Abstract:
Methods and apparatus are provided for generating a decision trees using linear discriminant analysis and implementing such a decision tree in the classification (also referred to as categorization) of data. The data is preferably in the form of multidimensional objects, e.g., data records including feature variables and class variables in a decision tree generation mode, and data records including only feature variables in a decision tree traversal mode. Such an inventive approach, for example, creates more effective supervised classification systems. In general, the present invention comprises splitting a decision tree, recursively, such that the greatest amount of separation among the class values of the training data is achieved. This is accomplished by finding effective combinations of variables in order to recursively split the training data and create the decision tree. The decision tree is then used to classify input testing data.

Description:
CROSS-REFERENCE TO RELATED APPLICATION(S)  
       [0001]     This application is a continuation of U.S. application Ser. No. 09/562,552, filed on May 2, 2000, the disclosure of which is incorporated by reference herein. 
     
    
     FIELD OF THE INVENTION  
       [0002]     The present invention is related to data classification and, more particularly, to methods and apparatus for generating decision trees using linear discriminant analysis.  
       BACKGROUND OF THE INVENTION  
       [0003]     The data classification problem is well known in the literature for its numerous applications in customer segmentation, customer targeting, and marketing. A supervised data classification problem is defined as follows. A set of records is given, each of which contains a set of feature variables and another predefined variable which is referred to as the class variable. This set of records is referred to as the training data. This training data is used to construct models in which the class variable may be predicted in a record in which the feature variables are known but the class variable is unknown. In order to perform this prediction, a model is constructed in which the feature variables are related to the class variable.  
         [0004]     Numerous techniques are known which relate the feature variables to the class variable. In order to perform the classification, numerous techniques are known for creating the relationship between the feature variables and the class variables. The techniques for building the relationships include, for example, DNF (Disjunctive Normal Form) Rules, decision trees, and Bayes classifiers, among others, see, e.g., Agrawal R., Ghosh S., Inielinski T., Iyer B., and Swami A., “An Interval Classifier for Database Mining Applications,” Proceedings of the 18th VLDB Conference, Vancouver, British Columbia, Canada 1992; Apte C, Hong S. J., Lepre J., Prasad S., and Rosen B, “RAMP: Rules Abstraction for Modeling and Prediction,” IBM Research Report RC 20271, June 1995; Quinlan J. R., “Induction of Decision Trees,” Machine Learning, Volume 1, Number 1, 1986; Shafer J., Agrawal R., and Mehta M., “SPRINT: A Scaleable Parallel Classifier for Data Mining,” Proceedings of the 22nd VLDB Conference, Bombay, India, 1996; Mehta M., Agrawal R., and Rissanen J., “SLIQ: A Fast Scaleable Classifier for Data Mining,” Proceedings of the Fifth International Conference on Extending Database Technology, Avignon, France, March 1996, the disclosures of which are incorporated by reference herein.  
         [0005]     In decision trees, which is the classification model technique to which the present invention is directed, a goal is to construct a hierarchical decomposition of the data. This hierarchical decomposition is used to create partitions of the data in which one of the class variables dominates. Tn order to predict the class variable for a record in which only the feature variables are known, the hierarchical partition in which the record lies is found. The dominant partition is used in order to predict the class variable. Decision trees are further discussed, for example, in J. C. Shafer, R. Agrawal, M. Mehta, “SPRINT: A Scalable Parallel Classifier for Data Mining,” Proc. of the 22th Int&#39;l Conference on Very Large Databases, Mumbai (Bombay), India, September 1996; M. Mehta, R. Agrawal and J. Rissanen, “SLIQ: A Fast Scalable Classifier for Data Mining,” Proc. of the Fifth Int&#39;l Conference on Extending Database Technology, Avignon, France, March 1996; and M. Mehta, J. Rissanen, and R. Agrawal, “DML-based Decision Tree Pruning,” Proc. of the 1 st Int&#39;l Conference on Knowledge Discovery in Databases and Data Mining, Montreal, Canada, August, 1995, the disclosures of which are incorporated by reference herein.  
         [0006]     In order to explain this further, the following example is presented. Consider a set of records comprising the following two feature variables: (i) age, and (ii) whether or not the person is a smoker; and the class variable comprising an indication of whether or not the person suffers from lung cancer. The distribution of records is such that smokers of higher age are more likely to suffer from lung cancer. A decision tree may partition the data in the following way. In the first phase, there are two branches formed; one corresponding to people who smoke, and another corresponding to people who do not. The branch corresponding to the people who smoke is more likely to have a higher percentage of lung cancer patients. At the second level, the tree is divided into two branches; one corresponding to the people with a higher age, and another corresponding to the people with a lower age. Thus, there are a total of four nodes at the lower level of the tree, and each of these nodes has a much lower number of records, but with different proportions of the class variable. Thus, for a given record, in which the class variable is not known, the class variable can be better predicted by finding the percentage of the class variables in the node in which the current record fits. Thus, this above-described method is one in which the decision trees are created by splits (i.e., division of the tree into branches) using a single variable.  
         [0007]     However, it would be highly advantageous to have methodologies in which the splits are created by linear combinations of variables which discriminate between the different nodes effectively.  
       SUMMARY OF THE INVENTION  
       [0008]     The present invention provides methods and apparatus for generating a decision trees using linear discriminant analysis and implementing such a decision tree in the classification (also referred to as categorization) of data. The data is preferably in the form of multidimensional objects, e.g., data records including feature variables and class variables in a decision tree generation mode, and data records including only feature variables in a decision tree traversal mode. Such an inventive approach, for example, creates more effective supervised classification systems. In general, the present invention comprises splitting a decision tree, recursively, such that the greatest amount of separation among the class values is achieved. This is accomplished by finding effective combinations of variables in order to recursively split the data and create the decision tree.  
         [0009]     More specifically, the present invention provides methodologies employing recursive discriminant analysis to create the combinations of features which are used to create the recursive splits. For example, consider a case where the feature variables comprise age, salary, and height in respective units. One possible example of a predicate or condition which is used to create a split may be: 30 age+25 salary+height  302 . Thus, all the records which satisfy this predicate will have some distribution of the class variable, whereas all records which violate this predicate will have a different distribution.  
         [0010]     It is also clear that different ways of performing the split will have different levels of discrimination as far as the nearest neighbor is concerned. Clearly, as mentioned above, it is unlikely that the particular kind of split where a single attribute or variable is used is the best possible split. A good split is one in which there is greater separation of the classes. For example, if in the above case the class variable corresponded to whether or not a person suffers from lung cancer, then a good split would be one in which one of the branches corresponds to all the people who have lung cancer, while the other branch corresponds to all the people who do not. This is because such a split would result in the greatest level of discriminatory power.  
         [0011]     Accordingly, in one aspect of the present invention, a method of generating at least a portion of a decision tree structure from one or more multidimensional data objects comprises the following steps. First, data associated with the one or more multidimensional data objects is represented as a node of the decision tree structure. Next, a condition is determined for dividing the data at the node into at least two subsequent nodes based on a discriminant measure which substantially maximizes separation between classes associated with the data. Lastly, the data is divided based on the condition, wherein the resulting nodes form at least a portion of the decision tree structure. The condition determining and data dividing steps may be repeated at each subsequent node.  
         [0012]     It is to be appreciated that the discriminant measure may be determined in accordance with a Fisher&#39;s discriminant technique. In such case, the data is divided at a split plane determined to be perpendicular to a direction determined according to the Fisher&#39;s discriminant technique, as will be explained below. The split plane may be determined to be a plane where an entropy measure is substantially optimized. The entropy measure may be determined in accordance with a gini index, as will be explained below. It is to be appreciated that other discriminant measures and/or other entropy measures may be employed in accordance with the present invention.  
         [0013]     In another aspect of the present invention, a method of performing data classification comprises the steps. First, at least a portion of a decision tree structure is generated from one or more multidimensional training data objects in accordance with the methodology described above. Then, the decision tree structure is used to predict one or more class variables associated with one or more multidimensional testing data objects in which only one or more feature variables are known. The class variable may be found by using tree traversal on the decision tree structure. A branch may be selected to traverse based on an entropy measure. For example, the branch yielding the lowest entropy measure may be selected as the branch to traverse.  
         [0014]     These and other objects, features and advantages of the present invention will become apparent from the following detailed description of illustrative embodiments thereof, which is to be read in connection with the accompanying drawings. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0015]      FIG. 1  is a block diagram illustrating a hardware implementation suitable for employing data classification using decision tree construction methodologies according to an embodiment of the present invention;  
         [0016]      FIG. 2  is a flow diagram illustrating an overall data classification procedure according to an embodiment of the present invention;  
         [0017]      FIG. 3  is a flow diagram illustrating an overall process for construction a decision tree according to an embodiment of the present invention;  
         [0018]      FIG. 4  is a flow diagram illustrating a procedure for using a decision tree to decide the class variable for a record in which only the feature variables are known according to an embodiment of the present invention;  
         [0019]      FIG. 5  is a flow diagram illustrating a procedure for choosing the correct branch of a decision tree while finding which nodes are relevant to a given record according to an embodiment of the present invention;  
         [0020]      FIG. 6  is a flow diagram illustrating a procedure for splitting up data at a given node of a decision tree according to an embodiment of the present invention; and  
         [0021]      FIGS. 7A through 7D  are diagrams illustrating an example of decision tree construction according to an embodiment of the present invention.  
     
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS  
       [0022]     The following description will illustrate the invention using an exemplary data processing system architecture. It should be understood, however, that the invention is not limited to use with any particular system architecture. The invention is instead more generally applicable to any data processing system in which it is desirable to perform supervised data classification via decision tree construction using linear discriminant analysis in order to achieve more effective classification or categorization results.  
         [0023]     Referring initially to  FIG. 1 , an exemplary architecture suitable for employing the present invention is shown. As illustrated, an exemplary system comprises a computer system  10 . The computer system  10  may be a separate system used to process input data. The computer system  10  may alternatively be part of a server which is coupled to multiple client devices via a network (e.g., Internet, private network, a local area network, or some other suitable network), wherein the client devices communicate with the server in order to access a database resident on the server. The methodologies of the invention may operate on all or portions of the data associated with the database. In any case, the system  10  comprises a central processing unit (CPU)  20 , a memory  30  and a disk  40 . The memory  30  may comprise a cache  32  and a main memory  34 . In a training mode, the data input to the computer system  10  includes a training data set, including records with feature variables and class variables, which is used to generate a decision tree. In a testing mode, the data input is a testing data set, including records with only feature variables and no class variables, which is the subject of the classification or categorization operation. Since an input data set may be very large, data sets are preferably stored on the disk  40 . The calculations for creating a decision tree from a training data set and using the decision tree to perform supervised data classification on a testing data set, according to the invention, are performed in the CPU  20  in association with the cache  32  and the main memory  34  of the memory  30 . The cache  32  is preferably present in the system in order to speed up calculations. The output of system  10  in training mode is the decision tree. In the testing mode, the output is the classification results, e.g., the class variable associated with the input record.  
         [0024]     Accordingly, in this illustrative embodiment, the decision tree construction and implementation operations of the present invention are executed in association with the computer system  10 . All or a portion of the results generated in association with the computer system may be presented on a display (not shown) to a system user, if so desired. Further, in one embodiment, software components including program instructions or code for performing the methodologies of the invention, as described herein, may be stored in one or more memory devices described above and, when ready to be utilized, loaded in part or in whole and executed by the CPU.  
         [0025]     Referring now to  FIG. 2 , a flow diagram is shown illustrating an overall data classification or categorization procedure for performing training and testing of a data set according to an embodiment of the present invention. The overall process begins at block  200 . In the overall process, the first step comprises creating a decision tree from training data, including feature variables and a class variable, which is then used for the purpose of categorization. This is accomplished in step  210 . Details of this process are explained below in the context of  FIG. 3 . Once the decision tree is constructed using the training data, in step  220 , a test record to be categorized is received by the system. This record is categorized by employing the decision tree, in step  230 . Steps  220  and  230  are repeated for each record of the testing data set. The process ends at block  240  when all test records have been classified. Classification results may be stored and/or output to a system user.  
         [0026]     Referring now to  FIG. 3 , a flow diagram is shown illustrating an overall process for construction a decision tree according to an embodiment of the present invention. It is to be appreciated that the process of  FIG. 3  corresponds to step  210  of  FIG. 2 . Unlike the existing decision tree construction techniques described above, the present invention uses repeated discriminant analysis in order to construct the tree, as will be explained in detail below. The inventive methodology uses a top down construction of the decision tree in order to build the nodes successively. The top down construction is facilitated by the use of a tracking parameter which we refer to as LIST.  
         [0027]     The process begins at block  300 . It is to be understood that the input to the process is the data set including the feature variables and the class variable. A “node” in this decision tree corresponds to a subset of the data which has been created by the various split points at the various levels. The starting point node is the entire data set, and the decision tree (and lower nodes) are built by creating further splits in the data. For example, Age 30 is a predicate or condition which creates a split, and divides a given node into two depending upon the age. Of course, as explained herein and below in further detail, the present invention does not rely on single attributes (e.g., Age) to create splits, but finds particular directions which are more relevant.  
         [0028]     Thus, in step  310 , the LIST is initialized to the root node of the tree. This initialization of the LIST is used to pick successive lower level nodes. In step  320 , the next node on LIST is selected. This node is then deleted from LIST. In step  330 , the condition or predicate which divides the data in the node with the greatest level of discrimination is selected. Step  330  will be further explained below in the context of  FIG. 6 . In step  340 , the two nodes of the tree corresponding to these two branches are constructed. For each node, we project the data into a lower dimensional (feature) space which is the plane perpendicular to the dividing condition. This is done in step  350 . It is to be understood that the terms dimension and feature may be used interchangeably. The process of projecting the data onto a plane perpendicular to a given direction changes the data representation. The projected data contains one dimension less, and its attribute set is different. An example of this projection used for decision tree construction according to the invention will be explained below in the context of  FIGS. 7A through 7D . In step  360 , for all nodes which have more than a certain number of points, we add them to LIST, if any. It is to be appreciated that the term “point” refers to a database record. Note that in step  320 , LIST was reduced by one node. So, after this iteration, the LIST size could either increase or decrease. It is to understood that LIST may increase or decrease in a given iteration since, in step  360 , no node, or one or more nodes may be added to LIST, whereas, in step  320 , LIST is reduced by one node. Thus, at this point, the LIST is checked to determine whether or not it is empty. If LIST is not empty, then the process returns to step  320  for the next iteration where the descendants of the next node on LIST are examined. If the LIST is empty, then the process returns the entire decision tree, which has thus been constructed. This is done in step  370 . The process ends at block  380 .  
         [0029]     Referring now to  FIG. 4 , a flow diagram is shown illustrating a procedure for using a decision tree to decide the class variable for a record in which only the feature variables are known according to an embodiment of the present invention. It is to be appreciated that the process of  FIG. 4  corresponds to step  230  of  FIG. 2 . The process begins at block  400 . In step  410 , a variable referred to as Current Node is set to the root node. In step  420 , the correct branch to traverse at the current node is found. A detailed process for finding the correct branch to traverse is explained below in the context of  FIG. 5 . In step  430 , the process determines if the descendant branch node is a leaf node. A leaf node is a node which has no descendants. If so, then in step  440 , the process reports the distribution of the class variables in the leaf node. This distribution of class labels indicates the relative probabilistic distributions of the various classes. If the descendant branch is not a leaf node, then the current node is set to the descendant branch node, in step  450 , and the process returns to the step  420 . Thus, given an input record with a set of feature variables, the process of  FIG. 4  outputs the class variable.  
         [0030]     Referring now to  FIG. 5 , a flow diagram is shown illustrating a procedure for choosing the correct branch of a decision tree to traverse while finding which nodes are relevant to a given record according to an embodiment of the present invention. It is to be appreciated that the process of  FIG. 5  corresponds to step  420  of  FIG. 4 . The process begins at block  500 . The input to the process is the possible branch nodes. In step  510 , the process calculates the “entropy” of each node which is a child of the current node. It is to be appreciated that, in order to calculate a level of discrimination, the present invention employs a concept referred to as “entropy.” The entropy of a set of branches of a decision tree refers to the level of separation of the various class variable distributions under that node. For example, let us consider a class variable which has two states. In such a case, let p i  be the percentage of the data which takes on a particular state from node i, where i varies from 1 though n, for each of the n possible branches of that node. Then, one measure of the entropy is referred to as the “gini index.” The gini index for the node from which the n branches emanate is defined as: . Note that the lower the value of the gini index, the greater the amount of discrimination, and the higher the value of the gini index, the less the level of discrimination. It is to be appreciated that the gini index is further described in M. Mehta, R. Agrawal and J. Rissanen, “SLIQ: A Fast Scalable Classifier for Data Mining,” Proc. of the Fifth Int&#39;l Conference on Extending Database Technology, Avignon, France, March 1996, the disclosure of which is incorporated by reference herein. It is to be further appreciated that other entropy measures may be employed, see, e.g., Agrawal R., Ghosh S., Imielinski T., Iyer B., and Swami A., “An Interval Classifier for Database Mining Applications,” Proceedings of the 18th VLDB Conference, Vancouver, British Columbia, Canada 1992, the disclosure of which is incorporated by reference herein. The node with the least entropy is picked as the most desirable branch. This is returned as the output in step  520 . The process ends at block  530 .  
         [0031]     Referring now to  FIG. 6 , a flow diagram is shown illustrating a procedure for splitting up data at a given node of a decision tree according to an embodiment of the present invention. It is to be appreciated that the process of  FIG. 6  corresponds to step  330  of  FIG. 3 . The process begins at block  600 . In step  610 , a measure referred to as the “Fisher&#39;s discriminant” of the data is calculated in the current node. The “Fisher&#39;s discriminant” is used to perform splits recursively such that the maximum level of discrimination among the classes is attained. Use of Fisher&#39;s discriminant is well known and is described in the prior art literature, e.g., Chakrabarti S., Dom B., Agrawal R., and Raghavan P., “Scalable Feature Selection, Classification and Signature Generation for Organizing Large Text Databases into Hierarchical Taxonomies,” VLDB Journal, 7(3), pp. 163-178, 1998, the disclosure of which is incorporated by reference herein.  
         [0032]     Fisher&#39;s discriminant method finds a direction along which the relative class separation as measured by the ratio of the interclass to intraclass variance is maximized. More specifically, let X and Y be two sets of database records corresponding to a two class problem, and let (X) and (Y) be the respective centroids. A centroid is obtained by averaging all the d-dimensional records in the set and is a vector containing d dimensions or features. Further, let the covariance matrix for the sets of points X and Y be denoted by M(X) and M(Y). The covariance matrix for a set of points is a d d matrix (d is the number of dimensions), in which the entry (i, j) is the covariance between dimensions i and j. Let M=(½) (M(x)+M(y)) and =(X)−(Y). Then, the Fisher&#39;s discriminant (or direction) is given by M −1 , where M −1  is the inverse of the matrix. By using the Fischer&#39;s discriminant repeatedly, in accordance with the invention, splits may be found recursively in a way so that, in each case, there is the maximum amount of discrimination among the different nodes. It is to be appreciated that other discriminant measures may be employed.  
         [0033]     Thus, once the Fisher&#39;s direction has been determined in step  610 , the split plane is determined as perpendicular to the Fisher&#39;s direction. This is done in step  620 . The plane is made to pass through one of the points in the database. This is done by choosing the point for which the corresponding gini index (or other entropy measure) is optimized so as to maximize class separation. This is done by picking the database points one by one and calculating the measure for each possibility. We pick the optimum value out of these possibilities. Accordingly, in step  630 , the data is split into two nodes in a way that maximizes discrimination between classes. Thus, given data at a given node, the process of  FIG. 6  yields the best possible data split at the node. The process of  FIG. 6  is then repeated to determine further nodes until the decision tree is completed. It is to be appreciated that, as per the process described in the context of  FIG. 3 , the decision tree is complete when LIST is empty.  
         [0034]     Referring now to  FIGS. 7A through 7D , diagrams are shown illustrating an example of decision tree construction according to an embodiment of the present invention.  FIG. 7A  shows a two-dimensional data set, where the feature set consists of two possible attributes: age and salary. There are two classes in the data set, which are respectively marked by “x” and “o” in  FIG. 7A . In order to illustrate the process better, an artificial example has been intentionally created in which there are four natural clusters (close groups of points) in the data marked A, B, C, D. In the first step, Fisher&#39;s discriminant is found. This is illustrated in  FIG. 7A . This defines the split plane as the plane which is perpendicular to this vector (shown by dotted line). Note that this plane is 1-dimensionality lower than the whole space. Since this is a 2-dimensional problem, the split plane is a 1-dimensional line, and is indicated by the dotted line in  FIG. 7A . Note that if this was a 3-dimensional problem, the split plane would have been 2-dimensional. Furthermore, the Fisher&#39;s direction shown in the diagram is somewhat hypothetical (for the purpose of illustration), and may not correspond with a precise mathematical analysis of the diagram shown here. The split plane divides the data into two sections, one containing the clusters A, B and the other containing the clusters C, D. The corresponding decision tree is shown in  FIG. 7C . At this stage, each of these two groups of data are treated independently as shown in  FIG. 7B . In the next step, each of these groups of data are (independently) projected onto the 1-dimensional split plane. Both independent projections are shown in  FIG. 7B . At this stage, the O-dimensional split point which maximizes the separation is found. The corresponding split and the decision tree is illustrated in  FIG. 7D .  
         [0035]     Accordingly, as explained in detail above, the present invention provides methodologies for creating a decision tree which provides effective discrimination between the nodes at every level of the tree. The invention creates splits at every level of the tree which maximize the level of separation between the different classes. This results in a decision tree which is more effective and useable for performing effective classification.  
         [0036]     Although illustrative embodiments of the present invention have been described herein with reference to the accompanying drawings, it is to be understood that the invention is not limited to those precise embodiments, and that various other changes and modifications may be affected therein by one skilled in the art without departing from the scope or spirit of the invention.