Definition 2.2:22 (Finite state machines, after Fogel 1997d) The finite state machines are defined as the ordered 5-tuples
2.3. Rough sets
2.3.2. Rough sets with real-valued attributes
If a problem is originally defined for real valued attributes, then, before the rough set theory can be used, some clustering and discretization of continuous attributes should be performed. Let this process be denoted as a transformation described by a vector function
: card (C) {1, 2, …, }card (C), where is called the discretization factor. The discretization factor simply denotes the number of clusters covering the domain of each individual attribute q C. Theoretically, this factor could be different for different attributes, but without the loss of generality, we assume its constancy over the set of attributes. Then, the discretization of any individual attribute q C, can be denoted as a transformation defined by a scalar function : {1, 2, …, }. In this case, one obtains the classical form of indiscernibility relation, defined as (Cyran and Stańczyk 2007a):
C
y q C f
x
q
f
y
q
I
x 0 Λ , , , , (2.3:2) Below, it will be shown that majority (however, not all) of notions defined in the theory of rough sets de facto do not demand the strong version of indiscernibility relation I0 defined by equation (1) (or by (2), if the discretization is required). From a formal point of view, what is really important, is the fact, that the indiscernibility relation has to be a relation of equivalence, i.e. it must be reflexive, symmetric and transitive. From practical point of view, objects indiscernible in a sense of the rough set theory, should be such objects, which are close in a real-valued space.
Hence, the exact form of the indiscernibility relation, as proposed by the classical theory of rough sets, as well as by its generalization VPRSA, is not actually required to create a coherent logical system. Some researchers (Järvinen 2001, Skowron and Stepaniuk 1996, Doherty and Szałas 2004, Słowiński and Vanderpooten 1997, 2000, Gomolińska 2002) go further in this generalizing tendency, resigning from the requirement of equivalence relation.
However, working with such generalizations is often not natural in problems, such as classification, when notion of abstract classes, inherently involved in equivalence relation, is of great importance. Therefore, the author has proposed such modification of the indiscernibility relation, which is particularly useful in many pattern recognition problems, which deal with a space of continuous attributes and which are defined in terms of equivalence relation.
To introduce formally this modification, let us change the notation of indiscernibility relation to be dependent on a family of sets of attributes, instead of being dependent simply on a set of attributes. By the family of sets of attributes, we understand a subset of a power set, based on the set of attributes, such, that all elements of this subset (these elements are subsets of the set of attributes) are mutually disjoint, and their union is equal to the considered set of attributes. This allows to introduce a structure to, originally unstructured, set of attributes, which the relation depends on.
Let C = { C1, C2, …, CN } denotes the introduced above family of disjoint sets of attributes Cn Q such that unstructured set of attributes C Q is equal to the union of members of the family C, i.e. C = UCnC, Cn. Then, let the indiscernibility relation be dependent on C instead of being dependent on C. Observe that both C and C contain the same collection of single attributes, however C includes additional structure as compared to C. If this structure is irrelevant for the problem considered, it can be simply ignored and one can obtain, as a special case, the classical version of indiscernibility relation I0. However, it is also possible to obtain other versions of this modified relation for which the introduced structure is meaningful.
Let us denote by I (without any subscript) an arbitrary relation, having mentioned above properties, reserving subscripts for denoting particular forms of I. The exact form of I, defined as I0 in (1) or (2), is not required for processing the rough information, except for some notions, which will be discussed later.
Definition 2.3:7 (Modified indiscernibility relation, after Cyran 2008b)
The modified indiscernibility relation I1(C) UU is such form of a relation I (in general different from I0), which is defined as
y Cn Clus
x Cn
Clus
y Cn
I
x 1 C C, , , , (2.3:3) where x, y U, and Clus(x,Cn) denotes the number of a cluster, that the element x belongs to.
▬
Theorem 2.3:1 (Generality of modified indiscernibility relation after Cyran 2008b)
The modified indiscernibility relation I1 is a generalized version of the classical form I0 of the indiscernibility relation known in CRSA.
Proof (after Cyran 2008b)
Note, that the cluster analysis is required in continuous vector spaces defined by sets of real valued conditional attributes Cn C. Note also, that there are two extreme cases of relation I1, obtained when family C is composed of exactly one set of conditional attributes C, and when family C is composed of card (C) sets, each containing exactly one conditional attribute q C. The classical form I0 of the indiscernibility relation can be obtained as the latter extreme special case of the modified indiscernibility relation I1, because then clustering and discretization is performed separately for each continuous attribute. Hence,
1
:
,
,
.0 n n
C q
n
n C q Clus x q f x q
q I
C I
n
C
Λ C (2.3:4)
what ends the proof.
■
In other words, the classical form I0 of the indiscernibility relation can be obtained as a special case of modified version I1 if we assume that family C is composed of such subsets Cn, that each contains just one attribute, and the discretization of each continuous attribute is based on separate cluster analysis as required by a function applied to each of attributes qn.
One can easily verify (by confrontation of the general form of indiscernibility relation I with presented below notions) that the following constructs form a logically consistent system, no matter what is the specific form of the indiscernibility relation. In particular it is true for such forms of relation I1 defined by (3), which is different from classical form I0, defined for discrete and continuous types of attributes in (1) and (2) respectively.
From Definition 9 it follows that set Z is elementary, when all elements x Z are C-indiscernible, i.e. they belong to the same abstract class [x]I (C) of relation I(C). If C = Q then Z is elementary set in S. C-elementary set is therefore the atomic unit of knowledge about universe U with respect to C. Since C-elementary sets are defined by abstract classes of relation I, it follows that any equivalence relation, in particular I1 can be used as I.
Definition 2.3:8 (C-definable sets, after Pawlak 1991)
If a set X is a union of C-elementary sets then X is C-definable, i.e. it is definable with respect to knowledge KC.
▬
Note, that a complement, a product, or an union of C-definable sets are also C-definable set (notion). Therefore the indiscernibility relation I(C), by generating knowledge KC , defines all what can be accurately expressed with the use of set of attributes C. Two information systems S and S’ are equivalent if they have the same elementary sets. Then the knowledge KQ is the same as knowledge KQ’. Knowledge KQ is more general than knowledge KQ’ iff
I(Q‟) I(Q), i.e. when each abstract class of the relation I(Q‟) is included in some abstract class of I(Q). C-definable sets, as unions of C-elementary sets are also defined for any equivalence relation I.
Definition 2.3:9 (C-rough set X, after Pawlak 1982, 1991, 1995a)
Any set being the union of C-elementary sets is a C-crisp set, any other collection of objects in universe U is called a C-rough set.
▬
A rough set contains a border, composed of elements such, that based on the knowledge generated by indiscernibility relation I, it is impossible to distinguish whether or not the element belongs to the set. Each rough set can be defined by two crisp sets, called lower and upper approximation of the rough set. Since C-crisp sets are unions of C-elementary sets, and C-rough set is defined by two C-crisp sets, therefore the notion of C-rough set is defined for any equivalence relation I, in particular for I1 different than I0.
Definition 2.3:10 (C-lower approximation of rough set X U, after Pawlak 1982, 1995a) The lower approximation of a rough set X is composed of those elements of universe, which belong for sure to X, based on indiscernibility relation I. Formally, C-lower approximation of a set X U, which is denoted as CX, is defined in the information system S = <U, Q, v, f > as CX = { x U: [x]I (C) X }.
▬
Definition 2.3:11 (C-upper approximation of rough set X U, after Pawlak 1982, 1995a) The upper approximation of a rough set X is composed of those elements of universe, which perhaps belong to X, based on indiscernibility relation I. Formally, C-upper approximation of a set X U, denoted as C¯ X is defined in the information system S = <U, Q, v, f > as C¯ X = { x U: [x]I (C) X Ø }.
▬
Definition 2.3:12 (C-border of rough set X U, after Pawlak 1982, 1995a)
The border of a rough set is the difference between its upper and lower approximation.
Formally, C-border of a set X, denoted as BnC (X) is defined as BnC (X) = C¯ X – CX.
▬
Definition 2.3:13 (C-positive region of the set X U, after Pawlak 1991)
C-positive region of a set X, i.e. such region whose elements can be classified as for sure belonging to X, is denoted as PosC (X) and defined as C-lower approximation of X.
▬
Definition 2.3:14 (C-negative region of the set X U, after Pawlak 1991)
C-negative region of X, denoted as NegC (X), contains all elements of universe U, which for sure do not belong to X, i.e. In other words, it is a complement of C-upper approximation of X, NegC (X) = U - C¯ X.
▬
Note, that both CX = { x U: [x]I (C) X } and C¯ X = { x U: [x]I (C) X Ø } are C-crisp sets, so they can be defined for arbitrary equivalence relation I, such as for example relation I1. Moreover, since BnC (X) is a difference of two C-crisp sets, its definition is also based on arbitrary equivalence relation I. The positive region (similarly like C-lower approximation of X) can be defined for arbitrary I, and the negative region, as a difference of U (which does not depend on I) and a C-crisp set C¯ X, is also based on arbitrary relation of equivalence I.
Note also, that the indiscernibility relation I generates in any information system S some topology which describes four different topological types of rough sets. These types are: sets roughly definable, sets internally indefinable, sets externally indefinable and sets totally indefinable.
Definition 2.3:15 (Sets roughly C-definable, after Pawlak 1995a)
Set X is roughly C-definable iff PosC (X) Ø and NegC (X) Ø, i.e. universe U contains some elements which for sure belong to X and some element which for sure do not belong to X.
▬
Definition 2.3:16 (Sets internally C-indefinable, after Pawlak 1995a)
Rough set X is called internally C-indefinable iff its positive region is empty, but negative region is not empty, i.e. when PosC (X) = Ø and NegC (X) Ø.
▬
Definition 2.3:17 (Sets externally C-indefinable, after Pawlak 1995a)
Rough set X is called externally C-indefinable iff its positive region is not empty, but negative region is empty, i.e. when PosC (X) Ø and NegC (X) = Ø.
▬
Definition 2.3:18 (Sets totally C-indefinable, after Pawlak 1995a)
Rough set X is called totally C-indefinable iff both positive and negative regions of X are empty, i.e. when PosC (X) = NegC (X) = Ø.
▬
It is easy to observe, that all notions defined in Definitions 15-18 as being declared by specific positive and negative regions, can be defined for any relation I, in particular for modified relation I1.
Notions of a rough set theory, applicable for a separate set X, are generally applicable also for families of sets X = { X1, X2, …, XN }, where Xn U, and n = 1, …, N. Examples of such notions are given below in Definitions 19-23.
Definition 2.3:19 (C-lower approximation of family of sets, after Mrózek 1998)
The lower approximation of a family of sets is a family of lower approximations of sets belonging to family considered. Formally, CX = {CX1, CX2, …, CXN}.
▬
Definition 2.3:20 (C-upper approximation of family of sets, after Mrózek 1998)
The upper approximation of a family of sets is a family of upper approximations of sets belonging to family considered. Formally, C¯ X = {C¯ X1, C¯ X2, …, C¯ XN}.
▬
Definition 2.3:21 (C-border of family of sets, after Mrózek 1998)
The boundary of the family of sets X is a union of boundaries of sets belonging to the family considered, i.e. BnC (X) = UXnX, BnC (Xn).
▬
Definition 2.3:22 (C-positive region of the family of sets, after Mrózek 1998)
The positive region of the family of sets X is a union of positive regions of sets belonging to the family considered, i.e. PosC (X) = UXnX, PosC (Xn).
▬
Definition 2.3:23 (C-negative region of the family of sets, after Mrózek 1998)
The negative region of the family of sets X is defined as NegC (X) = U - UXnX, C¯ Xn.
▬
Note, that concepts defined in Definitions 19-23, as families, differences and unions of C-crisp sets, are based on arbitrary relation of equivalence I. The theory of rough sets not only defines, as presented above, a framework of coherent notions, used for representation of uncertain knowledge, but also gives tools for associating the objects with numerical uncertainty measures.
Therefore, it follows that the last class of concepts considered in various models of the rough set theory described in this monograph, is a class of coefficients which indicate the accuracy and the quality of the approximation space.
Definition 2.3:24 (C-accuracy of approximation of a set: C (X), after Pawlak 1995a)
C-accuracy of approximation of a nonempty set X, denoted as C (X), is given by the ratio of lower and upper approximation of X, i.e., C (X) = card [PosC (X) ] / card (C¯ X).
▬
The accuracy of approximation defined in Definition 24 satisfies 0 C (X) 1. Using this coefficient, it is possible to give alternative definitions of crisp and rough sets, as presented in Definition 25. Another coefficient measuring the uncertainty in rough set theory is called quality of approximation defined in Definition 26.
Definition 2.3:25 (Roughness of a set, after Pawlak 1995a)
When C (X) = 1 then the considered set X is C-crisp in a system with knowledge KC
generated by the indiscernibility relation I(C). Similarly, if C (X) < 1, then X is called C-rough set.
▬
Definition 2.3:26 (C-quality of approximation of a set: C (X), after Pawlak 1995a)
C-quality of approximation of a set X, denoted as C (X) is defined as
C (X) = card [PosC (X)] / card (U).
▬
Interesting comparison of C-quality of approximation and Dempster-Shafer theory of evidence is given by Skowron and Grzymała-Busse (1994). In the context considered here, it is important to observe that the notions presented in Defintions 24-26 as numerical ratios of numbers associated with notions defined for any I, they are also meaningful for arbitrary relation I. Other notions, which are based on a notions of the upper and/or the lower approximation of a family of sets X, with respect to a set of attributes C, include: C-accuracy of approximation of a family of sets, C-quality of approximation of a family of sets. This latter coefficient is especially interesting for the application presented in the subsequent section, since it is used as an objective function in a procedure of optimization of the feature extractor. For this purpose, the considered family of sets is a family of abstract classes generated by the decision attribute d being the class of the image to be recognized (see.
Section 2.4 for the exemplary application). Here, let us define this coefficient for an arbitrary family of sets X.
Definition 2.3:27 (C-quality of approximation of a family of sets X, after Mrózek 1998) C-quality of approximation of a family of sets X, denoted by C (X) is defined as
C (X) = card [PosC (X)] / card (U).
▬
The analysis of concepts presented above indicates, that they do not require any particular form of the indiscernibility relation (like for example the classical form referred to as I0).
They are defined for any form of the indiscernibility relation (satisfying reflexity, symmetry and transitiveness), denoted by I (in particular I1, which is very usful in continuous space) and therefore they are strict analogs of classical notions defined with the assumption of original form of indiscernibility relation I0 defined by equations (1, 2).
Finally, let us discuss some of the notions of rough set theory that cannot be used in a common sense with the modified indiscernibility relation I1 defined by (3). Let us start with, the so called, basic sets which are abstract classes of relation I({q}) defined for singe attribute q. These are simply sets composed of elements indiscernible with respect to single attribute q.
Obviously, this notion loses its meaning when I1 is used instead of I0, because abstract classes generated by I0({q}) are always unions of some abstract classes generated by I0(C), however abstract classes generated by I1({q}) not necessarily are unions of abstract classes generated by I1(C). Therefore the conclusion that knowledge K{q} generated by I0({q}) is always more general than knowledge KC generated by I0(C), no longer holds when I1 is used instead of I0.
Similarly, notions of reducts, relative reducts, cores and relative cores no longer are applicable in their classical sense, since their definitions are strongly associated with single attributes. Joining these attributes into members of family C, destroys the individual treatment of attributes, required for these notions to have their well known meaning.
However, as long as the rough set theory is used in the continuous attribute space, to the extent not going beyond notions described Definitions 8-27, the modified I1 version should be considered more advantageous, as compared to the classical form I0. In particular, this is true in processing of the knowledge obtained from the holographic ring-wedge detector (given as the illustrative example in section 2.4), when the quality of approximation of family of sets plays the major role in the quality of recognition.