for any fuzzy relation that determines the degree to which two objects are indiscernible. We consider some special cases of FIR.
reflexive:$R(x,x) = 1$symmetric:$R(x,y) = R(y,x)$
The following equations are the tolerance relations on a quantitative attribute $a$, $R_a$, proposed by (R. Jensen and Q. Shen, 2009).
eq.1:$R_a(x,y) = 1 - \frac{|a(x) -
a(y)|}{|a_{max} - a_{min}|}$eq.2:$R_a(x,y) = exp(-\frac{(a(x) - a(y))^2}{2
\sigma_a^2})$eq.3:$R_a(x,y) =
max(min(\frac{a(y) - a(x) + \sigma_a}{\sigma_a},
\frac{a(x) - a(y) + \sigma_a}{\sigma_a}), 0)$BC.IND.relation.FRSTFor a qualitative (i.e., nominal) attribute $a$, the classical manner of discerning objects is used, i.e., $R_a(x,y) = 1$ if $a(x) = a(y)$ and $R_a(x,y) = 0$, otherwise. We can then define, for any subset $B$ of $A$, the fuzzy $B$-indiscernibility relation by
$R_B(x,y) = \mathcal{T}(R_a(x,y))$,
where $\mathcal{T}$ is a t-norm operator, for instance minimum, product and Lukasiewicz t-norm. In general, $\mathcal{T}$ can be replaced by any aggregation operator, like e.g. the average.
In the context of FRST, according to (A. M. Radzikowska and E. E. Kerre, 2002) lower and upper approximation are generalized by means of an implicator $\mathcal{I}$ and a t-norm $\mathcal{T}$. The following are the fuzzy $B$-lower and $B$-upper approximations of a fuzzy set $A$ in $U$
$(R_B \downarrow A)(y) = inf_{x \in U} \mathcal{I}(R_B(x,y), A(x))$
$(R_B \uparrow A)(y) = sup_{x \in U} \mathcal{T}(R_B(x,y), A(x))$
The underlying meaning is that $R_B \downarrow A$ is
the set of elements necessarily satisfying the
concept (strong membership), while $R_B \uparrow A$
is the set of elements possibly belonging to the
concept (weak membership). Many other ways to define the
approximations can be found in
BC.LU.approximation.FRST. Mainly, these
were designed to deal with noise in the data and to make
the approximations more robust.
Based on fuzzy $B$-indiscernibility relations, we define the fuzzy $B$-positive region by, for $y \in X$,
$POS_B(y) = (\cup_{x \in U} R_B \downarrow R_dx)(y)$
We can define the degree of dependency of $d$ on $B$, $\gamma_{B}$ by
$\gamma_{B} = \frac{|POS_{B}|}{|U|} = \frac{\sum_{x \in U} POS_{B}(x)}{|U|}$
A decision reduct is a set $B \subseteq A$ such that $\gamma_{B} = \gamma_{A}$ and $\gamma_{B'} = \gamma_{B}$ for every $B' \subset B$.
As we know from rough set concepts (See
A.Introduction-RoughSets), we are able to
calculate the decision reducts by constructing the
decision-relative discernibility matrix. Based on (Tsang
et al, 2008), the discernibility matrix can be defined as
follows. The discernibility matrix is an $n \times n$
matrix $(c_{ij})$ where for $i,j = 1, \ldots, n$
1) $c_{ij}= {a \in A : 1 - R_{a}(x_i, x_j) \ge \lambda_i}$ if $\lambda_j < \lambda_i$.
2) $c_{ij}={\oslash}$, otherwise.
with $\lambda_i = (R_A \downarrow R_{d}x_{i})(x_i)$ and $\lambda_j = (R_A \downarrow R_{d}x_{j})(x_{j})$
Other approaches of discernibility matrix can be read at
BC.discernibility.mat.FRST.
The other implementations of the FRST concepts can be
seen at BC.IND.relation.FRST,
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