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: apn-bs@yandex.ru
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
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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
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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
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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
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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
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(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
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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).
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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