Using Decision Boundary to Analyze Classifiers

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1 Using Decision Boundary to Analyze Classifiers Zhiyong Yan Congfu Xu College of Computer Science, Zhejiang University, Hangzhou, China Abstract In this paper we propose to use decision boundary to analyze classifiers. Two algorithms called decision boundary point set (DBPS) and decision boundary neuron set (DBNS) are proposed to obtain the data on the decision boundary. Based on DBNS, a visualization algorithm called SOM based decision boundary visualization (SOMDBV) is proposed to visualize the high-dimensional classifiers. The decision boundary can give an insight into classifiers, which cannot be supplied by accuracy. And it can be applied to select proper classifier, to analyze the tradeoff between accuracy and comprehensibility, to discovery the chance of over-fitting, to calculate the similarity of models generated by different classifiers. Experimental results demonstrate the usefulness of the method. 1. Introduction Classification is an important problem in machine learning, and it has many applications in real world. There are many classifiers now [1], whose performance is usually estimated by accuracy. Accuracy is the proportion of correct predictions to all predictions [1]. But accuracy is a raw performance score, and it cannot give much insight to classifiers [2]. Accuracy cannot tell which of the data are classified right and which are not, and it is unable to reveal the relative positions of the data predicted correct and incorrect. Data sets in real world are mostly highdimensional. Users usually get an intuitive insight by high-dimensional data visualization algorithms, which are unsupervised. The class boundary cannot be clearly visualized by these algorithms [3]. Because there is no powerful tools users cannot understand classifiers very well. [2] proposes to use decision region connectivity to analyze high-dimensional classifiers, which can be used to analyze convexity of decision region. The algorithm is independent of the dimension of the data set. [3] proposes an algorithm named SVMV to visualize classification results of Support Vector Machine(SVM) [4] using self-organizing mapping(som) [5]. The algorithm can clearly visualize SVM classification boundary and the distance between data and classification boundary in a 2-D map. But the algorithm substitutes weight matrix of SOM for input data of SVM decision function, which limits its application to other classifiers. The method for using decision boundary to analyze classifiers is proposed in this paper. Decision boundary is the distinguishing boundary classifier uses to predict data, so the predicting labels of the data on the two sides of boundary are different. Two algorithms are provided to obtain the data on the classifier s decision boundary. The first algorithm named decision boundary point set (DBPS) is used to get the points near the decision boundary of classifiers. The second algorithm named decision boundary neuron set (DBNS) is used to get the neurons of SOM near the decision boundary of classifiers. Based on DBNS an algorithm named SOM based decision boundary visualization (SOMDBV) is proposed to visualize the decision boundary of high-dimensional classifiers in a 2-D SOM map. In the next section, the procedures of DBPS, DBNS and SOMDBV are described. In section 3, analysis of classifiers using decision boundary is given. In section 4, experiments are performed to demonstrate the usefulness of the algorithms and analysis proposed. Conclusion is drawn in section 5. We assume the output of classifiers is discrete class label, not the probability of input data to belong to some class, although the latter one can be transformed to the former easily. 2. Decision boundary algorithms In this section, we describe the details of following three decision boundary algorithms, DBPS algorithm, DBNS algorithm and SOMDBV algorithm.

2 A model will be obtained after a classifier is trained on the training data set. When the new data is coming, the model will be used to predict the labels of the data. That is the normal usage of classifiers. Some classifiers behave like a white box, and can provide users with comprehensible results. For example, RIPPER [6], a well-known rule-based algorithm, learns set of rules, and the obtained rules give users a good understanding. But some other classifiers behave like a black box, and users are unable to understand what they have learned! SVM is an example of this kind of classifiers. The knowledge obtained by the trained SVM model is hidden in the decision function, which is complicated and abstract for user to understand. Users even do not know what has happened in the latter case. However, every classifier predicts the labels of data according to some guide lines. For example, RIPPER predicts the labels according to the rule set it has learned, while SVM predicts the labels according to the decision function it has trained. The guide lines, in spite of their forms, form decision boundaries in the input data space. The procedure of prediction can be seen as finding the relation between input data and boundaries. Using a trained classifier to classify data is equal to using the decision boundary of the same classifier to partition data. If the decision boundary of the classifier is obtained and visualized, users will have an insight into the classifier, which will help users to select proper classifier. The forms of the knowledge which classifiers adopt to construct decision boundary are diverse, so acquiring the analytical equations of the decision boundary is an exhausting task. Instead we obtain some sample points on the decision boundary to analyze classifiers. DBPS algorithm is used to obtain the sample point set near the decision boundary in the input space, while DBNS algorithm is used to obtain the sample neurons near the decision boundary on the 2-D SOM map, and SOMDBV makes use of DBNS to visualize the decision boundary on the 2-D SOM map DBPS algorithm There are two methods for obtaining the points on the decision boundary. The first one is internal method, which uses classifiers internal forms of knowledge to obtain the points on the boundary. For example, using the decision function of SVM we can compute the points on the boundary. The second one is external method, which uses some approximation methods to get the points near the boundary. The first method s advantage is that it can generate accurate points on the boundary, and its disadvantage is that every classifier needs its own implementation of this method because classifiers forms of knowledge are diverse. While the second method can be applied to more classifiers, but the points generated are not as accurate as the first one. DBPS algorithm adopts the second method, and generates the points near the boundary. DBPS uses binary search to calculate the points of intersection of decision boundary and connection between two data points predicted as different by classifiers. The detail of DBPS can be seen in Algorithm 1. Users can control the precision of the points by adjusting iter_no and toler. Algorithm 1* The Decision Boundary Point Set Algorithm generates the point set near the boundary X is the set of sample points. B is the set of decision boundary points. c(x) is the classifier function. iter_no is the limit of iterations numbers. toler is the tolerance of the boundary. for all x X do if c(x) == a then Xa Xa {x} else Xb Xb {x} for all xa Xa do for all xb Xb do B B {DBP (xa, xb, c, iter_no, toler)} function DBP (x1, x2, c, iter_no, toler) x_bound (x1 + x2) / 2 for i = 1 : iter_no do if distance(x1, x2) / 2 < toler then break if c(x_bound) == c(x1) then x1 x_bound else x2 x_bound x_bound (x1 + x2) / 2 return x_bound * The function distance(x1, x2) is trivial, so we do not describe the procedure of this function here.

3 If the classifier is high-dimensional, it needs highdimensional data visualization algorithm to visualize the obtained decision boundary point set DBNS algorithm There are two methods for visualizing the decision boundary of a high-dimensional classifier. The first one is to use DBPS algorithm to obtain the decision boundary point set in input space which is visualized by some high-dimensional data visualization method. The second one is to project the input data onto some low-dimensional map and calculate the point set on the decision boundary in the map space. SOMDBV algorithm adopts the second method, and DBNS algorithm is used to obtain the neurons near the decision boundary on the 2-D SOM map. SVMV algorithm uses the decision function of SVM to calculate the distance between the neuron and the classification boundary [3]. In DBNS algorithm, classifiers employ the weights of neurons as input to predict the labels of the data projected to these neurons. [7] adopts interpolation to get extended weight matrix, which avoids high computational complexity. We adopted the same process to obtain the neurons near the decision boundary. The method used to get the neurons near the boundary is the external method in 2.1, which is the same as DBPS algorithm. The topology of SOM used in this paper is rectangle grid, the algorithm can be applied to other topologies easily, however. As seen in Figure 1(a), if four neurons of the rectangle are predicted the same labels, then we suppose there is no neurons inside the rectangle which are near the boundary. Otherwise we use the interpolation to get neurons e, f, g, h, i, then partition the rectangle to 4 smaller ones. And we continue partitioning the small rectangles whose labels are not the same until the times achieve the number user given (Figure 1(b)). At last the center neuron of the rectangle is selected as the one near the decision boundary (Figure 1(c)). (a) (b) (c) Figure 1. Three cases of finding the neurons near the decision boundary. (a) predictions are the same; (b) predictions are not the same; (c) the last step. The detail procedure of DBNS algorithm can be seen in Algorithm2. Algorithm 2* The Decision Boundary Neuron Set Algorithm finds the neuron set near the boundary N is the neurons of the SOM, whose size is m n. B is the set of neurons near the decision boundary. c(x) is the classification model. iter_no is the limit of iterations numbers. for i = 1 : m-1 do for j = 1 : n-1 do N[] {N(i,j), N(i+1,j), N(i+1,j+1), N(i,j+1)} B B GetDBNeuron(N[], c, iter_no) function GetDBNeuron(N[], c, iter_no) dbn = {} if c(n[]) are not the same then if iter_no == 1 then dbn {GetCenterNeuron(N[])} else N2[][] Partition(N[]) for i = 1:4 do dbn dbn GetDBNeuron(N2[i], c, iter_no-1) return dbn * The function GetCenterNeuron(N[]) and function Partition(N[]) are trivial, so we do not describe procedures of these two functions here SOMDBV algorithm The SOMDBV algorithm adopts the second method in 2.2. It first projects the data onto 2-D SOM map, then uses DBNS algorithm to obtain the neurons near the decision boundary, and at last display the labels of the data, classifier s predictions of each neuron and the neurons near the decision boundary. The procedure of SOMDBV algorithm is as follows: 1) Classifier is trained on data set X to get the classification model C. 2) SOM algorithm is trained on the same data set X to get the weights W. 3) C is used to classify the W, and gives predictions L. 4) DBNS algorithm is used to get the neuron set N near the decision boundary.

4 5) Input data set X, classifier s predictions L and decision boundary neuron set N are displayed on the 2- D SOM map. 3. Applications of decision boundary What decision boundary can be used to analyze is as follows: 1) The distance between the data and decision boundary is clearly understood by users, which cannot be provided by accuracy. This will help user to select proper classifier. The classifier with boundary in the middle of the data belonging to different class is usually better than the classifier with boundary near data of one class and far from data of other class. It is also able to tell users in which region of the data space the classifier makes incorrect predictions. If users know the region which the new data is likely to fall into and there are several classifiers, they may be able to choose the proper classifier. 2) There is the tradeoff between accuracy and comprehensibility in data mining models [8]. The visualization of decision boundary is able to give an insight into the classifiers with high accuracy which usually results in low comprehensibility. The visualization will help users to analyze the tradeoff between accuracy and comprehensibility. 3) Over-fitting is struggled to avoid by classier users. Visualization of decision boundary can give insight to over-fitting. Given the same accuracy, the generalization of classifiers with complicated decision boundary is usually not as good as the ones with simpler decision boundary. This can help users to select the classifier with higher generalization, or set the proper parameters for classifier to obtain a more general model. 4) Decision boundary can be adopted to define the similarity of two models obtained by different classifiers. For example, the proportion of the region two classifiers predict the same labels to the whole region the data fall into may be a measure of models similarity. Then we can conclude two models are the same in the case of some given similarity. Given the similarity, one model may be able to be transformed into the model trained by another classifier, which can overcome the drawback of some classifiers. For example, trained artificial neural network (ANN) can be transformed into rule set by extracting rules from ANN, which improves the comprehensibility of the trained ANN with high accuracy [9]. The method for calculating similarity can be used to calculate the fidelity for extracting rules for ANN [9]. 5) Diversity among the base classifiers is important when constructing a classifier ensemble [10]. The decision boundary can be used to calculate the diversity. For example, the integral of the difference between two classifiers decision boundaries may be a measure of diversity, which reflects the difference of partitions of the data space between two classifiers. 4. Experimental results In this section, two experiments are performed to demonstrate the usefulness of proposed algorithms. Classifiers we used are RIPPER and SVM, and WEKA [11] is used for the implementation of these two classifiers. Gaussian kernel with parameter gamma is used as the kernel function of SVM. Implementation of SOM is from MATLAB SOM Toolbox. For SOM, the total iteration numbers are 1000, and the topology is grid topology. The size of SOM is Experimental results of DBPS DBPS algorithm is used to generate the decision boundary point sets of RIPPER and SVM for diamond data. Diamond data is two-class simulation data with 2 dimensions, whose boundary is diamond, and its two diagonals length is Each class has 100 data points generated randomly. The results are shown in Figure 2, where cross symbols denote the data inside the diamond, while star symbols denote the data outside the diamond, and line between data of different classes is decision boundary. The decision boundary generated by RIPPER is like a cross, while the one generated by SVM is like a diamond as seen in Figure 2. The decision boundary generated by SVM is almost in the middle of the data of different class, while the decision boundary by RIPPER is close to data of one class and far from the data of other one. The position of decision boundary by SVM is more proper than that by RIPPER, so SVM will be the proper one for diamond data. At the same time, the shape of decision boundary generated by RIPPER is more regular, which can be understood better by users. The decision boundary by SVM is more complicated, and it is more difficult for user to understand it. In this experiment, there is the tradeoff between a powerful model with high accuracy and a transparent model with high comprehensibility.

5 (a) (a) (b) Figure 2. Decision boundary point set (a) by RIPPER; (b) by SVM using Gaussian kernel with gamma = Experimental results of SOMDBV The data set for SOMDBV algorithm is Johns Hopkins University Ionosphere database, which is from UCI machine learning repository [12]. The data set contains 351 records with 34 dimensions, of which 225 records are labeled Good, and 126 are labeled Bad. The results are shown in Figure 3, where square symbols denote the data of class Bad, and triangle symbols denote the data of class Good. Cross symbols denote the neurons predicted Bad, and dot symbols denote the neurons predicted Good. The line of Figure 3 denotes the decision boundary. As the analysis of 4.1, SVM in Figure 3(b) is more proper than RIPPER. (b) (c) Figure 3. Visualization of the Ionosphere data set (a) by RIPPER; (b) by SVM using Gaussian kernel with gamma = 2; (c) by SVM using Gaussian kernel with gamma = 20.

6 As seen in Figure 3(b) and Figure 3(c), the decision boundary generated by SVM using Gaussian kernel with gamma being 20 is more complicated than that generated by SVM with gamma being 2. So the SVM using Gaussian kernel with gamma being 20 is more likely to over-fit the data. And this conclusion agrees with the experience and the common sense. The number of neurons predicted the same labels by RIPPER and SVM with gamma being 2 is larger than that by SVM with gamma being 20 and SVM with gamma being 2. So although the two SVM models are generated by the same classifier using different parameter, their similarity is less than that of SVM with gamma being 2 and RIPPER. 5. Conclusion and future work In this paper, a novel method for using decision boundary to analyze classifiers is proposed. Two algorithms are proposed to obtain data on decision boundary in different spaces. DBPS algorithm is used to obtain point set on decision boundary in input data space, while DBNS algorithm is used to obtain neuron set on decision boundary on 2-D SOM map. SOMDBV algorithm using DBNS algorithm is proposed to visualize the decision boundary of highdimensional classifiers. With the help of decision boundary, users can get an insight into the classifiers. Decision boundary can be used to select proper classifier, to reveal the tradeoff between accuracy and comprehensibility, to detect over-fitting, to calculate the similarity of classifiers and to calculate diversity in ensemble learning. This paper has not supplied calculation method for obtaining similarity and diversity. This work will be done in future, and the decision boundary will be used to analyze extracting rules from ANN and ensemble learning. Acknowledgements [3] X. Wang, S. Wu, and Q. Li, SVMV a novel algorithm for the visualization of SVM classification results, Advances in Neural Networks - ISNN 2006, Springer-Verlag, Berlin Heidelberg, 2006, pp [4] Vapnik, V.N., The nature of Statistical Learning Theory, Springer, Berlin Heidelberg, [5] Kohonen, T., Self-Organizing Maps, Springer, Berlin Heidelberg, [6] W. Cohen, Fast effective rule induction, Proceedings of the 12th International Conference on Machine Learning, Morgan Kaufmann, Tahoe City, CA, 1995, pp [7] S. Wu, and W.S. Chow, Support vector visualization and clustering using self-organization map and support vector one-class classification, Proceedings of IEEE International Joint Conference on Neural Networks, Portland, USA, 2003, pp [8] U. Johansson, L. Niklasson, and R. König, Accuracy vs. comprehensibility in data mining models, Proceedings of 7th International Conference on Information Fusion, Stockholm, Sweden, 2004, pp [9] R. Andrews, J. Diederich, and A.B. Tickle, Survey and critique of techniques for extracting rules from trained artificial neural networks, Knowledge-Based Systems, Elsevier, Amsterdam, 1995, pp [10] E.K. Tang, P.N. Suganthan, and X. Yao, An analysis of diversity measures, Machine Learning, Springer, Berlin Heidelberg, 2006, pp [11] Witten, I.H., and E. Frank, Data Mining: Practical machine learning tools and techniques, 2nd Edition, Morgan Kaufmann, San Francisco, [12] P.M. Murphy, and D.W. Aha, UCI Repository of machine learning databases [ Irvine, CA: University of California, Department of Information and Computer Science This paper is supported by 863 plan (No. 2007AA01Z197), and National Natural Science Foundation of China (No ). References [1] S.B. Kotsiantis, I.D. Zaharakis, and P.E. Pintelas, Machine learning: a review of classification and combining techniques, Artificial Intelligence Review, Springer, Berlin Heidelberg, 2006, pp [2] O. Melnik, Decision region connectivity analysis: a method for analyzing high-dimensional classifiers, Machine Learning, Kluwer, Netherlands, 2002, pp

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