The authors [2-5] have studied and presented the quantitative method of linguistic variables and linguistic threshold by fuzzy set. Chien-Hua Wang, Chin-Pang Tzong proposed an algorithms for mining fuzzy association rule [2]. In this paper, we extend the algorithms proposed in [2] for number data and linguistic variables by using hedge algebras.
Data mining with the approach of association rules is one of important aspects in the field of data mining.
Many authors have presented various methods, algorithms of data mining by association rules with numerical support and confidence value. However, in reality, these values are natural linguistic ones. Besides, importance value of each item is evaluated not only by quantity, frequency of occurrence in each transaction but also by the qualitative evaluation of administrators (for those items) by natural language. And hedge algebra have met the requirements for directly processing calculation on linguistic value (without fuzzification, but with direct calculation based on qualitative semantic function and flexible calculation). Thus, it is necessary to establish a method of data mining by association rules with hedge algebra, in which the input is qualitative transactional database and qualitative evaluation table of those database items and the support, confidence values are also natural language ones. 2.1
Let I = I1, I2, . . . , Im be a set of items. Let D, the task-relevant data, be a set of database transactions where each transaction T is a set of items, such is T ✓ I. Each transaction is associated with an identi er, called TID. Definition 1. An association rule has the form of X ! Y , where X ✓ Y ✓ I, and X \ Y = ✓ .
I, Definition 2. The support of association rule X ! Y the probability that X [ Y exists in a transaction in the database D.
support(X ! Y ) = |X \ Y | |N | Definition 3. The confidence of the association rule X ! Y is the probability that X [ Y exists given that a transaction contains X, i.e.
confidence (X ! Y ) = support(X [ Y ) = |X \ Y | support(X) |X| Where: |X| is the number of transactions, including X; |X \ Y | is the number of transactions, including X and Y ; N is the total of transaction database.
Mining the association rules of the database is finding all of the rules that have the degree of support and confidence greater than degree of support minsup and confidence minconf determined by the available user. 2.2
Let X be a linguistic variable and X be a set of its terms, called a term-domain of X. E.g. if X is the rotation speed of an electrical motor and linguistic hedges used to describe its speed are V ery, M ore, P ossibly, Little, denoted correspondingly for short by V, M, P and L, then X = {f ast, V f ast, M f ast, LP f ast, Lf ast, P f ast, Lslow, slow, P slow, V slow, ...} U 0, W, 1 is a term-domain of X.
It can be considered as an abstract algebra AX = (X, C, H, ), where H is a set of linguistic hedges, which can be regarded as one-argument operations, is called a semantics-based ordering relation on X and W, 0, 1 is a set of constants in X with fast and slow being primary terms of X and W, 0, 1 being additional elements in X interpreted as the neutral, the least and the greatest ones, respectively. Denote by hx the result of applying an h 2 H to x 2 X and by H(x) the set of all u 2 X generated algebraically from x by using hedges in H, i.e. H(x) = u: u = hn...h1x, h1, ..., hn 2 H.
It is natural that there is a demand to transform fuzzy sets defined on a
real interval [a, b], which represents the meaning of terms in a term-domain X,
into [a, b] or, for normalization, into [
Let AX = (X, C, H, ) be a linear HA. An f m: X ! [
X, the length of x is denoted by |x|, and defined Proposition 1. The fuzziness measure (f m) and the fuzziness measure of hedge h, denoted by µ (h), 8 h 2 H, with the following properties: 1) f m(hx) = µ(h) ⇥ f m(x) with 8 x 2 X; 2) f m(c+) + f m(c ) = 1; 3) P 4) P 5) P q i p,i6=0 f m(hic) = f m(c), c 2 { c+, c }; q i p,i6=0 f m(hix) = f m(x); q i 1 µ(hi) = ↵ , P1 j p µ(hj ) = , ↵ + 3
+ Calculate the fuzzy of variable X: f m(X); + Identify fuzzy approximately of X: I(X) = [a, b]; + The fuzzy average value of the variable X: gt(X) = Step 2: Handling qualitative table: A set of m items with their importance evaluated by d managers + Calculate the fuzzy of linguistic variables; + Calculate the average o↵uzzy approximatelyqualitativeterms for all items. kdt⇠tb(j) = 1 d ⇥
d X (a(j)i, b(j)i) ; (has the form: [aj , bj ]) i=1 + Calculate the average of fuzzy value for each item: gtdt⇠tb(j) = aj + bj 2 ; where: aj and bj are the values of kdt⇠tb(j), which kdt⇠tb(j) = [aj , bj ]
+ Transform the quantitative valueas Aj (j = (1, m)) as X variables in HA
(X 2 X), determined as follows:
Xsl = (Xsl, Gsl, Hsl, ), with: Gsl = {High, Low}, (High = H, Low = L);
c+ = {H}; c = {L}; Hs+l = {V ery, M ore}; Hsl = {Less, P ossibly}; (with
V ery > M ore; Less > P ossibly)
- Selection: Dom(sl); fm(H); fm(L); fm(V); fm(M); fm(L); fm(P);
- Identify fuzzy approximately of X is I(X), with X 2 X
- Transform the quantitative value of item into [
With each Aj 2 [
Step 5: Filter out all items in D⇠ , such that: satisfied frequent item of minimum support: sup(item) minsup.
Step 6: Establish Fuzzy FP-tree: establish Header table; establish FP-tree
+ Find out of all frequent itemsets (denote by n-itemset) from FP-tree; + Calculate the qualitative of n-itemset.
+ Using the formula (5) - in step 4: sup(n itemset) = Step 9: Export rules, calculate the confidence and check with minconf. Using the following substeps: + Check the association rules from result of step 8, each n-itemset with items (A1, A2, ..., An), (n = 2, M ): A1^ ... ^ Ai 1^ Ai+1...An ! Ai; (i = 1, M ) + Calculate the fuzzy confidence value of each possible fuzzy association rule as: conf (A ! B) = sup(A [ B) sup(A) ; (6) + Select the satisfied fuzzy association rule of minimum confidence.
During use of HA for fuzzy transaction database and quantify of linguitics, we view each element of HA is a fuzzy region. So, the process of creating fuzzy region based on the structure of HA will simple, intuitive, and more cient. 4
In this section, an example is given to illustrate the proposed algorithm. Input: includes three data follows: 1. The data set includes six quantitative transactions, as show in Table 2. 2. The importance of the items is evaluated by three managers as shown in Table 3.
3. A pre-defined linguistic minimum support value min s and linguistic minimum confidence value min c.
Step 1: Identify minsup, minconf from the pre-defined threshold linguistic
Identify parameters in HA: X = (X, G, H, ), with: G = {Low, High}; c+ = High (denoted by H); c = Low (denoted by L); H+ = {V ery, M ore}, H = {Less, P ossibly}; (with: V ery > M ore; Less > P ossibly) with: f m(L) = 0.3; f m(H) = 0.7; f m(V ) = f m(M ) = f m(L) = f m(P ) = 0.25; Identify fuzzy degree and fuzzy approximately of X:
With the variable X contains c = “Low”: + f m(V L) = 0.25 ⇥ 0.3 = 0.075 ) I(V L) = [0, 0.075] ) I(V L)T B = 3.75% + f m(M L) = 0.25 ⇥ 0.3 = 0.075 ) I(M L) = [0.075, 0.15] ) I(M L)T B = 11.25% + f m(P L) = 0.25 ⇥ 0.3 = 0.075 ) I(P L) = [0.15, 0.225] ) I(P L)T B = 18.75% + f m(LL) = 0.25 ⇥ 0.3 = 0.075 ) I(LL) = [0.225, 0.3] ) I(LL)T B = 26.25%
Similar, with the variable X contains c+ = “High”: + f m(LH) = 0.25⇥ 0.7 = 0.175 ) I(LH) = [0.3, 0.475] ) I(LH)T B = 38.75% + f m(P H) = 0.25 ⇥ 0.7 = 0.175 ) I(P H) = [0.475, 0.65] ) I(P H)T B = 56.25% + f m(M H) = 0.25 ⇥ 0.7 = 0.175 ) I(M H) = [0.65, 0.825] ) I(M H)T B = 73.75% + f m(V H) = 0.25 ⇥ 0.7 = 0.175 ) I(V H) = [0.825, 0.1] ) I(V H)T B = 91.25% - Select minsupport with linguistic thresholds as “Less Low” (denoted by LL) - Select minconf with linguistic thresholds as “More High” (denoted by MH) minsup = minsup(LL) = 26.25% minconf = minconf (M H) = 73.75% Step 2: Handling qualitative table: A set of m items with their importance evaluated by 03 managers.
Identify parameters in HA: Denote: I: Important; uI: UnImportant; O: Ordinary; VI: Very Important; VuI: Very UnImportant;
Xqt = (Xqt, Gqt, Hqt, ), with: Gqt = {I mportant, U nI mportant}; c+ = I mportant; c = U nI mportant; Hq+t = {V ery, M ore}; Hqt = {Less, P ossibly}; (with: V ery > M ore; Less > P ossibly).
Let: Wqt = 0.5; f m(I ) = 0.4; f m(uI ) = 0.6; f m(V ) = 0.3; f m(M ) = 0.2; f m(L) = 0.3; f m(P ) = 0.2;
Should have: f m(V I ) = 0.3 ⇥ 0.4 = 0.12 ) I (V I ) = [0.88, 1]; f m(V uI ) = 0.3 ⇥ 0.6 = 0.18 ) I (V uI ) = [0, 0.18]; f m(O) = 0.5 ) I (O) = [0.25, 0.75];
Table 3 is converted into Table 4, where kdt⇠tb is the average of fuzzy approximately qualitative; gtdt⇠tb is the average of fuzzy value. + fuzzy approximately of support: ([0.6, 1] ⇥ 3.04)/6 = [0.304, 0.51]; + fuzzy value of support: (0.304 + 0.51)/2 = 0.41 = 41%.
Step 5: Filter out all items in D⇠ . Such that: satisfied frequent item of minimum support: sup(item) minsup.
If: sup(item) < minsup (with: minsup = 26.25%, result at Step 1) Then: remove item in table 8.
Substep 7.1: Find out of all frequent itemsets (denote by n-itemset) from FP-tree (see Table 11) 2-item 3-item F.PH, B.PH: 1.52; F.PH, E.PH: 1.52; B.PH, E.PH: 2.28 F.PH, B.PH, E.PH: 1.52
itemset kdt⇠tb gtdt⇠tb F.PH, B.PH (0.693, 0.92) 81% F.PH, E.PH (0.6, 0.92) 76% B.PH, E.PH (0.693, 1) 85% F.PH, B.PH, (0.693, 0.92) 81%
E.PH
Itemset Support Minsup = 26.25% F.PH, E.PH 19% unselected F.PH, B.PH 21% unselected E.PH, B.PH 32% selected F.PH, B.PH, 21% unselected E.PH 5
The paper is an extension of the evaluation of fuzzy association rules was
researched by Chien-Hua Wang and Chin-Pang Tzong [