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This section covers the fundamental concepts behind fuzzy extensions to rough set theory. For those readers interested only in the basic ideas of rough set-based feature selection, s 3 through 6 are more directly of interest. The reliance on discrete data for the successful operation of rough set theory can be seen as a signi cant drawback of the approach. Indeed this requirement of rough set theory implies an objectivity in the data that is simply not present [178]. For example, in a medical dataset, values such as Yes or No cannot be considered objective for a Headache attribute as it may not be straightforward to decide whether a person has a headache to a high degree of accuracy. Again, consider an attribute Blood Pressure. In the real world this is a real-valued measurement, but for the purposes of RST must be discretized into a small set of labels Normal ,

This variable specifies which character set is acceptable to the client, for example:

Domain Controller The name of the printer object has been changed. Due to attribute replication, only the name change needs to be replicated to all other domain controllers, not the entire object and all of its other attributes.

HTTP_ACCEPT_CHARSET = iso-8859-1,*,utf-8

This variable specifies which encoding schemas are acceptable to the client, for example:

High, and so forth. Subjective judgments are required for establishing boundaries for objective measurements. A better way of handling this problem is the use of fuzzy-rough sets. Subjective judgments are not entirely removed as fuzzy set membership functions still need to be de ned. However, the method offers a high degree of exibility when dealing with real-valued data, enabling the vagueness and imprecision present to be modeled effectively.

Update Sequence Numbers (USNs)

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In this case, the client accepts gzip (compressed) files Thus, a CGI script can compress a large page by using gzip compression and send it, and the Web client will be able to decompress it

This variable specifies which language is acceptable to the client, for example:

Another concept you need to understand is Update Sequence Numbers (USNs). The Active Directory uses USNs, which are 64-bit numbers, in order to keep track of changes that occur to objects in the Active Directory. When an object is changed, its USN is updated so that all other domain controllers have an outdated USN for that object. By using USNs and updating them when database changes occur, domain controllers know they must receive replication data in order to update their databases. Each domain controller maintains a USN table that lists all of the USN numbers. When replication occurs, the USN table is checked for outdated USNs. Only the highest USN is stored to ensure that only the most current data is saved in the database. When you examine an Active Directory object s properties, such as a user account, you can click the Object tab and see the original USN and the current USN for the object, as shown in Figure 13-2.

In the same way that crisp equivalence classes are central to rough sets, fuzzy equivalence classes are central to the fuzzy-rough set approach [86,359,401]. For typical applications, this means that the decision values and the conditional values may all be fuzzy. The concept of crisp equivalence classes can be extended by the inclusion of a fuzzy similarity relation S on the universe, which determines the extent to which two elements are similar in S. For example, if S (x, y) = 0.9, then objects x and y are considered to be quite similar. The usual properties of re exivity ( S (x, x) = 1), symmetry ( S (x, y) = S (y, x)), and transitivity ( S (x, z) S (x, y) S (y, z)) hold. The transitivity requirement may not actually be useful when modeling real world situations. For example, let x, y, and z be a large box, a medium-sized box, and a small box. In this case x and y are somewhat similar (e.g., their similarity is 0.5). It may also be the case that y and z are somewhat similar to each other (e.g., their similarity is 0.5). But, in this example, x and z are not similar (e.g., their similarity is 0.1), and hence transitivity does not hold. However, the properties of transitive fuzzy relations are often desirable from a mathematical viewpoint and are used here. Using the fuzzy similarity relation, the fuzzy equivalence class [x]S for objects close to x can be de ned: [x]S (y) = S (x, y) The following axioms should hold for a fuzzy equivalence class F [132]:

This variable specifies what client software the requesting system is running and what operating system it is running on, for example:

HTTP_USER_AGENT = Mozilla/404 [en] (WinNT; I)

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