Rough sets

The theory of rough sets [55] has recently emerged as another major math-ematical tool for managing uncertainty that arises from granularity in the domain of discourse - that is, from the indiscernibility between objects in a set. The intention is to approximate a rough (imprecise) concept in the domain of discourse by a pair of exact concepts, called the lower and upper approximations. These exact concepts are determined by an indiscernibility relation on the domain, which, in turn, may be induced by a given set of attributes ascribed to the objects of the domain. The lower approximation is the set of objects definitely belonging to the vague concept, whereas the up-per approximation is the set of objects possibly belonging to the same. These approximations are used to define the notions of discernibility matrices, dis-cernibility functions, reducts, and dependency factors, all of which play a fun-damental role in the reduction of knowledge. Figure 2.6 provides a schematic diagram of a rough set. Let us now present some requisite preliminaries of rough set theory.

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 {a}.

An attribute a in A can be regarded as a function from the domain U to some value set Vn.

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Fig. 2.6 Lower and upper approximations in a rough set.

With every subset of attributes B C A, one can easily associate an equiv-alence relation IB on U:

IB = {(x,y) € U : for every a e B, a(z) = a(y)}.

If X C t/, the sets {x 6 £7 : [x]^B C X} and {z e t/ : [x]^B H X 7^ 0}, where [X]B denotes the equivalence class of the object x € U relative to IB, are called the B-lower and B-upper approximations of X in «S and denoted by BX and BX, respectively.

X (C U) is B- exact or B-definable in <S if BX = BX. It may be observed that BX is the greatest B-definable set contained in X, and BX is the smallest B-definable set containing X.

Let us consider, for example, an information system < U, {a} > where the domain U consists of the students of a school, and there is a single attribute a - that of "belonging to a class." Then U is partitioned by the classes of the school.

Now consider the situation when an infectious disease has spread in the school, and the authorities take the two following steps.

1. If at least one student of a class is infected, all the students of that class are vaccinated. Let B denote the union of such classes.

2. If every student of a class is infected, the class is temporarily suspended.

Let B denote the union of such classes.

Then B C B. Given this information, let the following problem be posed:

• Identify the collection of infected students. Clearly, there cannot be a unique answer. But any set I that is given as an answer must contain B and at least one student from each class comprising B. In other words, it must have B as its lower approximation and B as its upper approximation.

• I is then a rough concept or set in the information system < U, {a} >.

Further, it may be observed that any set /' given as another answer is roughly

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equal to /, in the sense that both are represented (characterized) by B and B.

The effectiveness of the theory of rough sets has been investigated in the domains of artificial intelligence and cognitive sciences, especially for repre-sentation of and reasoning with vague and/or imprecise knowledge, data clas-sification and analysis, machine learning, and knowledge discovery [56] . Their role in data mining is elucidated in Section 2.6, with particular reference to rough clustering in Section 6.5.4.

2.2.7 Wavelets

Application of wavelets have had a growing impact in signal and image pro-cessing over the last two decades. But wavelet is by no means a new theory, and it existed in mathematics since 1909 when Haar discovered the Haar transform. Since then, mathematicians have been working on wavelets, and

"wavelet analysis" used to be called "atomic decomposition" for a long time [57]. The wave in physics is defined as a disturbance propagated in media, typically as an oscillating function of time or space such as a sinusoid. The wavelet can be considered a snapshot of a wave oscillating within a short window of time or space. As a result, mathematically, the wavelet can be considered as a function which is both oscillating and localized.

Representation of a signal using sinusoids is very effective for stationary signals, which are statistically predictable and are time-invariant in nature.

Wavelet representation is found to be very effective for nonstationary signals, which are not statistically predictable and time-varying in nature.

Variation of intensity to form edges is a very important visual characteristic of an image. From signal theoretic perspective, discontinuities of intensities occur at the edges in any image and hence it can be prominently visualized by the human eye. The time and frequency localization property of wavelets makes it attractive for analysis of images because of discontinuities at the edges.

Wavelets are functions generated from one single function called the mother wavelet by dilations (scalings) and translations (shifts) in time (frequency) domain. If the mother wavelet is denoted by ij)(t), the other wavelets tpa'b(t) for a > 0 and a real number b can be represented as

(2 ' ³²⁾

where a and 6 represent the parameters for dilations and translations in the time domain. The parameter a causes contraction in time domain when a < 1 and expansion when a > 1. In Fig. 2.7, we illustrate a mother wavelet and its contraction and dilation.

We discuss further details of wavelet transformation and its properties in Section 3.8.3, and we describe how it can be applied for efficient image com-pression. Its application to data clustering is provided in Section 6.5.3.

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Fig. 2.7 (a) Mother wavelet V(0- (b) V> (*/«): 0 < a < 1. (c) ^(t/a): a > I.