IEICE TRANS. FUNDAMENTALS, VOL.E91–A, NO.6 JUNE 2008

1463

PAPER

Self-Organizing Map with False-Neighbor Degree between Neurons
for Eﬀective Self-Organization
Haruna MATSUSHITA†a) , Student Member and Yoshifumi NISHIO†b) , Member

SUMMARY
In the real world, it is not always true that neighboringhouses are physically adjacent or close to each other. in other words,“neighbors” are not always “true neighbors.” In this study, we propose
a new Self-Organizing Map (SOM) algorithm, SOM with False-Neighbor
degree between neurons (called FN-SOM). The behavior of FN-SOM is investigated with learning for various input data. We confirm that FN-SOM
can obtain a more eﬀective map reflecting the distribution state of input
data than the conventional SOM and Growing Grid.
key words: self-organizing maps, clustering, feature extraction, visualization

1.

Introduction

Since we can accumulate a huge amount of data in recent
years, it is important to investigate various clustering methods [1]. The Self-Organizing Map (SOM) has attracted attention for its clustering properties. SOM is an unsupervisedneural network introduced by Kohonen in 1982 [2] and isa simplified model of the self-organization process of thebrain. SOM obtains statistical features of input data and applies them to a wide field of data classifications. We canobtain the map reflecting the distribution state of input datausing SOM. In the learning algorithm of SOM, a winner,which is a neuron with the weight vector closest to the input
vector, and its neighboring neuron are updated, regardlessof the distance between the input vector and the neighboring neuron. For this reason, if we apply SOM to clustering of the input data which include some clusters located at
distant locations, there are some inactive neurons between
clusters. Because inactive neurons are on a part without the
input data, we are misled into thinking that there are some
input data between clusters.
Meanwhile, in the real world, it is not always true thatneighboring houses are physically adjacent or close to eachother. For example, a case that the next-door house is at
the top of a mountain whereas my house is at the foot (asFig. 1(a)), and another case that there is a river, which does
not have a bridge, between my house and my next-doorhouse (as Fig. 1(b)). This means that “neighbors” are notalways “true neighbors.” In addition, the relationship between neighborhoods is not fixed, but keeps changing with
Manuscript received October 26, 2007.
Manuscript revised February 8, 2008.
†
The authors are with the Department of Electrical and Electronic Engineering, Tokushima University, Tokushima-shi, 770–
8506 Japan.
a) E-mail: haruna@ee.tokushima-u.ac.jpb) E-mail: nishio@ee.tokushima-u.ac.jp
DOI: 10.1093/ietfec/e91–a.6.1463

(a)

(b)

Fig. 1 What are the “neighbors”? The houses B and C is A’s next-door
neighbors on the left and on the right, respectively. (a) The house B is at the
top of a mountain. (b) The river between A and B does not have a bridge.

time. It is important to change the neighborhood relationship flexibly according to the situation.
On the other side, the synaptic strength is not constantin the brain. So far, the Growing Grid network was proposed
in 1985 [3]. Growing Grid increases the neighborhood distance between neurons by increasing the number of neurons.
However, there is not much research changing the synaptic
strength even though there are algorithms which increase thenumber of neurons or consider rival neurons [4], [5].
In this study, we propose a new SOM algorithm, SOM
with False-Neighbor degree between neurons (called FNSOM). False-neighbor degrees are allocated between adjacent rows and adjacent columns of FN-SOM. We find theneuron q which has never become the winner, and the neurons, which is the most distant from q in a set of direct topological neighbors of q, are said to be “false neighbors” ofq. The initial values of all of the false-neighbor degrees are
set to zero, however, they are increased with learning, andthe false-neighbor degrees act as a burden of the distance
between map nodes when the weight vectors of neurons are
updated. FN-SOM changes the neighborhood relationshipmore flexibly according to the situation and the shape ofdata.
We explain the learning algorithm of FN-SOM in detail in Sect. 4. The learning behaviors of FN-SOM for 2dimensional input data and 3-dimensional data, which havesome clustering problem, are investigated in Sect. 5. In addition, we apply FN-SOM to a real world data set, Iris data.
Learning performance is evaluated both visually and quantitatively using three measurements. Furthermore, the results
are compared with those obtained by the conventional SOM
and Growing Grid. We can confirm that there are few inactive neurons using FN-SOM, and FN-SOM can obtain themost eﬀective map reflecting the distribution state of input

c 2008 The Institute of Electronics, Information and Communication Engineers
Copyright IEICE TRANS. FUNDAMENTALS, VOL.E91–A, NO.6 JUNE 2008

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data in the three algorithms. In addition, we investigate thecluster structure of the learned map by calculating distances
between neurons in order to confirm the clustering ability of
FN-SOM. As the results, we confirm that FN-SOM can obtain more eﬀective clustering map than SOM. In Sect. 6, inorder to utilize the false-neighbor degree eﬀectively, we propose a simple visualization method using the false-neighbor
degrees. We apply this method to the result of the iris data,
and we confirm that the method using false-neighbor degrees can obtain more eﬀective clustering result than SOM.
2.

Self-Organizing Map

First, we explain the learning algorithm of the conventional
Self-Organizing Map (SOM). SOM consists of n × m neurons located at a regular low-dimensional grid, usually a 2D n × m rectangular grid. The basic SOM algorithm is iterative. Each neuron i has a d-dimensional weight vector
wi = (wi1 , wi2 , · · · , wid ) (i = 1, 2, · · · , nm). The initial valuesof all the weight vectors are given over the input space at
random. The range of the elements of d-dimensional input
data x j = (x j1 , x j2 , · · · , x jd ) ( j = 1, 2, · · · , N) are assumed
to be from 0 to 1.
(SOM1) An input vector x j is inputted to all the neurons at
the same time in parallel.
(SOM2) Distances between x j and all the weight vectors arecalculated. A winner, denoted by c, is the neuron with theweight vector closest to the input vector x j ;c = arg min{wi − x j },i(1)where  ·  is the distance measure, Euclidean distance.
(SOM3) The weight vectors of the neurons are updated aswi (t + 1) = wi (t) + hc,i (t)(x j − wi (t)),(2)

where t is the learning step. hc,i (t) is called the neighborhood
function and is described as a Gaussian function;ri − rc 2,(3)
hc,i (t) = α(t) exp −2σ2 (t)where ri − rc  is the distance between map nodes c and ion the map grid, α(t) is the learning rate, and σ(t) corresponds to the width of the neighborhood function. Both α(t)and σ(t) decrease with time, in this study, we use following
equations;α(t) = α0 (1 − t/tmax ), σ(t) = σ0 (1 − t/tmax ),(4)

where α0 and σ0 are the initial values of α and σ, respectively, and tmax is the maximum number of the learning.
(SOM4) The steps from (SOM1) to (SOM3) are repeated
for all the input data.
3.

Growing Grid

We explain an overview of the Growing Grid. The networkof Growing Grid consists of nm neurons located at a rectangular n × m grid. Each neuron has a d-dimensional weightvector wi as the conventional SOM. A winning frequency γi
is associated with each neuron and is set to zero initially.
An input vector x j is inputted to all the neurons, and awinner c is found according to Eq. (1). The weight vectorsof the neurons are updated according towi (t + 1) = wi (t) + hGc,i (t)(x j − wi (t)),(5)

where hGc,i (t) is the neighborhood function of Growing
Grid;
⎞
⎛
⎜⎜⎜ dg 2 (c, i) ⎟⎟⎟⎟⎠ ,⎜hGc,i (t) = α0 exp ⎝−
(6)
2σ0 2
where α0 is a constant learning rate, and σ0 is a constantwidth parameter. dg (c, i) is the distance on the grid between
a winner c and each neuron i and is calculated by city-block
distance (which is also known as L1 -norm). At each learning
step, the winning frequency of c is incremented by γc new =γc old + 1.
After n × m × λg number of learning steps have been
performed, we determine the neuron q which has become
the winner most frequently;
q = arg max{γi }.
i(7)

We find the neuron f which is with the most diﬀerent weightvector in 1-neighbor of q. We insert a new row (or column)
between q and f . The weight vectors of the new neurons areinterpolated from their neighbors which does increase thedensity of weight vectors in the vicinity of wq . The number nof rows (or m of columns) are increased, then all the winningfrequencies are reset. We continue with the next round oflearning unless nm ≥ nmmax is fulfilled.
After the growth process is finished, we fine-tune theweight vectors using a decreasing learning rate. We per
= n × m × λ f steps according to Eq. (5) using
form tmax 
α(t ) = α0 (α1 /α0 )t /tmax . t denotes the learning step in thefine-tuning phase.
4.

SOM with False-Neighbor Degree (FN-SOM)

We explain a new SOM algorithm, SOM with FalseNeighbor Degree between neurons (FN-SOM). Falseneighbor degrees of rows Rr (1 ≤ r ≤ n−1) are allocated between adjacent rows of FN-SOM with the size of n × m grid(as Fig. 2). Likewise, false-neighbor degrees of columns Ck(1 ≤ k ≤ m − 1) are allocated between adjacent columns of
FN-SOM. In other words, R1 means the false-neighbor degree between neurons of the 1st row and the 2nd row, andC4 is the false-neighbor degree between neurons of the 4thcolumn and the 5th column. The initial values of all of thefalse-neighbor degrees are set to zero, and the initial values of all the weight vectors are given over the input spaceat random. Moreover, a winning frequency γi is associatedwith each neuron and is set to zero initially.

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(a)
Fig. 2 A false-neighbor degree of row Rr (1 ≤ r ≤ n − 1) and column Ck(1 ≤ k ≤ m − 1). Neurons of FN-SOM are located at a n × m rectangular
grid.

Learning Ste p(FN-SOM1) An input vector x j is inputted to all the neurons at the same time in parallel.
(FN-SOM2) Distances between x j and all the weight vectors are calculated, and the winner c is found according to
Eq. (1).
(FN-SOM3) Increment of the winning frequency of winnerc by γc new = γc old + 1.
(FN-SOM4) The neighboring distances between the winnerc and the other neurons are calculated. For instance, for two
neurons s1 , which is located at r1 -th row and k1 -th column,
and s2 , which is located at r2 -th row and k2 -th column, theneighboring distance is defined as the following measure;
⎞2
⎛
r2 −1
⎟⎟⎟
⎜⎜⎜
⎜
d f (s1 , s2 ) = ⎜⎝⎜|r1 − r2 | +Rr ⎟⎟⎠⎟
r=r1

⎞2
⎛
k2 −1
⎟⎟⎟
⎜⎜⎜
+ ⎜⎜⎝⎜|k1 − k2 | +Ck ⎟⎟⎟⎠ ,

(8)

k=k1

2 −1
Rr means the sum
where r1 < r2 , k1 < k2 , namely, rr=r1
of the false-neighbor degrees between the rows r1 and r2 ,2 −1
Ck means the sum of the false-neighbor degrees
and kk=k1
between the column k1 and k2 .
(FN-SOM5) The weight vectors of the neurons are updatedaswi (t + 1) = wi (t) + hF c,i (t)(x j − wi (t)),(9)

where hF c,i (t) is the neighborhood function of FN-SOM:d f (c, i)
hF c,i (t) = α(t) exp − 2.
(10)2σ (t)(FN-SOM6) If nm
i=1 γi ≥ λ is satisfied, we find the falseneighbors and increase the false-neighboring degree, according to steps from (FN-SOM7) to (FN-SOM10). If not,we perform step (FN-SOM11). In other words, we consider
the false-neighbors every time when the learning steps are
performed for λ input data.
Considering False-Neighbors(FN-SOM7) We find a set of neurons S which have never(b)

Fig. 3 Increment the false-neighbor degree. (a) q and its false-neighbor
fq are in the 3rd row and in the 2nd and 3rd column, respectively. Then, thefalse-neighbor degree C2 between columns 2 and 3 is increased by Eq. (13).
(b) q and fq are in the 2nd column and in the 4th and 3rd row, respectively.
Then, the false-neighbor degree R3 between rows 3 and 4 is increased by
Eq. (14).

become the winner:
S = {i | γi = 0}.

(11)

If the neurons, which have never become the winner, do notexist, namely S = ∅, we return to (FN-SOM1) without considering the false-neighbors.
(FN-SOM8) A false-neighbor fq of each neuron q in S ischosen from the set of direct topological neighbors of q denoted as Nq 1 . fq is the neuron whose weight vector is most
distant from q:
fq = arg max{wi − wq }, q ∈ S , i ∈ Nq 1 .
i(12)(FN-SOM9) A false-neighbor degree between each q andits false-neighbor fq , Rr or Ck , is increased. If q and fq are
in the r-th row and in the k-th and (k + 1)-th column (asFig. 3(a)), the false-neighbor degree Ck between columns k
and k + 1 is increased according to
Ck new = Ck old +

n+m
.
2nm

(13)

In the same way, if q and fq are in the k-th column and in the(r + 1)-th and r-th row (as Fig. 3(b)), the false-neighbor degree Rr between rows r and r + 1 is also increased accordingto
Rr new = Rr old +

n+m
.
2nm

(14)

The increment (n + m)/2nm is the average of 1/n and 1/m.
This is because Eqs. (13) and (14) are performed for all q
in the set S , therefore, Ck or Rr is increased at most n or mtimes for every λ inputs.
(FN-SOM10) The winning frequency of all the neurons arereset to zero: γi = 0.
(FN-SOM11) The steps from (FN-SOM1) to (FN-SOM10)
are repeated for all the input data.

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(a)

(b)

(c)

(d)

Fig. 4 Learning results of three algorithms for Target data. (a) Input data. (b) Conventional SOM.
(c) Growing Grid. (d) FN-SOM.

5.

Experimental Results

We apply FN-SOM to various input data and compare FNSOM with the conventional SOM and Growing Grid.
5.1 For 2-dimensional DataFirst, we consider 2-dimensional input data as shown in
Fig. 4(a). The input data is Target data set, which has a clustering problem of outliers [6]. The total number of the input
data N is 770, and the input data has six clusters which include 4 outliers. All the input data are sorted at random.
Both the conventional SOM and FN-SOM have nm =100 neurons (10 × 10). Growing Grid starts learning witha 2 × 2 neurons, and new rows and columns are inserted aslong as the number of neurons is less than nmmax = 100.
We repeat the learning 20 times for all input data, namely
tmax = 15400. The parameters of the learning are chosen as
follows;(For SOM)

Table 1 Quantization error Qe, Topographic error T e and Neuron
utilization U for Target data.

α0 = 0.1, σ0 = 0.9, λg = 30, α1 = 0.005, λ f = 100,
(For FN-SOM)
α0 = 0.3, σ0 = 4, λ = 3000,
where we use the same α0 and σ0 to SOM and FN-SOM forthe comparison and the confirmation of the false-neighbor
degree eﬀect.
The learning results of the conventional SOM and theGrowing Grid are shown in Figs. 4(b) and (c), respectively.
We can see that there are some inactive neurons between
clusters. The other side, the result of FN-SOM is shownin Fig. 4(d). We can see from this figure that there are justa few inactive neurons between clusters, and FN-SOM canobtain the more eﬀective map reflecting the distribution stateof input data than SOM and the Growing Grid.
Furthermore, in order to evaluate the learning performance of FN-SOM in comparison with the conventional

Growing Grid
0.02370.24550.8137

FN-SOM
0.0191
0.04420.9100

SOM and Growing Grid, we use the following three measurements to evaluate the training performance of the threealgorithms.
Quantization Error Qe: This measures the average distance between each input vector and its winner [2];
N

Qe =

1
x j − w̄ j ,
N j=1

(15)

where w̄ j is the weight vector of the corresponding winnerof the input vector x j . Therefore, the small value Qe is moredesirable.
Topographic Error Te: This describes how well the SOM
preserves the topology of the studied data set [7];α0 = 0.3, σ0 = 4,
(For Growing Grid)

SOM
0.02070.07400.8100

Qe
Te
U

N

Te =

1
u(x j ),
N j=1

(16)

where N is the total number of input data, u(x j ) is 1 if thewinner and 2nd winner of x j are NOT 1-neighbors eachother, otherwise u(x j ) is 0. The small value T e is more desirable. Unlike the quantization error, it considers the structureof the map. For a strangely twisted map, the topographic
error is big even if the quantization error is small.
Neuron Utilization U: This measures the percentage ofneurons that are the winner of one or more input vector in
the map [5];
nm

U=

1
ui ,
nm i=1(17)

where ui = 1 if the neuron i is the winner of one or more
input data. Otherwise, ui = 0. Thus, U nearer 1.0 is moredesirable and larger value of U means fewer inactive neurons.

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(a)

(b)

(c)

(d)

Fig. 5 Learning results of three algorithms for Hepta data. (a) Input data. (b) Conventional SOM.
(c) Growing Grid. (d) FN-SOM.

The calculated three measurements are shown in Table 1. The quantization error Qe of FN-SOM is the smallest
value in the three algorithms, and by using FN-SOM, Qe
has improved 7.7% from using the conventional SOM. Thismeans that there are few errors between the input data andthe neurons of the learned map, in other words, FN-SOM
can be most reflecting the distribution state of the input data.
The neuron utilization U of FN-SOM is also the best value
in the three algorithms. It means that 90% of the neurons of
FN-SOM are the winner of one or more input data, namely,
there are few inactive neurons. On the other hand, the topographic error T e of FN-SOM is the smallest value although
Qe and U are the best values. It means that FN-SOM selforganizes most eﬀectively with maintenance of top quality
topology. We consider these obtained results. The inactive
neurons are on a part without the input data. Therefore, if
the inactive neurons are few, more neurons can self-organize
the input data of clusters. In consequence, the distance between each input data and its neurons will be small, namely
the quantization error Qe will be a small value. Furthermore, the neurons of FN-SOM are not aﬀected by the falseneighbors, so, the neurons can learn more distant for the
distant input data, than SOM learning. For these reasons,
FN-SOM obtains the best results.

Table 2 Quantization error Qe, Topographic error T e and Neuron
utilization U for Hepta data.
Qe
Te
U

SOM

Growing Grid

FN-SOM

0.03600.16980.65000.04090.15090.68630.03060.14620.8000

Table 3 Quantization error Qe, Topographic error T e and Neuron
utilization U for Iris data.
SOM
Growing Grid
FN-SOM
Qe
0.03740.04520.0323Te0.24000.22670.1667U0.72000.70480.8200The quantization error Qe of FN-SOM is the smallest valueand has improved 15.0% from using the conventional SOM.
Nonetheless, the topographic error T e of FN-SOM is also
the smallest value, and it has improved 13.9% from using
the conventional SOM and 3.1% from using Growing Grid.
The neuron utilization U of FN-SOM is also the best value
in three SOMs, and it has improved 23.1% from using theconventional SOM and 16.6% from using Growing Grid.
From these results, we can say that FN-SOM is the mosteﬀective.
5.3 For Iris Data

5.2 For 3-dimensional Data
Next, we apply the three algorithms to 3-dimensional input
data, Hepta data set [6], as shown in Fig. 5(a). The input
data has a clustering problem of diﬀerent variances. The
total number of the input data N is 212, and the input data
has seven clusters. All the input data are sorted at random.
We repeat the learning 70 times for all input data,
namely tmax = 14840. The input data are normalized andare sorted at random. The learning conditions are the sameused in Fig. 4.
The learning results of three algorithms are shownin Figs. 5(b), (c) and (d), respectively. We can see that
FN-SOM has the fewest inactive neurons between clusters.
Three map quality measures are shown in Table 2. Resultsof FN-SOM are the best values in all map quality measures.

Furthermore, we apply FN-SOM to the real world clustering problem. We use the Iris plant data [8] as real data. Thisdata is one of the best known databases to be found in thepattern recognition literature [9]. The data set contains threeclusters of 50 instances respectively, where each class refers
to a type of iris plant. The number of attributes is four as the
sepal length, the sepal width, the petal length and the petal
width, namely, the input data are 4-dimension. The threeclasses correspond to Iris setosa, Iris versicolor and Iris virginica, respectively. Iris setosa is linearly separable fromthe other two, however Iris versicolor and Iris virginica are
not linearly separable from each other.
We repeat the learning 100 times for all input data,
namely tmax = 15000. The input data are normalized andare sorted at random. The learning conditions are the same

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(a)

(b)

(c)

Fig. 6 Visualization results of Iris data. Label A, B and C correspond to Iris setosa, Iris versicolorand Iris virginica, respectively. (a) Conventional SOM. (b) FN-SOM. (c) FN-SOM using false-neighbor
degree.

used in Fig. 4.
The calculated quantization error Qe, the topographic
error T e and the neuron utilization U are shown in Table 3.
We confirm that the quantization error Qe and the topographic error T e of FN-SOM are the smallest value in thethree algorithms. Qe of FN-SOM has improved 13.6% fromusing the conventional SOM and T e of FN-SOM has also
improved 30.5% from using the conventional SOM. Thisis because the result of FN-SOM hardly has inactive neurons between Iris setosa and the other two, therefore, themore neurons can self-organize the data of Iris versicolorand Iris virginica. Furthermore, U of FN-SOM is also thebest value. From these results, we can confirm the eﬃciency
of FN-SOM.
Furthermore, we investigate the cluster structure of thelearned map in order to confirm the clustering ability of FNSOM. Figures 6(a) and (b) show distances between adjacentneurons of the learning results of the conventional SOM and
FN-SOM for Iris data, respectively. This figure thus visualizes the cluster structure of the map. Black circles on thisfigure mean large distance between neighboring map nodes.
Clusters are typically uniform areas of white circles. We cansee that the boundary lines of FN-SOM are clearer than theconventional SOM because FN-SOM has few inactive neurons. From these results, we can say that FN-SOM can beapplied to clustering and can obtain more eﬀective clustering result than SOM.
6.

Visualization Method Using False-Neighbor Degrees

FN-SOM learns not only input vectors but also the falseneighbor degree between the neurons. In order to utilize
the false-neighbor degree eﬀectively, we propose a simple
visualization method using the false-neighbor degrees.
Figure 6(c) shows not only the distances but also thefalse-neighbor degrees. The false-neighbor degree reflects
the size of circles. The bigger the false-neighbor degree between the neurons, the bigger the circle is shown. From
this figure, we can see the boundary line and cluster relation
more clearly.
It is a very simple visualization method using the falseneighbor degrees. The FN-SOM holds the possibility of re-alizing a more eﬀective visualization method.
7.

Conclusions

In this study, we have proposed a new SOM algorithm, SOM
with False-Neighbor degree between neurons (called FNSOM). False-neighbor degrees are allocated between adjacent rows and adjacent columns of FN-SOM. The initialvalues of all of the false-neighbor degrees are set to zero,
however, they are increased with learning, and the falseneighbor degrees act as a burden of the distance between
map nodes when the weight vectors of neurons are updated.
We have applied FN-SOM to 2-dimensional data, 3dimensional data and Iris data, and we have investigatedthe learning behaviors of FN-SOM. Furthermore, the results
were compared with those obtained by the conventional
SOM and Growing Grid. We have confirmed that the quantization error and the topographic error of FN-SOM were thesmallest value among the three algorithms. Moreover, theneuron utilization of FN-SOM was the largest value among
the three algorithms. From these results, we have confirmed
the eﬃciency of FN-SOM.
Besides, in order to confirm the clustering ability of
FN-SOM, we have investigated the cluster structure of themap by calculating the distances between neurons. As
the results, FN-SOM has obtained more eﬀective clustering
map than SOM, although investigating the clustering ability
quantitatively is our future work.
In addition, in order to utilize the false-neighbor degreeeﬀectively, we have proposed a simple visualization method
using the false-neighbor degrees. We have confirmed the effectiveness of the proposed method and that FN-SOM holds
the possibility of realizing a more eﬀective visualization
method.
Acknowledgment
This work was supported by Grant-Aid for JSPS Fellows(19-7384).
References[1] J. Vesanto and E. Alhoniemi, “Clustering of the self-organizing map,”

MATSUSHITA and NISHIO: SELF-ORGANIZING MAP WITH FALSE-NEIGHBOR DEGREE BETWEEN NEURONS

1469

IEEE Trans. Neural Netw., vol.11, no.3, pp.586–600, 2002.
[2] T. Kohonen, Self-organizing Maps, Berlin, Springer, 1995.
[3] B. Fritzke, “Growing grid — A self-organizing network with constantneighborhood range and adaptation strength,” Neural Processing Letters, vol.2, no.5, pp.9–13, 1995.
[4] L. Xu, A. Krzyzak, and E. Oja, “Rival penalized competitive learningfor clustering analysis, RBF net, and curve detection,” IEEE Trans.
Neural Netw., vol.4, no.4, pp.636–649, 1993.
[5] Y. Cheung and L. Law, “Rival-model penalized self-organizing map,”
IEEE Trans. Neural Netw., vol.18, no.1, pp.289–295, 2007.
[6] A. Ultsch, “Clustering with SOM: U*C,” Proc. Workshop on SelfOrganizing Maps, pp.75–82, 2005.
[7] K. Kiviluoto, “Topology preservation in self-organizing maps,” Proc.
International Conference on Neural Networks, pp.294–299, 1996.
[8] D.J. Newman, S. Hettich, C.L. Blake, and C.J. Merz, UCI Repositoryof Machine Learning Database, 1998, http://www.ics.uci.edu/˜mlearn/
MLRepository.html[9] R.A. Fisher, “The use of multiple measurements in taxonomic problems,” Annual Eugenics, no.7, part II, pp.179–188, 1936.

Haruna Matsushita
was born in Tokushima, Japan, in 1982. She received the B.E.,
M.E. degrees from Tokushima University, Tokushima, Japan, in 2005 and 2007. She is currently working towards Ph.D. degree at the same
university. Her research interests include theory
and application of Self-Organizing Maps. She isa student member of the IEEE.

Yoshifumi Nishio
received B.E., M.E.,
and Ph.D. degrees in electrical engineering from
Keio University, Yokohama Japan, in 1988,
1990, and 1993, respectively. In 1993, he joined
the Department of Electrical and Electronic Engineering at Tokushima University, Tokushima
Japan, where he is currently an Associate Professor. From May 2000 he spent a year in the
Laboratory of Nonlinear Systems (LANOS) at
the Swiss Federal Institute of Technology Lausanne (EPFL) as a Visiting Professor. His research interests include analysis and application of chaos in electrical circuits, analysis of synchronization in coupled oscillatory circuits, development of analyzing methods for nonlinear circuits, theory and applicationof cellular neural networks, and neural network architecture. He was the
Chair of the IEEE CASS Technical Committee on Nonlinear Circuits andSystems (NCAS) during 2004–2005, the Steering Committee Secretaryof the IEICE Research Society of Nonlinear Theory and its Applications(NOLTA) during 2004–2007, and is currently the Secretary/Treasurer of
the IEEE CASS Shikoku Chapter. He is serving as an Associate Editor
for the IEEE CAS Magazine, the IEEE CASS Newsletter, and the RISP
Journal of Signal Processing, and the Editor for the IEICE Fundamentals
Review. He is a senior member of the IEEE and the RISP.