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.
Downloaded by [National Chiao Tung University ] at 02:29 01 May 2014
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
Downloaded by [National Chiao Tung University ] at 02:29 01 May 2014
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.
In 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:Downloaded by [National Chiao Tung University ] at 02:29 01 May 2014
whe 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:
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
Downloaded by [National Chiao Tung University ] at 02:29 01 May 2014
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
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:
That is, the salary of the em ployee whose E mp-IDs S23 is about 43,700.
Downloaded by [National Chiao Tung University ] at 02:29 01 May 2014
Ta b le 4 . A fuzzy re lation contains null value s
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.
REFERENCES
Che n, S. M. 1994. Using fuzzy reasoning te chnique s for fault diagnosis of the J-85 je t engines. Proceed in g Th ird N atio n al Con feren ce on Scien ce an d
Downloaded by [National Chiao Tung University ] at 02:29 01 May 2014
GENERATING FUZZY RULES 723
Tech n olo gy of N atio n al D efen se, Taoyuan, Taiwan, Republic of China, pp. 29] 39.
Hart, A. 1985. The role of induction in knowledge elicitation. E xp ert Syst. 2:24] 28.
Hunt, E. B., J. Martin, and P. J. Stone . 1966. E xp erien ce in in d u ctio n. Ne w York:
Acade mic Pre ss.
Jeng, B., and T. P. Liang. 1993. Fuzzy indexing and retrie val in case-base d systems. Pro ceedin gs 1993 Pan Pacific C on feren ce o n In fo rm atio n System s, Taiwan, Re public of China, pp. 258] 266.
Kande l, A. 1986. F u zzy m ath em atic al tec h n iq u es w ith applic atio n s. Reading, MA:
Addison-W esley.
Pe terson, J. L. 1981. Petri n ets, th eory an d th e m od elin g of system s. Engle wood Cliffs, NJ: Pre ntice -Hall.
Q uinlan, J. R. 1979. Discove ring rules by induction from large colle ction of example s. In E xp ert system s in th e m ic ro electro n ic age, ed. D. Michie . Edinburgh: Edinburgh Unive rsity Pre ss, pp. 168] 201.
Q uinlan, J. R. 1987. Decision tree s as probabilistic classifers. Pro ceedin gs 4th In tern atio n al Wo rksh o p o n Ma ch in e L earn in g. Los Altos, CA: Morgan Kauffman, pp. 33] 37.
Safavian, S. R., and D. Landgre be. 1991. A surve y of decision tree classifier methodology. IE E E Tran s. Syst. Ma n C ybern et. 21:660] 674.
Sudkam p, T., and R. J. Hamme ll II. 1994. Inte rpolation, comple tion, and le arning fuzzy rule s. IE E E Tran s.Syst. Ma n Cybern et. 24:332] 342.
Wang, L. X., and J. M. Me nde l. 1992. Generating fuzzy rules by le arning from example s. IE E E Tran s.Syst. Ma n Cybern et. 22:1414] 1427.
Yeh, M. S., and S. M. Che n. 1995. Gene rating fuzzy rules from relational database systems. Pro ceed in gs 6th In tern atio n al Co n feren ce o n In fo rm atio n Ma n agem en t, Taipe i, Taiwan, Re public of China, 219] 226.
Yuan, Y., and M. J. Shaw. 1995. Induction of fuzzy decision tree s. Fu zzy Sets Syst. 69:125] 139.
Z adeh, L. A. 1965. Fuzzy sets. In fo rm.C on trol 8:338] 353.
Z adeh, L. A. 1975. The conce pts of a linguistic variable and its application to approxim ate reasoning I .s . In fo rm.Sci. 8:199] 249.