2007 International Symposium on Nonlinear Theory and itsApplications
NOLTA'07, Vancouver, Canada, September 16-19, 2007

Gray Scale Display of Input Data Using Shooting SOM
Masato Tomita, Haruna Matsushita and Yoshifumi Nishio
Department of Electrical and Electronic Engineering, Tokushima University
2–1 Minami-Josanjima Tokushima 770-8506, JAPAN
Email: {tomita, haruna, nishio}@ee.tokushima-u.ac.jp
Abstract—The Self-Organizing Map (SOM) is popular algorithm for unsupervised learning and is widely applied for many applications. In the previous study, we haveproposed a new type of SOM algorithm, which is called
Shooting SOM (SSOM) algorithm. The important featureof SSOM is that the neurons move like aiming at a target,
namely, only some neurons near the cluster move toward
the cluster to hit the area where input data are concentrated and 1-neighborhood neurons of the winner neuron
get away a fraction of an inch from the cluster. Becauseof this feature, SSOM tends to self-organize each cluster
along the figure of each cluster. We investigate the behaviorof SSOM and apply SSOM to data visualization problems.
1. Introduction
In data mining, data visualization is one of the mostimportant issues and is very useful for many applications, such as engineering applications covering. For instance, areas like pattern recognition, image analysis, process monitoring and control, and fault diagnosis. Then,the Self-Organizing Map (SOM) attracts attentions for thevisualization methods in recent years. SOM is popular
tools for visualization and clustering of high-dimensional
data [1]. SOM is unsupervised neural network introduced
by Kohonen in 1982 [2] and is a model simplifying selforganization process of the brain. SOM maps multidimensional data onto a 2-dimensional grid. SOM can classify
input data according to similarities and patterns which are
obtained by the distance between neurons and is applied towide fields of data classifications. Although many methods
to visualize the relationship of input data by using SOM
have been proposed [3]-[7], it seems to be very diﬃcult toconstruct a simple method using SOM for universal input
data. On the one hand, in the world, the amount and complexity of data increase from year to year and control oflarge-volume data attracts a lot of attention. Therefore, it
is important to investigate various visualization methods ofinput data.
In the previous study, we have proposed a new type SOM
algorithm, which is called Shooting SOM (SSOM) algorithm [8]. The important feature of SSOM is that the neurons move like aiming at a target, namely, only some neurons near the cluster move toward the cluster to hit the area
where input data are concentrated, 1-neighborhood neuronsof the winner neuron move slightly from the cluster, andother neurons do not move at all from the initial position.
Because of this feature, SSOM tends to find near the centerof the area where input data are concentrated.
In this study, by using SSOM, we propose a gray scale
display method of input data. The proposed display method
is that the shading is reflected by the number of becoming the winner. By learning the input data, we think thatthe proposed method can simplify and visualize the information of the input data that it is diﬃcult to judge by appearance. We investigate the behavior of SSOM and apply
SSOM to the proposed display method. The eﬃciencies ofSSOM are confirmed by several simulation results.
In the Section II, we explain the learning algorithm ofSSOM algorithm. In the Section III, the behavior of SSOM
is explained with some simulation results. In the Section
IV, we apply SSOM to the proposed display method.
2. Shooting SOM (SSOM)
In the previous research, we have proposed SSOM. In
this section we introduce SSOM algorithm in order to makeclear new points of the proposed application method.
We explain the learning algorithm of SSOM. The learning algorithm of SSOM is largely similar to the conventional SOM. In SSOM, m neurons are arranged as aregular 2-dimensional grid. The range of the elementsof d-dimensional input data x j = (x j1 , x j2 , · · · , x jd ) ( j =1, 2, · · · , N) are assumed.
(SSOM1) The initial values of all the weight vectors wi =(wi1 , wi2 , · · · , wid ) (i = 1, 2, · · · , m) of the neurons are given
between 0 and 1 in a reticular pattern.
(SSOM2) An input data x j is inputted to all the neurons at
the same time in parallel.
(SSOM3) We find the winner neuron c by calculating thedistances between x j and wi according to;c = arg min{kwi − x j k}.
i(1)

In other words, the winner neuron c is the neuron with theweight vector nearest to the input vector x j . In this study,
Euclidean distance is used for Eq. (1).
(SSOM4) We measure whether the winner neuron or 1neighborhood of the winner neuron or otherwise and show
in Fig. 1.

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Winner neuron

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1-neighborhood of winner neuron

Figure 1: Winner neuron and 1-neighborhood neurons of
the winner neuron.

Furthermore, we determine the update rate Ai for eachcases as follows;
1,if i = c−a,if i is 1-neighborhood of c(2)Ai = 
 0,otherwise.
where a is the constant number and usually set the smaller
value. Namely, if the neuron i is the winner neuron, theneuron i is updated like the winner neuron of the conventional SOM. If the neuron i is 1-neighborhood neuron, theneuron i get away a fraction of an inch from the input data
because a is negative number. And if the neuron i is otherwise, the neuron i is not updated from initial position.
(SSOM5) The weight vectors of all the neurons are updatedas;wi (t + 1) = wi (t) + Ai × α(t)(x j − wi (t)),(3)
where t is the learning step and α(t) is the learning rate. α(t)decrease with time according to the following equations;α(t) = α(0) (1 − t/T ) ,(4)

where T is the maximum number of the learning.
(SSOM6) The steps from (SSOM2) to (SSOM5) are repeated for all the input data, namely, from j = 1 to j = N.
3. Learning Results
3.1. 2-dimensional input data

3.2. 3-dimensional input data
We consider the 3-dimensional input data of 1800 points
whose distribution is as Fig. 2(2a). 1200 points are distributed within a sphere with radius 0.3. 600 points aredistributed within a cube, 0.3 on a side. We consider theconventional SOM and the proposed SSOM with 400 neurons (20 × 20). The parameters for the learning of the conventional SOM and SSOM are chosen as follows;α(0) = 0.9, σ(0) = 1/16, a= −0.003,(6)

We execute the learning for all input data 3 times. As
we can see from Fig. 2(2b) and Fig. 4(2c), SSOM caneﬀectively self-organize each cluster, just like in the 2dimensional input data.
4. Gray Scale Display
In this study, we apply SSOM to the proposed display
method for the simulation results. And we apply the conventional SOM to the conventional display method.
4.1. Display method

We consider the 2-dimensional input data of 1200 points
whose distribution is as Fig. 2(1a). 800 points are distributed within a circular range from 0.1 to 0.5 horizontallyand from 0.1 to 0.5 vertically. 400 points are distributed
within a rectangular range from 0.7 to 0.9 horizontally andfrom 0.7 to 0.9 vertically. We consider the conventional
SOM and the proposed SSOM with 400 neurons (20 × 20).
The parameters for the learning of the conventional SOM
and SSOM are chosen as follows;α(0) = 0.9, σ(0) = 1/16, a= −0.005,

We execute the learning for all input data 3 times. The
simulation results of SSOM is shown in Fig. 2(1c). SSOM
self-organizes only near the center of the area where theinput data are concentrated.
Let us examine the behavior of SSOM in more detail. As
we can see from Fig. 2(1b) and Fig. 2(1c), the conventional
SOM self-organizes input data of two clusters with all neurons. However, we can see that SSOM moves like aiming at
a target from learning process. Only some neurons near thecluster move toward the cluster to hit the area where input
data are concentrated, 1-neighborhood neurons of the winner neuron move slightly from the cluster, and other neurons do not move at all from the initial position. Therefore,
only neurons near a cluster self-organize the cluster. In addition, in the interior of the cluster, some neurons which
distribute the outside of the cluster can not move into the interior of the cluster because only neurons of the interior canself-organize the input data of the cluster. Consequently,
SSOM can eﬀectively self-organize each cluster along theshape of the cluster.

(5)

We propose the gray scale display method as follows.
The proposed method is that the shading is reflected by thenumber of becoming the winner as shown in Fig. 3. For
example, because the neuron 5 becomes a winner 5 times,
the shading of the circle color is the deepest. In contrast,
because neurons 7 and 8 have never been a winner, theshading of these circles is the lightest. Namely, the morenumber of becoming the winner is deeper color. Therefore,
we can visually see the positional relationship of input data
and rough density.

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k : the number of becoming winner.

Figure 3: Gray scale display method

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y0.5
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input data. (a) Input data. (b) Learning result of the conventional SOM. (c) Learning result of SSOM.

4.2. Example 1 (Result in Fig. 2)
We apply the gray scale display method to the simulation
results of Fig. 2. The results are shown in Fig. 4. As we cansee from Fig. 4(1a) and Fig. 4(2a), the conventional display
method using the conventional SOM can only visualize theboundary line between clusters as usual. However, we cansee that the proposed gray scale method using SSOM visualizes the positional relationship of input data as well as thenumber of clusters from Fig. 4(1b) and Fig. 4(2b). Therefore, not only visualizing the shape of each cluster, the proposed method can visualize the rough density by learning.
4.3. Example 2 (Lorenz model)
In this subsection, we apply the proposed gray scale display to the Lorenz chaotic attractor. We prepare the 2-Figure 4: (1) 2-dimensional input data. (2) 3-dimensional
input data. (a) Learning result of the conventional SOM.
(b) Learning result of SSOM.

dimensional and the 3-dimensional input data of the Lorenz
attractor as shown in Fig. 5(1a) and Fig. 5(2a). Fig. 5(b)to Fig. 5(c) show the learning results of the conventional
SOM and SSOM, respectively. By using these learning results and the proposed gray scale display, we obtained theresults of the gray scale visualization as shown in Fig. 6.
From the results, we can not visualize the shape and therelationship for the conventional SOM. However, for theproposed display method and SSOM, we can see the relationship of input data and the shape of each cluster.
5. Conclusions
In this study, we have discovered the new property of theShooting SOM (SSOM) proposed in the previous study andproposed the gray scale display method. We have explained
the diﬀerences between the conventional SOM and SSOM

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Figure 6: (1) 2-dimensional Lorenz model. (2) 3dimensional lorenz model. (a) Simulation result of the conventional SOM. (b) Simulation result of SSOM.

(2b)

1References

10.8[1] T. Kohonen, “The Self-Organizing Maps,” Neurocomputing, vol. 21, pp. 1-6, 1998.

0.5

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[2] T. Kohonen, Self-Organizing Maps, Berlin, Springer,
vol. 30, 1995.

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01
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Figure 5: (1) 2-dimensional. (2) 3-dimensional. (a) Input data. (b) Learning result of the conventional SOM.
(c) Learning result of SSOM.

have investigated its behavior. Furthermore, we have applied SSOM to visualize the relationship of input data andhave confirmed its eﬃciency by combining the best properties of SSOM and the proposed display method.
In the future, we try to extend to higher dimensional
input data because we think that the proposed method
can show maximize the benefit by learning more than
3-dimensional input data. We try to discover new applications of SSOM in diverse fields such as sound dataprocessing.
Acknowledgments[3] J. Vesanto and E. Alhoniemi, “Clustering of the SelfOrganizing Map,” Neural Networks, vol. 11, pp. 586600, 2000.
[4] D. Alahakoon, S. Halgamuge and B. Srinivasan, “Dynamic Self-Organizing Maps with Controlled Growthfor Knowledge Discovery,” Neural Networks, vol. 11,
pp. 601-614, 2000.
[5] J. Laaksonen, M. Koskela and E. Oja, “PicSOM-SelfOrganizing Image Retrieval With MPEG-7 ContentDescriptors,” Neural Networks, vol. 13, pp. 841-853,
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[6] K. Simunic and J. Novak, “Combining Visualization
and Interactive Clusterung for Exploring Large Document Pools,” Proc. of VIIP’04, 2004.
[7] C. Hsu, “Generalizing Self-Organizing Map for Categorical Data,” Neural Networks, vol. 17, pp. 294-304,
2006.
[8] M. Tomita, H. Matsushita and Y. Nishio, “ShootingSOM and its Application for Clustering,” Proc. ofNOLTA’06, pp. 199-202, 2006.

This work was partly supported by Yamaha Music Foundation.

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