Introduction to Fuzzy Databases

Transcription

1 Introduction to Fuzzy Databases Adnan Yazici Dept. of Computer Engineering, Middle East Technical University, 06531, Ankara/Turkey

2 Fuzzy Information in Databases Fuzzy information or fuzzy data can appear for different reasons. 1. First, it may be due to the imprecision of real data. For example, a sensor data may be a distribution, rather than a precise value. 2. Second, information can arise from subjective judgments. For instance, data about the quality of public schools, the safety of the neighbor hood, etc. can capture the soft boundaries between qualitative descriptions such as poor, fair, good, excellent, etc. 3. Third, information that a user is interested in may not be precise. For example, a college senior may be interested in finding a university that has a good graduate engineering program and low living cost. Formulating a query using a threshold such as annual living cost is less than x dollars will exclude those universities whose annual living cost is slightly above x, but whose graduate engineering program is excellent.

3 Fuzzy Information in Databases Enterprise Precise Vague Data Certain Imprecise Query Language Crisp Not Crisp Examples??

4 Fuzzy Information in Databases The superior modeling power of the semantic and object oriented data models make it possible to represent uncertainty and impreciseness in databases at various db modeling levels: The attribute level The linkage (or relevance) of an attribute to the object The object (or tuple) level The class hierarchy level The relationship between objects The relationship between classes

5 Fuzzy Information in Databases Fuzzy logic has been used to extend database systems in various areas: 1. Processing imprecise queries on data that is precise 2. Processing imprecise queries on data that is imprecise 3. Storing and updating information/data that is imprecise by nature Imprecise information in databases can be classified into three types: 1. Similarity-based approaches 2. Possibility-based approaches 3. Partial membership of a tuple in a relation.

6 Similarity Relation A similarity relation is a mapping, s: D D [0,1], such that for x,y,z D, the following rules hold: s(x, x) = 1 (reflexivity), s(x, y) = s(y, x) (symmetry), s(x, z) max y D ( min (s(x, y), s(y, z))) (max-min transitivity).

7 Example Let for a domain D, we have D = {a,b,c}. We define relationship s for domain D and the equivalence classes induced by s. a b c s a b c Similarity Relations [1,0.8) : {a}, {b}, {c} [1.0,0) : {a,b}, {c} [0,0] : {a,b,c} The Equivalence classes induced by s An Example Similarity Matrix and Corresponding Equivalence Classes

8 Fuzzy Information in Databases 1. A Similarity-Based Relational Database Assessments Site-id site-1 site-2 site-3 Severity [critical, severe] [good] [good, average] The Assessments Relation of Similarity-Based Relational Model

9 Fuzzy Information in Databases critical severe poor average good critical severe poor average good

10 Fuzzy Information in Databases 2. A Possibility-Enhanced Relational Database Name Age Salary Ali 25 {0.8/40K, 0.9/42K, 1.0/43K} Ng {0.8/34, 0.9/35, 1.0/36} 55K Srikanth {0.9/33, 1.0/35, 0.6/36} {0.8/62K, 0.9/64K, 1.0/65K}

11 Fuzzy Information in Databases 3. Partial membership of a tuple in a relation. There are relations whose membership is in [0,1] has a gray area. For example, the relation R, Endangered-species may include wild animals, lions, whose quantity has been significantly decreasing over the years but have not been officially declared as endangered species. A weighted tuple t in a relation R is a tuple associated with its membership degree in R, denoted μ R (t). Endangered-species name quantity weight Panda Eagle Lions 1, ,000 50,

12 Fuzzy Information in Databases Imprecise Queries: The imprecision of information retrieved can be due to two reasons: 1. The information stored in the system is imprecise; 2. The query posted by the user is imprecise in nature (called imprecise queries). A query is imprecise if it includes at least one of the following components: 1. Imprecise conditions, (i.e., annual family income is low) 2. Imprecise operators, (i.e., countries whose top three imported goods are about equal to the top three exported goods of Turkey. Others, about the same as, slightly less than, significantly less than, somewhat greater than, etc.) μ About-the-same (I 3 (c), E 3 (Turkey)) = I 3 (c) E 3 (Turkey)) / 3 3. Imprecise quantifiers, (i.e., companies whose customers are mostly from government agencies. Others, few, some, quite a lot, etc.) μ mostly (r), where r = company s customers from government agencies / all customers of a company

13 Fuzzy Database Modeling Conceptual database models EER, IFO, SDM, IRIS, UML, etc. Logical database models 2 nd Generation DB Models Network, Hierarchic, and Relational 3rd Generation DB Models Non-1NF (NF 2 ), Object-Relation Model, Object-Oriented Database Models, and Deductive Database Models Next generation Models Intelligent Database Systems, Deductive Object-Oriented models Ceng-708, FDB Modeling

14 Fuzzy Database Modeling Physical database models Data organization Index Structures Domain specific databases Active db models Spatial db models Temporal db models Spatiotemporal db models Multimedia (Video) db models Wireless sensor network db models

15 Uncertainty in Databases (research issues) Uncertainty in Relational DB Model Various uncertain and imprecise information in other models such as Conceptual EER, IFO, etc. Logical...Non-1NF (NF 2 ), OO, Deductive OO, active db, spatio-temporal dbs and others. Information retrieval models Uncertainty in Querying

16 Extended NF 2 Model Complex and uncertain data together should be dealt with at both logical and physical database levels. Atomic, relation-valued, set-valued, range-valued, and fuzzy-valued attributes are treated in a uniform fashion. The tuples of relation-valued attributes should be physically represented in a chain of clustered form for efficiency. The implementation of the model includes support for database schema definition, insertion, deletion, update, and querying the database. The SQL like language should be supported. The language should provide users with new query capabilities based on conditions that may involve preferences and describe more or less acceptable items in addition to crisp queries.

17 Definitions of Some of the Related Concepts Definition: An NF 2 schema may contain any combination of simple or higher order attributes on the right hand side of the rules whereas a 1NF scheme can contain only atomic attributes. (An attribute A j is a higher order attribute if its schema appears on the left-hand side of a rule; otherwise it is simple.) Two new restructuring operators (other than the basic algebraic operators) are the Nest and Unnest (as well as Pack and Unpack).

18 Example For example, the Department relation can be defined as a nested structure using the following forms: DEPT = (DNo, DName, Instructors, Students, Courses) Instructors = (IName, ISSN) Students = (SNumber, SName, Contact-Person) Courses = (CNo, Prerequisites, CCategory) Contact-Person = (Name, TelNum)

19 Hierarchy Tree for Department Relation DEPT DNO DNAME INSTRTUCTORS STUDENTS COURSES INAME ISSN SNUMBER SNAME C-PERSON CNO PREREQUISITES CCATEGORY NAME TELNUM

20 Extending NF 2 Data Model for Uncertain Information Definition: Let SchR be a relation schema of relation R with attributes (A 1,A 2,, A n ). Each attribute A j may be: simple or set-valued or fuzzyvalued or range-valued or relation-valued; which are all defined below. Again D 1, D 2,..., D n is a finite set of domains. Let r, an instance of R, be composed of a set of ordered k-tuples of the form <a 1,a 2,...,a n >, which is a subset of (D 1 xd 2 x...xd n ). The domains, D j (1 j k), can be one of the following:

21 The Domains D j is the domain of an atomic-valued attribute. D j is the domain of a null-valued attribute. Domain D j is extended to the domain D j = D j {unk, dne, ni}. D j is the domain of an incomplete (range)-valued attribute whose values may be atomic or an interval, [a j1 -a j2 ]. Both values are taken from the domain D j. D j is the domain of a fuzzy-valued attribute and subtends a set of fuzzy linguistic terms. A fuzzy attribute value is a nonempty subset of D j and represented as [a j1, a j2,...,a jm ]. D j is the domain of a set-valued attribute represented as {a j1, a j2,...,a jm }. Any value of this attribute is a subset of the power set of D j. D j is the domain of a relation-valued (composite) attribute. Any value of this attribute, a j, is a tuple of the form <a j1,a j2,...,a jm > which is an element of (D j1 xd j2 x...xd jm ), where 1<m and 1 j k.

22 Employee Relation Employee Name SSN TelNo Salary Degrees Languages Age Univ. GPA Year G. Dark dne [ ] ITU {Fr., Eng.} [MidYng, Yng] W. Fisher [ ] E.U {Ger.} [MidYoung] T. Smith ni [3700] METU {Ger,Fr,Eng} [VeryYng,Yng] Y.Tomson [ ] CWRU {Eng.} [Young]

23 The Extended NF 2 : The extended NF 2 model includes the extension of the basic operators such as Selection, Projection, Cartesian product, Union, and Difference, as well as two new restructuring operations, Merge and Unmerge The sorts (types) are more complex than those of the relational data model. τ = dom fdom ndom idom <B 1 : τ 1,,B m : τ m > {τ s } where τ s fdom idom {τ} and B 1,,B m are distinct attributes. The set of values of sort τ (i.e. the interpretation of τ), denoted as t[τ], is defined as follows: t[dom] = dom, t[fdom] = {[v 1,...,v j ] i: 1 i j: v i t[dom]}, t[ndom] = {v i i: 1 i j: v i t[{unk,dne,ni} dom}]}, t[idom] = {[v 1 -v j ] i: 1 i j: v 1 v i v j, v i t[dom]}, t[{τ s }] = {{v 1,...,v j } i,j: 1 i j: v i t[τ s ]. If t[{τ s }]= {}, then v i = dne}, and t[<b 1 :τ 1,,B m :τ m >]={<B 1 :τ 1,,B m :τ m > i: 1 i m: v i t[τ i ]}.

24 Restructuring Operators The Merge Operator A level value, L j (A j ) is chosen based on the following rule: Lj(Aj) = 0 or 1 if (Aj AttA Aj:{τj}) (Aj RelA) 0 λ 1 otherwise, where A j :{τ j } means that attribute A j is set-valued. Definition: Suppose we have relation S with sort; Sort(S) = <A 1 :τ 1,,A i :τ i,, A k :τ k,..., A n :τ n >, where the scheme SchS:(A 1,, A i,, A k,, A n ) and 1 i, k n. Then for instances s of S, we have the following: R = MERGE(S)[(A i,...,a k ) B] WITH L(A 1 )= λ 1,...,L(A n )= λ n = {<A 1 :x 1,, A i-1 :x i-1, B:y, A k+1 :x k+1,a n :x n > y = <A i :x i,...,a k :x k > <A 1 :x 1,...,A n :x n > s(s)}}. t [R]= t [SchR-(Ai,...,Ak) B] t [B]= {tj[b] ti s(s) tj s(s) tj[schr-b] ti[schr-b]} where 1 k n, 1 i,j k and produces a relation r with scheme SchR=SchR-(A i,..,a k ) B, where B is a sub-relation with attributes (A i,...,a k ) and does not occur in the scheme of resulting S, and is collecting similar values.

25 The Query Language Query Syntax SELECT <Attribute List> FROM <Relation Name > MERGE <Relation Name> WITH LEVEL <Level Value List> WHERE <Condition List> The LEVEL value, Lj(Aj), list can be specified as follows: CASE 1: Whole set of attributes has the same level-value: LEVEL (Attribute 1, Attribute 2,, Attribute n ) = Level-Value CASE 2: All the attributes have a different level-value: LEVEL (Attribute 1 ) = Level-Value 1, LEVEL (Attribute 2 ) = Level- Value 2 LEVEL (Attribute n ) = Level-Value n CASE 3: Hybrid model: LEVEL (Attribute 1 Attribute m ) = Level-Value 1, LEVEL (Attributem+1 Attributes) = Level-Value2 LEVEL (Attributes+1 Attributen) = Level-Valuet

26 Extension to SQL MERGE must be followed by a relation name that exists in the database. Every MERGE must have associated LEVEL condition lists. LEVEL values must be in [0, 1] range. MERGE can have more than one LEVEL condition list. Attributes that have the same level value can be grouped together in parenthesis. Relation-valued attributes must have a nested select statement in parenthesis. Every higher order attribute specified with SELECT must have a nested SELECT statement.

27 ExSQL The syntax of the more detailed select statement (for fuzzy querying case) is as follows: (SELECT <Attribute-Name> {,<Attribute-Name>} FROM <Relation Name>-itm = MERGE (<Relation Name>)[ ] WITH <LEVEL (<Attribute Name> {, <Attribute Name>}) =Level-Value> {, <LEVEL (<Attribute Name> {, <Attribute Name> } ) =Level-Value>}] WHERE <Attribute Name Operator Constant-Value> {, <Attribute Name Operator Constant-Value>})

28 Fuzzy Querying Algorithm: Given an extended NF 2 relation scheme SchR:(A 1,A 2,...,A n ) with a set of identifying attributes X (consisting of (a set of) prime attributes of R), apply the Merge operator over the relation scheme of relation R. Case1: If all members of X having level value 1(one) are merged, then; All crisp attributes (including the ones that can take null values) and set-valued attributes whose level values are 0 (zero) result in M-sets, Others do not change their types. Case2: If members of X having level values 0 (zero) are merged, then fuzzy values may be obtained. Crisp, null and M-Set valued attributes that have level value 0 (zero) result in M-Sets, the others (e.g. fuzzy-valued) do not change their types. if the level value is between 0 (zero) and 1 (one), then F-sets are obtained (only for fuzzy-valued attributes, the level values between 0 (zero) and 1 (one) may be specified), Attributes with level value 1 (one) do not change their types.

29 Example: Employee relation. Employee Name SSN TelNo Salary Degrees Languages Age Univ. GPA Year G. Dark dne [ ] ITU {Fr., Eng.} [MidYng, Yng] W. Fisher [ ] E.U {Ger.} [MidYoung] T. Smith ni [3700] METU {Ger,Fr,Eng} [VeryYng,Yng] Y.Tomson [ ] CWRU {Eng.} [Young]

30 Example Query Query: Retrieve the Name and SSN of the more or less young employees who get around $3500 of salary and whose GPAs are greater than (SELECT Name, SSN FROM Employee-itm = MERGE (Employee)[] WITH Level (Name, SSN, TelNo) = 0.0, (SELECT Degrees, Age, Salary FROM Employee = MERGE (Employee) [ ] WITH Level (Age) = 0.8 (SELECT GPA FROM Degrees = MERGE (Degrees) [] WITH Level (Univ, GPA, Year) = 0.0 WHERE GPA > 3.0) WHERE Age = Young AND Salary = )) Answer: Name: {T. Smith} and SSN: {1857-3} another query statement (similar to the one given above) can be written by changing only the level value for the Age attribute to 0.5. Answer: Name: {W. Fisher, T. Smith} and SSN: {1832-1, }

31 Resulting NF 2 relation Employee-itm Name SSN TelNo Salary Degrees Languages Age Univ. GPA Year {W. Fisher, T.Smith} {1832-1, } { , ni} [ ] {E.U., METU} {3.40, 3.54} {1990, 1991} {Ger,Fr,Eng} [MidYoung, VeryYoung, Young] {GDark} {1788-1} {dne} [ ] {ITU} {3.24} {1988} {Fr., Eng.} [MidYng, Yng] {Y.Tomson} {1978-4} { } [ ] {CWU} {2.80} {1991} {Eng.} [Young]

32 Similarity Relation for Age attribute s age VeryYng Young MidYng Old VeryOld VeryYng Young MidYng Old VeryOld D = {Very-Young, Young, Mid-Young, Old, Very-Old} age

33 Conclusion One direction of current research on databases is the incorporation of complex and uncertain data into database models and systems. Here we presented the extended NF 2 data model, an example for the extended SQL-like query language, permitting representation and manipulation of complex and uncertain data as well as precise data. The model supports arbitrarily deep nested relations.