Major modifications of rough sets (VPRSM, DRSM, Near sets)

As it has been presented above, since the first publication by Pawlak (1982, 1991) of the rough set theory (RST) as an information retrieval system generating rules, which describes uncertain knowledge in a way alternative to fuzzy sets methodology (Zadeh 1965), many modifications of the RST have been proposed. The most notable of them include Variable Precision Rough Set Model (VPRSM) published by Ziarko (1993), Dominance Rough Set Approach (DRSA) introduced by Greco, Matarazzo and Slowinski (Greco et al. 1999a), and Near Set Theory (NST) developed by Peters (2007).

The first mentioned modification (VPRSM) is dedicated for large data sets, where inconsistencies, tolerated to some extent, can be advantageous. The second (DRSA) is appropriate for attributes with inherent preference order and not necessarily discretized.

Finally, the latter (NST), by using affinities between perceptual objects and perceptual granules, provides a basis for perceptual information systems useful in science and engineering. It is also worthwhile to notice that there exists methodology which incorporates Ziarko's idea of variable precision to DRSA methodology resulting in Variable Consistency Dominance Rough Set Approach (VCDRSA) (see Greco et al. 2001).

The crucial notion in the VPRSM is the coefficient describing the level of uncertainty. It specifies, whether the element x  U belongs to a set X  U when indiscernible relation I(C) generates the knowledge K_C in information system S.

Definition 2.3:6 (Uncertainty level, after Ziarko 1993)

The uncertainty level coefficient is a function denoted by _X^C (x) and defined as

_X^C (x) = card { X  [x]_I(C) } / card { [x]_I(C) }.

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Defined above coefficient is also referred to as a rough membership function of an element x, due to similarities with membership function known from the theory of fuzzy sets.

This function gave base for the generalization of rough set theory called rough set model with variable precision (Ziarko 1993). This model assumes that lower and upper approximations are dependent on additional coefficient , such that 0    0.5, and are defined as CX = { x  U: _X^C (x)  1 -  } and C¯ X = { x  U: _X^C (x) >  } respectively. The boundary in this model is defined as BnC (X) = { x  U:  < _X^C (x) < 1 -  }. It is easy to observe that the classical rough set theory is the special case of variable precision model with  = 0.

Since X  U, CX  CX  C¯ X  C¯ X, it follows that VPRSA is a weaker form of the theory as compared to classical model, and therefore, it is often preferable in analysis of large information systems with some amount of contradicting data. The membership function of an element x can be also defined for a family of sets X as _X^C (x) = card {(

UXnX, Xn )  [x]I(C)} / card {[x]I(C)}. If all subsets Xn of the family X are mutually disjoint, then x  U, _X^C (x) =  XnX, _Xn^C (x). It is evident that the definition of the rough membership function of the element _X^C (x) assumes only the existence of classes of equivalence of the relation I, and the VPRSA formally differs from classical model only in the definition of the lower and the upper approximation by the use of this coefficient.

Therefore, all rough set-based notions are defined for arbitrary I also in this generalized model.

While the VPRSA was addressing challenges with information processing in large data repositories, the next considered modification of rough set theory, the DRSA, is the response to multicriteria classification problem. The most influential modification encountered in DRSA as compared to CRSA is the change of indiscernibility relation, which is an equivalence relation, to a dominance relation. Using this modification DRSA is able to take into account preference orders in the description of objects by condition and decision attributes.

This is significant improvement, since the well-known methods of knowledge discovery and machine learning do not use the information about preference orders in multicriteria classification. However, taking this information into account can be important in many practical problems, which involve evaluation of objects on preference ordered domains.

Therefore, when dealing with such multicriteria classification DRSA often outperforms CRSA, which is not able to make use of this important information – the new model proposed by the author called quasi-dominance rough set approach (refer to Cyran 2009d) addresses the preference order not resigning from the indiscernibility relation, as it is explained in detail in section 2.3.3).

In DRSA like in CRSA, the rough approximation of the partition of information system is a starting point for induction of the IF-THEN decision rules. However, the syntax of these rules is adapted to represent preference orders. The DRSA keeps almost all the best properties of the CRSA: it analyses only facts present in data and possible inconsistencies are not corrected. Moreover, this approach does not need any prior discretization of continuous-valued attributes. In fact, the only known drawback of DRSA is impossibility of using the (relative) value reducts, what motivated the author to propose a hybrid approach, the QDRSA, keeping possibility to use (relative) value reducts and taking into account the

preference order not resigning from the indiscernibility equivalence relation (see section 2.3.3).

Detailed description of DRSA applicable to multicriteria classification and other multicriteria decision problems such as choice and ranking problems is given in Greco et al.

(1999b). This latter paper shows that within DRSA heterogeneous information can be effectively processed. The heterogeneity include in this context qualitative and quantitative information, which is ordered and non-ordered and processed using crisp and fuzzy evaluations, as well as ordinal, quantitative and numerical non-quantitative scales of preference.

The applications of DRSA vary from such areas as market analysis, where the usefulness of the DRSA and its advantages over the CRSA are presented on a real study of evaluation of the risk of business failure (Greco et al. 1998) to bioinformatics, where DRSA is applied in the search for signatures of natural selection operating at molecular level (Cyran 2010).

Remarkably, DRSA can be applied in conjunction with VPRSM-based concepts, what has been demonstrated by Greco et al. (2001) in Variable Consistency model of DRSA (VCDRSA).

After presenting VPRSM and DRSA, let us focus on NST. This theory, proposed by Peters (2007), was introduced in a context of perception-based approach to studying the nearness of observable objects in a continuum of physical world. The near sets are disjoint sets of such objects that resemble each other, where resemblance between disjoint sets occurs whenever there are observable similarities between the objects in the sets.

In order to determine the similarity between perceptual objects, it is required to compare lists of values, which describe the objects. In other words, a list of such feature values defines an object‟s description. Hence, comparison of object descriptions provides a basis for NST, whose goal is to offer an efficient framework to group together objects that are perceived as similar based on their descriptions. In particular NST is useful in analysis of digital images perceived as disjoint sets of points (Peters 2009, Peters and Ramanna 2009, Pal and Peters 2010).

The near sets methodology starts with choosing the appropriate method to describe observed objects. This task is accomplished by the selection of probe functions, which represent features of observable objects. Foundations of probe functions were introduced by Pavel (1993). In NST, a notion of a probe function is used as a mapping from an object to a real number, which represents value of an observable feature (Peters 2007). By using probe functions, near sets offer an ideal framework for solving problems based on human perception.

The NST understands perception as a combination of the meaning in psychophysics (Hoogs et al. 2003, Bourbakis 2002) with a view found in Merleau-Ponty‟s (1945) work.

Psychophysics considers perception of an object, which effects human knowledge about an object, as depending on sense inputs that are the source of signal values, called stimularions, in the cortex of the brain. According to this view, the transmissions of sensory inputs to cortex cells correspond to the probe functions defined in terms of mappings of sets of sensed objects to sets of real-values representing signal values.

This view assumes that the magnitude of each cortex signal value represents a sensation that is a source of object feature values assimilated by the mind. It is based on observation that perception in animals can be modeled as a mapping from sensory cells to brain cells. In particular, visual perception is modeled as a mapping from stimulated retina sensory cells to visual cortex cells. Such mappings, representing probe functions, measure observable physical characteristics of objects in the environment. Therefore a probe function in NST provides a basis for what is commonly known as feature extraction (Guyon et al. 2006) since the sensed physical characteristics of an object can be clearly identified with object characteristic features.

When considering modifications and improvements of the classical rough set approach (CRSA) defined by Pawlak (1991) it may be of some interest to discuss the relation between the given enhanced approach and the original CRSA. Basically there are two kinds of this relation: the first is when the modified approach is more general than the CRSA and then the CRSA is a special case of it, and the second is when the modified approach uses the inspiration from CRSA but in fact it defines a new methodology which cannot be reduced to the CRSA.

The example of the first type is VPRSM, because CRSA is a special case of VPRSM with precision parameter set to one. Also the modified indiscernibility relation, as defined by Cyran (2008b) is more general than the original one, since the latter is a special case of the first. Contrary to these examples, the DRSA is such enhancement which cannot be reduced to classical rough sets: it is inspired by the notions present in RST, but the introduction of dominance relation for preference-ordered attributes (called criteria) instead of equivalence relation present in CRSA is the reason why CRSA cannot be derived from DRSA as its special case.

In this context, the NST is of special type. On one hand, Peters (2007) has proved that near sets are the generalization of rough sets, as each rough set is a near set, and not each near set is a rough set. On the other, the extension of the approximation space (Peters et al.

2007), which is a fundamental notion for RST practically leads to the resignation from the essence of this notion in NST (Peters and Wasilewski 2009). Therefore, although formally NST is a generalization of RST, in practice, it approaches the information granules from a different perspective, which is more focused on search for affinities with the use of tolerance relation, than on defining the approximation space using the equivalence relation.

There have been proposed also other modifications of RST, mainly changing the equivalence relation to the weaker similarity relation (Słowiński and Vanderpooten 2000), or defining the equivalence relation in continuous attribute space without the need of discretization. Introduction of the structure into the set of conditional attributes together with the application of cluster analysis methodology for this purpose has been proposed by Cyran (2008b). This problem is further described in section 2.3.2. The applicability of the latter modification for the problem which was primarily solved with the use of CRSA (see Cyran and Mrózek 2001) has been demonstrated in the case study presented in section 2.4. It is also worth to say that the domain of possible applications of the modified indiscernibility relation extends to all problems with continuous attributes.