Generating Fuzzy Queries from Weighted Fuzzy Classifier Rules
Antonio C. S. Branco Alexandre G. Evsukoff Nelson F. F. Ebecken COPPE/Federal University of COPPE/Federal University of COPPE/Federal University of Rio de Janeiro, Brazil Rio de Janeiro, Brazil Rio de Janeiro, Brazil [email protected] [email protected] [email protected]
Abstract will return all the records that match the query sentence. However, it is frequently desirable to rank those records, This work presents a methodology to generate such that the manager is able to define priorities. automatically a set of fuzzy queries that are translated Neural networks and Bayesian classifiers are also from a set of fuzzy rules learned from a data set. A rule frequently found in most of DM suites. Such models may simplification approach is proposed, which allows a be coded into DBMS, via PMML, to retrieve a ranked list better interpretability of the fuzzy query sentence. The of records. Nevertheless, neural networks and Bayesian fuzzy query sentences are generated in such a way that classifiers models are not linguistically understandable, they can be processed by a standard fuzzy query system. such that managers cannot directly validate the knowledge The methodology is applied to a benchmark data set extracted by the DM algorithm. considering a scenario of target selection in a direct Fuzzy queries have emerged in the last 25 years to deal market application. The results obtained by the current with the necessity to soften the Boolean logic in relational approach are compared with the results of a standard a databases. A fuzzy query system is an interface to human decision tree induction algorithm. users to get information from database using (quasi) natural language sentences [1][2][3]. Many fuzzy queries implementations have been proposed, resulting in slightly 1. Introduction different languages such as SQLf [4], FQUERY [5] and Summary SQL [6][7], among others. Although some The integration of Data Mining (DM) tools with Data variations according to the particularities of different Base Management Systems (DBMS) is now more than implementations, the answer to a fuzzy query sentence is trend, it is a reality. The major DBMS vendors have a generally a list of records, ranked by the degree of already integrated DM solutions within their products. On matching. the other hand, the main DM suites have also provided the The focus of attention of the fuzzy query research has integration of DM models into DBMS trough modeling been on the expressive power of the fuzzy query languages such as the Predictive Model Markup Language language, since the query is usually provided by the user. (PMML). It is thus a fact that future research on new DM Nevertheless, the definition of a query to retrieve the tools and methods must consider their integration with records according to their features’ values is a very DBMS. difficult task. Not due to the lack of expressive power of On the application side, data intensive industries, such the query language, but due to the difficulty on defining as insurance, accounting and telecommunications, among the concept itself. others, need frequently to retrieve their costumers for Fuzzy queries sentences are structured definitions of different marketing relationship actions. Defining a query fuzzy concepts. Under this assumption, fuzzy queries can sentence that captures the main features of a subset of be automatically generated by fuzzy rule based classifiers. records is a non trivial task. Generally, however, past Fuzzy rule based classifiers are a very active research experiences can be used to guide the new actions, such field [8][9][10]. The weighted fuzzy rule based approach that queries generation can thus be regarded as a DM task. allows better classification results since it allows to define Structured Query Languages (SQL) provides a more precisely the decision boundary among the classes structured description of the retrieved records. A standard [11][12]. The translation of weighted fuzzy rules into SQL sentence representing a concept can be translated fuzzy queries is not straightforward since most of fuzzy from rules generated by an algorithm for decision-tree or queries languages do not support weights. rule induction. The translation of machine learning rules In this work, it is proposed a methodology to translate into SQL sentences is straightforward: for each class, each a set of rules computed by a weighted fuzzy classifier into rule corresponds to a sentence (in the WHERE clause), fuzzy queries, such that they can be coded into most of and all rules related to the same class are aggregated with available fuzzy query languages. The whole procedure is a disjunctive (OR) operator. A standard query, however, sketched in Figure 1. The left side of the figure shows an
existing fuzzy query system (denoted Fuzzy SQL) used to matching approach [8]. Under this approach, the output of process the fuzzy queries connected to a DBMS. The the fuzzy classifiers is computed in two steps: proposed methodology is showed in the right side of the 1. for each input, compute partial outputs as the figure. It is considered that a labeled database exists from degree of matching of the observed input value to which a training set may be selected to generate a set of each class, fuzzy rules by the fuzzy classifier. The set of rules 2. compute the final output by the aggregation of all generated by the fuzzy classifier may be very large, partial outputs. containing many useless rules. The fuzzy rule base is thus The following subsections describe the main steps of pruned and then translated into a set of fuzzy query the fuzzy rule-based classifier approach. sentences. The fuzzy query system is used to retrieve an ordered set of records from a new and unlabelled 2.1. Fuzzification database, corresponding to a given class. In a general application, the input variables may be numeric (discrete or continuous) or nominal. Fuzzy sets Fuzzy SQL Rule Base Translation allow a unified representation for nominal and numeric variables as fuzzy sets. Fuzzification is thus an important Fuzzy Queries Fuzzy Rule issue in fuzzy query generation since it provides the Base Unlabelled numeric-to-linguistic interface that allows dealing with Data Base Training Set numeric values as linguistic terms.
X X X Y 1 2 K n 1 Generally, each input variable x can be described x (1) x (1) x (1) y (1) i 1 2 K n 1 Fuzzy x (2) x (2) x (2) y (2) 1 2 K n 1 Classifier using ordered linguistic terms in a descriptor set M M O M M x ( N ) x ( N ) x ( N ) y ( N ) Labeled 1 2 K n 1 Ai = {Ai1,K, Ain } . When the variable is nominal, the Data Base i descriptor set is the set of possible values for the variable Figure 1. The proposed approach. (or a combination of them). When the variable is numeric, The remainder of this paper is organized as follows. the meaning of each term Aij ∈ A i is given by a fuzzy Next section presents the weighted fuzzy rule-based set defined on the variable domain. classifier and the translation of fuzzy rules into fuzzy For a single variable input x (t) , the fuzzification queries. Section 3 describes the learning algorithm of i vector u (t) = u (t), ,u (t) is computed by the fuzzy fuzzy rules and the pruning procedure. Section 4 presents i ( i1 K ini ) the results of the application of the proposed methodology sets in the fuzzy partition of the input variable domain as: for a set of benchmark data sets and a case study using the (1) ui (t) = (µ A (xi (t)),K, µ A (xi (t))) . Credit Card data set. The fuzzy queries results are i1 ini compared to standard queries generated by a standard An easy way to parameterize fuzzy sets is to use decision-tree induction algorithm. Finally some triangular membership functions that are completely conclusions are drawn. determined by the centers of triangles, which may be considered as prototypes values for the corresponding 2. Fuzzy rule-based classifier fuzzy sets (see Figure 2). µ Consider the standard classification problem where (x) A1 A2 A3 A4 A5 input variables are presented as a p-dimensional vector x p in the input variable domain X1 × X 2 ×K× X p = X and the output variables represented by the classes’ set C = {}C1,K,Cm . The solution to the classification p1 p2 p3 p4 p5 problem is to assign a class label Ck ∈ C to an observation x(t)∈ X p , where t represents a record in the Figure 2: Example of fuzzy partition. database. The fuzzification vector generalizes the information Fuzzy classifiers attempt to compute, for each class, a contained in the input variable and is computed in the fuzzy membership value µC (x(t)) that correspond to the same way if the variable is numeric or nominal. For k nominal variables, there is no fuzziness and the degree of matching of the observation x(t) to the class fuzzification vector is a binary vector, indicating which Ck . nominal value is present on the record. This section describes the weighted fuzzy rule-based In a general-purpose fuzzy classifier, multidimensional classifier approach, which is based on the fuzzy pattern fuzzification may also be considered and computed by
fuzzy clustering methods. Nevertheless, multidimensional A rule base is defined for each input variable and used fuzzy sets often do not represent linguistic concepts and to compute partial outputs by fuzzy inference. Fuzzy rules are not supported by some fuzzy query languages. with more than one variable in the antecedent allow Multidimensional fuzzy sets are thus out of the scope of representing interactions between input variables but the this paper. resulting rule base can grows exponentially for large problems. A trade off between the size and the number of 2.2. Fuzzy rules rules in the rule base is a complex optimization problem that is out of the scope of this work. A fuzzy rule relates input linguistic terms Aij ∈ Ai to 2.3. Fuzzy inference the classes Ck ∈ C in rules like: i (2) The fuzzy classifier output is represented by the class if xi (t) is Aij then class is Ck with cf = ϕ jk . membership vector y(t) = µ (x(t)), , µ (x(t)) . i ()C1 K Cm where ϕ jk ∈ []0,1 is a confidence factor that represents Each component µ (x(t)) is the membership of a given the rule certainty. Ck The rule (2) describes a flexible constraint on the input record x(t) to the class Ck . values of the variable xi that can be related to the class The vector y (t) = µ (x (t)), , µ (x (t)) is the i ( C1 i K Cm i ) (or concept) C . For instance, if the variable x k i partial output membership vector whose components are represents the “salary”, then an example of the rule could the classes’ membership values considering only the be: “if the customer’s salary is high then the customer’s information in the input variable i . class is Gold”. The confidence factor represents how The output of each sub-model is computed by much of this relation is true, for instance 95%. The rule’s composition-projection inference: certainty factor may be considered as a relative quantifier y (t) = u (t) Φ . (3) [13]. The above example could be interpreted as “Most of i i o i the high salary customers are Gold class customers”, The composition-projection operation computed by the where “most” is the linguistic quantifier that represent standard max-min composition operator as: µ (x (t)) = max min µ (x (t)), Φ (A , C ) . (4) 95% of the records in the database. Linguistic quantifiers Ck i ( ( Aij i i ij k )) can be represented as a fuzzy set defined over the unit j=1...ni interval and have been used to define fuzzy summaries in Equation (4) computes the degree of matching of the the language Summary SQL [6]. fuzzification vector ui (t) with the prototype of the class i In this work, the confidence factor ϕ jk represents how Ck , represented by the corresponding column of the rule base matrix Φ . much the term Aij ∈ Ai is linked to the class Ck ∈C in i i The final output is computed by the aggregation of all the model defined by the rule (2). A value ϕ jk > 0 means partial outputs by an aggregation operator p that the observation of the term Aij is related with the H : [0,1] → [0,1] as: i µ (x(t)) = H(µ (x (t)), , µ (x (t))) . (5) occurrence of the class Ck in ϕ jk of the records. Ck Ck 1 K Ck p A set of rules (or a rule base) for each input variable The best aggregation operator must be chosen defines a sub-model that is represented by the matrix Φ according to the semantics of the application. A t-norm i operator, such as the “minimum”, gives good results to as shown in Table 1. In such matrix, the lines j = 1...ni express that all partial conclusions must agree. In are related to the terms in the input variable descriptor set classification problems, the final decision is computed by Ai and the columns k = 1...m are related to classes in a decision rule. The most usual decision rule is the “maximum rule”, where the class is chosen as the one the set C , such that Φ (A ,C ) = ϕ i . i ij k jk with greatest membership value (Figure 3). Table 1: Rule base weights’ matrix. u1(t) y1(t) x1(t) A1 Φ1 Φi C1 K Cm y(t) H Ai1 Φi (Ai1,C1) K Φi (Ai1,Cm ) M M u p (t) y p (t) x (t) A Φ M M O M p p p
A Φ (A ,C ) Φ (A ,C ) ini i ini 1 K i ini m Figure 3. The weighted fuzzy classifier.
The translation of fuzzy rules in to fuzzy queries is thus equivalent to the classifier result. The fuzzy query presented next. sentence for the class Ck is: 2.4. Writing fuzzy queries from weighted fuzzy SELECT where u (t) = µ (x (t)) and v (t) is the correct ((x1 is A11 AND w11k is TRUE) OR ...OR ij Aij i k (x1 is A1n1 AND w1n1k is TRUE)) membership of the sample t to the class C . k The aggregation between rules is computed by an Considering p input variables, each one with n fuzzy aggregation operator (5), which is a conjunctive operator sets and m classes, there will be p.n.m rules. Equation (AND) between sentences. The fuzzy query evaluation is (8) must be computed for every rule in the fuzzy rule base, in order the model to be complete. In a probabilistic framework, equation (8) for a procedure must consider the most important rules to nominal input variable xi is computed as: generate a compact set of sentences in the fuzzy query. N(A ∩C ) (9) 3.2. Rule Base Pruning Φ (A ,C ) = ij k . i ij k N Aij Fuzzy rules’ pruning and simplification is a very active where N is the number of samples in the training research area [15][16]. In this work, fuzzy rules’ pruning (Aij ∩Ck ) is necessary to allow a more compact set of query set that have the value Aij for the variable xi and are sentences. The pruning procedure is based on the values classified as class C ; and N is the total number of of two indexes: k Aij The Horizontal index ( I H ), computed as: samples that have the value Aij for the variable xi . m ⎛ ⎞ (12) The conditional probability of the class Ck , given the ∑ ⎜ max ()Φi (Aij , Ck ) − Φi (Aij , Ck )⎟ k=1⎝ k=1..m ⎠ observation of the value Aij , computed from Bayes rule I (x , A ) = H i ij m −1 is: The Vertical index ( I ), computed as: N N (10) V (Aij ∩Ck ) Ck ni ⎛ ⎞ (13) P(A | C ).P(C ) N N ⎜ ⎟ ij k k Ck ∑ max ()Φi (Aij , Ck ) − Φi (Aij , Ck ) P(C | A ) = = ⎜ j=1..n ⎟ k ij P(A ) N j=1⎝ i ⎠ ij Aij IV (xi, Ck ) = N ni −1 where N is the total number of samples classified as The horizontal index IH ∈[]0,1 measures how much Ck the fuzzy set Aij of the variable xi is able to discriminate class Ck and N the total number of samples in the training set. among the classes. The vertical index IV ∈[0,1] measures Comparing equations (8) and (10), it can be seen that how much a class Ck is detected according to the the rule base weights are an estimation of the a posteriori possible values of the variable x . Fuzzy rules can be probability of the class occurrence given by the i observation of the value A , and: pruned by selecting a threshold values for one of these ij indexes. Φi (Aij ,Ck ) ≈ P(Ck | Aij ) . (11) For general-purpose classifiers, the horizontal index Equality in (11) is achieved for nominal variables. For should be used since it allows selecting the fuzzy rules continuous input variables, the result for rules’ output that will result in a better classification performance. For provides the interpolation of the conditional probabilities, fuzzy queries, however, the queries are executed for a weighted by the membership to the corresponding fuzzy given class independently to retrieve the best set. representatives’ records of that class. Thus, the vertical The final output is an aggregation of all input variables index should be used to select the rules that best to compute the final class membership. When this discriminates the records of a given class. aggregation is computed by a t-norm operator (like the 3.3. Fuzzy Query Evaluation minimum or product) the final class membership is a rough estimation of the joint conditional probability of The fuzzy query evaluation is based on the standard each class, given the observation of all input variables. metrics for evaluation of information retrieval systems: The expression in (8) has been referred as the Precision and Recall. The Precision metric is defined as ∑count , and has been used to compute the value of the number of relevant records selected as a fraction of relative quantifiers [13]. Linguistic quantifiers can be the total number of selected records. The Recall metric is processed by some fuzzy query languages such as defined as the number of relevant records selected as a Summary SQL [7] to compute fuzzy summaries. In this fraction of the total number of relevant records. work, the fuzzy rules are translated into fuzzy queries in The Precision and Recall metrics can be computed such a way that they can be processed by any fuzzy query directly from the confusion matrix, presented in Table 2, language with an equivalent result to the classifier. which is similar to the one usually used to evaluate The rule base weights are computed for all possible classification systems. The rows represent the results of rules. There are generally a large number of possible the query system and the columns represent the true rules, many of that are useless for the classification, which information about the selected records. makes difficult the interpretation of the classifier model The values in the Table 2 are the standard ones: TP and the corresponding query sentence. A pruning stands for the number of true positive samples, FP is the number of false positive samples, TN is the number of true negative samples and FN is the number of false parameters and the results in Table 3 are the average negative samples. values of the 10-fold cross validation for both algorithms. Bold values indicate the best values for each benchmark. Table 2: Confusion matrix The FWC will have a better accuracy than the decision Relevant Not Relevant tree induction algorithm when numerical variables are Selected TP FN predominant and variables are roughly statistically Not Selected FP TN independent [14]. The NB algorithm performs better when the variables are independent and probabilities’ The Precision and Recall metrics are computed as: densities may be well approximated by the normal TP (14) probability distribution function. Nevertheless, in general, P = . the classifiers’ performance depends mainly on the (TP FN) + characteristics of the problem. TP (15) R = . Table 3: Classifiers’ results (TP + FP) Precision and Recall rates are both desired, but it is Benchmark # Vars # Class FWC J4.8 NB generally very difficult to achieve high values of both Abalone 8 3 44.00 39.70 42.25 metrics simultaneously: as the Recall rate increases, the Balance Scale 4 3 36.46 43.20 36.48 Precision usually decreases and vice-versa. Credit Card 15 2 13.63 14.20 23.27 Differently from a standard SQL query, a fuzzy query Prima indians 8 2 31.38 25.90 23.69 returns all the records in the database, even those Ionosphere 34 2 6.27 8.55 29.91 associated to a very small membership value. In a Breast Cancer 9 2 5.76 8.40 5.75 practical application, it is necessary to set a membership Glass 9 7 47.64 29.40 51.87 threshold or the maximum number of returned records. Wine 13 3 5.62 7.90 2.80 These parameters must be set to run the query. The Precision and Recall metrics, when computed over The results in Table 3 were obtained without rule base the training set, are useful to give the user an insight of pruning. Although, fuzzy rule base pruning may slightly the membership threshold to be set in the fuzzy query. improve the performance of the FWC [12][16], it may Moreover the user can adjust the threshold to their own generate “roles” in the model. In this work rule base needs, based on the results of the training set, which is an pruning is used only for the query generation. impossible using a standard SQL query. 4.2. Fuzzy Query Generation 4. Results and Discussion In order to evaluate the fuzzy queries generated by the proposed approach, the Credit Card approval data set was The evaluation of the current approach is performed used. This data set contains 690 examples, each with 6 under two perspectives. The fuzzy weighted classifier is numeric and 9 categorical attributes. For confidentiality evaluated from a set of benchmark classification data sets reasons, the attributes’ descriptions have been omitted. and results are compared with the J4.8 decision tree The records are originally classified into two classes: induction and the Naïve Bayes algorithms, both computed within the Weka suite [17]. The evaluation of the fuzzy “approved” ( C1 ) or “rejected” ( C2 ), respectively with query generated by the proposed approach is performed 45.3% and 54.7% of apriori probability. The data set was under a target marketing scenario, using the Australian randomly divided into a training data set and a testing Credit Card data set. The fuzzy query results are data set containing respectively 460 and 230 records with compared with the results of the J4.8 decision tree the same class distribution. TM induction algorithm. As a “proof of concept”, the Fuzzy Query tool [20] is used to implement the fuzzy queries generated by the 4.1. Fuzzy Weighted Classifier Evaluation method. The software is a Win32-based application designed to run in conjunction with any ODBC-compliant A fuzzy weighted classifier, as described above, was database management systems. generated for a set of benchmarks data sets. For all data The fuzzy rule based classifier computes a solution sets the fuzzy classifier was generated using n = 5 fuzzy with all rules. The number of rules generated by the membership function of all numerical variables. classifier is very large, which makes difficult the Table 3 presents, for each benchmark data, the number interpretation. The pruning procedure presented above of variables, the number of classes and the error rates for was thus applied to the fuzzy rule base. The original rule the Fuzzy Weighted Classifier (FWC), for the J4.8 base with 75 rules for each class was thus reduced to 10 algorithm and for the Naïve Bayes (NB) algorithm. The rules by selecting the rules where IV > 0.5 . The decision J4.8 and NB algorithms were computed with all default tree induced by the J4.8 algorithm has 30 rules. The simplified rule base was translated into a fuzzy The Precision and Recall measures computed by the query for class C1 . The resulting fuzzy query is very fuzzy weighted query (Fuzzy SQL) with two different simple and represents the “good customer” model thresholds values over the testing set are shown in Table generated by the fuzzy classifier in query sentences as 4. The Precision and Recall metrics computed by the J4.8 presented in section 2.4. The understanding of the queries algorithm over the same testing set are also shown in generated in this example is not possible since the Table 4. It can be seen that the fuzzy query results over interpretations of variables are not available for this the testing set correspond roughly to the ones computed benchmark. from the training set. Moreover the selection of the threshold allows the user to choose between a greater 4.3. Fuzzy Query Evaluation Precision and Recall value, while the J4.8 result is fixed. The relative importance of Precision and Recall for a Direct marketing has become one of potential certain application depends on the user. Experienced users applications of data mining techniques, due to the great can work with high recall and low precision, as they are volume of available data from transactional databases capable to reject irrelevant information. Novice users, on [18][19]. The objective of direct marketing is to contact the other hand, need higher precision because they lack customers to offer products and services according to the judgment of the experienced users. their specific needs. In a direct marketing campaign, the Table 4: Fuzzy query evaluation in the testing set potential customers must be selected from the entire database in order to maximize the profit and minimize the Precision (%) Recall (%) losses due to an overcharge of marketing material sent to Fuzzy SQLδ = 0.25 92.30 60.60 wrong customers. Customers’ selection or generally Fuzzy SQLδ = 0.20 79.27 88.88 “target” selection is thus an important issue in direct J4.8 89.10 75.20 marketing and several data mining techniques may be applied to model customer profile according to the In most of applications in data intensive industries, the objective of the campaign. number of returned records is very large. Thus it is The evaluation of the fuzzy queries generated by the necessary to select the more suitable record for a given proposed approach is done for a target marketing application. As it can be seen from Figure 4, the precision scenario, using the Credit Card data set. In this data set, of the records returned by the fuzzy query with grater the relevant class for the application discussed in this membership values is very large. section is the class C1, which represents credit approvals, The decision tree induced by the J4.8 algorithm could such that the fuzzy query designed for the class C1 should be used to construct a standard SQL query to retrieve not select a record from the class C . potential customers in the data set. Nevertheless, the 2 query result would not be ordered as the fuzzy query The Precision and the Recall metrics as a function of result. the membership values of the returned records for the training set are shown in Figure 4. The results show that, 5. Conclusions as expected, as grater the membership value is, the higher is the Precision of the selected records, but the lower is This work has shown a methodology that automatically the Recall. An optimal threshold value can thus be chosen generates fuzzy queries from a training data set. The to allow high Precision and Recall rates. In this fuzzy queries are translated from a set of fuzzy weighted application, the optimal threshold value must be chosen classifier rules. A pruning procedure to simplify the fuzzy around 0.25. rule base and the resulting set of fuzzy queries was also proposed. Precision x Recall This methodology has been implemented in a general- 100 purpose fuzzy query tool, which can be connected to any 80 ODBC-compliant database management system. The fuzzy queries are able to retrieve an ordered list of 60 the records according to the fuzzy rule based model (%) 40 learned by the fuzzy classifier. The Precision and Recall analysis of the selected records of the fuzzy query allow 20 Precision Recall the user to select an optimal threshold for other data. 0 The proposed methodology was developed focusing 0.00 0.10 0.20 0.30 0.40 0.50 direct marketing application, but may be useful in other Membership kind of application such as case based reasoning or information retrieval. Figure 4: Precision and Recall for the training set. The future directions of this work are to apply this IEEE Transactions on Fuzzy Systems, Vol. 9, No. 4, methodology in unstructured data for application to text pp. 506 – 515, 2001. mining and web mining. [12] H. Ishibuchi and T. Yamamoto. Rule Weight Acknowledgments Specification in Fuzzy Rule-Based Classification Systems, IEEE Transactions on Fuzzy Systems, Vol. The authors are grateful to the Brazilian Research 13, No. 4, pp. 428 – 435, 2005. Agencies FINEP, CNPq and CAPES for the financial support for this research. [13] L. A. Zadeh. 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