Issue 40

L. Zou et alii, Frattura ed Integrità Strutturale, 40 (2017) 137-148; DOI: 10.3221/IGF-ESIS.40.12 140 where t * is dimensionless the equivalent   retains a stress unit. Neighborhood Rough Set Theory Founded by Pawlak, rough set theory [14] aims to find the inner links of the massive, imprecise, incomplete and uncertain data, it has become an important tool to study granular computing theory nowadays [15]. However, the tradition rough set just works in discrete spaces and it can’t deal directly with the numerical data that widely existed in the practical application. When dealing with the numerical data, discretization is first done to transform the numerical value into the symbol value [16, 17]. This transformation inevitably brings about information loss and the computing results usually depend largely on the effect of discretization algorithm. To deal with this problem, a neighborhood rough set model is proposed based on the definitions of δ neighborhood and neighborhood relations in metric spaces [18, 19]. Several foundation definition of neighborhood rough set theory including neighborhood   , lower and upper approximations, dependency degree, significance of the attribute, reduction and core are first introduced here. Definition 1 neighborhood   U is a non-empty finite set in the real number space, i x U   , the  -neighborhood of xi is defined as ( ) { , ( , ) } i i x x U x x       (7) where  is a metric function, 1 2 3 , , x x x U   , it satisfied 1 2 ( , ) 0 x x   , 1 2 ( , ) 0 x x   if and only if 1 2 x x  , 1, 2 2 1 ( ) ( , ) x x x x    and 1, 3 1 2 2 3 ( ) ( , ) ( , ) x x x x x x      . The family of neighborhood granules { ( ) | } i i x x U   forms an element granule system for a given metric space , U    . We have , ( ) i i x U x      and ( ) x U x U     . A neighborhood relation N can be written as a relation matrix ( ) ( ) ij n n M N r   , where 1 ij r  if ( ) j i x x   or 0 ij r  otherwise. Definition 2 Lower and upper approximations The lower and upper approximations of X in terms of relation N for a given , U N   are defined as { | ( ) , }, i i i NX x x X x U     (8) { | ( ) , }, i i i NX x x X x U       (9) The boundary region of X is , BNX NX NX   (10) Definition 3 Dependency degree The dependency degree of the decision attribute D to the condition attribute B is defined as | | ( ) , | | B B N D D U   (11) It is obvious that 0 ( ) 1 B D    . If ( ) 1 B D   , we say D completely depend on B otherwise D is  -depend on B. Definition 4 Significance of the attribute Given a neighborhood decision table , , , , U C D V f   , B ⊆ C,a ∈ C-B, the significance of a to B is defined as ( , , ) ( ) ( ), B a B SIG a B D D D      (12) Definition 5 Reduction Given a neighborhood decision table , , , , U C D V f   , B ⊆ C, we say the subset of attributes B is a reduction of C if ( ) ( ) B C D D    and , ( ) ( ) B B b b B D D       . Definition 6 Core