Self-organizing map clustering based on continuous multiresolution entropy

Abstract

The detection of changes in the parameter values of a nonlinear dynamic system is a branch of study with multiple applications. In this paper, we explore a variant of an automatic detector and clustering of slight parameter variations in nonlinear dynamic systems proposed by Torres et al. [Automatic detection of slight changes in nonlinear dynamical systems using multiresolution entropy tools, Int. J. Bifurc. Chaos 11(4) (2001) 967–981]. The new method takes the advantages of the continuous multiresolution entropy to localize slight changes in the parameters, and uses self-organizing maps to quantify and cluster these changes. We discuss the performance of this method while applied to automatic segmentation of natural and synthetic diphthongs in the presence of additive noise. Our results show the potentiality of the proposed method.

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Physica A 361 (2006) 337–354
Self-organizing map clustering based on
continuous multiresolution entropy
H.M. Torresa,1, J.A. Gurlekiana,1, H.L. Rufinerb,2,
M.E. Torresc,2,?
aLaboratorio de Investigaciones Sensoriales, Consejo Nacional de Investigaciones Cientı ´ficas y Te ´cnicas,
Instituto de Neurociencias Aplicadas, Hospital de Clı ´nicas, Buenos Aires, Argentina
bLaboratorio de Ciberne ´tica, Fac. Ingenierı ´a, Universidad Nacional de Entre Rı ´os, Oro Verde,
Entre Rı ´os, Argentina
cLaboratorio de Sen ˜ales y Dina ´micas no Lineales, Fac. Ingenierı ´a, Universidad Nacional de Entre Rı ´os,
Oro Verde, Entre Rı ´os, Argentina
Received 1 April 2005; received in revised form 4 May 2005
Available online 22 June 2005
Abstract
The detection of changes in the parameter values of a nonlinear dynamic system is a branch
of study with multiple applications. In this paper, we explore a variant of an automatic
detector and clustering of slight parameter variations in nonlinear dynamic systems proposed
by Torres et al. [Automatic detection of slight changes in nonlinear dynamical systems using
multiresolution entropy tools, Int. J. Bifurc. Chaos 11(4) (2001) 967–981]. The new method
takes the advantages of the continuous multiresolution entropy to localize slight changes in
the parameters, and uses self-organizing maps to quantify and cluster these changes. We
discuss the performance of this method while applied to automatic segmentation of natural
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www.elsevier.com/locate/physa
0378-4371/$-see front matter r 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.physa.2005.05.073
?Corresponding author. Tel.: +543434975078/77x122; fax: +543434975100/01x105.
E-mail address: metorres@ceride.gov.ar (M.E. Torres).
1This research has been carried out with the support of the Consejo Nacional de Investigaciones
Cientı´ficas y Te ´ cnicas, Argentina.
2Supported by Universidad Nacional de Entre Rı´os (UNER) and Agencia Nacional de Promocio ´ n
Cientı´fica y Tecnolo ´ gica, Argentina, under Project PICT-2002 11-12700, and PID-UNER 07F/6060.
sinc(i) Laboratory for Signals and Computational Intelligence (http://fich.unl.edu.ar/sinc)
H. M. Torres, J. Gurlekian, H. L. Rufiner & M. E. Torres; "Self-organizing map clustering based on continuous multiresolution entropy"
Physica A: Statistical Mechanics and its Applications. Vol. 361, No. 1, pp. 337--354, 2006.

Page 2

and synthetic diphthongs in the presence of additive noise. Our results show the potentiality of
the proposed method.
r 2005 Elsevier B.V. All rights reserved.
Keywords: Wavelet transform; Continuous multiresolution entropy; Self-organizing maps; Nonlinear
systems; Speech segmentation
1. Introduction
It is known that slight parameter variations in nonlinear dynamic systems can
produce important changes in the system’s behavior. If the system dynamics can be
mathematically described, there is a wide range of tools that can be used to
characterize its performance [1,2]. When the model equations are not known, and
only temporal data sequences acquired by the system under study are available,
techniques such as Lyapunov exponents [3,4], phase space reconstruction [4] and
correlation dimension [5] can be used. Unfortunately, they require a great amount of
data for their estimation, and signal stationarity is commonly assumed, which is not
always true in real applications [6,7].
A technique that allows the temporal localization of slight parameter changes in
the law governing the subjacent dynamic of a signal, coming from a nonlinear
dynamic system, has been proposed in Ref. [8] and is known as the continuous
multiresolution entropy (CME). It computes the entropy evolution by means of
sliding windows at each scale of the continuous wavelet transform (CWT) of the
given signal. The CME has been applied to non-stationary time series, with the
advantage that it does not require a large amount of data and it has low
computational cost.
CME is sensitive to changes in the dynamic complexity, displaying statistical
variations in the multiresolution entropy. In Ref. [8], an automatic detection
algorithm for the temporal localization of slight parameter changes at the governing
complex system was presented. This algorithm was based on the time series CME
analysis.
In the present work we propose a new technique that combines the CME with
Kohonen’s self-organizing maps (SOM), which shows to be useful not only to detect
the above-mentioned parameter changes, but also to perform a sort of clustering of
the given data. This is motivated by the SOM capacity to cluster and visualize high-
dimensional data, looking for a subjacent structure in the input data. SOMs have
been widely used for clustering in different applications, including speech signal
processing [9]. The performance and robustness of the new method proposed here is
tested in the presence of noise in numerical simulations and applied to vowel
segmentation in diphthongs, which are the phoneme sequences that present more
difficulties in continuous speech [10,11].
This paper is organized as follows: In Section 2, we present a brief summary of
basic concepts of entropy, continuous wavelet transform, entropy temporal
evolution, continuous multiresolution entropy and self-organizing maps. In Section
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H. M. Torres, J. Gurlekian, H. L. Rufiner & M. E. Torres; "Self-organizing map clustering based on continuous multiresolution entropy"
Physica A: Statistical Mechanics and its Applications. Vol. 361, No. 1, pp. 337--354, 2006.

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3 we describe how we combine these tools in order to perform the desired clustering.
Experiments and results are presented and discussed in Section 4. Conclusions are in
Section 5.
2. Definitions
2.1. Entropy evolution
The Shannon entropy of a random variable x is defined as [12]
Z
where fðxÞ is the density function of x, and the integration spreads only over the
region where fðxÞa0, with the agreement that ulnðuÞ ¼ 0 if u ¼ 0.
In order to define the temporal entropy evolution (TEE), we consider the Shannon
entropy of the random variable x as seen through a sliding temporal window Wtof a
given length L 2 R. For t 2 R, let xWtðuÞ be the function x restricted to interval
Wt¼ ½t ? L=2;t þ L=2? ? R and we define HL
HL
xðtÞ ¼ ?
R
Hy¼ ?
R
fðxÞ lnðfðxÞÞdx ,(1)
xðtÞ as [13,14]
Z
fðxWtÞ lnðfðxWtÞÞdxWt.(2)
In a similar way, the parametric entropy or q-entropy, which depends on a real
parameter qa1, known as Tsallis entropy [15], is given by [16,17]
Z
As before, we define the temporal q-entropy evolution (TqEE) Hq;L
Z
Hq
x¼ ðq ? 1Þ?1
R
fðxÞ½1 ? fðxÞq?1?dx .(3)
xðtÞ as [18]
Hq;L
xðtÞ ¼ ðq ? 1Þ?1
R
fðxWtÞ½1 ? fðxWtÞq?1?dxWt.(4)
2.2. Continuous wavelet transform
In order to analyze a signal xðtÞ 2 L2ðRÞ, the continuous wavelet transform (CWT)
[19,20] splits it up making inner products with a collection of functions
ja;bðtÞ ¼ jaj?1=2jð½t ? b?=aÞ, which are dilated and translated versions of a given
mother wavelet jðtÞ:
dxða;bÞ ¼ jaj?1=2
R
a
where jnindicates the complex conjugated of j and jðtÞ is an oscillatory function
whose Fourier transform FðoÞ satisfies
Z
Z
xðtÞjnt ? b
??
dt , (5)
Cj¼ 2p
R
joj?1jFðoÞj2doo1 . (6)
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H. M. Torres, J. Gurlekian, H. L. Rufiner & M. E. Torres; "Self-organizing map clustering based on continuous multiresolution entropy"
Physica A: Statistical Mechanics and its Applications. Vol. 361, No. 1, pp. 337--354, 2006.

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The CWT provides a natural tool for the time-frequency analysis because each ja;bis
predominantly located in a certain region of the time-frequency plane with a central
frequency that is inversely proportional to the scale a.
2.3. Continuous multiresolution entropy
Let D ¼ fdxða;bÞ; ða;bÞ 2 R2g be the set of the CWT of xðtÞ as defined in Eq. (5).
At each fixed scale a, and for t 2 R, we consider the temporal window Wa;tof length
L 2 R; defined by Wa;t¼ ½t ? L=2;t þ L=2?.
We define the continuous multiresolution entropy (CME), corresponding to the
Shannon entropy, as [14]
Z
and the one corresponding to the Tsallis entropy, the continuous multiresolution q-
entropy (CMqE), as [14]
Z
where da;t
xis dxða;bÞ restricted to b 2 Wa;tfor each scale a.
In this way a multiresolution measure is obtained. As the temporal variable t
evolves in R, at a fixed scale a, the CMEL
xða;tÞ represents the temporal evolution of
the wavelet coefficients entropy in the sliding temporal window Wa;t.
CMEL
xða;tÞ ¼ ?
R
fðda;t
xÞ lnðfðda;t
xÞÞdda;t
x, (7)
CMqEL
xða;tÞ ¼ ð1 ? qÞ?1
R
fðda;t
xÞ½1 ? ðfðda;t
xÞÞq?1?dda;t
x, (8)
2.4. Kohonen self-organizing maps
An SOM [9] is a non-supervised neural net model that allows to group and
visualize high-dimensional data. The SOM aim is to find a subjacent structure in the
input data.
An SOM defines a map from an N-dimension input space to a D-dimensional
neural arrangement, where, for j ¼ 1;...;J, we note as zjthe input vector and each
neuron i has associated a reference vector wi¼ ½wi
being the number of neurons, which has to be specified at the design stage. An
example for D ¼ 1 is shown in Fig. 1.
During an SOM training process, two stages can be distinguished: the first one is
the ordering of the reference vectors wiand the second one is the convergence of such
vectors. Both stages are iterative. At the ordering stage, for each iteration
v ¼ 0;...;V, which will be indicated as a super index in wiðwv
as:
(1) Initialize the reference vectors w0
iin a random way.
(2) At each iteration v, a zjvector is randomly chosen from the input data and its
similarity with the reference vectors wv?1
i
Euclidean distance. The neuron c which best approximates zjis called the ‘winner
1;...;wi
N?, with i ¼ 1;...;M, M
iÞ, can be formulated
is evaluated using, for example, the
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H. M. Torres, J. Gurlekian, H. L. Rufiner & M. E. Torres; "Self-organizing map clustering based on continuous multiresolution entropy"
Physica A: Statistical Mechanics and its Applications. Vol. 361, No. 1, pp. 337--354, 2006.

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neuron’ and it is given by
c ¼ argmin
i
kzj? wv?1
i
k .(9)
(3) The reference vectors are updated to wv
i:
wv
i¼ wv?1
i
þ hv; c
i, called ‘neighborhood function’, defines the map topology, and
connects the neurons in the arrangement. Usually hv; c
RDand ri2 RDare the location vectors of nodes c and i, respectively, in the
arrangement. Here we have considered a Gaussian neighborhood function, hv; c
?
where sðvÞ corresponds to the neighborhood width and aðvÞ 2 ½0;1Þ is the learning
coefficient. Both aðvÞ and sðvÞ are monotonically decreasing functions of algorithm
iterations. In our case
i½zj? wv?1
i
? .(10)
The function hv;c
i
¼ hðkrc? rik;vÞ, where rc2
i, as
hv; c
i
¼ aðvÞexp ?krc? rik2
2s2ðvÞ
?
,(11)
sðvÞ ¼ 1 þ ½sð0Þ ? 1?½V ? v?
V
,(12)
and
aðvÞ ¼ að0Þ 1 ?v
V
hi
, (13)
where sð0Þ and að0Þ must be defined in the design stage. This adaptation rule makes
the reference vectors in the neighborhood of c move toward the input vector. The
approximation degree is given by the neighborhood function hv;c
decreases as the patterns are presented.
(4) Repeat steps (2)–(3) a fixed number of times V.
For the convergence stage, the steps (2)–(3) are repeated for the new values of V,
sð0Þ and að0Þ.
i, which gradually
ARTICLE IN PRESS
Fig. 1. Scheme of a one-dimensional SOM array, with input zj¼ ½zj
vector w1¼ ½w1
1zj
2?, and four elements with reference
1w1
2?, w2¼ ½w2
1w2
2?, w3¼ ½w3
1w3
2? and w4¼ ½w4
1w4
2?.
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H. M. Torres, J. Gurlekian, H. L. Rufiner & M. E. Torres; "Self-organizing map clustering based on continuous multiresolution entropy"
Physica A: Statistical Mechanics and its Applications. Vol. 361, No. 1, pp. 337--354, 2006.

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When the training is finished, the vectors wV
function of the input data. Another important characteristic of the SOM is its
generalization capacity; it means that if a new input is presented to the SOM it can
be identified by its closest neuron based on the training achieved from inputs
previously seen.
iapproximate the probability density
3. Self-organizing map clustering based on continuous multiresolution entropy
In this section, we summarize the algorithm proposed in this work for automatic
detection and clustering of parameter changes in the underlying dynamics of signal
xðtÞ. Let us assume that either one signal or a set of signals, xðtÞ, holds the parameter
changes we want to detect. Below we describe the steps corresponding to the CME-
based algorithm (see Fig. 2). In a similar way it can be performed using the CMqE. It
involves the following steps:
(1) Let xðtÞ be the temporal evolution of a given signal and let us consider its discrete
evolution ^ xðkÞ, obtained by a regular sampling, ^ xðkÞ ¼ xðkDÞ, where k 2 Z and D
is the sampling rate.
(2) For
i ¼ K=2;ðK=2Þ þ ðK ? 2LÞ;ðK=2Þ þ 2ðK ? 2LÞ;...;imax
the temporalwindow
Wi;k
of
½i ? ðK=2Þ;i þ ðK=2Þ?. Let ^ xiðkÞ stand for ^ xðkÞ at the temporal window Wi;k
centered i.
(3) For each ^ xiðkÞ, the corresponding wavelet coefficient sets Diare obtained, with
Di¼ fd^ xiðj;kÞg. At this step a suitable wavelet has to be chosen.
(4) For each scale j the entropy is evaluated by sliding windows of length L and a
shift of m. In this way the CME is calculated, obtaining CMEL
discrete version of Eq. (7). In order to see the computational details, please refer
to Ref. [21].
(5) With the matrices CMEL
CMEL
(6) For each scale j, the corresponding CMEL
mean and unit standard deviation, obtaining Zðj;k0Þ, the statistically normalized
matrix associated with CMEL
^ x. The matrix Z qualitatively reveals the occurrence
weconsider
Wi;k¼
length
K 2 Z,defined by
^ xiðj;k0Þ by using the
^ xi,the matrix CMEL
^ x1 ...CMEL
^ xis formed by concatenation:
^ ximax?.
^ xðj;k0Þ is statistically normalized to zero
^ x¼ ½CMEL
^ x0CMEL
^ xi ...CMEL
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Fig. 2. Schemes of the stages of the proposed method, which are explained in the text.
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H. M. Torres, J. Gurlekian, H. L. Rufiner & M. E. Torres; "Self-organizing map clustering based on continuous multiresolution entropy"
Physica A: Statistical Mechanics and its Applications. Vol. 361, No. 1, pp. 337--354, 2006.

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of a slight parameter change in the underlying nonlinear dynamical system by
mean of a jump (up or down) at all the scales. However, this information that
appears to be redundant does not show the same intensity in the different scales.
(7) Each column of Z is used like an input vector to train an SOM.3At this point, we
have to select the topology, dimensions and training parameters of the map. In
general, no a priori rule exists to fix these parameters, and for each case there
exists an optimal set of them. The dimensions of the map and neuron disposition
in the arrangement are the more critical items. Once the SOM is trained,
temporal evolution of the winner neuron will be considered as the system output.
In such a way, changes in the winner neuron represent variations in the system
parameters, and for parameters of similar values, neighbor winner neurons will
be obtained. We have to emphasize that for different training data set and/or
SOM parameters different output configurations will be obtained, since these
depend on the training, although the data represent the same configurations of
the system.
In this work, a linear arrangement ðD ¼ 1Þ of neurons was defined, since it allows an
easy visualization and analysis of results. The number of neurons was empirically
determined for each of the test signals in order to assure a minimum of neurons equal
to the number of steady parameter sets that we want to identify. Extra neurons were
added to capture at least one transitional parameter set between steady parameter
sets. An excessive number of neurons would allow the network to capture more
information of the system, resulting in more than one neuron for each parameter set.
The training parameters are chosen to make the SOM to converge. In general, it is
accepted that this is obtained with small values of learning coefficients and great
number of iterations.
A possible criterion to select the wavelet could be to select one with the best
localization properties. For example, Mexican Hat is acknowledged to have the best
temporal localization properties [19] or Morlet wavelet is useful in locating
singularities [20]. Nevertheless, even if these properties give good hints about which
wavelet would be the best suited, the ultimate wavelet is, in most cases, empirically
selected and the best one might change from signal to signal. The signal’s frequency
range to be analyzed and the number of voices N (i.e., the length of the logarithmic
partition of the frequency range) must be established at this point. This frequency is
normalized, as is usual in wavelet analysis. If the frequency range is established as
[fmin;fmax], then aN?1
0
¼ fmax=fmin. As usual in the ‘‘quasi-continuous’’ wavelet
decomposition, in order to compute the CWT (Eq. (5)), we consider a dyadic
partition of the scales ða ¼ 2j; j 2 ZÞ and a uniform partition of the temporal
variable ðt ¼ kD; k 2 ZÞ, which gives rise to a quasi-continuous time-scale plane
representation d^ xiða ¼ 2j, t ¼ kDÞ ¼ d^ xiðj;kÞ of the sampled data ^ xiðkÞ. If a complex
wavelet is used, for example the Morlet wavelet of order 5, the coefficients dxðj;kÞ are
complex and then ðjd^ xðj;kÞj2Þ has to be considered as matrix D, i.e., the time-scale
ARTICLE IN PRESS
3In the present work, the SOM-PAK [22] software was used in order to carry out the experiments
described in Section 2.4.
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H. M. Torres, J. Gurlekian, H. L. Rufiner & M. E. Torres; "Self-organizing map clustering based on continuous multiresolution entropy"
Physica A: Statistical Mechanics and its Applications. Vol. 361, No. 1, pp. 337--354, 2006.

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matrix containing the squared magnitude of the CWT. In this paper we have
computed the CWT using the time-frequency toolbox (TFTB).4for Matlab.5
4. Results and discussion
In this section, some preliminary applications of the automatic detector based on
the CME–CMqE and SOM described in the previous section are first illustrated
through a simulated example corresponding to logistic map. Then we present the
results with synthesized and real speech diphthong signals.
4.1. Logistic equation
In this section we introduce the results obtained with our method applied to a well-
known nonlinear system, the logistic map, given by
xðk þ 1Þ ¼ aðkÞxðkÞ½1 ? xðkÞ? .
In order to perform the experiments, a signal has been generated according to
Eq. (14). Parameter aðkÞ varies slightly from a0¼ 3:545 to af¼ 3:562 by steps of
Da¼ 0:001. This variation is smoothed by dividing the sampling region
k 2 ½1;9 ? 104?
n ¼ 1;2;...;18, and defining the parameter variation aðkÞ in each interval according
to
?
where a1;nand a2;nare the initial and final parameter values in the nth interval, kc;nis
the central point of the change in In, and r is a fixed change radius. Setting a1;1¼ a0,
we obtain a2;1¼ a0þ Da¼ a1;2 and a2;18¼ af. Observe that this means that the
major parameter changes in each interval is concentrated around kc;nwith a radius r,
even if the parameter itself changes very slowly in each interval In, as can be seen
in Fig. 3 for the initial interval ðn ¼ 1Þ, where the parameter a varies, according to
Eq. (15), for a1;1¼ 3:545, a2;1¼ 3:546. This slight variation of the parameter allows
to test the segmenting and the clustering capacity of the method. We fixed r ¼ 100
and kc;n¼ 5000 þ ðn ? 1Þ ? 104=2? for all n (see Fig. 4(a)).
For the range of frequencies, from fmin¼ 0:1 to fmax¼ 0:45, the corresponding 40
CWT scales using a Mexican Hat wavelet have been evaluated, with window length
K ¼ 5000. Then, the CME corresponding to a sliding window, with L ¼ 500 samples
and shifted every m ¼ 100 samples, has been obtained. To estimate the probabilities,
20 bins were used. With the purpose of avoiding border effects, the first window of
(14)
insubintervals
In¼ ½2501 þ ðn ? 1Þ104=22501 þ n104=2?
for
aðkÞ ¼a1;nþ a2;n
2
þa2;n? a1;n
p
arctank ? kc;n
r
?
, (15)
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4TFTB has been developed by Francois Auger, Olivier Lemoine, Paulo Gonc -alves and Patrick Flandrin,
with the support of the CNRS (France) and Rice University (1995–1996). It is freely available on the
WWW at http://perso.wanadoo.fr/francois.auger/tftb.html or at http://gdr-isis.org/Applications/tftb/
iutsn.univ-nantes.fr/auger/tftb.html. It includes a reference manual and a tutorial.
5MatlabsThe MathWorks, Inc., v. 7. http://www.mathworks.com.
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H. M. Torres, J. Gurlekian, H. L. Rufiner & M. E. Torres; "Self-organizing map clustering based on continuous multiresolution entropy"
Physica A: Statistical Mechanics and its Applications. Vol. 361, No. 1, pp. 337--354, 2006.

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the CME was not taken into account. Finally, the matrix CME has been normalized.
With these data, a linear SOM with M ¼ 35 neurons was trained using a Gaussian
neighborhood. For the ordering stage, we used an initial learning coefficient
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0123456789
x 104
3.545
3.55
3.555
3.56
a(k)
k
Scale(j)
k
0123456789
x 104
x 104
0.1
0.2
0.3
0.4
0123456789
0
10
20
30
Winning Neuron
k
(a)
(b)
(c)
Fig. 4. (a) Time evolution of test signal parameter a, generated with Eq. (15); (b) CME and (c) the winner
neuron. Clearly, each parameter value a corresponds to a neuron.
0 100020003000 40005000
n
6000 7000800090001000
3.5
3.5002
3.5004
3.5006
3.5008
3.501
a[n]
Fig. 3. Parameter evolution a according to Eq. (15), with a1;1¼ 3:545, a2;1¼ 3:546, r ¼ 100 and
kc;1¼ 5000.
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H. M. Torres, J. Gurlekian, H. L. Rufiner & M. E. Torres; "Self-organizing map clustering based on continuous multiresolution entropy"
Physica A: Statistical Mechanics and its Applications. Vol. 361, No. 1, pp. 337--354, 2006.

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að0Þ ¼ 0:005, an initial neighborhood sð0Þ ¼ 35 and a number of iterations
V ¼ 500:000. For the convergence stage, we used að0Þ ¼ 0:0015, sð0Þ ¼ 2 neurons
and V ¼ 700:000 iterations.
Results are presented in Fig. 4. Comparing the first and third plots, we can see that
the method not only detects the change in the parameter but also clusters them,
assigning a neuron in particular to the parameter value. The previous experiments
have been run also using CMqE, for a value of q ¼ 1:8, obtaining similar results.
4.2. Speech signals—diphthongs
The phoneme segmentation consists of the division of a speech emission in
different phonetic units, that is to say phonemes that form part of it [23,24]. The
objective here is to insert a temporal mark indicating the beginning and the end of
each phoneme, together with the label of the identified phoneme. Different
techniques were used in order to carry out this task [25–29,2]. A preliminary
attempt to segment speech using entropy was presented in Ref. [30]. In spite of this,
there are situations in which the problem has not yet been solved, such as in the
presence of noise or in real-time applications.
Automatic segmentation of vowels in diphthongs is one of the most difficult tasks
due to the lack of a clear acoustic boundary during transitions between vowels. In
Fig. 5 main resonances6of the vocal tract can be observed as peaks—called
formants—superimposed over the Fourier power spectrum of a vowel /i/. The first
three steady state formants over time constitute a way to characterize isolated
vowels. As diphthongs are essentially a combination of two vowels, a corresponding
set of formant transitions or glides appears between them (see Fig. 8). Fluctuations
on continuous speaking rate and adjacent phoneme context produce nonlinear
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Energy (dB)
50
F1
F0
F2
F3
30
10
-10
-30
-50
02468 10
Frequency (kHz)
Fig. 5. Power spectrum, and its envelope of a Spanish vowel /i/ pronounced in sustained form, where the
formant frequencies have been highlighted.
6Estimated from an autoregressive model applied to a time windowed speech signal.
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Physica A: Statistical Mechanics and its Applications. Vol. 361, No. 1, pp. 337--354, 2006.

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variations on both durations of steady state segments and transitional segments. In
fast speech, steady state segments of closed vowels /i/ and /u/ tend to disappear and
only transitions and the open vowel formants remain available. Considering the 14
vocalic combination pairs, those composed by the pairs /ei/ and /ou/ are the most
difficult ones to separate, due to the proximity of their formant frequencies. In
running speech, vowels /i/ and /u/ are produced as semi-consonants [j] and [w] where
these vowels reduce their duration, intensity and move their formant frequencies
toward those corresponding to /e/ and /o/, respectively. Fig. 6 shows the movement
of vowel /i/ produced as a semi-consonant [j] in both [je] and [ej] diphthongs. As a
reference the F1–F2area of the isolated vowels /i/ and /e/ can be observed in the
background. Clearly, F2of /i/ is lowered and overlaps with F2of /e/. F1of /i/ tends
to increase and overlaps with F1 of /e/. We considered both diphthongs within a
syllable in a word, and those diphthongs are produced by word fusion extracted
from continuous speech under a variety of prosodic conditions.
4.2.1. Artificially generated diphthongs
In order to verify the capacity of our method to reveal the parameter change of a
system, we synthesized the speech emissions corresponding to diphthong [je]. We
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250300 350 400450500
550
2000
2200
2400
2600
2800
3000
3200
3400
3600
Frequency of F1 in Hz
Frequency of F2 in Hz
/i/
/e/
[j] in [ej]
[j] in [je]
/i/
/e/
[j] in [ej]
[j] in [je]
Fig. 6. Comparison between isolated vowels and the positional variants of /i/, in [je] and [ej] diphthongs
(adapted from [32]).
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sinc(i) Laboratory for Signals and Computational Intelligence (http://fich.unl.edu.ar/sinc)
H. M. Torres, J. Gurlekian, H. L. Rufiner & M. E. Torres; "Self-organizing map clustering based on continuous multiresolution entropy"
Physica A: Statistical Mechanics and its Applications. Vol. 361, No. 1, pp. 337--354, 2006.

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used a Klatt [31] synthesizer in order to generate 10 signals, using 3 formants and
frequency parameters, time and amplitude taken at random, with the premise that
the generated waves would correspond to the Spanish diphthong [je]. For all the
signals the energy was fixed at 60dB and the fundamental frequency was kept at
150Hz. In Table 1 the parameters used are shown and Fig. 7(a) is a diagram of the
parameters given to the synthesizer in order to obtain the diphthong numbered 1.
Steady state formants were added before initial transition time and after final
transitional time, to complete a total duration of 200ms. Furthermore, the
transitions are smoothed by the synthesizer.
In these experiments our purpose was to detect the beginning and the end of the
transition from one vowel to another. With this in mind, 50 CWT scales have been
calculated using a Mexican Hat wavelet, from fmin¼ 0:1 to fmax¼ 0:5. For the CME
calculation, a sliding window with L ¼ 500 (0.05s) samples was used, with a shift
m ¼ 50 (0.005s) samples, and in the probability estimation, 10 bins were used. After
normalizing the data obtained with the CME, a linear SOM with M ¼ 5 neurons
was trained using a Gaussian neighborhood. For the ordering stage, we used an
initial learning coefficient að0Þ ¼ 0:005, an initial neighborhood sð0Þ ¼ 5 and a
number of iterations V ¼ 500:000. For the convergence stage, we used að0Þ ¼ 0:0015,
sð0Þ ¼ 1 neurons and V ¼ 700:000 iterations. Here we used all the signals to train
and test the SOM.
The detection time of the beginning of the parameter change has a root mean
square error (RMSE) of 0.0076s, with respect to the real ones, with a standard
deviation (SD) of 0.0065s. At the end of the transition an RMSE of 0.011s, and an
SD of 0.011s were obtained. See Fig. 7(c) for stimuli number 1 output. Taking into
account the fixed fundamental frequency of 150Hz, which is a cycle duration of
0.0066s, we can say that the results obtained in average do not go beyond two cycles.
4.2.2. Diphthongs extracted from natural speech
Going ahead with our ‘‘leit motiv’’, we proceed with the segmentation of
diphthongs extracted from continuous speech emissions. Samples were extracted, as
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Table 1
Parameters entered into the synthesizer to generate the test diphthongs
SignalInitial/final time (s)
F1(KHz)
F2(KHz)
F3(KHz)
1
2
3
4
5
6
7
8
9
0.070/0.140
0.065/0.140
0.065/0.125
0.075/0.140
0.070/0.135
0.060/0.095
0.075/0.115
0.060/0.085
0.055/0.105
0.055/0.115
.250/.400
.330/.400
.215/.400
.266/.530
.250/.530
.250/.550
.280/.500
.280/.500
.200/.370
.200/.400
3.0/2.0
2.5/2.0
3.0/1.7
3.0/2.0
3.0/2.0
3.0/2.0
3.0/2.0
3.0/2.0
3.0/2.0
3.0/2.0
3.5/2.6
3.0/2.6
3.5/2.3
3.5/2.6
3.5/2.6
3.5/2.6
3.5/2.6
3.5/2.5
3.5/2.6
3.5/2.6 10
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H. M. Torres, J. Gurlekian, H. L. Rufiner & M. E. Torres; "Self-organizing map clustering based on continuous multiresolution entropy"
Physica A: Statistical Mechanics and its Applications. Vol. 361, No. 1, pp. 337--354, 2006.

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in the previous example, from a prosodic database [33] containing most (97%) of the
Spanish syllables in all allowed positions within a word and in both stressed and
unstressed conditions. Diphthongs [ej] appeared 74 times in total, [je] appeared 199
times, [ow] appeared 32 times and [wo] appeared 9 times.
In Fig. 8 we present a portion of the speech waveform excerpted from the sentence
[kaDa kamjon karGa entre kinse j Bejnte mil pesos]7(‘‘Each truck loads between
fifteen and twenty thousand pesos’’). Phoneme labels, wide band FFT spectrogram
and the first three formants are also displayed. We can appreciate that the variation
in the formants F2and F3in between 3.3 and 3.4s, i.e., when the diphthong [ej] is
pronounced, is very smooth.
The corresponding CME was obtained. CWT was calculated using a Mexican Hat
wavelet, with 40 scales, fmin¼ 0:1 and fmax¼ 0:5. In CME analysis, we used a sliding
window of L ¼ 512 samples (0.032s) width and an m ¼ 100 samples (3.1ms) shift.
For the probability estimation, 10 bins were used. With the obtained CME matrices,
ARTICLE IN PRESS
Scale(j)
00.020.040.060.080.10.120.14 0.160.180.2
0.1
0.2
0.3
0.4
0.5
00.02 0.04 0.060.080.1 0.12 0.140.16 0.180.2
0
2
4
6
Winner Neuron
Time (sec.)
0 0.020.040.06 0.080.1 0.120.140.160.18 0.2
0
1000
2000
3000
4000
Frequency (Hz.)
F3
F2
F1
(a)
(b)
(c)
Fig. 7. (a) Diagram of the formant evolution used for the synthesis of the stimuli number 1; (b) CME and
(c) the winner neuron. The dashed lines correspond to the beginning and end of the change of the
parameters. The solid line corresponds to the segmentation performed by our method.
7[B;D;G] are representing the approximate allophones of phonemes /b,d,g/.
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H. M. Torres, J. Gurlekian, H. L. Rufiner & M. E. Torres; "Self-organizing map clustering based on continuous multiresolution entropy"
Physica A: Statistical Mechanics and its Applications. Vol. 361, No. 1, pp. 337--354, 2006.

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an SOM was trained in a similar way as the previous example, but with only three
neurons.
In Fig. 9 an example of typical outputs for the pair [ej] is shown. The net activates
the neuron number 1 before the presence of /e/, and the neuron number 3 for [j]. In
the transition, the winner neuron is number 2. The dashed lines correspond to the
segmentation found in the database, i.e., it corresponds to a human labeler mark,
and the solid lines correspond to the detection found with our method. With the aim
of testing the performance of the method proposed here, we have calculated the
RMSE and the SD between the segmentation marks in the database and the time
instant given by the middle point in the transition from neuron number 1 to number
3 (that is to say, the middle point of the neuron number 2). The values obtained are
shown in Table 2. Taking into account that the considered speech signals had a
fundamental frequency of 200Hz, in average the RMSE does not exceed four cycles,
i.e., 20ms, relative to manual labels.
4.2.3. Robustness to additive noise
In order to explore the robustness of our method in the presence of additive noise,
we have performed the same experiments as in Section 4.2.2 but adding noise to the
signals corresponding to diphthong [ej]. Noisy signals were used both for training
and testing stages. Two kinds of noises have been used: white and babble.8The
signal noise ratio (SNR) is a measure of the strength of a wanted signal relative to the
ARTICLE IN PRESS
2.93 3.1 3.23.33.4
-0.2
0
0.2
0.4
sejBejn
Time (sec)
Frequency (Hz)
2.933.1 3.2 3.33.4
0
1000
2000
3000
4000
(a)
(b)
Fig. 8. (a) Speech waveform portion and phoneme labels (on top), and (b) wide band FFT spectrogram,
and the first three formants, corresponding to the phone sequences [se j Bejn], showing diphtongs [ej].
8The Signal Processing Information Base (SPIB), of Rice University, Houston, USA, http://
spib.rice.edu/spib.html, has been used.
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H. M. Torres, J. Gurlekian, H. L. Rufiner & M. E. Torres; "Self-organizing map clustering based on continuous multiresolution entropy"
Physica A: Statistical Mechanics and its Applications. Vol. 361, No. 1, pp. 337--354, 2006.

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amount of background noise and it is defined as
SNR ¼ 10log10
PðSignalWithoutNoiseÞ
PðNoiseÞ
, (16)
where Pð:Þ indicates the power of the corresponding signal, and it is expressed in
decibels (dB). Observe that SNR ¼ 0dB if PðSignalWithoutNoiseÞ ¼ PðNoiseÞ means
as much noise power as signal, which means less SNR value than the regular
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Table 2
RMSE and SD obtained by comparing the label marks of the database and the ones generated by our
method
[ej] [je][ow][wo]
RMSE (in s)
SD (in s)
0.020
0.017
0.023
0.019
0.017
0.017
0.016
0.017
Scale (j)
0.020.040.06 0.08 0.10.120.14
0.1
0.2
0.3
0.4
0.5
Scale (j)
0 0.020.040.060.08 0.10.120.14
0.1
0.2
0.3
0.4
0.5
0 0.020.040.060.08 0.10.120.14
0
1
2
3
4
Winner Neuron
Time(sec.)
(a)
(b)
(c)
Fig. 9. (a) Scalogram, (b) CME and (c) winner neurons of diphthong [ej] in the context ‘‘intente ´ influir’’
(‘‘I tried to influence’’). The dashed line corresponds to the segmentation found in the database, and the
solid line corresponds to the detection found with our method.
H.M. Torres et al. / Physica A 361 (2006) 337–354
351
sinc(i) Laboratory for Signals and Computational Intelligence (http://fich.unl.edu.ar/sinc)
H. M. Torres, J. Gurlekian, H. L. Rufiner & M. E. Torres; "Self-organizing map clustering based on continuous multiresolution entropy"
Physica A: Statistical Mechanics and its Applications. Vol. 361, No. 1, pp. 337--354, 2006.