This paper presents an autonomous approach fiir the clustering algorithm based on a mountain function proposed by Yager and Filev. It intends to answer the ...
An Autonomous Approach to the Mountain-Clustering Method P. J. Costa Branco, N.Lori and J. A. Dente Instiluto Superior T6cnico Laboratorio de Mecatrdnica - CAUTL AV.Rovisco Pais, 1096 Lisboa Codex, Portugal1 FaxN". 351-1-8482987
Abstract
M' denotes the initial "mountain function" obtained from s data points. The term e-alxk-Ni'p measures The term
This paper presents an autonomous approach fiir the clustering algorithm based on a mountain function proposed by Yager and Filev. It intends to answer the parameter selection problem and attenuate the effects of the granularity of the griding in algorithm's performance using a cluster realocation procedure. The solving of those problems has greatly enhanced the possibility of achieving an autonomous mountain-clustering process. The proposed clustering approach is explained in detail and examples of its performance are analyzed
a "membership grade" of x k in cluster candidate
pondered by its distance I x k - N i I p and a decaying parameter a. The usedp norm is the Euclidean distance (p = 2). The second step consists on finding the mountain vertice
N;
called a ''mountain function" and each
hi'
M' . The process goes on until the
(3) being the parameter 6 related with the assumed clusters number. The previous algorithm, although simple, presents two
problems concerning the parameters c1 , p , and 6 calculation. The first one is that the precision and/or number of clusters strongly depends on grid number and the second is that they don't have an exact definition being set by the operator in a trial-and-error basis. To correct that, solutions are proposed creating an autonomous mountain method including definition of parameters a, p, and a cluster reallocation procedure to eliminates the strong influence of the chosen grid's granularity. Two examples based on Yager
density function is
N j mountain point
M ~ ( N ,=) - $ e - u l r k - N ~ l ~
0-8186-7126-2/95$4.00 0 1995 IEEE Proceedings of ISUMA-NAFIPS '95
as specified by
mountain M 2 instead of relation (3) is found
(each grid node) has a density value calculated by (1).
k=l
k f : ( N r ) and
2
This operation eliminates the effect of the cluster NI*over the other mountain nodes, localizing the second better cluster and so on. The parameter p is the mountain elimination parameter, being its function similar to a but now related with clusters localization. The final step is return to step one but with the new
The technique of approximate clustering via the mountain method proposed by Yager and Filev [ 11 presents a quick and simple technique to help us identify a data set structure [ 2 ] . Although simple, the method presents two main problems: the parameter's selection of the algorithm doesn't have an exact definition being fixed by the operator in a triiil-anderror basis; the second problem is the strong influence of the chosen grid's granularity on the algorithm's performance. To surpass that, solutions are introduced for automatically compute the parameters of the algorithm (a, p and S> and a cluster adaptation procedure based on a defined cluster's center of mass is introduced. The mountain method can be divided in a three step clustering. In the first step, the n-dimensional data space is partitioned by an arbitrary grid and a density value (A4) is
(Ni). The
having the highest: function value
calculates a new mountain function expression (2).
1: Introduction
attributed to each grid node
Ni,
(1)
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and Filev results are presented to visualize the algorithm's evolution and an experimental data set is utilized to demonstrate its potential use on estimating adequate fizzy initialization data structures in fuzzy model structures.
2: Autonomous parameter definition
The terms c m ( i e k ) and expressions (7) and (8).
2.1: Parameter a For a bi-dimensional data (n = 2), d, and d, are the grid length in x,y directions as seen on Figure 1. The building parameter a is responsible for the construction of the initial mountain. Instead of using the same a value in all directions (in this case x and y ) , a vector calculated by (4).
of components (a,,
a,)
(7)
sin( i o k )=
(Yk
- Niy
r
Substituting equations (7-8) in (6), the new ponderation term is expressed by (9) and the initial "mountain function" by equation (10).
is
1
1
are calculated using
(4) The solutions are justified by the necessity of an exponential
e
analyse with range d, in x direction and dy in y direction. With small grid lengths, the exponential term has to decay more fast because data is supposed more concentrated. In the other side, if data in some direction is sparse, the calculated grid lenghts are larger and the exponential term decays
-I
slowly. The term e-"'xk-N'12 is rewritten taking into account the angle between the data (angle
iek)
(Xk,Yk)
and the grid node
(9) '
I
I
k=l
N,
2.2: Parameter p
as Figure 1 shows.
The definition of parameter significance of point
4
(xL,yk)
is concerned with the
on the election of a certain
principal cluster. Thus, if ( x k , y k ) is highly considered on the election of the principal cluster, it shouldn't be so when the second cluster is defined and so on. Hence, the principal cluster for k iteration ( M i ) must now have the associated parameter p depending on its location relative to the considered grid node (being a cluster too but with a smaller indice). These considerations introduce a parameter vector
L-L' Fig. 1. Grid lengtb's and the angle between cluster
Ni and a
p ( k ) with two components, in the same way as 6 , becoming different in each iteration as the reference cluster
point t X k , Y k ) . The new ponderation term ( 6 ) between point
(xk ,Yk)
the grid vertice i is now a function of parameters
( M i )changes. Equations (1 1-12) express the
and
same idea as
(4) for parameter a.The term N ; k ) , is the x component of the principal cluster considered in k iteration. The other term
a,, a y ,
their angle and distance r ( 5 ) between the grid vertice and the considered point.
Nixis the component x for the i cluster.
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All points (m) in the window are weighted using the exponential ponderation term ct as before. Equations (14-15) compute the center of mass defining the new coordinates
(qx r. ) for i cluster in directions x and y .
’ ‘U
m C
The final reduction equation is expressed by (13) for each iteration.
r, =: zx
e-axr
xk
k=l m
C
e-QXr
k=l m
__ k=l
5Y --
5
p Y r
k=l (15) The iterative center-of-mass search procedure consists on moving the clusters position from grid’s vertices to their mass centers, and redefining their windows. It suffers an halt if the difference between the cluster location and its center-of-mass becomes irrelevant. This occurs when the sum over all center increments become less than a small number E. In our examples ~=0.0001.
2.3: Parameter 6 The purpose of parameter 6 was to limit the number of clusters and establish the low enough mountain height. The autonomous algorithm stops when all grid nodes have been extracted or, when negative indices appear in the discounted clusters, the extraction process continues until all resulted clusters are negative ( M;
ePayryk
Figure 3 shows the reallocation example for 3 clusters
< 0).
NI,
N2 and N 3 , initially associated with grid vertices. Each corresponding window moves in direction of the real clusters as seen in the figure. Altlhough, only the window with some
3: The cluster reallocation procedure As the algorithm begins with a fixed grid, all clusters are centered on its vertices. To correct that situation, the clusters are reallocated getting closer to cluster’s center of mass. Following it’s explain this cluster reallocation procedure. For each identified cluster there is an associated window, as illustrated by Figure 2. The dimensions of the cluster
N3). Because inside the window associated with cluster AT2 there are no points, it stays in its
points moves ( N , and
initial position and is eliminated as a valid cluster.
window are the grid partitions d, and d y , being the associated cluster its center. The center of mass of this window is calculated having as axis origin the associated cluster. Therefore, all distances into the window are referred to it.
I
I
Fig. 3. Illustrative example of cluster adaptation for 3 different clusters
4: Examples
- = d x ’ Z
The proposed autoiiomous clustering algorithm is compared with the results obtained by the original “mountain method’’ to the isosceles triangle and bridge cases. There are three types of data in figures 5,s and 16: the triangles the discarded clusters; the circles the adapted clusters; and the black points, in the end of arrow way, the
dx -
Fig. 2. Example illustrating each cluster with its associated window.
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final clusters after reallocation process. The fEst two examples compare the results presented by Yager and Filev [ 11 with those obtained applying the autonomous mountain method. The last example shows an application of this clustering algorithm to hzzy structure identification of an experimental electromechanical data.
4.1: Case 1 For the triangle case, it was used 5 partitions in each axis. First, the initial mountain function is computed using the
0.4 0.2
calculated vector parameter 6 and the equation (IO) (see Figure 4). After, the mountain is reduced (pelling process)
0
-
-0.2 0
using the vector parameter P(k) and the expression (13). From all cluster candidates, in this case 25 grid vertices, only 14 are selected (marked by triangles and circles). Each cluster candidate is then realocated by the cluster adaptation process as indicated in Figure 5 by a set of arrows. It's noted that some clusters (triangules) are not adapted because there is no data into their associated window and therefore they are discarded. The other clusters marked with circles are then adapted and converge to real clusters center. Figure 6 shows the clusters calculated using the original method. A considerable added performance using the autonomous mountain method can be verified, mainly taking into acount that the clusters are not now attained to the grid as in the original method.
1
1.5
2
proposed autonomous mountain clustering technique. 2 1.8 1.6 1.4 1.2 1 0.8
0.6 0.4
0.2
I
,,I ', ,' 1 '. ,
0.5
Fig. 5. Isosceles triangle example with application of the
I
*
*. 40 y 0 0 0 0
P
----
eg q
aq
RI
e
*I
Fig 6 . Isosceles triangle example used by the original mountain method.
4.2: Case 2 The second example is related with the bridge case. Using the autonomous algorithm with 5 partitions, the initial mountain function is computed (Figure 7.) After the mountain pelling, the cluster's adaptation process realocates the selected ones conceiving three principal clustes (see Figure 8), instead of two as shown in Yager's results (Figure 9). The clusters in the corners have, of course, highest mountain indices than that in the middle bridge. The third cluster can be interpreted has if we were looking at the figure from a long distance and the two cluster at comer were fused.
5
Fig. 4. Mountain function for isosceles triangle case.
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4.2: Case 3 Clustering techniques can estimate the linguistic structure of fuzzy models by calculating the rules constituting the system and consequently the number of membership functions and its localization. Figure 10 is an experimental data set composed by two normalized variables: angular speed of an induction motor (0) and its stator current magnitude (4. The data composes a set of examples of the system behaviour and the objective is to identifjl the relations between the two variables o and I. ,1
81
.
,I ..........:....................i..........i
-1
6
.
I
I
1.................!...
~
- I
.......... ;..........j.......... 4,............;........... ...........r .........
..
I
I
Fig 7. Mountain function for bridge case using the autonomous algorithm. 0
1 ,!I
j
j
4
6
0 ........-.p........:..........1..........+ .......... ;....................
-0.5
-12
0
2
8
IO
12
Motor speed [ O ]
Fig. 10. Data set used to identify the initial fuzzy structure of an electromechanical system.
0
0.5
1
1.5
2
First, its calculated the mountain function using 8 partitions for each variable. Some mountain indices (M) indices can be negative during the pelling process as seen in Figure I 1, and they are setted to zero . 'The mountain is reduced to identify the other clusters stopiing the process when all indices become zero after pelling (Figures 12 to 15)
2.5
Fig. 8. Bridge example with application of the proposed clustering technique showing the three identified clusters.
Fig. 9. Bridge example used by original mountain method.
Stator current I
"
U
-
Angular speed o
Fig. i I. Initial mountain function
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0
Fig. 12. First mountain reducion zeroing the negative indices.
Fig. 15. Last mountain reducion.
,-
,- I \
0
Fig. 13. Second mountain reducion.
Fig. 15. Identified clusters expressing the relations between the two variables.
5: Conclusion Within the context of the clustering process developed by Yager and Filev [l], our approach has demonstrated to be simple and efficient in resolving the problems of parameter dependency of the original method. We present solutions creating an autonomous mountain method including defmition of parameters and eliminating the last parameter 6 0
6: References
Fig. 14. Third mountain reducion.
After cluster adaptation process, six clusters (rules) are identified relating o with I initializing an initial fuzzy model for the electromechanical system (Figure 16.).
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[I]
Ronald R. Yager and Dimitar P. Filev, "Approximate Clustering Via the Mountain Method", IEEE Trans. on Syst. Man and Cyb., vol. 24, No. 8, August 1994.
[2]
Fuzzy Models for Pattern Recognition, J. C . Bezdek and S. K. Pal, Eds. New York: IEEE Press, 1992.
