
## 2 Perspective Of Methods

To dissection of different attributes effects on the dilution, we'll use some of the main approaches in the data engineering field, as well associated with intelligent computational and approximate reasoning methods which based on them a new algorithm SONFIS will be proposed. 

## 2.1 Rough Set Theory (Rst)

The rough set theory introduced by Pawlak (Pawlak1981, 1982) has often proved to be an excellent mathematical tool for the analysis of a vague description of object. The adjective vague referring to the quality of information means inconsistency, or ambiguity which follows from information granulation. An information system is a pair *S=< U, A >*, where U is a nonempty finite set called the universe and A is a nonempty finite set of attributes. An attribute a can be regarded as a function from the domain U to some value set Va . An information system can be represented as an attribute-value table, in which rows are labeled by objects of the universe and columns by attributes. With every subset of attributes B⊆ A, one can easily associate an equivalence relation B I on U: 
{( , ) : , ( ) ( )} B *I x y U for every a B a x a y* =∈ ∈ = (1)

$\mathbf{a}=\mathbf{a}\times\mathbf{b}$ 2. 

Then, B aB a I I ∈ = I . 

If X U⊆ ,the sets {: } x *Ux X* [ ]B ∈ ⊆ and {: } [ ]B *xU x X* ∈ ≠ I ϕ , where [ ]B x denotes the equivalence class of the object x ∈U relative to B I , are called the B-lower and the B-upper approximation of X in S and denoted by BX and BX , respectively. Consider 12 n U = {x , x , ...,x } and 12 n A = {a , a , ...,a } in the information systemS= U, A p f . By the discernibility matrix *M(S) of S* is meant an n*n matrix such that 

$$c_{_{i j}}=\left\{a\in A\,:\!a(x_{_i})\neq a(x_{_j})\right\}$$
 c a A ax ax *ij i j* =∈ ≠ { :( ) ( )} (2)
A discernibilty function s f is a function of m Boolean variables 1... m a a corresponding of attribute 1... m a a , respectively, and defined as follows: 

$$(2)$$
$$f_{s}\left(a_{1},...,a_{m}\right)=\wedge\{\vee(c_{\bar{y}}):i,j\leq n,j\prec i\,,c_{\bar{y}}\neq\varphi\}$$
$$({\mathfrak{I}})$$

1 ( ,..., ) { ( ) : , , , } s m ij ij f *a a c i j n j ic* =∧ ∨ ≤ ≠ p ϕ (3)
where ( ) ij ∨ c is the disjunction of all variables with ij a c ∈ .Using such discriminant matrix the appropriate rules are elicited (Pal et al, 2004).Indiscernibility relation (similarity) reduces the data by defining of equivalence classes, under the present attributes. Reduction of attributes (conditions) can be utilized by keeping of the attributes that maintain the similarity relation. The underway of idea behind finding out of reducts is the preserving of general information state with minimum attributes, and so superfluous attributes are eliminated. In equation (3), it can be proved that {a a i i 1 ,..., r}is a reduct in *S iff* 1 ... i i r a a ∧ ∧ is a prime implicant of s f .In our work, rule induction procedure was implemented within the ROSETA software system, a rough set tool-kit for knowledge discovery and data mining (φhm et al,1998). 

## 2.2 Neuro-Fuzzy Inference System (Nfis)

There are different alternatives of fuzzy inference systems. Two well-known fuzzy modeling methods are the Tsukamoto fuzzy model and Takagi– Sugeno–Kang (TSK) model. In the present study, we focus on the TSK. The TSK fuzzy inference systems can be impl anted in the form of a Neuro-fuzzy network structure .In this study, we have employed an *adaptive neurofuzzy inference system* (Jang et al, 1997), within the general structure of SONFIS. 