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MATHEMATICAL METHOD IN PATTERN RECOGNITION Multidimensional Data Visualization Based on the Minimum Distance Between Convex Hulls of Classes A. P. Nemirko St. Petersburg Electrotechnical University “LETI”, St. Petersburg, Russia e-mail: [email protected] Abstract—The problem of data visualization in the analysis of two classes in a multidimensional feature space is considered. The two orthogonal axes by which the classes are maximally separated from each other are found in the mapping of classes as a result of linear transformation of coordinates. The proximity of the classes is estimated based on the minimum-distance criterion between their convex hulls. This criterion makes it possible to show cases of full class separability and random outliers. A support vector machine is used to obtain orthogonal vectors of the reduced space. This method ensures the obtaining of the weight vector that determines the minimum distance between the convex hulls of classes for linearly separable classes. Algorithms with reduction, contraction, and offset of convex hulls are used for intersecting classes. Experimental studies are devoted to the application of the considered visualization methods to biomedical data analysis. Keywords: multidimensional data visualization, machine learning, support vector machine, biomedical data analysis DOI: 10.1134/S1054661818040247 INTRODUCTION Despite the rapid development of the neural network approach to pattern recognition, there remains a wide field of problems characterized by the description of classes in a multidimensional feature space and the search for solutions in it. This is especially the case in biology and medicine. It is important for the researcher to know how many classes intersect and to try to construct the best separating surface in order to solve these problems. In a multidimensional space, the area of intersection of classes is invisible, and the decision rules are constructed based on some theoretical hypotheses. However, there are often cases where a more detailed study of the intersection area is of particular importance, for example, in the case of a high cost of medical diagnostic errors or errors in the detection of outlier points that do not fit into a description of some biological species. The problem of an adequate mapping of the intersection area to a 2D space arises. Otherwise, this problem can be called the visualization of classes on a plane. The following statistical methods for dimension reduction and visualization are used for this purpose: principal component analysis (PCA) [1] and the method of mapping to the plane [2–4] based on Fisher’s discriminant analysis (FDA) [5]. Unfortunately, these methods do not always give an exhaustive picture of the intersection of classes and do not always reflect cases of complete separation or random outli- Received June 10, 2018 ers [2]. Errors that manifest themselves in additional experimental points in the area of intersection of classes occur in the mapping of multidimensional classes to a plane due to information losses. The problem arises to find such a mapping of classes to the plane that the number of experimental points that fall into the intersection area on the plane is the same as in the multidimensional space (a smaller number is impossible). In this paper, the intersection of classes is considered as the intersection of their convex hulls. Therefore, the class intersection area is considered as the intersection of their convex hulls, both in the multidimensional space and in the plane. The minimum distance between their convex hulls is considered as a space transformation criterion. Then the visualization problem is formulated as follows. Find a subspace of dimension 2 in orthogonal projections onto which the minimal distance between the convex hulls of the classes is maximal. USE OF RECOGNITION PROCEDURES Let x i , i = 1,2,..., N be the vectors of the training set X in the n-dimensional feature space. They belong to one of the two classes ω1, ω2 . The linear recognition problem at the learning stage is to find the hyperplane g(x) = wT x + w0 = 0 which optimally classifies all the vectors of the training set, where w is the weight vector and w0 is the scalar threshold. Here, w is sought so that the hyperplane g(x) , which is perpendicular to w , best separates the classes (in the sense of the learning criterion). A straight line perpendicular to g(x) is called the ISSN 1054-6618, Pattern Recognition and Image Analysis, 2018, Vol. 28, No. 4, pp. 712–719. © Pleiades Publishing, Ltd., 2018. MULTIDIMENSIONAL DATA VISUALIZATION BASED (a) 713 N w= ∑λ y x i =1 i i i w0 = wT x i − yi , A B where λi are the Lagrange multipliers. For linearly separable classes, other algorithms also solve the problem of finding the minimum distance between convex hulls of classes (nearest point problem, NPP): the SK algorithm (Schlesinger–Kozinec algorithm) [9] and the MDM algorithm (Mitchell– Dem’yanov–Malozemov algorithm) [10]. These algorithms can also be used to generate a weight vector for class visualization. D (b) CASE OF LINEARLY INSEPARABLE CLASSES A B D Fig. 1. To the definition of the proximity of two convex hulls to each other: (a) D is the minimum distance between A and B and the corresponding weight vector, (b) D is the minimum penetration depth between A and B and the corresponding weight vector. separating line (or axis). In the direction of the separating axis, i.e., toward w , classes are the least close to each other. Thus, the procedure of finding w through the definition of g(x) can be considered as a class separability criterion. For optimal g(x) , w is also optimal. Therefore, it can be considered as the first axis for visualizing classes on a plane. The second axis is sought in the plane perpendicular to w by the learning criterion (the same or another). Further, the recognition procedure of the support vector machine is mainly considered [6, 7]. It is known that in the case of disjoint classes the support vector machine (SVM) calculates the minimum distance between convex hulls of classes and the corresponding weight vector w [8]. The problem of finding the optimal separating hyperplane is formulated as follows for the SVM in the case of linearly separable classes ⎧⎪ w 2 → min ⎨ T ⎪⎩ yi (w x i + w0 ) ≥ 1, λi > 0, i = 1,2,… , N , (1) where yi is the class indicator (+1 for ω1 and –1 for ω2 ) for each x i . This is the problem of convex quadratic programming (with respect to w , w0 ) in a convex set taking into account the set of linear inequalities. Its solution has the following form: PATTERN RECOGNITION AND IMAGE ANALYSIS In the case of linearly inseparable classes A and B , it is possible to use the criterion of minimum penetration or intersection depth of classes D, which is used in collision detection problems, instead of the minimum-distance criterion between their convex hulls Conv(A) and Conv(B) [11, 12]. This criterion is defined as the minimum value by which it is necessary to offset B in any direction, so that Conv(A) does not intersect B . Finding such a direction gives the required penetration vector w . Following the same procedure, it is possible to conclude that in the case of separable classes the distance between them can be defined as the minimum distance by which it is necessary to offset B in any direction so that Conv(A) and B begin to intersect (Fig. 1). Many methods for calculating penetration depth use the Minkowski sum, which for two sets A and B is defined as follows: A ⊕ B = {a + b : a ∈ A, b ∈ B}. Then for two sets of nodes of convex hulls A and B it is possible to form a configuration space (configuration space obstacle, CSO) denoted as S in the form S = A ⊕ (− B). The use of a configuration space makes it possible to replace the calculation of the penetration depth D, as the distance between the two nearest points of the convex hulls, with the distance from the origin to the nearest point on Conv(S) as shown in Fig. 2. There are algorithms for finding the penetration depth for 2D and 3D cases [11, 12], but the penetration depth calculation for the nD case is a difficult and computationally complex task. This problem is greatly simplified if projections of convex hulls (or classes themselves) to a direction in a multidimensional space are considered. In the general case, the minimum distance between convex hulls and the penetration depth of one hull into another at their intersection can be found by trying different directions in the multidimensional space and by examining the extreme points of the projections of the classes on these directions. Let Vol. 28 No. 4 2018 714 NEMIRKO A and B be the projections of classes onto some direction in a multidimensional space. If the degenerate variant of the full containment of one set into another is excluded, all the variants of the relative arrangement of classes on their projections onto some direction can then be represented in the form of Fig. 3. According to Fig. 3, the proximity measure (including the intersection measure) D can be defined as the following procedure: B 8 A 6 A-B 4 D (0,0) 2 D 0 −2 −4 −6 −8 −6 −4 −2 0 2 4 6 8 Fig. 2. To the definition of the penetration depth through the Minkowski sum: ∗ is class A, o is class B, “stars” are the Minkowski difference S, where S = A ⊕ (− B), + is the origin (0, 0), and D is the penetration depth. Here, A = {(1, 3); (8, 2); (7, 5); (1, 6)}, B = {(3, 6); (4, 7); (7, 3); (8, 7)}. a1 = min( A); a2 = max( A); b1 = min(B ); b2 = max(B); if a1 < b1 {case 1} D = a2 − b1 else {case 2} D = b2 − a1 end. If the sets intersect, D will have a negative sign. The minimum value of D for two classes can be found by determining such direction w in a multidimensional space for which D is minimal. Case 1 (a) A B amin bmin amax bmax (b) B A amin amax bmin bmax Case 2 B (a) A bmin amin bmax amax (b) B bmin A bmax amin amax Fig. 3. Variants of the location of the projections of classes A and B on some direction in a multidimensional space: (a) the classes intersect, (b) the classes do not intersect. PATTERN RECOGNITION AND IMAGE ANALYSIS Vol. 28 No. 4 2018 MULTIDIMENSIONAL DATA VISUALIZATION BASED Many classification methods solve the problem of inseparable classes by either minimizing classification errors or by procedures for minimizing such errors. In the general case, the resulting weight vector does not coincide with the penetration depth vector required to obtain a visual pattern on the plane. In the SVM, this problem is solved by minimizing classification errors, which is also not the best solution in terms of the penetration depth minimization criterion. Equation (2) is used instead of (1) for linearly inseparable classes in the SVM method. N ⎧1 2 w C + ξi → min ⎪ w,w0,ζi 2 i = 1 ⎪ ⎪ T ⎨ yi (w x i + w0 ) ≥ 1 − ξi , i = 1,2,…, N , ⎪ξi ≥ 0, i = 1,2,…, N ⎪ ⎪⎩ ∑ (2) where the variables ξi ≥ 0 indicate the error value at x i , i = 1,2,..., N objects, and the factor C is the method-setting parameter that makes it possible to adjust the ratio between the maximization of the width of the separating margin and minimization of the total error. The pattern for solving this problem is similar to the solution of the problem for the case of linearly separable classes. USE OF MODIFIED SVM METHODS Improved solutions of the problem are proposed irrespective of class intersection conditions. They are implemented by transforming convex hulls into reduced convex hulls (RCHs) [13] and scaled convex hulls (SCHs) [14], which reduces the problem to the analysis of linearly separable classes. ⎧⎪ R(X, μ) = ⎨v : v = ⎪⎩ Convex Hull Scaling [14] The SCH of the set X = {x i , x i ∈ R d , i = 1,2,..., k} with nonnegative reduction factor λ ≤ 1 denoted by S (X, λ) is defined as the following expression: ⎧⎪ S (X, λ) = ⎨v : v = λ ai x i + (1 − λ)m, ⎪⎩ i =1 k ⎫⎪ ai = 1, 0 ≤ ai ≤ 1⎬ , ⎪⎭ i =1 which can also be rewritten as k ∑ ∑ ⎧⎪ S (X, λ) = ⎨v : v = ⎪⎩ ∑ ∑ For inseparable classes, i.e., when the convex hulls of classes intersect in the feature space, the RCH method is used to transform them to the form of complete separation [13]. The RCH of the set X denoted by R(X, μ) with an additional constraint on each factor ai that bounds it from above by nonnegative number μ < 1 is defined as follows [8]: PATTERN RECOGNITION AND IMAGE ANALYSIS i = 1, k ∑ a (λx i =1 i i + (1 − λ)m), ⎫⎪ 0 ≤ ai ≤ 1⎬ , ⎪⎭ ∑ x is the centroid. For the given λ, every point λ∑ a x + (1 − λ)m of S (X, λ) is a conwhere m = (1/k ) k i =1 i i i i vex combination of the centroid m and the point ∑ k i i ⎫⎪ = 1, x i ∈ X, 0 ≤ ai ≤ μ⎬ . ⎪⎭ i =1 The smaller μ, the smaller the RCH size. Therefore, initially inseparable convex hulls can be transformed to become separable by selecting an appropriate reduction factor μ. It is known that for an inseparable case, finding the maximally soft margin between two classes is equivalent to finding the pair of nearest points between two RCHs by selecting an appropriate reduction factor [15]. The complexity of computing the RCH increases with decreasing μ. In addition, the number of extreme points and the RCH shape change with the change in the parameter μ. The SCH method does not have these shortcomings. i =1 ⎫⎪ ai x i , 0 ≤ ai , ai = 1, x i ∈ X ⎬ . i =1 i =1 ⎭⎪ k i =1 i Reduction of Convex Hulls ⎧⎪ = ⎨v : v = ⎩⎪ ∑a x , ∑a ∑a Conv(X) k k k Let the elements of one class be X = {x i , x i ∈ R d , i = 1,2,..., k} . Then the convex hull generated by the training set of one class is defined as follows: 715 k ax i =1 i i from the original convex hull Conv(X); i.e., ∑ k it lies on the linear segment connecting a x and i =1 i i the centroid m (Fig. 4) Thus, initially overlapping convex hulls can be reduced and become separable if λ is selected appropriately. Once they become separable, it is possible to find a classifier with the maximum gap between the two SCHs using the nearest-point algorithm. This strategy is the same as within the RCH method in [13] and [16]. Therefore, it can be considered as a variant of SVM classifiers. However, unlike the RCH, the SCH has the same form of the resulting convex hull and the Vol. 28 No. 4 2018 716 NEMIRKO number of extreme points as the original convex hull, which leads to an easier search for a pair of nearest points between SCH classes. Convex Hull Offset A similar offset convex hull (OCH) procedure can be proposed for intersecting classes, as a result of which all elements of one class are offset by a constant value in the direction of the difference vector between their centroids. The problem with separated classes is then solved, after which the reverse offset is performed. Let xi, i = 1, 2, …, N be vectors in the n-dimensional feature space of the training set X . They belong to one of the two classes ω1, ω2, which are linearly inseparable, and n1, n2 are the number of class members ω1, ω2, respectively. Then n1 n2 ∑ ∑ M1 = 1 x(1) M2 = 1 x(2) i , i n1 i =1 n2 i =1 are centroids of classes, m= M1 − M2 M1 − M2 is the displacement vector. Assume that MT1 m > MT2 m , and we offset the first class relative to the second one. Then the new position of the vectors x1i is x1i new = x1i + km, where k is the offset factor selected proportional to MT1 m − MT2 m . The offset is directed along the m axis. After determining the weight vector, the inverse transformation is carried out: x1i new = x1i − km. All the considered methods that use the reduction and offset of convex hulls depend on the coordinates of the class centroids and, therefore, are only approximate methods for estimating the penetration depth. EXPERIMENTS The degree of intersection of classes after their mapping to the plane was estimated by the number g of members of the training samples of both classes that fall into the intersection area, i.e., g = (n1 + n2 )/(N1 + N 2 ) , where n1, n2 is the number of points of the first and second classes that fall into the intersection area of convex hulls and N1, N 2 is the number of members of the training sample of the first and second classes. It is obvious that 0 < g < 100. It is assumed that the minimum g corresponds to the minimum D for intersecting classes. v m v + (1 )m Fig. 4. To the definition of a SCH. Each point of the SCH is the convex combination of the centroid m and the corresponding point of the original convex hull v for the reduction factor λ. Two classes of Fisher’s irises were used to visualize 4D data in the first experiment [17]: Iris virginica and Iris versicolor. Each class consists of 50 samples measured by four features: the length and width of the sepal and the length and width of the petal. It was shown earlier [18] that the potentially achievable minimum class intersectability in this training sample is 1% Fig. 5. The results of class intersection after their mapping to the plane using different algorithms are presented in Table 1. The paper uses the modified Platt’s SMO Algorithm for SVM classifiers [19, 20]. These results show that the SVM method is the best for visualizing the two-class problem given in the multidimensional feature space among the three methods under consideration. It yields the minimal intersection of the classes when they are mapped to a plane. However, in this method parameters should be selected in each individual case. The second problem under consideration was breast cancer diagnosis. The data were taken from the Breast Tissue database [21]. They consist of 106 samples of breast tissue measured by nine parameters of Table 1. Intersection of classes on a plane for different algorithms Algorithm N1 + N2 n1 n2 g% PCA 50 + 50 2 6 8 FDA 50 + 50 1 2 3 SVM 50 + 50 1 0 1 PATTERN RECOGNITION AND IMAGE ANALYSIS Vol. 28 No. 4 2018 C2 MULTIDIMENSIONAL DATA VISUALIZATION BASED 1.0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 3 (a) 717 (b) 2.5 3.0 3.5 4.0 4.5 5.0 5.5 2 1 0 C1 1 2 3 6.0 3.5 (c) 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 2.5 2.4 2.3 2.2 2.1 2.0 1.9 3.0 2.5 2.0 1.5 1.0 (d) 3.9 3.8 3.7 3.6 3.5 3.4 3.3 3.2 3.1 3.0 2.4 2.3 2.2 2.1 2.0 1.9 Fig. 5. Visualization of 4D data by Fisher’s irises: (a) visualization using the PCA method, (b) the result of applying the SVM method, (c) the intersection area from graph (b), and (d) the result of using coordinate-wise search after the SVM method. The versicolor class is given in all panels on the right. 3.0 2.5 2.0 1.5 1.0 0.5 0 0.5 1.0 1.5 2.0 4 3 4 3 2 1 0 1 2 3 2 1 0 1 2 3 4 4 2 5 1 0 1 2 3 Fig. 6. Mapping of classes of mammary neoplasms to the plane using the PCA method. Fibroadenoma class (left) and carcinoma class (right). The abscissa axis is the first weight vector and the ordinate is the second. It can be seen that the convex hulls of the classes intersect. Fig. 7. Mapping of classes of mammary neoplasms to the plane using the SVM method. The mutual arrangement of classes and axes is the same as in Fig. 6. Classes are completely linearly separable. tissue impedance. The data were verified for six classes of mammary neoplasms, of which two classes were selected for our experiments: 21 cases of breast carcinoma (malignant tumor) and 15 cases of fibroadenoma (benign tumor). The initial data were normal- ized to the mean value and variance. The result of the application of the PCA algorithm to these data is shown in Fig. 6. The result of data visualization using the SVM algorithm given in Fig. 7 showed their complete linear separability. PATTERN RECOGNITION AND IMAGE ANALYSIS Vol. 28 No. 4 2018 718 NEMIRKO (a) (b) 8 6 4 2 0 2 4 6 8 10 8 6 4 2 0 2 4 6 8 10 5 0 5 10 15 20 5 (c) 8 6 4 2 0 2 4 6 8 10 0 5 10 15 5 0 5 10 15 Fig. 8. Mapping of B (right) and M (left) classes to the plane by the PCA method: (a) visualization of classes and their convex hulls, (b) mapping of points of class B that fall into the intersection area, and (c) mapping of the points of class M that fall into the intersection area. The use of the penetration depth criterion to visualize the intersection area of classes does not always lead to a decrease in the class intersectability. This especially concerns the cases of their strong intersection. Consider the data for the problem of breast cancer diagnosis by nine cytological features [22]. These data consist of 683 cases: 444 cases of benign tumor B (benign) and 239 cases of malignant cancer M (malignant). The signs are integers in the range from 1 to 10. The elimination of duplicate points led to their reduction to 454 points (236 for benign and 213 for malignant). Visualization of these data using PCA, FLD, and SVM procedures gave almost the same degree of class intersection (g = 13%). Below are the results of data processing using PCA (Fig. 8) and SVM (Fig. 9). 2 0 CONCLUSIONS The criterion of proximity of convex hulls D of classes can be used to map the class intersection area from a multidimensional space to a plane. For disjoint classes, this criterion consists in minimizing the distance between convex hulls. For intersecting classes, it is transformed into minimization of the degree of their mutual intersection D. The D criterion is automatically satisfied if the SVM method is used for linearly separable classes. For linearly inseparable classes, it is advisable to use the SVM method with RCH, SCH, and OCH transformations as approximate solutions. The last one is the simplest. However, instead of SVM, it is acceptable to use other NPP algorithms. In the general case, search procedures should be used to find optimal values of D. The SVM method has proven to be the best of the three analyzed mapping methods. It yielded the minimal intersection of classes when they were mapped to a plane. It is advisable to focus further work on improving the methods for calculating penetration depth for classes defined in a multidimensional space and reducing their computational complexity. 2 ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research, project nos. 18-07-00264 and 18-29-02036. 4 6 8 REFERENCES 10 12 14 25 20 15 10 5 Fig. 9. Mapping of classes and their convex hulls to the plane obtained by the SVM method: B class (left) and M class (right). 0 1. I. T. Jolliffe, Principal Component Analysis, 2nd ed. (Springer-Verlag, New York, 2002). 2. A. P. 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S. Keerthi, S. K. Shevade, C. Bhattacharyya, and K. R. K. Murthy, Improvements to Platt’s SMO Algorithm for SVM Classifier Design, Technical Report CD99-14, Control Division, Dept. of Mechanical and Production Engineering, National University of Singapore, 1999. Available at: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.46.8538 (accessed May 2018). 20. S. Theodoridis and K. Koutroumbas Pattern Recognition, 4th ed. (Academic Press, 2009). 21. Breast Tissue Data Set. UCI Machine Learning Repository. Available at: http://archive.ics.uci.edu/ml/datasets/breast+tissue (accessed April 2018). 22. Breast Cancer Wisconsin (Original) Data Set. UCI Machine Learning Repository. Available at: https://archive.ics.uci.edu/ml/datasets/breast+cancer+wisconsin+(original) (accessed May 2018). Translated by O. Pismenov Anatolii Pavlovich Nemirko. Graduated from St. Petersburg Electrotechnical University “LETI” in 1967. Since 1986, has worked as a professor at the Department of Bioengineering Systems at the same university. Received doctoral degree in 1986 and academic title of professor in 1988. Scientific interests: pattern recognition, processing and analysis of biomedical signals, intelligent biomedical systems. Author of more than 300 scientific publications, including 90 papers and five monographs. Board member of the International Association for Pattern Recognition and member of the editorial board of Pattern Recognition and Image Analysis journal. Vol. 28 No. 4 2018