Information Sciences 179 (2009) 2325–2332
Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
Extracting complex linguistic data summaries from personnel database via simple linguistic aggregations Zheng Pei a,*, Yang Xu b, Da Ruan c, Keyun Qin b a b c
School of Mathematics and Computer Engineering, Xihua University, Chengdu, Sichuan 610039, China Department of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China Belgian Nuclear Research Centre (SCKCEN), Boeretang 200, B-2400 Mol, Ghent University, Gent, Belgium
a r t i c l e
i n f o
Article history: Received 16 December 2008 Accepted 20 December 2008
Keywords: Linguistic data summary Personnel database Fuzzy statement The LOWA operator Genetic algorithms The 2-tuple linguistic representation model
a b s t r a c t A linguistic data summary of a given data set is desirable and human consistent for any personnel department. To extract complex linguistic data summaries, the LOWA operator is used from fuzzy logic and some numerical examples are also provided in this paper. To obtain a complex linguistic data summary with a higher truth degree, genetic algorithms are applied to optimize the number and membership functions of linguistic terms and to select a part of truth degrees for aggregations, in which linguistic terms are represented by the 2-tuple linguistic representation model. Crown Copyright 2009 Published by Elsevier Inc. All rights reserved.
1. Introduction An abundance of data is often beyond human cognitive and comprehension skills in reality. Natural language is thus used to communicate information by human beings. A linguistic data summary, which is expressed by a sentence or a small number of sentences in a natural language, would be desirable and human consistent, e.g., in a personnel department, linguistic data summary ‘‘almost all younger and well qualified employees are well paid” seems useful. To obtain linguistic data summaries from database is not a trivial act, but requires some more intelligent techniques [18,22,23,26–29,32,33]. Linguistic data summaries have been extensively studied in [19,20,24,34]. In this paper, from the fuzzy logical point of view, we discuss extracting linguistic data summaries from personnel database. In Section 2, we extract a simple linguistic data summary as a starting point. In Section 3, we then extract a complex linguistic data summary. In Section 4, we optimize the number and membership functions of linguistic terms and obtain a complex linguistic data summary with a higher truth degree based on genetic algorithms (GAs) are discussed. We conclude the paper in Section 5. 2. Linguistic data summary Yager presented an approach to the linguistic data summary [29]:
* Corresponding author. Tel.: +86 02887600760; fax: +86 02887600764. E-mail address:
[email protected] (Z. Pei). 0020-0255/$ - see front matter Crown Copyright 2009 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2008.12.018
2326
Z. Pei et al. / Information Sciences 179 (2009) 2325–2332
V is a quality (attribute) of interest, e.g., age, salary, etc.; Y = {y1, y2, . . . , yn} is a set of objects (employees) that manifest quality V, V(yi) (i = 1, . . . , n) are values of quality V for each object yi; D = {V(yi)ji = 1, . . . , n} is a set of data (the ‘‘database” on question). A linguistic data summary of a data set consists of: (1) summarizer S, e.g., young; (2) a quantity in agreement Q, e.g., most; (3) truth T. Hence, a linguistic data summary can be expressed, e.g., most of employees are young is T. It can be formalized by Qy’s are S is T, in which Q is a fuzzy linguistic quantifier, S is a summarizer (property), and T is a truth degree.
most ðQÞ of employees ðy’sÞ are young ðSÞ is T:
ð1Þ
Eq. (1) is equal to decide the valuation of fuzzy statement are(Qy’s, S). Many researchers used T(are(Qy’s, S)) = a, a 2 [0, 1] to express the valuation [19,20]. In this paper, Q = {q1, q2, . . . , qm}, S = {s1, s2, . . . , sl} and T = {t1, t2, . . . , tk}, for each qm0 2 Q ; sl0 2 S, and tk0 2 T are linguistic terms, respectively, e.g., most, young, and very true. For the issue on how to decide the order and indexes of linguistic terms, we refer to [3,10–14]. 2.1. Fuzzy sets of linguistic terms Summarizer sl0 2 S is a linguistic expression semantically represented by a fuzzy set on D = {V(yi)ji = 1, . . . , n}. For example, if sl0 is young and Dsl0 ¼ f1; . . . ; 90g, then the fuzzy set of sl0 is ls 0 : Dsl0 ! ½0; 1. In this paper, the set Y of objects is a finite set, l and sl0 2 S has a fuzzy set on Dsl0 . From the logical point of view, the fuzzy sets of qm0 2 Q and tk0 2 T are different from the fuzzy set of sl0 [21,25,31]. In fact, for the classical universal quantifier ", numbers of objects are emphasized, i.e., ("u)p(u) means ‘‘all u such that p(u)”. The situation of the truth degree is similar to that of the quantifier. Let P(Y) = {AjA # Y} be the power set of Y. Define a binary relation on PðYÞ : A B () j A j¼j B j, where jAj is the potency of A and ‘‘” is an equivalence relation on P(Y), denote PðYÞ ¼ PðYÞ= . The fuzzy sets of qm0 2 Q and t k0 2 T can be defined as following: For each fuzzy linguistic quantifier qm0 2 Q , define
lqm0 : PðYÞ ! ½0; 1:
ð2Þ
For each fuzzy linguistic truth degree t k0 2 T, define
ltk0 : ½0; 1 ! ½0; 1:
ð3Þ
Remark 1. Generally, the type of each fuzzy set of Eq. (2) needs to satisfy some properties, e.g., ‘‘most” is a non-decreasing function, etc. As an example, the following fuzzy sets are given
1 if j A jP 1; l8 ðAÞ ¼ 0 if A ¼ ;; 8 if j A jP a; > : 0 if j A j6 b;
l9 ðAÞ ¼
1 if A ¼ Y; 0 otherwise; 8 jAj1 if 1 6j A j6 3; > < 2 labout3 ðAÞ ¼ 5jAj if 3 6j A j6 5; > : 2 0 otherwise:
In which a, b such that a, b 2 (0, j Yj), and decided by experts or deciders. Fuzzy linguistic truth degrees Eq. (3) are already defined by many researchers [21,25,31]. 2.2. Extracting a simple linguistic data summary Based on Eq. (1), extracting the linguistic data summary is equal to deciding Q, S, and T of Eq. (1). Let a simple linguistic data summary be ‘‘Qy’s are S is T”. Then Q, S and T can be obtained automatically by the following steps: Fixing a summarizer sl0 2 S (it can be one or several) and a level (threshold) h, this can be done by experts or deciders. Let
Dhs 0 ¼ l1 s 0 ðVðyi ÞÞ ¼ fVðyi Þ j ls 0 ðVðyi ÞÞ P hg: l
l
l
ð4Þ
Selecting a fuzzy linguistic quantifier qm0 2 Q . According to Eq. (4), qm0 can be selected such that
lqm0 ðAÞ ¼ maxflq1 ðAÞ; lq2 ðAÞ; . . . ; lqm ðAÞg; Dhs 0 g. l
ð5Þ
where A ¼ fyi j Vðyi Þ 2 If lqm0 ðAÞ is not only one, then one qm0 can be selected by deciders or has a maximal index. Selecting a fuzzy linguistic truth degree tk0 2 T. From the logical point of view, the more objects satisfying statement with the quantifier is, the higher the truth degree t k0 2 T is. On the other hand, the bigger lqm0 ðAÞ is, the more objects satisfying the statement with the quantifier is. Hence,
2327
Z. Pei et al. / Information Sciences 179 (2009) 2325–2332
ltk0 ðlqm0 ðAÞÞ ¼ maxflt1 ðlqm0 ðAÞÞ; lt2 ðlqm0 ðAÞÞ; . . . ; ltk ðlqm0 ðAÞÞg:
ð6Þ
Example 2. A personnel database is in Table 1. 1.8 of y1 means the salary of the employee y1 is 1.8 thousand dollar. Let Sage = {young(y), middle age(ma)}, Ssalary = {low(l), high(h)}. Select Q = {several(s), about half(ah), most(m)}, T = {approximately true(at), true(t), very true(vt)}. Their membership functions are as following:
8 8 if x 2 ½25; 30; if x P 45; > > : : 0 if x > 40; 0 if x 6 35; 8 8 1 if x 2 ½1:8; 2; 1 if x 2 ½3:3; 3:5; > > < < ll ðxÞ ¼ 12 ð2:5 xÞ if x 2 ð2; 2:5; lh ðAÞ ¼ x 2:3 if x 2 ½2:3; 3:3Þ; > > : : 0 if x P 2:5: 0 x < 2:3; 8 8 0 if j A j< 4 > 1 > > > < 1 j A j 2 if j A j2 ½4; 6; < 2 ðj A j 1Þ if j A j2 ½1; 3; 2 ls ðAÞ ¼ 2 13 j A j if j A j2 ð3; 6; lah ðAÞ ¼ > > 4 12 j A j if j A j2 ð6; 8; > : > : 0 if j A j> 6; 0 if j A j> 8; 8 ( 1 if j A j2 ½10; 12; > < 1 if x 2 ½0:5; 1; lm ðAÞ ¼ 14 ðj A j 6Þ if j A j2 ½6; 10Þ; lat ðxÞ ¼ 1 > x if x 2 ½0; 0:5Þ; : 2 0 if j A j< 6: 8 if x 2 ½0:8; 1; > 0 if x 2 ½0; 0:8Þ: : 0 if x 2 ½0; 0:5Þ: (1) Fixing a summarizer s0 = young 2 Sage and s00 = high 2 Ssalary. Let threshold h = 0.5. Then according to ly and lh, we obtain
D0:5 s0 ¼ fVðyi Þ j ls0 ðVðyi ÞÞ P 0:5g ¼ f25; 31; 35; 28; 34; 27g;
ð7Þ
D0:5 s00
ð8Þ
¼ fVðyi Þ j ls00 ðVðyi ÞÞ P 0:5g ¼ f2:8; 3:0; 3:5; 2:9; 3:1g;
As0 ¼ fyi j Vðyi Þ 2 D0:5 s0 g ¼ fy1 ; y3 ; y4 ; y5 ; y9 ; y10 g; As00 ¼ fyi j Vðyi Þ 2 D0:5 s00 g ¼ fy3 ; y4 ; y5 ; y6 ; y9 ; y10 ; y11 ; y12 g; (2) By ls, lah, lm and Eq. (9), we obtain
ð9Þ ð10Þ
ls ðAs0 Þ ¼ lm ðAs0 Þ ¼ 0; lah ðAs0 Þ ¼ 1,
maxfls ðAs0 Þ; lah ðAs0 Þ; lm ðAs0 Þg ¼ lah ðAs0 Þ;
ð11Þ
according to lat, lt, lvt and Eq. (11), we obtain
lat ðlah ðAs0 ÞÞ ¼ lt ðlah ðAs0 ÞÞ ¼ lv t ðlah ðAs0 ÞÞ ¼ 1;
ð12Þ
selecting a fuzzy linguistic truth degree which has a maximal index, i.e., the fuzzy linguistic truth degree is Hence, according to Eqs. (11) and (12), we obtain the following linguistic data summary:
lv t ðlah ðAs0 ÞÞ.
about half of employees are young is very true:
ð13Þ
(3) Similar (2), we also obtain the following linguistic data summary
most of employees have high salary is approximately true:
ð14Þ
3. Extracting a complex linguistic data summary based on either the Max operator or the LOWA operator Here, the so-called complex linguistic data summary has the form ‘‘Qy’s are S1 and and Sr is T.” Based on Eq. (4)–(6), it is easy to obtain ‘‘Q1y’s are S01 is T1”, , and ‘‘Qry’s are Sr is Tr”. Hence, for a complex linguistic data summary, the problem is how to combine {Q1, . . . , Qr} and {T1, . . . , Tr} to obtain Q and T, respectively. Table 1 Personnel database. VnY
y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
y11
y12
Age Salary .. .
25 1.8 .. .
48 2.0 .. .
31 2.8 .. .
35 3.0 .. .
28 2.8 .. .
51 3.0 .. .
37 2.3 .. .
43 2.5 .. .
34 3.5 .. .
27 2.9 .. .
53 3.0 .. .
45 3.1 .. .
2328
Z. Pei et al. / Information Sciences 179 (2009) 2325–2332 0
Method 1: Let summarizer srl0 2 Sr0 ðr0 ¼ 1; . . . ; rÞ, and h
Dsrr00 ¼ l1 0 ðVðyi ÞÞ ¼ fVðyi Þ j l r 0 ðVðyi ÞÞ P hr 0 g: s sr
ð15Þ
l0
l0
l0
Then Q can be selected as
lQ ðBÞ ¼ maxflQ 1 ðBÞ; lQ 2 ðBÞ; . . . ; lQ r ðBÞg; where B ¼ fyi j Vðyi Þ 2
Dhs11 g 0
\ \ fyi j Vðyi Þ 2
l
ð16Þ
Dhsrr0 g.
T can be selected as
l
lT ðlQ ðBÞÞ ¼ maxflT 1 ðlQ ðBÞÞ; lT 2 ðlQ ðBÞÞ; . . . ; lT r ðlQ ðBÞÞg:
ð17Þ
Method 2: Based on the LOWA operator [10], linguistic terms Q and T can be selected by
Q ¼ /ðQ 1 ; Q 2 ; . . . ; Q r Þ ¼ WBT ¼ C r fwk ; bk ; k ¼ 1; . . . ; rg ¼ w1 b1 ð1 w1 Þ C r1 fbh ; bh ; h ¼ 2; . . . ; rg; T
ð18Þ
0 ¼ /ðT 1 ; T 2 ; . . . ; T r Þ ¼ W B ¼ C fw0k ; bk ; k ¼ 1; . . . ; rg 0 0 0r1 0 0 0 ¼ w1 b1 ð1 w1 Þ C fbh ; bh ; h ¼ 2; . . . ; rg; 0 0T
0r
ð19Þ Pr
Pr
0 where W = [w1, . . . , wr] and W 0 ¼ ½w01 ; . . . ; w0r are weighting vectors such that wi ; w0i 2 ½0; 1; i¼1 wi ¼ 1 and i¼1 wi ¼ 1. P P 0 0 0 bh ¼ wh = rj¼2 wj , b0h ¼ w0h = rj¼2 w0j , B = {b1, b2, . . . , br} and B0 ¼ fb1 ; b2 ; . . . ; br g are vectors associated with {Q1, . . . , Qr} and {T1, . . . , Tr} such that
B ¼ rðfQ 1 ; . . . ; Q r gÞ ¼ fQ rð1Þ ; Q rð2Þ ; . . . ; Q rðrÞ g; B0 ¼ r0 ðfT 1 ; . . . ; T r gÞ ¼ fT r0 ð1Þ ; T r0 ð2Þ ; . . . ; T r0 ðrÞ g; where Qr(j) 6 Qr(i) and T r0 ðjÞ 6 T r0 ðiÞ ð8i 6 jÞ, r and r0 are permutations over {Q1, . . . , Qr} and {T1, . . . , Tr}, respectively. C2 and C0 2 are defined by
C 2 fki ; bi ; i ¼ 1; 2g ¼ k1 b1 ð1 k1 Þ b2 ¼ Q k 2 fQ 1 ; . . . ; Q r g; 0
0
0
C 02 fk0i ; bi ; i ¼ 1; 2g ¼ k01 b1 ð1 k01 Þ b2 ¼ T k0 2 fT 1 ; . . . ; T r g; 0
0
0
0
0
0
where b1 = Qj, b2 = Qi (j P i) and b1 ¼ T j0 ; b2 ¼ T i0 ðj P i Þ then k = min{m,i + round(k1(j i))} and k ¼ minfk; i þ 0 0 roundðk01 ðj i ÞÞg, where round is the usual round operation. In Eqs. (18) and (19), weighting vectors W and W0 can be computed by [30],
Q ðx; a; bÞ ¼
8 > ba
1
if a 6 x < b;
ð20Þ
where x, a, b 2 [0, 1]. Some examples of Q(r, a, b) are most, at least half and as many as possible, their parameters (a, b) are (0.3, 0.8), (0, 0.5), and (0.5, 1), respectively. By Q(r, a, b), weighting vectors W and W0 are
i i1 ; a; b Q ; a; b ; i ¼ 1; . . . ; r; r r i 0 0 i1 0 0 0 wi ¼ Q ; a ; b Q ; a ; b ; i ¼ 1; . . . ; r: r r wi ¼ Q
ð21Þ ð22Þ
Example 3. Continue Example 2, by Eqs. (13) and (14), combining {about half, most} and {approximately true, very true} are needed. Using Method 1, we obtain B ¼ As0 \ As00 ¼ fy3 ; y4 ; y5 ; y9 ; y10 g, so, ls ðBÞ ¼ 13 ; lah ðBÞ ¼ 0:5, lm(B) = 0, lat(lah(B)) = 1, lt(lah(B)) = lvt(lah(B)) = 0, and a complex linguistic data summary is
ðabout half of employees are young and have high salaryÞ is approximately true: Using Method 2, we select the linguistic quantifier ‘most’ and the weighting vectors W = W0 = (0.4,0.6), and 0 C 2 fki ; bi ; i ¼ 1; 2g ¼ 0:4 most3 0:6 about half 2 ¼ about half 2 ; C 02 fk0i ; bi ; i ¼ 1; 2g ¼ 0:4 very true3 0:6 approximately true1 ¼ true2 . Hence, the complex linguistic data summary is
ðabout half of employees are young and have high salaryÞ is true:
In Example 3, the conclusions of Method 1 and Method 2 are almost the same. In Method 1, by the logical relationship among fuzzy linguistic quantifiers, the fuzzy linguistic truth degrees and summarizer, B of Eq. (16) can be obtained. Since the truth degree of the complex linguistic data summary is maximal, the Max operator is used. In Method 2, the round()
Z. Pei et al. / Information Sciences 179 (2009) 2325–2332
2329
operator is used, and the round operator in combination of linguistic indexes is very sensitive to the values around 0.5 (round(0.499) = 0 and round(0.501) = 1), the difference is occurred between Method 1 and Method 2. From the simple computing point of view, Method 2 is better to aggregate linguistic terms. 4. Optimizing a complex linguistic data summary based on GAs Genetic algorithms (GAs) are search algorithms that use operations found in natural genetics to guide the trek through a search space [4,15]. A number of papers have been devoted to the automatic generation of the knowledge base of a fuzzy rule-based system (FRBS) using GAs [1,2,5–9,16]. In this section, we optimize the number and membership functions of linguistic terms and obtain a complex linguistic data summary with a higher truth degree based on GAs. 4.1. Optimizing the number and membership functions of linguistic terms Personnel database is an information system, in which employee (in Y) is an object and a quality (in V) is an attribute. In a real-world personnel database, generally, V(yi) (yi 2 Y) is a numerical one. In practice, a numeral value is hard to obtain for a good decision. Hence, a linguistic data summary is needed, the process is equal to classifying on Y, especial fuzzy classifying. Naturally, selecting the number and membership functions of linguistic terms is a problem. Here, suppose there exist L qualities (attributes) in V, and each domain of attribute is denoted by Dl R+,l = 1, . . . ,L, then each object (employee) yi 2 Y is understood as a point on space D1 DL, i.e., yi = (di1, . . . , diL), dil 2 Dl. Suppose that n0 (n0 < n) patterns yi = (di1, . . . , diL), i = 1, . . . , n0 are given as training patterns from M classes: class 1 (C1), . . ., class M (CM). The problem is to generate the number and membership functions of linguistic terms on Dl that divide the pattern space into M disjoint decision areas. Assuming each axis Dl is partitioned into Kl fuzzy subsets fAlkl j kl ¼ 1; . . . ; K l g, then D1 DL is divided into K1 KL fuzzy subspaces, and each fuzzy subspace can be expressed by an If–Then rule:
Rk1 kL : If di1 is A1k1 and and diL is ALkL ; then yi belonges to class C m with CF ¼ CF k1 kL :
ð23Þ
where Rk1 kL is a label of an If–Then rule. Alkl ðl ¼ 1; . . . ; LÞ are fuzzy subsets on Dl, Cm (m = 1, . . . , M) is the consequent, and CF k1 kL is the grade of certainty of the If–Then rule and determined by the following procedure, 1. For each class Cm and rule Rk1 kL , we have
aCm ¼
X
A1k1 ðdi1 Þ ALkL ðdiL Þ:
ð24Þ
yi 2C m
2. Selecting
CF k1 kL ¼ maxfaC1 ; ; aCM g:
ð25Þ
Remark 4. If CF k1 kL ¼ 0, the rule Rk1 kL is useless to classify yi, and the consequent of the rule Rk1 kL is modified by Cm = ;. If two or more aC m are equal to CF k1 kL , then the rule is not good to classify yi, and also Cm = ;. 0
0
Based on the rule set R, a new pattern y0 ¼ ðdi1 ; . . . ; diL Þ is classified by 1. Calculate ck1 kL for each rule Rk1 kL ,
ck1 kL ¼ A1k1 ðd0i1 Þ ALkL ðd0iL Þ CF k1 kL ;
ð26Þ
2. Find the class C m0 such that
cCm0 ¼ maxfck1 kL j Rk1 kL 2 Rg:
ð27Þ
If cC m0 ¼ 0 or C m0 ¼ ; of the rule, then y0 is left as an unclassified pattern, else assign y0 to the class C m0 determined by Eq. (27). The main components of optimizing the number and membership functions of linguistic terms based on GAs are described as follows [6,7]: (1) Encoding the solution: The two components of the solution to be encoded are the number of linguistic terms for variable (granularity) and the membership functions that define their semantics. (1) Number of labels (S1), in this paper, there are L variables (qualities), the number of labels per variable is stored into an integer array of length L. In this contribution, the possible values considered are the set {3, . . . , 9}. (2) Membership functions (S2), in this paper, we deal with triangular functions, a real number array of L 9 3 positions is used to store the membership functions.If sl is the granularity of variable l; sl 2 f3; . . . ; 9g; P 1lj ; P 2lj ; P 3lj are the definition points of the label j of the variable l, and S2l is the information about the fuzzy partition of variable l in S2, then a graphical representation of the chromosome is shown
2330
Z. Pei et al. / Information Sciences 179 (2009) 2325–2332
as following: S1 = (s1, s2, . . . , sL), S2l ¼ ðP 1l1 ; P2l1 ; P 3l1 ; ; P1lsl ; P2lsl ; P3lsl Þ; S2 ¼ ðS21 ; S22 ; ; S2L Þ; S ¼ S1 S2 . Uniform fuzzy partitions are denoted by ðV 1lj ; V 2lj ; V 3lj Þ for each variable. For general fuzzy partition, a variation interval is defined for each h i V 2 V 1 V 2 V 1 one of the membership function definition points [6], i:e:; P1lj 2 ½L1lj ; R1lj ¼ V 1lj lj 2 lj ; V 1lj þ lj 2 lj ; P 2lj 2 L2lj ; R2lj ¼ h i V 2 V 1 V 3 V 2 V 3 V 2 V 3 V 2 V 2lj lj 2 lj ; V 2lj þ lj 2 lj ; P3lj 2 L3lj ; R3lj ¼ V 3lj lj 2 lj ; V 3lj þ lj 2 lj . (2) Initial gene pool: The initial population is composed of four groups [6]: (1) Each chromosome will have the same number of labels in all its variables and the membership functions are uniformly distributed across the domain of variable. (2) Each chromosome can have a different granularity per variable (different values in S1) and the membership functions are uniformly distributed as in the first part. (3) Each chromosome will have the same number of labels in all its variables. Then a uniform fuzzy partition is built for each variable as in the first group and the variation intervals of all the definition points are calculated. Finally, a value for all the definition points is randomly chosen from the correspondent variation interval. (4) Each chromosome can have different number of labels per variable as in the second group and the membership functions are calculated in the same way as in the third group, a random value is in the variation interval. (3) Evaluating the chromosome: Maximizing the number of correctly classified pattern and minimizing the number of If– Then rule are the goals, so, we have
Minimize : f ðsÞ ¼ x1 DCPðsÞ þ x2 j s j;
ð28Þ
where s is a chromosome, DCP(s) is the number of unclassified patterns by s, jsj is the number of If–Then rules in s. The weights in Eq. (28) should be specified as 0 < x2 x1 [17]. f(s) is treated as the fitness function in GAs. (4) Genetic operators: Selection, crossover and mutation are as follows: Selection: Let the current population be W. The selection probability P(s) of chromosome s is
PðsÞ ¼ ðfmax ðWÞ f ðsÞÞ=
X
ðfmax ðWÞ f ðs0 ÞÞ;
ð29Þ
s0 2W
where fmax(W) = max{f(s)js 2 W}. Crossover: Two different crossover operators are considered [15]. (1) Crossover when both parents have the same granularity level per variable, in this case, the crossover operator in S2 and maintaining the parent S1 values in the t 1 w 3 w offspring. If ðSv2 Þt ¼ ððP 111 Þv ; . . . ; ðP3LsL Þv Þ and ðSw 2 Þ ¼ ððP 11 Þ ; . . . ; ðP LsL Þ Þ are to be crossed, the following four offspring are generated:
ðSv2 w Þtþ1 ¼ ððP111 Þv w ; . . . ; ðP3LsL Þv w Þ; ðP ilsl Þv w ¼ dðP ilsl Þv þ ð1 dÞðPilsl Þw ; 1 ðSv2 w Þtþ1 ¼ ððP111 Þv w ; . . . ; ðP3LsL Þv w Þ; ðP ilsl Þv w ¼ ð1 dÞðP ilsl Þv þ dðPilsl Þw ; 2 ¼ ððP111 Þv w ; . . . ; ðP3LsL Þv w Þ; ðP ilsl Þv w ¼ maxfðPilsl Þv ; ðPilsl Þw g; ðSv2 w Þtþ1 3 ¼ ððP111 Þv w ; . . . ; ðP3LsL Þv w Þ; ðP ilsl Þv w ¼ minfðPilsl Þv ; ðPilsl Þw g; ðSv2 w Þtþ1 4 where i = 1, 2, 3. This operator uses a parameter that is either a constant or a variable whose value depends on the age of the population [15].(2) Crossover when the parents encode different granularity levels, in this case, the crossover operator in S1 and S2. Let Sv = ((s1)v, . . . , (sl)v, (sl+1)v, . . . , (sL)v, (S21)v, . . . , (S2l)v, (S2(l+1))v, . . . , (S2L)v) and Sw = ((s1)w, . . . , (sl)w, (sl+1)w, . . . , (sL)w, (S21)w, . . . , (S2l)w, (S2(l+1))w, . . . , (S2L)w) be the individuals to be crossed at point l, the two resulting offspring be Sv1 w ¼ ððs1 Þv ; . . . ; ðsL Þw ; ðS21 Þv ; . . . ; ðS2l Þv ; ðS2ðlþ1Þ Þw ; . . . ; ðS2L Þw Þ; Sv2 w ¼ ððs1 Þw ; . . . ; ðsL Þv ; ðS21 Þw ; . . . ; ðS2l Þw ; ðS2ðlþ1Þ Þv ; . . . ; ðS2L Þv Þ. Mutation: Two different operators are used. (1) Mutation on S1, in this case, once a new value s0l 2 f3; . . . ; 9g at the point l of S1 is selected, a uniform fuzzy partition for this variable is stored in its corresponding zone of S2. (2) Mutation on S2: Let ðSv2 Þt ¼ ððP111 Þv ; . . . ; ðPilsl Þv ; . . . ; ðP3LsL Þv Þ and the element ðPilsl Þv be selected for this mutation (the domain of ðP ilsl Þv is ½ðPilsl Þvl ; ðP ilsl Þvr ), the result is a vector ðSv2 Þtþ1 ¼ ððP 111 Þv ; . . . ; ððP ilsl Þv Þ0 ; . . . ; ðP 3LsL Þv Þ, and
ððPilsl Þv Þ0
(
¼
ðPilsl Þv þ Dðt; ðPilsl Þvr ðPilsl Þv Þ if e ¼ 0; Pilsl Þv þ Dðt; ðPilsl Þv ðPilsl Þvl Þ
if e ¼ 1:
with t being the current generation, e being a random number in {0, 1}, and the function D(t,y) is t b Dðt; yÞ ¼ yð1 r ð1T Þ Þ, with r being a random number in the interval [0, 1], T the maximum number of generations and b a parameter chosen by users [6]. 4.2. Obtaining a complex linguistic data summary with a higher truth degree To make the complex linguistic data summary with a higher truth degree, parts of {Q1, . . . , Qr} and {T1, . . . , Tr} can be selected to obtain Q and T, respectively. In this subsection, GAs are used to select parts of {Q1, . . . , Qr} and {T1, . . . , Tr}. To avoid the loss of information, the 2-tuple linguistic representation model is used to represent Q r0 and T r0 (r0 = 1, . . . , r).
2331
Z. Pei et al. / Information Sciences 179 (2009) 2325–2332
Definition 5 [14]. Let si 2 S be a linguistic term. Then its equivalent 2-tuple linguistic representation is obtained by means of the function h as:
h : S ! ðS ½0:5; 0:5ÞÞ;
hðsi Þ ¼ ðsi ; 0Þ:
Definition 6 [14]. Let {s0, s1, . . . , sg} be a linguistic term set and b 2 [0, g] a value supporting the result of a symbolic aggregation operator. Then the 2-tuple that expresses the equivalent information to b is obtained with the following function:
K : ½0; g ! ðS ½0:5; 0:5ÞÞ; i ¼ roundðbÞ; si ; KðbÞ ¼ ðsi ; aÞ; with a ¼ b i; a 2 ½0:5; 0:5Þ: where round() is the usual round operation, si has the closest index label to b and a is the value of symbolic translation. Definition 7 [12]. Let A = {(r1, a1), . . . , (rm, am)} be a set of 2-tuples to be aggregated. Then the extended LOWA operator, /e, is defined as T
e
m
/ ððr1 ; a1 Þ; . . . ; ðr m ; am ÞÞ ¼ WB ¼ EC fwi ; ðrrðiÞ ; arðiÞ Þ;
i ¼ 1; . . . ; mg ¼ K
m X
! 1
wi K ððr rðiÞ ; arðiÞ ÞÞ
¼K
i¼1
m X
! wi brðiÞ ;
i¼1
ð30Þ where br(i) = K1((rr(i), ar(i))), r is a permutation over {1, . . . , m} such that rr(j) 6 rr(i) ("i 6 j). We just discuss how to obtain a part of {(T1, 0), . . . , (Tr, 0)} based on GAs for aggregating a complex linguistic data summary with a higher truth degree. (1) Encoding the solution: The length of each chromosome is r, and
S ¼ t1 t2 tr ;
and 8r 0 2 f1; . . . ; rg; t r0 2 f0; 1g;
ð31Þ
(2) Initial gene pool: By Eq. (31), the initial population is randomly selected in 2r solutions as usual [7]. (3) Evaluating the chromosome: For a solution s = t1, t2, . . . , tr and fixed a linguistic quantifier Q(r, a, b) which is defined by Eq. (20), using the operator /e which is defined by Eq. (30), we obtain
0 ðT j ; aj Þ ¼ /e ðsÞ ¼ EC fwi ; ðT rðiÞ ; arðiÞ Þ j i ¼ 1; . . . ; r; and t rðiÞ ¼ 1g ¼ K@ m
m X
1 wi K1 ððrrðiÞ ; arðiÞ ÞÞA;
ð32Þ
i¼1;trðiÞ ¼1
where wi is decided by Eq. (20), in this paper, suppose Q(r, a, b) is fixed, Tj 2 {T1, . . . , Tr} and ar(i) = 0. According to Eq. (32), the fitness function is obtained as following:
0
1
m X
Maximize : f ðsÞ ¼ K@
wi K1 ððr rðiÞ ; 0ÞÞA ¼ ðT j ; aj Þ;
ð33Þ
i¼1;t rðiÞ ¼1
(4) Genetic operators: Selection is defined as following: Let the current population be W. Then the selection probability P(s) of chromosome s is
, PðsÞ ¼ ðf ðsÞ fmin ðWÞÞ
X
ðf ðs0 Þ fmin ðWÞÞ;
ð34Þ
s0 2 W 0
where fmin(W) = min{f(s)js 2 W}, assume the index of fmin(W) is j0 , then f ðsÞ fmin ðWÞ ¼ j þ aj j aj0 . The generation of the offspring population is put into effect by using the classical binary multi-point crossover and uniform mutation operators [9,16].Obviously, based on these genetic operators and fitness function Eq. (33), the optimal solution can be obtained, the optimal solution makes a complex linguistic data summary with a higher truth degree. Correspondingly, using /e, aggregating {(Q1, 0), . . . , (Qr, 0)} can be obtained.
5. Conclusions From the formalization point of view, a linguistic data summary presented by Yager is equal to a fuzzy statement with a fuzzy linguistic quantifier. In this paper, based on the structure and valuation of a fuzzy statement, the method
2332
Z. Pei et al. / Information Sciences 179 (2009) 2325–2332
to extract Q, S and T of a linguistic data summary is discussed. For a complex linguistic data summary, the LOWA operator is used to obtain Q, S and T. In practice, optimizing a complex linguistic data summary is needed. In this paper, a formal optimizing method based on GAs is given, in which linguistic terms are represented by the 2-tuple linguistic representation model. Acknowledgements The authors thank professor Francisco Herrera for improving this paper significantly. This work is supported by the excellent young foundation of Sichuan Province (Grant No. 06ZQ026-037) and SZD, the National Natural Science Foundation of China (Grant No. 60875034), the Special Research Funding to Doctoral Subject of Higher Education Institutes in China (Grant No. 20060613007). References [1] R. Alcala, J. Alcala-Fdez, F. Herrera, J. Otero, Genetic learning of accurate and compact fuzzy rule based systems based on the 2-tuples linguistic representation, International Journal of Approximate Reasoning 44 (2007) 45–64. [2] J. Casillas, O. Cordón, M.J. Del Jesus, F. Herrera, Genetic feature selection in a fuzzy rule-based classification system learning process for highdimensional problems, Information Sciences 136 (2001) 135–157. [3] O. Cordón, F. Herrera, I. Zwir, Linguistic modeling by hierarchical systems of linguistic rules, IEEE Transactions on Fuzzy Systems 10 (1) (2002) 2–19. [4] O. Cordón, F. Gomide, F. Herrera, F. Hoffmann, L. Magdalena, Ten years of genetic fuzzy systems: current framework and new trends, Fuzzy Sets and Systems 141 (2004) 5–31. [5] O. Cordón, F. Herrera, L. Magdalena, P. Villar, A genetic learning process for the scaling factors, granularity and contexts of the fuzzy rule-based system data base, Information Sciences 136 (2001) 85–107. [6] O. Cordón, F. Herrera, P. Villar, Generating the knowledge base of a fuzzy rule-based system by the genetic learning of the data base, IEEE Transactions on Fuzzy Systems 9 (4) (2001) 667–674. [7] O. Cordón, F. Herrera, A proposal for improving the accuracy of linguistic modeling, linguistic rules, IEEE Transactions on Fuzzy Systems 8 (3) (2000) 335–344. [8] O. Cordón, F. Herrera, Hybridizing genetic algorithms with sharing scheme and evolution strategies for designing approximate fuzzy rule-based systems, Fuzzy Sets and Systems 118 (2001) 235–255. [9] O. Cordón, F. Herrera, A three-stage evolutionary process for learning descriptive and approximative fuzzy logic controller knowledge bases from examples, International Journal of Approximate Reasoning 17 (4) (1997) 369–407. [10] F. Herrera, E. Herrera-Viedma, Aggregation operators for linguistic weighted information, IEEE Transactions on System, Man, and Cybernetics. Part A: Systems Humans 27 (1997) 646–656. [11] F. Herrera, E. Lopez, M.A. Rodriguez, A linguistic decision model for promotion mix management solved with genetic algorithms, Fuzzy Sets and Systems 131 (2002) 47–61. [12] F. Herrera, L. Martínez, A 2-tuple fuzzy linguistic representation model for computing with words, IEEE Transactions on Fuzzy Systems 8 (6) (2000) 746–752. [13] F. Herrera, L. Martínez, A model based on linguistic 2- tuples for dealing with multigranularity hierarchical linguistic contexts in Multiexpert DecisionMaking, IEEE Transactions on Systems, Man and Cybernetics. Part B: Cybernetics 31 (2) (2001) 227–234. [14] F. Herrera, L. Martínez, The 2-tuple linguistic computational model: advantages of its linguistic description, accuracy and consistency, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 9 (2001) 33–48. [15] F. Herrera, M. Lozano, J.L. Verdegay, Fuzzy connectives based crossover operators to model genetic algorithms population diversity, Fuzzy Sets and Systems 92 (1997) 21–30. [16] F. Herrera, M. Lozano, J.L. Verdegay, A learning process for fuzzy control rules using genetic algorithms, Fuzzy Sets and Systems 100 (1998) 143–158. [17] H. Ishibuchi, K. Nozaki, N. Yamamoto, H. Tanaka, Selecting fuzzy if–then rules for classification problems using genetic algorithms, IEEE Transactions on Fuzzy Systems 3 (3) (1995) 260–270. _ [18] J. Kacprzyk, S. Zadrozny, Computing with words in intelligent database querying: standalone and Internet-based applications, Information Sciences 134 (2001) 71–109. _ [19] J. Kacprzyk, S. Zadrozny, Linguistic database summaries and their protoforms: towards natural language based knowledge discovery tools, Information Sciences 173 (2005) 281–304. [20] J. Kacprzyk, R.R. Yager, Linguistic summaries of data using fuzzy logic, International Journal of General Systems 30 (2001) 133–154. [21] Y. Liu, E.E. Kerre, An overview of fuzzy quantifiers: (I) Interpretations, Fuzzy Sets and Systems 95 (1998) 1–21. [22] J.M. Mendel, Computing with words and its relationships with fuzzistics, Information Sciences 177 (2007) 988–1006. [23] Z. Pei, G. Resconi, A.J. Van Der Wal, K.Y. Qin, Y. Xu, Interpreting and extracting fuzzy decision rules from fuzzy information system and their inference, Information Sciences 176 (2006) 1869–1897. [24] Z. Pei, J. Du, L. Yi, Y. Xu, Obtaining a complex linguistic data summaries from database based on a new linguistic aggregation operator, in: J. Cabestany, A. Prieto, D.F. Sandoval (Eds.), IWANN 2005, LNCS 3512, Springer-Verlag, Berlin, Heidelberg, New York, 2005, pp. 771–778. [25] Z. Pei, The algebraic properties of linguistic value ‘‘Truth” and its reasoning, in: Lecture Notes in Artificial Intelligence 4529 (IFSA2007), Springer, 2007, pp. 436–444. [26] G. Raschia, N. Mouaddib, SAINTETIQ: a fuzzy set-based approach to database summarization, Fuzzy Sets and Systems 129 (2002) 137–162. [27] E. Trillas, On the use of words and fuzzy sets, Information Sciences 176 (2006) 1463–1487. [28] I.B. Turksen, Meta-linguistic axioms as a foundation for computing with words, Information Sciences 177 (2007) 332–359. [29] R.R. Yager, A new approach to the summarization of data, Information Sciences 28 (1982) 69–86. [30] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Transactions on Systems, Man and Cybernetics 18 (1988) 183–190. [31] M.S. Ying, Linguistic quantifiers modeled by Sugeno integrals, Artificial Intelligence 170 (2006) 581–606. [32] L.A. Zadeh, Toward a generalized theory of uncertainty (GTU) – an outline, Information Sciences 172 (2005) 1–40. [33] L.A. Zadeh, Is there a need for fuzzy logic?, Information Sciences 178 (2008) 2751–2779 [34] L. Zhang, Z. Pei, Extracting fuzzy linguistic summaries based on including degree theory and FCA, in: 12th International Fuzzy Systems Association World Congress (IFSA 2007), LNAI 4529, Mexico, 2007, pp. 273–283.
