Concept Analysis in Web Informatics- 5th GrC Model - Using Ordered Granules

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1 Concept Analysis in Web Informatics- 5th GrC Model - Using Ordered Granules Tsau Young (T. Y.) Lin Department of Computer Science San Jose State University San Jose, California 95192, USA tylin@cs.sjsu.edu Abstract 5th GrC model is the formal model specified into the category of sets It is a theory of ordered granules,namely, granules are ordered subsets of the universe, We extract a 5th GrC model from a set of web pages. A granule is a high frequent sequence of keywords, It is a tuple in a relation and naturally carries some concept expressed in web pages. The concept analysis in this paper is about true human concepts that are expressed in web documetns. 1 Introduction The term granular computing(grc) was proposed in 1996, rouhgly by Lin and Zadeh. Since then,many models have been studied. 12 years later, in GrC2008, we proposed to use one of the formal model(8th GrC Model) as the Formal GrC Model. It is a catgeory theory based model. Somewhat a surpise, in the level of abstract category theory, this model was the category model of database (except the cardinality ) that we started in 1990 [4]. However, the semantics are very different: In database a tuple is a representation of an entity while in GrC, a tuple is an ordered collection of data that represent some higher level knowledge. Based on this model, all other modelcan be derived. In this paper, we will be interested in the case, the category is the category of sets. This is the first GrC model that is intrinsically beyond set theory, that is, a granule is more than a set. We illustrate the idea in Web Informatics. 2 Web Informatics A document or a web page can be viewed as a list of linearly ordered tokens or terms; to emphasize this view, we call it linear text. It is a knowledge representation of human idea or human thoughts. However, mathematically speaking, this is not a good representation, because The semantics in the linear text can only be revealed through human reading. In this section. We recall the construction that reveal authors thoughts mathematically. We may schematically summarize the flow of constructions; it is an ordered version of [10] IDEA [ human thoughts [ human thoughts ] (summarize) [ linear text ] [ ] simplicial complex ofconcepts Note that none of those is a mathematical map; it merely indicates information flows. A novel point in this combinatorial structure is the capability to reveal the idea hidden between lines, namely, the model can read between the lines (words). Let us consider the following 3 fictitious web pages Example 1 Ficitious web pages:. 1. Wall Street is a symbol for American finance industry. Most of the computer systems for those financial institue have employeed information flow security policy to protect privacy and security of the information. 2. Wall Street is a shorthand for US finance industry. Its E-security has applied security policy that was based on the ancient intent of Chinese wall. 3. Wall Street represents an abstract concpet of NYC finance industry. Its information security policy is Chinese wall. ]

2 2.1 Mining Associations on the Web First, we need to collect important keywords. Normally, we use tfidf formula to extract them. The formula can be found in any text on information retrieval; you may find it in our paper [9] [16]. Here we extract the collection based on common sesne. However, the important information are based those keywords that has high document frequency By counting and stemming, the keyword and keyword tuples can be extracted: Note that from now on keywords and keyword tuples really refer to those that have high frequency. 1. By stemming,the important words are: U = { Wall, Street, symbol, America, finance, industry, computer, systems, financial, institue, employ, information, flow, security, policy, protect, privacy, security, shorthand, US, security, apply, base, ancient, intent, China, wall, represent, abstract, concpet } 2. The keyword set are based on document frequency. we take high frequent keywprds as 1-nary relation: R 1 = { China, finance, industry, policy, security, street, wall }. 3. Keyword pairs are based on document frequency, namely, the number of doocuments that contain the keywords. In other words, any two nearby kewywords (distance is less than 10 tokens or within one sentence) appear more than two documents are included. Keyword pairs form a binary relation: R 2 U1 2 U2 2, where where Uj 2 = U, j =1, 2 R 2 = {(Wall, street); (Wall, finance); (Wall, industry); (street, finance); (street, industry); (finance, industry); (china, wall); (security, policy); (security, china); (policy, china) (security, wall); (policy, wall) } 4. Keyword triple forms a 3-nary relation: R 3 U1 3 U2 3 U3 3, where Uj 3 = U, j =1, 2, 3 R 3 = {(Wall, street, finance); (Wall, street, industry); (street finance, industry); (security, policy, china); (security, policy, wall) } 5. Keyword 4-tuples froms a4-nary relation: R 4 U1 4 U2 4 U3 4 U4 4,where Uj 4 = U, j =1, 2, 3, 4 R 4 = { (Wall, street, finance, industry); (security, policy, china, wall) } The tuples in these relations are granules fot U. Observe that the granules are n-tuples, not subsets (n =1, 2, 3, 4). For example (street, Wall), (industry, finance) are not meaningful so order is important. Let us fits this example into the formal GrC theory. 3 Formal GrC Models on the Web Let us fits this example into the formal GrC theory. Definition 1 A category consists of 1. A class of objects, and 2. A set Mor(X, Y) of morphisms for every ordered pair of objects X and Y, which satisfies certain properties. In the caetgory of sets, the objects are the set, the morphims are the mapping between sets. In web informatics, sets are the sets of keywords in web pages. The following is the formal defintion from [13], we will fit the example into this formal definition Definition 2 5th GrC Model 1. Let U = {U h j h h H, j h J h } be a given family of classical sets, called the universe. Note that distinct indices do not imply the sets are distinct. Ex U = {U, U1 2,U2 2,U1 3,U2 3,U3 3,U1 4,U2 4,U3 4,U4 4 }, where U j l k = U for all j, and l k 2. Let {U k l k K b H; L k b J k } be a family of Cartesian products of various lengths n, where n = L k is the cardinal number of L k that may be infinite. Here b means a sub-bag. Recall that a bag is a set that allows the repetition of some elements [?]. Ex {U1 1,U1 2 U2 2,U1 3 U2 3 U3 3,U1 4 U2 4 U3 4 U4 4 }, where U j l k = U for all j, and l k 3. Recall that an n-ary relation is a subset R n of a product space in the previous item. Ex R 1,R 2,R 3,R 4 are the selected relations 4. Let β = m M {R m } or = {R m m M} is a set n-tuples for various n or a set of n-relations; note that n is not fixed and can be infinite. (a) β is a set of tuples, that is, = R 1 R 2 R 3 R 4 = {China, finance, industry, policy, security, street, wall (Wall, street); (Wall finance); (Wall industry); (street finance); (street industry); (finance, industry); (china, wall); (security, policy); (security, china); (policy, china) (security, wall); (policy, wall) (Wall, street, finance); (Wall, street, industry); (street finance, industry); (security, policy, china); (security, policy, wall) (Wall, street, finance, industry); (security, policy, china, wall) }

3 (b) β is a set of relations, that is, = {R 1,R 2,R 3,R 4 } Then the pair (U,β) is the formal definition (with two possible interpretations of β) of 5th GrC Model. We may call both interpretations Relational GrC Model. Note that the granular structure (GrS) of 5th GrC model is (1) the collection of all tuples in a relational database (= a set of relations), if the sets U h j h are attribute domains or (2) a set of realtonal tables. Also note that 5th GrC Model is the relational structure (without functions) of the First Order Logic. Observe that 5th GrC Model may live in GrM, so we allow all the index sets be very large (infinite). So we adopt the following index convention: Convention for index. In computer science, the index often runs through a countable set. In this case, we often denoted it as follows: k =1, 2,...k,...without naming an index set. As this paper will include GrM, we will take the following convention: the lower case letter, say k, denotes the parameter that runs though a set, K, whose name is the corresponding cap letter. By extend the category of sets to abstract category, we have the following general GrC Model. Let CAT be a given category. Definition 3 Category Theory Based GrC Model 1. C = {C h j h H, j h J h }, called the universe, be a family of objects in the Category CAT. 2. There are families (which are bags) of Cartesian products {Cl k k K b H; L k b J k } of various lengths n, where n = L k is the cardinal number of L k that may be infinite. They are called product objects. 3. An n-ary relation object R j is a sub-object of a product object. 4. β be a family of n-ary relations (n is any cardinal number and could vary). The pair (C,β) is 8th GrC model. This is the final formal GrC model. We may refer to it as the Category Theory Based GrC Model 4 Knowledge Structures on the Web In last sectionm we have associated to the three fictitioud web pages a 5th GrC Model. In this section we shall describe its knowledge structures of this model. Let us first recall the concept of knowledge structure from [13]. 4.1 Four Structures of GrC Models A GrC Model has four structures: granular, quotient, knowledge and linguistic structures. Mathematically the quotient and the knowledge structures are the same. Linguistic structures are close to granular strcuture in PNL (precisiate natural language) processing. We shall explain the four structure as in [13]. 1. Granular Structure (GrS): It is the collection of all granules. 2. Quotient Structure (QS): If each granule is abstracted into a point and the intersections of granules are abstract to the interactions of points, then such a collection of points is called quotient structure. In the case of partition, the quotient structure is a classical set, called quotient set. 3. Knowledge structure: By giving each point in the quotient structure a meaningful unique symbol, then the named quotient structure is called knowledge structure. The knowledge structure provides an intuitive view of the quotient structure; Mathematically, the knowledge structure is the same as quotient structure. 4. Linguistic structure: By giving each granule in the granular structure a word that reflects its meaning. The interactions among these words are reflected in precisiated natural language. (in knowledge structure, the interactions among symbols are explicitly described mathematically) The linguistic structure is the domain of computing with words. The knowledge structure is the traditional knowledge representation theory. Example 2 The Granular Structures of the fictitious three web pages. 1. Its Granular Structure is the β in Section 3: β = {China, finance, industry, policy, security, street, wall (Wall, street); (Wall finance); (Wall industry); (street finance); (street industry); (finance, industry); (china, wall); (security, policy); (security, china); (policy, china) (security, wall); (policy, wall) (Wall, street, finance); (Wall, street, industry); (street finance, industry); (security, policy, china); (security, policy, wall) (Wall, street, finance, industry); (security, policy, china, wall) } 2. Since the granules are keyword tuples, we shall use them to name the granules. So keyword tuples are name of themselves. So the knowledge structure is β itself

4 4.2 Helpful Geometry - Simplical Complex Let us recall some geometry. A n-dimensional Euclidean space is a space in which elements can be addressed using the Cartesian product of n sets of real numbers. A point in n-dimensionl Euclidean space is called unit point, if its coordinates are all 0 but a single 1, (0,...,0, 1, 0,..., 0). These unit points will be called vertices; they are denoted v 0,...,v n. We will use them to illustrate the notion of simplical complex. First let us define the n-simplex and its faces: Definition 4 1. a geometric n-simplex, denoted by Δ(v 0,...,v n ), is a set of unit points {v 0,...,v n }, called vertices. 2. A q-subset of a n-simplex is called a q-face; it is a q- simplex Δ(v j0,...,v jq ) whose vertices are a subset of {v 0,..., v n } with cardinality q In an abstract n-simplex, the vertices can be any abstract objects,such as keywords. Let us examine some example in 3-Euclidean space. A 0- simplex Δ(v 0 ) consists of a vertex v 0 =(1, 0, 0), which is a point in the Euclidean space. A 1-simplex Δ(v 0,v 1 ), that consists of two points {v 0, v 1 }, is an open segment (v 0, v 1 ). Observe that it does not include its end points. A 2-simplex Δ(v 0,v 1,v 2 ), that consists of three points {v 0, v 1, v 2 }, is an open triangle,which does not include its edges and vertices. Next we will define the simplical complex ( [17]p.108): Definition 5 A simplicial complex C consists of a set {v} of vertices and a set {s} of finite nonempty subsets of {v}, called simplexes, such that Any set consisting of one vertex is a simplex. Closed condition: Any nonempty subset of a simplex is asimplex. Any simplex s containing exactly q +1vertices is called a q-simplex. A simplex is said to be maximal if it is not a face of any other simplex. Any set of n +1indepedent objects can be viewed as a set of abstract vertices. Such a complex is an abstract simpicial complex. Example 3 In Figure 1, we have a goemtric simplicial complex that consists of 12 vertices that are organized in the form of a 3- complex. Let us enumerate maximal simplexes: 1. The maximal 3-simplex Δ(a, b, c, d), and all its faces. 2. The maximal 3-simplex Δ(w, x, y, z) and all its faces. 3. The maximal 2-simplexes lying between two 3-simplexes: Δ(a, c, h), Δ(c, h, e), Δ(e, h, f), Δ(e, f, x), Δ(f,g,x), Δ(g, x, y) and their faces. Figure 1. A complex with 12 vertices. 4.3 Concept - Reading between words We would like to present some interesting phenomena. A keyword tuple, semantically, may have nothing to do with its individual keyword and subtuples. For example, 1. The keyword pair (2-tuple) Wall Street represents a concept that has nothing to do with Wall and Street 2. The keyword 4-tuple security policy of China wall represents the concept, Chinese wall security policy that has nothing to do with subsets China wall (physical wall) China (country name) and etc. These examples indicate that the strength of this approach is the ability to capture the concepts that are defined implicitly by some keyword tuples. In plain words, it can read between tokens. Let us formally define the keyword tuple that carries the un-worded concept. Definition 6 A n d -keyword tuple is an ordered keywords (of cardinailty n) that has a high frequency of co-occurrences (called SUPPORT) keyword tuple that are at most d tokens apart. In the case that d and n are understood, we abbreviate it simply as keyword tuple. 4.4 Concept Strucutre Previous section explains that a keyword tuple really carries some intrinsic concept that cannot be deduced from the individual keywords. So keywords and keyword tuples represent certain concepts. The collection of these concepts may represent some deep knowledge hidden in the web pages. To illustrate the geomeric structure of the concepts in the fictitious webpages, we will use alphabets to name these abstract vertices. From which we can picture geometricaly how these keywords and keyword tuples are fitted together and express some deep knolwedge.

5 1. Recall that keyword means high frequency words, so keyword set is the 1-nary relation R 1 = { China, c=finance, d=industry, x=policy, w=security, b=street, a=wall, z=wall }.. 2. Keyword pairs are based on document frequency. In other words, any two nearby kewywords (within the distance of 10 words in one sentence) appear more than two documetns are listed. In real documents, one could use the distance of 20 words within one paragraph. Keyword pairs is the relation: R 2 U U. R 2 = {(a=wall, b=street); (a=wall c=finance); (a=wall d=industry); (b=street c=finance); (b=street d=industry); (c=finance, d=industry); (y=china, z=wall); (w=security, x=policy); (w=security, y=china); (x=policy, y=china) (w=security, z=wall); (x=policy, z=wall) } 3. Keyword triple are based on the same critria as pairs. Keyword triples is:r 3 U U U R 3 = {(a=wall, b=street, c=finance); (a=wall, b=street, d=industry); (b=street c=finance, d=industry); (w=security, x=policy, y=china); (w=security, x=policy, z=wall) } 4. Keyword 4-tuples are based on the same critria. Keyword 4-tuples is the relation R 4 U U U U R 4 = { (a=wall, b=street, c=finance, d=industry); (w=security, x=policy, y=china, z=wall) } 5.. The picture of this example is the the same as Figure 1, except all those trianlges between two terahedra should be removed. 6. In Englishlanguage, the Cap and lower cases often used to differenciate between proper nounds and common nouns. In most of the cases, they should be treated differently. If Wall and wall are identified the two terahedra should meet, namely a and z should be regard as the same point. Now we shall introduce various concepts associated with this simplicial concepts Definition 7 An I-concept (I means intermediate) is the concept correspond to a simplex. An P-concept(P means primitive) is the concept correspond to a maximal simplex. A C-concept is the concept correspond to a connected component of the simplicial complex A pair concept, (Pconcept, I-concept), represents the P-concept that does not include sub-concept I-concept. The two maximal 3-simplex carry two P-concepts in this set of web pages. All the q-faces carry all related I-concepts. If wall and Wall are idenitfied, we have one C0concept, otherwise two C-concepts. This simplicial complex of the web pages show us that the complex well organize concepts in the web into a beautiful knowledge. 5 Some applicationa - Document Clustering Traditional clustering requires a document set should be regroupped into disjoint classes, namely, equivalence classes of documents. From the view of contents, this is an unreasonable requirment. Many documents are inter-related in some concepts and unrelated in others, So if we cluster the documennts based on concepts, overlappings are very natural. Our approach does not requring disjoint between clusters Based on previously defined concepts, we may define: Two documents are I/P/C-clustered together if both documents have the same I/P/C-concept. Recall that two keyword tuples, w 0 and w n, are in the same connected component if there is a finite sequence w 0,w 1,...w n of keywords tuples, such that any consecutive two keyword tuples have a common sub-keyword tuple. We will consider the P-concept clustering only. The first primitive concept is (Wall street finance industry); this cluster containss all three web pages. The second primitive concpet is (security policy china wall), onlly last two contain this concept. In certain situations, we may decide to ignore some I- concepts. we will call this method relative clustering. We will label each document by the given pair if the document contains any P-concept but not I-sub-concept. In plain words, two documents are not going to be clustered together if their only common topics are in the sub-concept. 6 Conclusion GrC is a powerful concept, In this paper, we use 5th GrC Model to explore the some deep knowledge hidden in the web. The GrS, β, often captures very deep knowledge. Let us examine some mathematics: 1. If we choose β to be the topology (the family of open sets and etc), then U is equipped with the knowledge of continuity. 2. If U is the complex polynomial ring of one variable, and choose β to be SPEC(U) (the collection of prime ideals) then the quotient structures is the complex plain (geometrically is the Eudlidean plain) with one genric point [14].

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