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GENERATING
FUZZY RULES
FROM RELATIONAL
DATABASE SYSTEMS
FOR ESTIMATING
NULL VALUES
SHYI-MING CHEN MING-SHIOW YEH
Published online: 29 Oct 2010.
To cite this article: SHYI-MING CHEN MING-SHIOW YEH (1997)
GENERATING FUZZY RULES FROM RELATIONAL DATABASE SYSTEMS FOR
ESTIMATING NULL VALUES, Cybernetics and Systems: An International
Journal, 28:8, 695-723
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GENERATING FUZZY RULES FROM
RELATIONAL DATABASE SYSTEMS
FOR ESTIMATING NULL VALUES
SHYI-MING CHEN
MING-SHIOW YEH
Department of Computer and Information Science National Chiao Tung University, Hsinchu, Taiwan, Republic of China
This pape r pre se nts a ne w algorithm for constructing fuzzy de cision tre e s from re lational database syste ms and gene rating fuzzy rule s from the constructed fuzzy de cision tre es. W e also pre se nt a me thod for de aling with the com ple te ne ss of the constructed fuzzy de cision tre e s. Base d on the gene rate d fuzzy rule s, we also pre se nt a m ethod for e stim ating null value s in re lational database syste ms. The propose d m ethods provide a use ful way to e stimate null value s in re lational database syste m s.
s .
Kande l 1986 pointed out that during the past decade the re search
s
fie lds of applie d computer science e .g., information processing, artifi-.
cial inte llige nce , knowle dge proce ssing in e xpert systems have estab-lished the ne ed for formulation of mode ls of impre cise information systems that would simulate human approxim ate reasoning. Since Z ade h s1965 propose d the fuzzy se t the ory, the the ory has bee n widely used. for repre senting and re asoning with impre cise and unce rtain
informa-This work was supporte d in part by the National Science Council, Re public of China, unde r grant NSC 87-2213-E-009-011.
Addre ss corre sponde nce to Profe ssor Shyi-Ming Che n, Ph.D., De partme nt of Com-pute r and Inform ation Science , National Chiao Tung Unive rsity, H sinchu, Taiwan, R.O .C.
Cybernetics and Systems: An International Journal, 28:695] 723, 1997
CopyrightQ1997 Taylor & Francis
0196-97 22 ¤97 $12.00 + .00 695
tion. Roughly spe aking, a fuzzy se t is a se t with fuzzy boundarie s. Le t A
v 4
be a fuzzy set of the unive rse of discourse V,Vs v1,v2, . . . ,vn , and let
w x s .
m A be the me mbe rship function of A, m A:Vª 0, 1 . The n m A vi
indicate s the de gre e of mem bership of vi in A, where 1F iF n. Traditionally, knowledge base s of rule-based system s are con-structe d by the proce ss of knowledge acquisition. Som e re searchers have concentrated on ge nerating rule s by le arning from examples sJe ng & Liang, 1993; Sudkamp & Hamme ll, 1994; W ang & Me nde l, 1992;
. s .
Ye h & Chen, 1995 . W ang and Mende l 1992 use a fuzzy associative me mory to construct a fuzzy rule base from num erical and linguistic information and apply it to the truck backer-upper control. Sudkamp
s .
and Hamme ll 1994 prese nte d a me thod for learning fuzzy rule s and use d the ge nerate d fuzzy rules to determ ine the mapping from input
s .
space to output space . Hart 1985 applied the induction of de cision tree s to knowledge acquisition for e xpert systems, where e xpe rts supply various e xample s to construct de cision tree s and the re sulting de cision tree s are used to gene rate rules.
s .
Safavian and Landgrebe 1991 have m ade a surve y of some me th-ods for constructing decision tre e s from colle ctions of example s. Yuan
s .
and Shaw 1995 pointe d out that although the de cision tree s ge nerate d by the se m ethods are useful in building knowle dge -based expe rt sys-tem s, the y often cannot properly e xpress and handle the vague ne ss and ambiguity associated with human thinking and perce ption. To ove rcome
s .
these drawbacks, Q uinlan 1987 sugge ste d a probabilistic me thod for constructing de cision tree s as probabilistic classifie rs, where
inaccura-s .
cie s of attribute values are treate d as noise . Yuan and Shaw 1995
s .
pointed out that the lim itation of Q uinlan’s work 1987 is that the types of unce rtaintie s arising in classification proble ms are not ne ce ssarily probabilistic, appe aring as random ne ss or noise. Thus, Yuan and Shaw s1995. pre sented a fuzzy decision tre e induction me thod, and they pointed out that fuzzy decision tree s re pre se nt classification knowle dge more natually and are more robust in tolerating im pre cise , conflict, and missing information.
s .
In this paper, we pre sent a fuzzy conce pt le arning system FCLS algorithm to construct fuzzy de cision tree s from re lational database systems and to ge nerate fuzzy rules from the constructe d fuzzy de cision tree s. Furthe rmore, we prese nt a method for dealing with the comple te -ne ss of the constructe d fuzzy de cision tre e s. Based on the ge -nerate d fuzzy rule s, we also prese nt a method for estimating null value s in
GENERATING FUZZY RULES 697 re lational database systems. The proposed methods provide a useful way to e stimate null value s in relational database systems.
BASIC CONCEPTS OF GENERATING FUZZY RULES
FROM RELATIONAL DATABASE SYSTEMS
The relational data mode l is the most popular data mode l of database systems used in comme rcial applications be cause it can be very e asily understood and imple me nte d. In the following, we will introduce the conce pts of gene rating fuzzy rules from relational database systems. An e xample of a re lation in a relational database system is shown in Figure 1, whe re A,B, . . . , and C are attribute s and the ir value s are in the
w x
inte rval 0, 10 . The relationship betwe en the attributes A,B, and C can
s .
be define d as Cs A*B
r
10. That me ans that the value s of attribute C are dete rmine d by the values of attributes A and B.In orde r to ge nerate fuzzy rule s from a re lational database system,
s .
we must use the concepts of fuzzy se ts Z adeh, 1965 and fuzzification sW ang & Me nde l, 1992 . The main purpose of the fuzzification process. is to transfe r the input crisp data into fuzzy data and incorporate the imprecision. If the attributes in a re lation of a re lational database
s .
system are considere d as linguistic variable s Z ade h, 1975 , the n we
s .
must partition the input dom ains i.e ., the domains of attributes into
s .
se veral fuzzy regions linguistic te rms in advance , whe re a linguistic variable is a variable whose value s are linguistic terms rather than nume rical value s. This m eans to de fine the corre sponding me mbership
s .
functions of the linguistic te rms fuzzy regions for e ach linguistic
s .
variable attribute . Le t X be a linguistic variable in a dom ainV and let
X1,X2, . . . ,Xn be the ir corre sponding linguistic terms, whe re the me m-be rship function curve s of the linguistic te rms X1,X2, . . . ,Xn are shown in Figure 2. From Figure 2, we can see that for e ve ry vg V, we can ge t one or two m embe rship grades corre sponding to differe nt
s . linguistic te rms. For the e xample in Figure 2, we can get m Xiv and
s . s . s . s . w x s . w x
m Xiq 1v , where m Xi v q m Xiq 1v s 1, m Xiv g 0, 1 , m Xiq 1v g 0, 1 , and 1F iF ny 1. W e also ensure that this partition must satisfy the
s .
« -comple tion Kandel 1986 . In this pape r, we le t « equal 0.5. This s .
me ans that the re exists at le ast one Xi such that m Xiv G 0.5 for eve ry
vg V. In the proce ss of constructing fuzzy de cision tree s, we transform
v s .4
the crisp value v into the singleton fuzzy se t Xi
r
m Xiv , where Xi is as . s . s .
linguistic te rm and m Xiv G 0.5. If m Xiv s m Xiq 1v s 0.5, we
F igu r e 1 . A re lation in a re lational database syste m.
v s .4
form v into Xi
r
m Xiv . For e xample , let us consider the re lation shown in Figure 1. Assume that we would like to de compose the value ss .
of the attribute A into three fuzzy re gions linguistic te rms : Low, Medium , and High, whe re the mem bership function curves of Low, Medium , High are shown in Figure 3, and V is the domain of the
v 4
attribute A. In this case , Low, Medium , H igh are calle d the fuzzy domain of the attribute A. Through the fuzzification proce ss, the re lation shown in Figure 1 became the fuzzy relation shown in Figure 4. If attribute X is an unfuzzifiable attribute , the n we le t e ach domain value ofX be a singleton fuzzy set. For e xample , if the education de gre e of an employee is B achelor, the n we let the mem bership value of B ache lor be e qual to 1.0; that is, the dom ain value Bachelor of the
v 4
attribute De gree is fuzzifie d into Bachelor
r
1.0 .In the following, we introduce the concepts of deriving fuzzy de ci-sion tree s from fuzzy re lations. In a fuzzy decici-sion tre e , a nonte rminal node is also calle d a de cision node. The re are two kinds of terminal
s .
nodes in a fuzzy de cision tree , i.e ., certainty factor CF node s, denoted
s .
by ` , and hypothe tical ce rtainty factor HCF nodes, denote d by .
F igu r e 2 . Fuzzy de compositi on of domain V.
GENERATING FUZZY RULES 699
F igu r e 3 . Me mbe rship function curve s of linguistic terms ``Low,’ ’ ``Me dium,’ ’ and ``High.’ ’
The certainty factor nodes and the hypothe tical ce rtainty factor node s are associate d with values be twe e n ze ro and one. The path from the
s
root node to each terminal node i.e ., certainty factor node or hypothe t-.
ical certainty factor node forms a fuzzy rule . Figure 5 shows an e xample of the fuzzy de cision tre e, whe re X,Y,Z are attributes in the
s .
re lational database and Xi,Yj,Zk 1F iF n, 1F jF m, 1F kF p are linguistic terms, re spe ctively. From Figure 5, we can se e that the value ofZ is de te rmine d by the value s ofX and Y. Note that ifX or YorZ is a linguistic variable, then Xi or Yj or Zk would be linguistic te rms, re spe ctively, whe re 1F iF n, 1F jF m, 1F kF p. In Figure 5, the certainty factor node CF indicates that the re are some tuples in thei
re lation shown in Figure 6 that satisfy the classification in a de gre e
w x
de noted by CF , where CFi ig 0, 1 . A hypothe tical certainty factor node HCF exists because the re are no tuples that satisfy the classificationx
and that are ge ne rate d by tre e growning. Consider the path
X1 Y1 Z2
Xª Yª Zª CF in the fuzzy de cision tree shown in Figure 5. Thisj
path indicate s that there is a fuzzy rule
s . s .
IFX is X1 andYis Y1 TH E NZis Z2 CFs CFj 1
F ig u r e 4. Fuzzification of the re lation shown in Figure 1.
in the fuzzy rule base. A null path is a path whose te rm inal node is a hypothetical certainty factor node . For example, in Figure 5, the path
X1 Y2 Z2
Xª Yª Zª HCF is a null path. A nonnull path is a path whosex
term inal node is a ce rtainty factor node . For example, in Figure 5, the
X1 Y1 Z1
pathXª Yª Zª CF is a nonnull path. Figure 7 shows an example ofi
a subtre e of a fuzzy de cision tre e constructe d by the fuzzy re lation shown in Figure 4.
In the following, we introduce the concepts of gene rating fuzzy rule s from the ge nerated fuzzy de cision tre e . Gene rally speaking, the form of a fuzzy rule is as follows:
s . s .
IFX is Xi ANDYis Yj THE NZ is Zk CFs c 2
whe re
1. X, Y, Z are linguistic variable s and Xi, Yj, Zk are linguistic te rms re prese nte d by fuzzy se ts.
s .
2. c is the ce rtainty factor CF value of the fuzzy rule indicating the
w x
de gre e of belie f of the rule, whe re cg 0, 1 .
Be cause in a fuzzy decision tree , the path from the root node to
s
e ach te rm inal node certainty factor node or hypothetical certainty .
factor node forms a fuzzy rule , after constructing a fuzzy de cision tre e , we can ge nerate fuzzy rules from the constructe d fuzzy decision tre e . Thus, if we have a fuzzy de cision tre e as shown in Figure 7, we can gene rate the following fuzzy rule s:
. . .
s .
IFAis Me dium andBis High THENC is High CFs 0 .50
s .
IFAis Me dium andBis Medium THENC is Medium CFs 0 .65
s .3
s .
IFAis Me dium andBis Medium THENC is Low CFs 0 .63
s .
IFAis Me dium andBis Low THE NCis Low CFs 0 .69 ,
. . .
whe re A and B are calle d ante cede nt attribute s, and C is called a conse que nt attribute.
F ig u re 5 . A fu zz y d e ci si o n tr e e . 701
F igu r e 6 . A fuzzy re lation.
A FUZZY CONCEPT LEARNING SYSTEM ALGORITM
s .
In the following, we pre sent a fuzzy conce pt le arning system FCLS
s .
algorithm based on H unt e t al. 1966 for constructing a fuzzy de cision tree from a re lational datasbase system and gene rate fuzzy rule s from the constructe d fuzzy de cision tre e.
F igu r e 7. A subtre e of a fuzzy de cision tre e.
GENERATING FUZZY RULES 703
s v 4.
Le t S be a se t of atrributes i.e ., Ss X,Y, . . . ,W that de te rmine
s
attribute Z i.e., the se t Scontains a set of ante ce dent attribute s, and Z .
is a conse que nt attribute . Se le cting differe nt e le me nts in the se t Sas a root node may re sult in constructing differe nt kinds of fuzzy de cision
s .
tree s. In the ID3 le arning algorithm, Q uinlan 1979 m akes use of the e ntropy function of information theory to choose the fe ature that le ads to the gre ate st re duction in the estimated e ntropy of information of the training instance s. In this paper, we use the conce pt of fuzziness of
s .
attribute FA to select an attribute in Sas a decision node that has the smallest FA, such that the numbe r of node s in the gene rated fuzzy de cision tre e can be reduce d.
Definition 1: Le t S be a set of attribute s that de te rmine attribute
v 4 s .
Z,Ss X,Y, . . . ,W , and let t Xj de note the value of the attribute X of
s .
the jth training instance i.e., the jth tuple of a relation in a re lational s . database; the n the fuzziness of the attribute X, denoted by FA X , is de fined by c s . 1y m
s
t X.
p
t
Xi j/
js 1 s . s . FA X s 4 cwhe re c is the numbe r of training instance s, Xi is any linguistic te rm of s s ..
the attribute X, and m Xit Xj indicates the de gre e of m embe rship that the value of the attribute X of the jth training instance be longs to the linguistic term Xi.
Example 1: A ssume that we have a relation that has only four tuples as shown in Figure 1, and assume that Figure 4 is the result of the s . fuzzification of Figure 1, then the fuzzine ss of the attribute A, FA A, can be evaluate d as follows:
s . ws . s . s . s .x
FA A s 1y 0 .5 q 1 y 0 .5 q 1 y 0 .75 q 1 y 1 .0
r
4s .
s 0 .31 5
Definition 2: E very certainty factor node in the path of a fuzzy de cision s .
tree is associated with a certainty factor CF value . The ce rtainty factor
value CF is de fined by
v s . s . s . 4 s .
CFs m in Avg F1 , Avg F2 , Avg F3 6
s . s . s .
whe re Avg F1 , Avg F2 , Avg F3 are the average value s of the linguistic term s F1, F2, and F3, respe ctively, and F1, F2, and F3 are on a path
F1 F2 F3
D1ª D2ª D3ª C F in the tre e, and
s s . m
s
t D.
p
Fi j i js 1 s . s . Avg Fi s 7 s s .whe re t Dj i re pre sents the value of the attribute Di of the jth tuple of
s
a re lation, s is the the numbe r of training instance s i.e ., the numbe r of .
tuples in the re lation in which the value of the attribute Di is the s s ..
linguistic term Fi, m Fit Dj i indicate s the de gre e of me mbe rship that the value of the attribute Di of the jth tuple of a re lation be longs to the linguistic term Di, and 1F iF 3.
Be fore we pre se nt the FCLS algorithm, we must fuzzify a re lation of a re lational database system into a fuzzy relation as de scribed pre viously. Le t S be a se t of ante ce dent attribute s and Z be a conse -quent attribute of a re lation in a re lational database system. In a re lational database system, a tuple in a relation forms a training instance. Le t T be a set of training instances. The FCLS algorithm is now pre sented as follows:
FCLS Algorithm
Step 1: Fuzzify the re lation into the fuzzy relation.
Step 2: Select an attribute among the set S of antece dent attributes that has the smalle st FA. Assume attribute X with the smalle st FA; the n partition the se t T of the traininhg instance s into subsets T1,T2, . . . , and Tn according to the fuzzy domain v X1,X2, . . . , Xn4of the attribute X, re spe ctively. Compute the
s . s .
ave rage value Avg Xi of Xi based on formula 7 , whe re 1F iF n.
GENERATING FUZZY RULES 705 Step 3: Le t the attribute X be the de cision node, and sprout the tre e
according to the fuzzy domain of the attribute X shown as follows:
X
X1 X2 ? ? ? Xn
whe re X1,X2, . . . ,Xn are linguistic terms re prese nte d by fuzzy
v 4
sets and the set X1,X2, . . . ,Xn is the fuzzy domain of the attribute X.
v 4
Step 4: Le t Ss Sy X , whe re y is the differe nce operator be twee n sets. Step 5: For i¤ 1 to n do v Le tT¤ Ti; If Ss B the n v
cre ate a decision node for consequent attribute Z.
partition the training instances T into T1,T2, . . . ,Tk
accord-v 4
ing to the fuzzy domain Z1,Z2, . . . ,Zk of the attribute Z. s .
compute the average value Avg Zi of Zi, whe re 1F iF k; s .
cre ate a terminal node for e very Ti with Avg Zi / 0 and compute the value CF associate d with the create d certaintyi
factor node for eve ry nonnull path in the tre e. 4
else
go to step 2.
In the following, we use an e xample to illustrate the fuzzy rule gene ration proce ss.
Example 2: Assume that in a re lational database system we have a re lation shown in Table 1.
From Table 1, we can see that the attribute Salary is de termined by
v
the attribute s De gre e and E xperience . In this case ,Ss De gree, E xpe -4
rience and Z s Salary, whe re the attribute s Degree and E
Ta b le 1 . A re lation in a re lational database syste m
E mp-ID De gre e Expe rie nce Salary
S1 Ph.D. 7.2 63,000 S2 Maste r 2.0 37,000 S3 Bache lor 7.0 40,000 S4 Ph.D. 1.2 47,000 S5 Maste r 7.5 53,000 S6 Bache lor 1.5 26,000 S7 Bache lor 2.3 29,000 S8 Ph.D. 2.0 50,000 S9 Ph.D. 3.8 54,000 S10 Bache lor 3.5 35,000 S11 Maste r 3.5 40,000 S12 Maste r 3.6 41,000 S13 Maste r 10 68,000 S14 Ph.D. 5.0 57,000 S15 Bache lor 5.0 36,000 S16 Maste r 6.2 50,000 S17 Bache lor 0.5 23,000 S18 Maste r 7.2 55,000 S19 Maste r 6.5 51,000 S20 Ph.D. 7.8 65,000 S21 Maste r 8.1 64,000 S22 Ph.D. 8.5 70,000
pe rie nce are called antecedent attribute s and the attribute Salary is called a consequent attribute. From Table 1, we can see that the value s
s . s . s .
of the attribute Degree are Ph.D. P , Master M , and B ache lor B , and the domains of the attribute s E xperience and Salary are from 0 to 10 and from 20,000 to 70,000, re spe ctively. In this e xample , we also let
v s . s .
the fuzzy dom ain of the attribute De gree be Ph.D P , Maste r M , s .4
B ache lor B and the fuzzy domain of the attribute s E xperience and
v s . s . s .
S a la ry b e h igh H , som e w h a t-h igh S H , m e d iu m M ,
s . s .4
some what-low SL , low L , re spectively. The m embership function curves of the se linguistic terms are shown in Figure 8.
First, we fuzzify the relation shown in Table 1. The result of fuzzification of Table 1 is a fuzzy re lation shown in Table 2.
GENERATING FUZZY RULES 707
F igu r e 8. Me mbe rship function curve s.
Ta b le 2 . A fuzzy re lation
Em p-ID De gre e Expe rie nce Salary
v 4 v 4 v 4 S1 Ph.D.r1.0 SHr0.9 Hr0.8 v 4 v 4 v 4 S2 Maste rr1.0 SLr0.5 SLr0.8 v 4 v 4 v 4 S3 Bache lorr1.0 SHr1.0 Mr0.5 v 4 v 4 v 4 S4 Ph.D.r1.0 Lr0.9 Mr0.8 v 4 v 4 v 4 S5 Maste rr1.0 SHr0.75 SHr0.8 v 4 v 4 v 4 S6 Bache lorr1.0 Lr0.75 Lr0.9 v 4 v 4 v 4 S7 Bache lorr1.0 SLr0.65 Lr0.6 v 4 v 4 v 4 S8 Ph.D.r1.0 Lr0.5 SHr0.5 v 4 v 4 v 4 S9 Ph.D.r1.0 SLr0.6 SHr0.9 v 4 v 4 v 4 S10 Bache lorr1.0 SLr0.75 SLr1.0 v 4 v 4 v 4 S11 Maste rr1.0 SLr0.75 SLr0.5 v 4 v 4 v 4 S12 Maste rr1.0 SLr0.7 Mr0.6 v 4 v 4 v 4 S13 Maste rr1.0 Hr1.0 Hr1.0 v 4 v 4 v 4 S14 Ph.D.r1.0 Mr1.0 SHr0.8 v 4 v 4 v 4 S15 Bache lorr1.0 Mr1.0 SLr0.9 v 4 v 4 v 4 S16 Maste rr1.0 SHr0.6 Mr0.5 v 4 v 4 v 4 S17 Bache lorr1.0 Lr1.0 Lr1.0 v 4 v 4 v 4 S18 Maste rr1.0 SHr0.9 SHr1.0 v 4 v 4 v 4 S19 Maste rr1.0 SHr0.75 SHr0.6 v 4 v 4 v 4 S20 Ph.D.r1.0 SHr0.6 Hr1.0 v 4 v 4 v 4 S21 Maste rr1.0 Hr0.55 Hr0.9 v 4 v 4 v 4 S22 Ph.D.r1.0 Hr0.75 Hr1.0
s .
The n, based on formula 4 , we can com pute the fuzzine ss of e ach
v 4
attribute in the set S,Ss Degre e , E xperience , shown as follows:
s . FA De gree s 0 s . ws . s . s . FA E xperience s 1y 0 .9 q 1 y 0 .5 q 1 y 1 .0 s . s . s . s . q 1 y 0 .9 q 1 y 0 .75 q 1 y 0 .75 q 1 y 0 .65 s . s . s . s . q 1 y 0 .5 q 1 y 0 .6 q 1 y 0 .75 q 1 y 0 .75 s . s . s . q 1 y 0 .7 q 1 y 1 .0 q 1 y 1 .0 s . s . s . s . q 1 y 1 .0 q 1 y 0 .6 q 1 y 1 .0 q 1 y 0 .9 s . s . s . q 1 y 0 .75 q 1 y 0 .6 q 1 y 0 .55 s .x q 1 y 0 .75
r
22 s . s 0 .23 8Afte r applying the FCLS algorithm, the fuzzy de cision tre e is constructe d as shown in Figure 9.
F igu r e 9. Fuzzy de cision tre e of Example 2.
GENERATING FUZZY RULES 709 Consequently, we can ge t 16 fuzzy rule s from Figure 9 shown as follows:
s .
Rule 1 : IF De gre e is P AND Expe rie nce is L TH EN Salary is M CFs 0 .70
s .
Rule 2 : IF De gre e is P AND Expe rie nce is L TH EN Salary is SH CFs 0.50
s .
Rule 3 : IF De gre e is P AND Expe rie nce is SL THE N Salary is SH CFs 0.60
s .
Rule 4 : IF De gre e is P AND Expe rie nce is M TH EN Salary is SH CFs 0.80
s .
Rule 5 : IF De gre e is P AND Expe rie nce is SH THE N Salary is H CFs 0 .75
s .
Rule 6 : IF De gre e is P AND Expe rie nce is H THE N Salary is H CFs 0 .75
s .
Rule 7 : IF De gre e is M AND Expe rie nce is SL TH EN Salary is SL CFs 0 .65
s .
Rule 8 : IF De gre e is M AND Expe rie nce is SL TH EN Salary is M CFs 0 .60
s .9
s .
Rule 9 : IF De gre e is M AND Expe rie nce is SH THE N Salary is M CFs 0.50
s .
Rule 10: IF De gre e is M AND Expe rie nce is SH THE N Salary is SH CFs 0 .75
s .
Rule 11: IF De gre e is M AND Expe rie nce is H THE N Salary is H CFs 0 .78
s .
Rule 12: IF De gre e is B AND E xpe rie nce is L THE N Salary is L CFs 0.88
s .
Rule 13: IF De gre e is B AND E xpe rie nce is SL THE N Salary is L CFs 0.60
s .
Rule 14: IF De gre e is B AND E xpe rie nce is SL THE N Salary is SL CFs 0.70
s .
Rule 15: IF De gre e is B AND E xpe rie nce is M THE N Salary is SL CFs 0 .90
s .
Rule 16: IF De gre e is B AND E xpe rie nce is SH THE N Salary is M CFs 0 .50 .
A ve ry important prope rty of the proposed FCLS algorithm for constructing a fuzzy decision tre e is that it te rminate s. The proof of this
s .
prope rty requires the following le mma Pe te rson, 1981 :
Lemma 1: In any infinite directe d tre e in which each node has only a finite numbe r of direct succe ssors, there is an infinite path le ading from the root.
s . Proof: See Pe te rson 1981, p. 97 .
Theorem 1: The fuzzy decision tre e constructed by the FCLS algorithm is finite .
Proof: The proof is by contradiction. Assume that the re e xists an infinite fuzzy decision tree . B e cause each node in the tree is associate d with an attribute of a re lation, and be cause the numbe r of succe ssors for e ach node X in the tre e is limite d by the numbe r of linguistic te rms sfuzzy re gions in the fuzzy domain of the attribute , by Le mma 1 the re is. an infinite path from the root node to e ither the ce rtainty factor node or the hypothe tical ce rtainty factor node . But in a fuzzy de cision tre e , the path length from the root node to e ither the certainty factor node or the hypothetical ce rtainty factor node is limite d by my 1, whe re m
is the num ber of attribute s in a relation of a re lational database system. This is a contradiction. Proving that an infinite fuzzy de cision tre e e xisted was incorre ct.
COMPLETENESS OF FUZZY DECISION TREES
In the following, we pre se nt a me thod for de aling with the complete ne ss of the constructe d fuzzy decision tre e created by the propose d FCLS algorithm. The main purpose of a fuzzy le arning algorithm is to use a
s .
se t of training instance s i.e., tuples of relations to learn or construct some fuzzy rules. If the training instance s to be learned do not contain all kinds of conditions, null paths will be produce d in the ge nerate d
s .
fuzzy de cision tree . Sudkamp and Hamme ll 1994 propose d the region growing method and the we ighte d average me thod to com plete the e ntries of fuzzy associative me mory. In the following, we will pre sent a me thod for comple ting the null paths in a fuzzy de cision tree based on
s .
Sudkamp and Hamm ell 1994 . A fter the fuzzy de cision tree has be come a completed fuzzy de cision tre e , the comple te fuzzy rule base will be gene rated from the tre e.
Le t a be a mapping function from linguistic te rms to ordinary numbe rs and le t b be a m apping function from ordinary numbe rs to
GENERATING FUZZY RULES 711 linguistic terms. For example, in Figure 8, we let
s . s . a L s 1 , b 1 s L s . s . a SL s 2 , b 2 s SL s . s . a M s 3 , b 3 s M s 10. s . s . a SH s 4 , b 4 s SH s . s . a H s 5 , b 5 s H
For eve ry path in the fuzzy de cision tre e cre ated by the proposed FCLS algorithm, if the re are some null paths, the n a hypothetical certainty factor node HCF is cre ated for each null path. In this case, the path from the root node to a hypothe tical ce rtainty factor node forms a virtual fuzzy rule . Furthe rmore , in orde r to minimize the e rror of the de gree of be lie f of the ge nerated virtual fuzzy rule s, we le t the value associate d with e ach hypothetical certainty factor node be equal to 0.5. Assume that Figure 10 is a subtree of a fuzzy decision tree , where Yiy 1,
Yi, and Yiq 1 are fuzzy value s in the fuzzy domain of Y; Yiy 1 and Yiq 1
are on the nonnull path with the rightmost value ZL and the leftm ost value ZR ofZ, whe re ZL and ZR are linguistic te rms and where Yi is on the null path of the tree . Furthe rmore , assume that we want to sprout the branch ZU de noted by the dotted line shown in Figure 10, whe re ZU is a linguistic term. Then,
Case 1: If we want to sprout the branch ZU de note d by the dotted line shown in Figure 11, then ZU can be e valuate d as follows:
s . s . a Z qa Z y 1
I
L R s . s . bt
/
, if a ZL y a ZR 2is an odd numbe r and CFLG CFR
s . s .
a ZL qa ZR q 1
í
b , if a s Z .y a s Z .ZUs
t
/
L R2
is an odd numbe r and CFL- CFR
s . s .
a ZL qa ZR
b , othe rwise
J
t
2/
s 11.
F ig u r e 1 0 . A subtre e of a fuzzy de cision tre e .
and we le t the associated value of the hypothe tical certainty node HCFU be e qual to 0.5.
Case 2: If we want to sprout the branch ZU denote d by the dotte d line shown in Figure 12, where the node Z of Y1 to Yiy 1 cannot sprout out any branche s, and assume that we have the following metaknowle dge :
The smalle r the value s of the attribute Y, the sm aller the value of the attribute Z,
the n ZU can be e valuated as follows : s .
1q a ZR
ZUsb
t
/
2 s 12.
s .
F igu r e 1 1 .A subtre e of a fuzzy de cision tre e case 1 .
GENERATING FUZZY RULES 713
s .
F igu r e 1 2 .A subtre e of a fuzzy de cision tre e case 2 .
Assume that we have the following me taknowle dge :
The smalle r the value s of the attribute Y, the gre ate r the value s of the attribute Z,
the n ZU can be e valuated as follows :
s . s . a ZK q a ZR s . ZUsb
t
/
13 2 s .whe re a ZK has the large st ordinary numbe r in the fuzzy domain of Z, and we le t the value of hypothetical ce rtainty node HCFU be e qual to 0.5.
Case 3: If we want to sprout the branch ZU de note d by the dotted line shown in Figure 13, whe re the decision node Z of Yiq 1 to Ym
cannot sprout out any branches, and assume that we have the following metaknowle dge:
The gre ater the values of the attribute Y, the smalle r the value s of the attribute Z,
the n ZU can be e valuated as follows : s .
1q a ZL
ZUsb
t
/
2 s 14.
s .
F igu r e 1 3 .A subtre e of a fuzzy de cision tre e case 3 .
Assume that we have the following me taknowle dge :
The gre ater the value s of the attribute Y, the greate r the values of the attribute Z,
the n ZU can be e valuated as follows :
s . s . a ZL q a ZK ZUsb
t
/
2 s . 15 s .where a ZK has the largest ordinary numbe r in the fuzzy dom ain ofz, and we le t the value of hypothe tical ce rtainty node HCFU be equal to 0 .5 .
This procedure will go on continuously until the re is no null path in the fuzzy decision tree .
Example 3: The assumptions are the same as in E xample 2, whe re Figure 9 is the constructed fuzzy de cision tre e of E xample 2. B y pe rforming the propose d me thod, the complete fuzzy de cision tree is constructe d as shown in Figure 14. Figure 14 shows a comple te fuzzy de cision tree de rive d from Figure 9 after performing the proposed me thod.
F ig u r e 1 4 . C o m p le te fu zz y d e ci si o n tr e e o f E xa m p le 3 . 715
Mª Mª
For the path De gre e E xpe rie nce Salary shown in Figure 14, based on case 1 of the propose d me thod, we can see that
Yis M , Yiy 1s SL , Yiq 1s SH , ZLs M, ZRs M .
< s . s .<
B ecause a ZL y a ZR s 0, we can see that
s . s . a ZL q a ZR ZUs b
t
/
2 s . s . a M q a M s bt
/
2 3q 3 s bt
/
2 s . s b 3 s MThat is, we can ge t a virtual fuzzy rule shown as follows:
Mª Mª Mª
s .
Degree E xperie nce Salary 0 .50 16
From Figure 14, we can se e that the re are thre e null paths in the tree , that is,
Mª Lª Lª
De gre e E xpe rie nce Salary 0 .50
Mª Mª Mª
s .
De gre e E xpe rie nce Salary 0 .50 17
Bª Hª SHª
De gre e E xpe rie nce Salary 0 .50
GENERATING FUZZY RULES 717 Thus, we can get three virtual fuzzy rule s shown as follows:
Rule 17 : IF De gree is M AND E xpe rie nce is L
s .
TH E N Salary is L CFs 0 .50 Rule 18 : IF De gree is M AND E xpe rie nce is M
s 18.
s .
TH E N Salary is M CFs 0 .50 Rule 19 : IF De gree is B AND E xpe rie nce is H
s .
TH E N Salary is SH CFs 0 .50
ESTIMATION OF NULL VALUES IN RELATIONAL
DATABASE SYSTEMS
In the following, we introduce the de fuzzification te chnique of fuzzy
s .
numbe rs. In Chen 1994 , we have prese nte d a de fuzzification te chnique
s .
of trape zoidal fuzzy numbers based on Kande l 1986 as shown in s .
Figure 15, where the defuzzification value DE F Zk of the fuzzy numbe r Zk is e and
s . s .
es aq bq cq d
r
4 19A triangular fuzzy numbe r can be thought as a spe cial case of a s . trape zoidal fuzzy numbe r. Thus, the de fuzzification value DE F Zk of
F igu r e 1 5. De fuzzification of a trape zoidal fuzzy numbe r.
F igu r e 1 6 . De fuzzification of a triangular fuzzy numbe r.
the triangular fuzzy num ber Zk shown in Figure 16 is e, whe re
s . s .
es aq bq bq d
r
4 20If v is a crisp value of a fuzzifable attribute Vin some tuple s of a
v s .. s s ..4
re lational database, the n we let Vs
r
m VS v , Vtr
m Vt v be fuzzified values of v, where VS and Vt are linguistic te rm s repre se nte d by fuzzys . s . s . s .
se ts, m VS v G m Vt v , and m VS v q m Vtv s 1.0 If v is a crisp value of a unfuzzifiable attribute Vin some tuple s of a relational database, the n
v s .4
we le t v
r
1.0 be the fuzzified value of v. In orde r to e stimate null values in a re lational database system , we must first m odify the fuzzifiedws s .. s s ..x
value of v into the form Fvs Vm
r
m Vm v , Vnr
m Vn v , whe re Vmand Vn are linguistic terms repre sented by fuzzy sets.
s . s .
Case 1: If V is a fuzzifiable attribute linguistic variable and m VS v / 1.0, then we le t
s . s . s . s .
Vm s VS, Vns Vt, m Vm v s m VS v , m Vn v s m Vt v
s . s .
Case 2: If V is a fuzzifiable attribute linguistic variable and m VSv s 1.0, then we le t
s . s . s .
Vm s Vns VS and m Vm v s m Vn v s m VS v s 1 .0
GENERATING FUZZY RULES 719
Case 3: If Vis an unfuzzifiable attribute , the n we le t
s . s .
Vm s Vns v and m Vm v s m Vn v s 1 .0
Example 4: Le t us conside r the me mbership function curves shown in Figure 8.
1. If Vs E xperience and v s 1.0, then, based on Figure 8, the
fuzzi-v s . s .4 ws . s .x
fied value of v is L
r
1.0 , SLr
0 and Fvs Lr
1.0 , Lr
1.0 . 2. If Vs E xperience and vs 5.5, then based on Figure 8, thefuzzi-v s . s .4 ws .
fied value of v is M
r
0.75 , SHr
0.25 and Fvs Mr
0.75 , sSHr
0.25 ..x3. IfVs E xperie nce and vs 3, the n based on Figure 8, the fuzzified
v s . s .4 ws . s .x
value of v is SL
r
1.0 , Mr
0 and Fvs SLr
1.0 , SLr
1.0 . 4. If Vs De gree and v s Master, then the fuzzified value of v isv sMaster
r
1.0.4and Fvs wsMaste rr
1.0 , Master. sr
1.0 ..xIn the following, we prese nt a me thod for e stimating null value s in re lational database systems.
Assume that xand yare crisp dom ain value s of attribute s X andY in some tuple s of a re lational database, respe ctively, and assume that z
ws s .. s s ..x
is a null value of attribute Z. Le t Fxs Xa
r
m Xa x , Xbr
m Xb x andw s .. s s ..x
Fys Yc
r
m Yc y , Ydr
m Yd x be the modified forms of the fuzzified values of x and y, respectively. Assume that the fuzzy rule base contains the following fuzzy rule s ge nerate d by the proposed FCLS algorithm: s . IFXis Xa andYis Yc TH E NZis ZM1 CFs C1 s . IFXis Xa andYis Yc TH E NZis ZM CFs C2 2 s . IFXis Xa andYis Yc TH E NZis ZM3 CFs C3 s 21. s . IFXis Xb andYisYd TH E NZis ZN1 CFs D1 s . IFXis Xb andYisYd TH E NZis ZN2 CFs D2whe re X and Y are ante ce dent attribute s; Z is a conse que nt attribute ;
Xa, Xb, Yc, Yd, ZM1, ZM2, ZM3, ZN1, and ZN2 are linguistic te rms re prese nte d by fuzzy sets. Then, the null value z can be evaluate d as follows: zs 3 s . 2 s . p i-1Ci= DE F ZMi p is 1Di= DE F ZNi s . s . s . s . m Xs x = m Yc y = p 3 q m Xb x = m Yd y = 2 C p D is 1 i is 1 i s . s . s . s . m Xa x = m Yc y q m Xb x = m Yd y s 22. s . s .
whe re DE F ZMi and DE F ZNi are defuzzified value s of the fuzzy se ts
ZMi and ZNi, respe ctively.
Example 5: Assume that a relational database system contains a re la-tion shown in Table 3, and assume that we want to e stim ate the null value of the attribute Salary shown in Table 3.
From Table 3, we can see that the tuple with E mp-IDs S23 has a null value in the attribute Salary. Based on the me mbe rship functions shown in Figure 8 and after performing the fuzzification process, Table 3 be come s Table 4.
ws . s .x
He nce, we can se e that Fxs Master
r
1.0 , Maste rr
1.0 andws . s .x
Fys M
r
0.75 , SLr
0.25 . Then, afte r executing the propose d FCLS algorithm and according to the ge nerate d fuzzy rule s 7, 8, and 18 shown in E xample s 2 and 3, the null value of the attribute Salary can be e stimate d, whe re rules 7, 8, and 18 are shown as follows:Rule 7 : IF De gre e is Maste r AND E xpe rience is SL
s .
TH E N Salary is SL CFs 0 .65
Rule 8 : IF De gre e is Maste r AND E xpe rience is SL
s 23.
s .
TH E N Salary is M CFs 0 .60
Rule 18 : IF De gree is Master AND E xperience is M
s .
TH E N Salary is M CFs 0 .50
GENERATING FUZZY RULES 721
Ta b le 3 . A re lation contains null value s
E mp-ID De gre e Expe rie nce Salary
S1 Ph.D. 7.2 63,000 S2 Maste r 2.0 37,000 S3 Bache lor 7.0 40,000 S4 Ph.D. 1.2 47,000 S5 Maste r 7.5 53,000 S6 Bache lor 1.5 26,000 S7 Bache lor 2.3 29,000 S8 Ph.D. 2.0 50,000 S9 Ph.D. 3.8 54,000 S10 Bache lor 3.5 35,000 S11 Maste r 3.5 40,000 S12 Maste r 3.6 41,000 S13 Maste r 10 68,000 S14 Ph.D. 5.0 57,000 S15 Bache lor 5.0 36,000 S16 Maste r 6.2 50,000 S17 Bache lor 0.5 23,000 S18 Maste r 7.2 55,000 S19 Maste r 6.5 51,000 S20 Ph.D. 7.8 65,000 S21 Maste r 8.1 64,000 S22 Ph.D. 8.5 70,000 S23 Maste r 4.5 Null s .
Based on formula 22 , the null value of the attribute Salary of the e mploye e whose E MP-IDs S23 shown in Table 3 can be e valuated as follows: s . s . s . 0 .5= DE F M 0 .6= DEF M q 0 .65 = DEF SL 1= 0 .75 = q 1= 0 .25 = 0 .5 0 .6q 0 .65 1= 0 .75 q 1 = 0 .25 0 .6= 45 ,000 q 0 .65 = 35 ,000 1= 0 .75 = 45,000 q 1 = 0 .25 = 1 .25 s 1= 0 .75 q 1 = 0 .25 s . s 43 ,700 24
That is, the salary of the em ployee whose E mp-IDs S23 is about 43,700.
Ta b le 4 . A fuzzy re lation contains null value s
E mp-ID De gre e E xpe rie nce Salary
v 4 v 4 v 4 S1 Ph.D.r1.0 SHr0.9 Hr0.8 v 4 v 4 v 4 S2 Maste rr1.0 SLr0.5 SLr0.8 v 4 v 4 v 4 S3 Bache lorr1.0 SHr1.0 Mr0.5 v 4 v 4 v 4 S4 Ph.D.r1.0 Lr0.9 Mr0.8 v 4 v 4 v 4 S5 Maste rr1.0 SHr0.75 SHr0.8 v 4 v 4 v 4 S6 Bache lorr1.0 Lr0.75 Lr0.9 v 4 v 4 v 4 S7 Bache lorr1.0 SLr0.65 Lr0.6 v 4 v 4 v 4 S8 Ph.D.r1.0 Lr0.5 SHr0.5 v 4 v 4 v 4 S9 Ph.D.r1.0 SLr0.6 SHr0.9 v 4 v 4 v 4 S10 Bache lorr1.0 SLr0.75 SLr1.0 v 4 v 4 v 4 S11 Maste rr1.0 SLr0.75 SLr0.5 v 4 v 4 v 4 S12 Maste rr1.0 SLr0.7 Mr0.6 v 4 v 4 v 4 S13 Maste rr1.0 Hr1.0 Hr1.0 v 4 v 4 v 4 S14 Ph.D.r1.0 Mr1.0 SHr0.8 v 4 v 4 v 4 S15 Bache lorr1.0 Mr1.0 SLr0.9 v 4 v 4 v 4 S16 Maste rr1.0 SHr0.6 Mr0.5 v 4 v 4 v 4 S17 Bache lorr1.0 Lr1.0 Lr1.0 v 4 v 4 v 4 S18 Maste rr1.0 SHr0.9 SHr1.0 v 4 v 4 v 4 S19 Maste rr1.0 SHr0.75 SHr0.6 v 4 v 4 v 4 S20 Ph.D.r1.0 SHr0.6 Hr1.0 v 4 v 4 v 4 S21 Maste rr1.0 Hr0.55 Hr0.9 v 4 v 4 v 4 S22 Ph.D.r1.0 Hr0.75 Hr1.0 v 4 v 4 S23 Maste rr1.0 Mr0.75, SLr0.25 Null
CONCLUSIONS
In this pape r, we have prese nte d a new algorithm for constructing fuzzy de cision tre es from re lational database systems and gene rating fuzzy rule s from the constructed fuzzy de cision tre es. W e also have pre sented a me thod for de aling with the comple te ness of the constructe d fuzzy de cision tre es. B ase d on the gene rated fuzzy rule s, we also pre sent a me thod for e stimating null value s in relational database systems. The propose d m ethod provides a useful way to e stimate null value s in re lational database systems.
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