A middle time recognition of epileptic seizures from geometrical patterns of EEG data
An approach for middle- time recognition of epileptic seizures from EEG data is proposed. The method considers sharp changes in the recorded data using geometrical patterns of the signal in phase-space. The approach was developed using experimental clinical EEG data recorded from ten patients and re...
Збережено в:
Дата: | 2002 |
---|---|
Автори: | , , , , , |
Формат: | Стаття |
Мова: | English |
Опубліковано: |
Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
2002
|
Назва видання: | Системні дослідження та інформаційні технології |
Теми: | |
Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/50249 |
Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
Цитувати: | A middle time recognition of epileptic seizures from geometrical patterns of EEG data / A. Makarenko, B. Oleksandruk, K. Schindler, F. Donatti, A. Villa, I. Tetko // Систем. дослідж. та інформ. технології. — 2002. — № 4. — С. 120-127. — Бібліогр.: 16 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraineid |
irk-123456789-50249 |
---|---|
record_format |
dspace |
spelling |
irk-123456789-502492013-10-09T03:06:05Z A middle time recognition of epileptic seizures from geometrical patterns of EEG data Makarenko, A. Oleksandruk, B. Schindler, K. Donatti, F. Villa, A. Tetko, I. Нові методи в системному аналізі, інформатиці та теорії прийняття рішень An approach for middle- time recognition of epileptic seizures from EEG data is proposed. The method considers sharp changes in the recorded data using geometrical patterns of the signal in phase-space. The approach was developed using experimental clinical EEG data recorded from ten patients and reliably predicted epileptic seizures in the ten-minute interval before the seizure onsets. An estimation of sensitivity and specificity of the proposed method is also provided. Запропоновано підхід до передбачення епілептичних припадків з ЕЕГ даних на середньотермінових інтервалах. Метод вивчає різкі зміни в отриманих даних використовуючи геометричну картину сигналу в фазовому просторі. Підхід развинено на основі використання реальних клінічних ЕЕГ даних, що записані у десяти пацієнтів, і показано передбачення епілептичних припадків за час до десяти хвилин перед припадком. Запропоновані також оцінки чутливості та особливостей запропонованого підходу. Предложен подход для предсказания эпилептических припадков из ЭЭГ данных на средневременных интервалах. Метод изучает резкие изменения в полученных данных используя геометрическую картину сигнала в фазовом пространстве. Подход развит на основе использования реальных клинических ЭЭГ данных записанных у десяти пациентов и показал предсказание эпилептических припадков за время до десяти минут перед припадком. Предложены также оценки чувствительности и особенностей предложенного подхода. 2002 Article A middle time recognition of epileptic seizures from geometrical patterns of EEG data / A. Makarenko, B. Oleksandruk, K. Schindler, F. Donatti, A. Villa, I. Tetko // Систем. дослідж. та інформ. технології. — 2002. — № 4. — С. 120-127. — Бібліогр.: 16 назв. — англ. 1681–6048 http://dspace.nbuv.gov.ua/handle/123456789/50249 519.6 en Системні дослідження та інформаційні технології Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України |
institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
collection |
DSpace DC |
language |
English |
topic |
Нові методи в системному аналізі, інформатиці та теорії прийняття рішень Нові методи в системному аналізі, інформатиці та теорії прийняття рішень |
spellingShingle |
Нові методи в системному аналізі, інформатиці та теорії прийняття рішень Нові методи в системному аналізі, інформатиці та теорії прийняття рішень Makarenko, A. Oleksandruk, B. Schindler, K. Donatti, F. Villa, A. Tetko, I. A middle time recognition of epileptic seizures from geometrical patterns of EEG data Системні дослідження та інформаційні технології |
description |
An approach for middle- time recognition of epileptic seizures from EEG data is proposed. The method considers sharp changes in the recorded data using geometrical patterns of the signal in phase-space. The approach was developed using experimental clinical EEG data recorded from ten patients and reliably predicted epileptic seizures in the ten-minute interval before the seizure onsets. An estimation of sensitivity and specificity of the proposed method is also provided. |
format |
Article |
author |
Makarenko, A. Oleksandruk, B. Schindler, K. Donatti, F. Villa, A. Tetko, I. |
author_facet |
Makarenko, A. Oleksandruk, B. Schindler, K. Donatti, F. Villa, A. Tetko, I. |
author_sort |
Makarenko, A. |
title |
A middle time recognition of epileptic seizures from geometrical patterns of EEG data |
title_short |
A middle time recognition of epileptic seizures from geometrical patterns of EEG data |
title_full |
A middle time recognition of epileptic seizures from geometrical patterns of EEG data |
title_fullStr |
A middle time recognition of epileptic seizures from geometrical patterns of EEG data |
title_full_unstemmed |
A middle time recognition of epileptic seizures from geometrical patterns of EEG data |
title_sort |
middle time recognition of epileptic seizures from geometrical patterns of eeg data |
publisher |
Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України |
publishDate |
2002 |
topic_facet |
Нові методи в системному аналізі, інформатиці та теорії прийняття рішень |
url |
http://dspace.nbuv.gov.ua/handle/123456789/50249 |
citation_txt |
A middle time recognition of epileptic seizures from geometrical patterns of EEG data / A. Makarenko, B. Oleksandruk, K. Schindler, F. Donatti, A. Villa, I. Tetko // Систем. дослідж. та інформ. технології. — 2002. — № 4. — С. 120-127. — Бібліогр.: 16 назв. — англ. |
series |
Системні дослідження та інформаційні технології |
work_keys_str_mv |
AT makarenkoa amiddletimerecognitionofepilepticseizuresfromgeometricalpatternsofeegdata AT oleksandrukb amiddletimerecognitionofepilepticseizuresfromgeometricalpatternsofeegdata AT schindlerk amiddletimerecognitionofepilepticseizuresfromgeometricalpatternsofeegdata AT donattif amiddletimerecognitionofepilepticseizuresfromgeometricalpatternsofeegdata AT villaa amiddletimerecognitionofepilepticseizuresfromgeometricalpatternsofeegdata AT tetkoi amiddletimerecognitionofepilepticseizuresfromgeometricalpatternsofeegdata AT makarenkoa middletimerecognitionofepilepticseizuresfromgeometricalpatternsofeegdata AT oleksandrukb middletimerecognitionofepilepticseizuresfromgeometricalpatternsofeegdata AT schindlerk middletimerecognitionofepilepticseizuresfromgeometricalpatternsofeegdata AT donattif middletimerecognitionofepilepticseizuresfromgeometricalpatternsofeegdata AT villaa middletimerecognitionofepilepticseizuresfromgeometricalpatternsofeegdata AT tetkoi middletimerecognitionofepilepticseizuresfromgeometricalpatternsofeegdata |
first_indexed |
2025-07-04T11:49:57Z |
last_indexed |
2025-07-04T11:49:57Z |
_version_ |
1836716966726860800 |
fulltext |
© A. Makarenko, B. Oleksandruk, K. Shindler, F. Donatti, A. Villa, I. Tetko, 2002
120 ISSN 1681–6048 System Research & Information Technologies, 2002, 4
TIДC
НОВІ МЕТОДИ В СИСТЕМНОМУ АНАЛІЗІ,
ІНФОРМАТИЦІ ТА ТЕОРІЇ ПРИЙНЯТТЯ РІШЕНЬ
UDC 519.6
A MIDDLE- TIME RECOGNITION OF EPILEPTIC SEIZURES
FROM GEOMETRICAL PATTERNS OF EEG DATA
A. MAKARENKO, B. OLEKSANDRUK, K. SCHINDLER,
F. DONATTI, A. VILLA, I. TETKO
An approach for middle- time recognition of epileptic seizures from EEG data is
proposed. The method considers sharp changes in the recorded data using
geometrical patterns of the signal in phase-space. The approach was developed using
experimental clinical EEG data recorded from ten patients and reliably predicted
epileptic seizures in the ten-minute interval before the seizure onsets. An estimation
of sensitivity and specificity of the proposed method is also provided.
1. INTRODUCTION
Epilepsy is one of the most common diseases and is observed in about 1% of the
population [1–3]. The prediction of epileptic seizure is important from many
points of view, e.g. the pharmaceutical design, localization of the epileptogenic
zone, recognition of neurological intracranial signals.
Recent information technologies provide new tools for such purposes. There
are many different methods for prediction of epileptic seizures hat are based on
spectral analysis, frequency plane, methods of dynamical systems and chaos theory,
correlation analysis and others [4–6]. However, the existing methods provide only
short time predictions before seizures (seconds in linear analysis and about 2–7
minutes in nonlinear methods). One of the most promising approaches in this field
[6] is based on phase space reconstruction using an extension of methods devel-
oped by F. Takens and P. Grassberger (see for description refs [7, 8]).
This article introduces a new algorithm based on the analysis of geometrical
structure of signals in phase space. This approach is in the mainstream of our ear-
lier recognition of HPLC data [9, 10]. and extracellular neuronal spikes in the
brain by phase space and correlation measure methods [11] and the complexity
measure of functions. The proposed approach makes possible to predict an appear-
ance of seizures in ten-minute interval before the onset of epileptic seizures.
2. EXPERIMENTAL DATABASE
The investigation was made on the background of clinical data with patients EEG
recorded at the Department of Neurology at the University of Bern. The EEG data
were recorded with the Biomedical Monitoring Systems Inc (Nicolet-BMSI,
A middle time recognition of epileptic seizures from geometrical patterns of EEG data.
Системні дослідження та інформаційні технології, 2002, 4 121
Madison, Wisconsin, USA). The Extended International 10–20 System
(32 channels) was used and the data were sampled at 200 Hz frequency. The data
were stored on a CD using HARMONY format. These raw data were translated
into ASCII text format using utilities developed by Dr. V.V. Volkovich and the
authors.
The data collected from ten patients were analyzed [12]. The analysis of
records was made at several different states: before seizure onset, during the
seizures, after the seizure and at a reference stage, at least one hour before or after
the seizures. There were about 80 records available for our analysis. The data
from all patients were analysed using the proposed method. However, the patients
had different sex, different localization of the epilepsy (left or right hemisphere)
and the epileptic diseases also had a different origin (mesial temporal sclerosis or
tumor). In addition, the provided data were of different duration ranging from 1 to
23 minutes of pre-ictal recordings. Therefore, in this particular study that is aimed
to introduce and to show a feasibility of a new approach, only data from two
patients (patient A and B) with the most similar characteristics that have
particularly long (12–19 minutes) records in the pre-ictal states were analysed.
After some preliminary visual inspection of data we selected electrode FP1 for all
results reported in this study. The results recorded from other electrodes were
similar but they are not considered in this article.
Some typical records of initial EEG are shown on Fig. 1.
Fig. 1. EEG signals recorded for a patient A in normal (panel A) and in the pre-ictal
(panel B) states several minutes before the seizures. The amplitudes of different electrode
signals are shown as a function of time. The labels of electrodes correspond to the
Extended International 10–20 System. Full recording time is 4 seconds for each panel.
Sampling frequency rate is 200 Hz
A Normal Stage B Seizure appearance
A. Makarenko, B. Olekcandruk, K. Shindler, F. Donatti, A. Villa, I. Tetko
ISSN 1681–6048 System Research & Information Technologies, 2002, 4 122
The left panel A shows an example of recording in the normal state. The
right panel contains data in pre-ictal state several minutes before the seizure and it
is characterized by an appearance of high-frequency components. We performed
an analysis of data in order to determine if the same effect can be observed and
quantified to reliably detect epileptic seizures and to monitor state of the patient in
automatic way.
3. METHODOLOGY
The main goal of this investigation was to detect which changes in the EEG sig-
nals could be used to predict epileptic seizures. Our basis hypothesis was to inves-
tigate the changes in the high-frequency components of the epileptic signal before
the seizure onset. Following the previous analysis that indicated a success of
analysis of signals in phase space [5, 6] we developed an original approach de-
scribed in Appendix A. While most of the previous methods were based on the
reconstruction of parameters of attractors (i.e., estimation of correlation dimensions,
embedding space, Lyapunov exponents, etc.) our approach directly considers
changes of the geometrical patterns of EEG signals in phase space. The signal
parameters were calculated by finite difference numerical methods (first differ-
ences for the first derivative, etc.). Examples of the signals in the normal and the
pre- ictal states are shown in the Fig. 2 and 3.
The patterns shown on both Figures were calculated using 5 sec (1000 points
per pattern) for channel FP1 as indicated in Appendix A.2. The geometrical
differences of signals on Figure 2 and 3 are evident. Therefore there was a need to
develop a method that could formalize the observed difference and can be used
for automatic monitoring of the patient state.
We developed a special numeric deviation index (DIC) that provided an
adaptive comparison of phase-space patterns in normal and the analyzed state by
considering the differences of individual patient parameters from the seizure-free
state (Appendix A.3).
FP1
electrode
A Differential dynamical
characteristic
B Integral dynamical
characteristic
Normal
Stage
d
iff
er
en
tia
l c
ha
r.
signal amplitude
si
gn
al
a
m
pl
itu
de
integral char.
Fig. 2. The signals from Fig. 1 (panel A) in phase space. The coordinates of points in the
left picture correspond to the values of the first derivative of signal (y-axis) and signal
amplitude (x-axis). The coordinates of points in the right picture correspond to the values of
the integral approximation of the signal (x-axis) and the signal amplitude (y-axis)
A middle time recognition of epileptic seizures from geometrical patterns of EEG data.
Системні дослідження та інформаційні технології, 2002, 4 123
4. CALCULATED RESULTS
The introduced index was calculated for all analyzed patients. Examination of data
indicated that the proposed index could be reliably used to detect the approaching
of seizures. An example of such analysis is illustrated on Fig. 4 and in Table.
T a b l e
Alarm level(index)
FP1 – CONTROL FP1 – PRESZ
Prediction status Time before the sei-
zure onset
0,125055285 0,587137577 635
0,249336978 0,513365047 630
0,256995382 0,94740823 625
0,203734454 0,327842294 620
0,461058119 0,439615508 615
0,263394124 1,265349021 610
0,230498485 1,118525366 605
0,157853545 1,133283503 600
0,135868341 8,143915624 Seizure 595
0,057482662 9,559343495 Seizure 590
0,168661222 11,84986324 Seizure 585
0,062955558 11,54262492 Seizure 580
0,261158555 12,33985246 Seizure 575
0,127643911 11,15360036 Seizure 570
0,090919277 13,14521231 Seizure 565
0,13670642 13,06123044 Seizure 560
0,226164765 14,83807468 Seizure 555
0,209961843 13,79559341 Seizure 550
0,108803929 14,25365139 Seizure 545
0,842898843 14,67213868 Seizure 540
0,099201891 14,03969703 Seizure 535
FP1
electrode
A Differential dynamical
characteristic
B Integral dynamical
characteristic
17 minutes
before
seizure
Fig. 3. The distribution of points in phase-space of the same patient from Figure 1 & 2 in
the pre-ictal state 17 minutes before the seizure. The difference between normal and the
analyzed state can be easily observed
A. Makarenko, B. Olekcandruk, K. Shindler, F. Donatti, A. Villa, I. Tetko
ISSN 1681–6048 System Research & Information Technologies, 2002, 4 124
The Table 1 show numeric values of the DIC calculated for the patient B. As
in the previous example a reliable detection of seizure was performed in ten-
minute time interval.
The second row indicates DIC values calculated at different times (indicated
in the last row) before the start of the seizure. The first row shows the values of
the same index calculated for an arbitrary period of the same length in the normal
state of the patient. There is a sharp increase (about 10 times in the magnitude) in
the value of the DIC index about 600 sec before the onset of the seizures.
Similar results were calculated for other analyzed patients and they will be
described in details in a separate study considering individual features of each
patient (Makarenko et al, in prep.). Thus the proposed approach represents a
promising method to detect epileptic seizures in the ten-minute interval before the
onset of seizures.
Some remarks should be posed about sensitivity and specificity of the
developed index. Some simple estimation of the sensitivity of the proposed
approach could be performed using the Shewhart control chart method [13]. This
method predicts an alarm signal whenever the average value of the observed
parameter (in our case of DIC, that is calculated each sec5=∆t ) measured over
time tNt ∆= * exceeds some threshold level kµ
∑
=
=
+=≥
Ni
i
h
DIX
N
Ky
N
hKy
,...,1
.1)(
,)( σ
µµ
Fig. 4. Dynamics of deviation index (DIC) before the seizure calculated using data
recorded with FP1 electrode for patient A. The seizure corresponds to time tic 190 (one
tic is equal to 5 sec). The DIC detects approaching of the seizure starting with 74th tic, i.e.
about 9 minutes before the seizure
Alarm seizure index
0
2
4
6
8
10
12
14
16
1 22 43 64 85 10
6
12
7
14
8
16
9
19
0
Time tics
A
la
rm
in
de
x
am
pl
itu
de
Normal
state
Before
seizure
A middle time recognition of epileptic seizures from geometrical patterns of EEG data.
Системні дослідження та інформаційні технології, 2002, 4 125
In this formula µ and 2σ corresponds to the mean and variance of the DIC
recorded in the normal state. The number N and parameter h represents
adjustable parameters that should be specified by the user. The larger values of
parameter N will provide more reliable detection of epileptic seizures and will
decrease number of false alarms. However, at the same time the larger values of
this parameter will require longer times to produce an «alarm signal» of the
system for the prediction of the onset of epileptic seizure, i.e., will decrease its
sensitivity. For example, let us fix the value of parameter N to be 12, i.e., the
detection of the alarm signal is done by considering a continuous record of 60 sec
duration. The increase of parameter h decreases the sensitivity of the method,
while the decrease of this coefficient increases probability of false positive errors
(false alarm signals). If we assume that the signal in the normal state is generated
according to the Gauss distribution, than the number of false alarm in the
Shewhart control chart method is given by Gauss cumulative distribution function
)(1 hφ− . For example, for h=3 one can expect to have only 1 error in 12 hours of
recording. Our analysis has indicated, that for such value 3=h the method
correctly predicted all seizures for all analyzed patients. At the same time, there
were no false alarms for the data recorded in the normal state of the patients.
However, the last result can be biased since only very short records (ca.
20 minutes) in the normal states of the patients were available for our analysis.
Thus, further analysis of data is required to better evaluate the performance of the
proposed method. It is possible that an assumption about the Gauss distribution of
DIC index in the normal state of patients is not correct. In this case, more
complex methods of data analysis, such as neural networks [14] or Group
Methods of Data Handling [15, 16] can be also used.
5. DISCUSSION
The results described in this paper indicate high predictive power of the proposed
approach. The time of seizure prediction (about 10 minutes from the seizure
onset) is sufficient in many cases for medical purposes, such as preparing of drugs
and treatment of patient by medical staff. These results are in good agreement
with another methodology based on the time- series analysis and phase space re-
construction [6]. Indeed, both methods take into account geometrical features of
the signals in phase space. The method proposed in the current article has similar
sensitivity and it makes possible to detect the epileptic seizures 10 minutes before
the seizures (ca 7 minutes in ref. [6]). A proper comparison of both approaches
requires that both methods will be be applied to the same datasets.
ACKNOWLEDGEMENT
This work is partially supported by INTAS 97-0173 and 00-0363 and SNSF 7-
IP-062620 grants. The authors are grateful to M. Kollar for their help with
preparing of EEG data and V.V. Volkovich for providing us the data conver-
sion utility.
A. Makarenko, B. Olekcandruk, K. Shindler, F. Donatti, A. Villa, I. Tetko
ISSN 1681–6048 System Research & Information Technologies, 2002, 4 126
APPENDIX A. ALGORITHM FOR DEVIATION INDEX CALCULATION (DIC).
The algorithm counts some measures of deviating of analyzed patient states from
its normal (without seizure) states. The algorithm consists from three stages: pre-
processing of signals, pattern building and evaluation DIC.
A.1. Preprocessing of signals in normal and current states of patient
Firstly the signals are normalized to new variables with zero mean value on
the selected time interval ∆t:
α+= ii xx~ ,
β+= ii cc~ ,
where
N
xi∑−=α ,
N
ci∑−=β , ictN ,200*∆= is the signal in the normal
state, ix is the signal in the analyzed state and ii xc ~,~ are the normalized signals.
Secondly we normalize the amplitudes of the signals on each time interval to
have the same average mean squared deviation:
ii xx ~*γ= ,
∑
= 2
ix
Nγ ,
ii cc ~*η= ,
∑
= 2
ic
Nη .
A.2. Construction of the geometrical pattern in phase space.
The two-dimensional pattern is constructed. The first axis corresponds to
dt
dx (i.e., differential characteristic that is estimated as txx ii ∆− + /)( 1 ) and another
corresponds to the sum of two nearest points 1++ ii xx , (i.e., the integral
characteristic). The total region is divided on the rectangle sub-regions and the
numbers of points in each sub-region is calculated. The calculated counts are
normalized on the analyzed number of considered time intervals.
A.3. DIC calculation
Firstly we calculate an average pattern for the normal state of the patient. In
order to calculate this pattern, available data from the normal state are subdivided
on n time records each of which has duration t (see above). The pattern of interest
is the average value of all n patterns. In addition to mean values, the standard
deviations are calculated for each point in the phase space. The weight of each
point in the phase space is calculated as
ji
jiw
,
,
dispersionwidth*height
1
+
= .
A middle time recognition of epileptic seizures from geometrical patterns of EEG data.
Системні дослідження та інформаційні технології, 2002, 4 127
The DIC is calculated as a difference between patterns in normal and in the
analyzed states as «»
∑ •−= 2
,
2
,, )ternCurrentPatternAveragePat( jijiji wDIC
REFERENCES
1. Fishman D., Goldberg J. R. What can you do about epilepsy? Dell Pub Co: New
York, 1991. — 147 p.
2. Penfield W., Jaspers H. Epilepsy and the functional anatomy of the human brain. —
Churchill: London, 1954. — 300 p.
3. Engel J. J. Seizure and epilepsy, F.A. Davis Company. — Philadelphia, 1989. — 215
p.
4. Lehnertz K., Widman G., Andrzejak R., Arnhold J., Elger C. E. Is it possible to an-
ticipate seizure onset by non-linear analysis of intracerebral EEG in human par-
tial epilepsies? — Rev Neurol (Paris), 1999. — 155. — P. 454–456.
5. Lehnertz K. Non-linear time series analysis of intracranial EEG recordings in patients
with epilepsy an overview // Int. J. Psychophysiol. — 1999. — 34. — P. 45–52.
6. Le Van Quyen M., Martinerie J., Navarro V. at all. Anticipation of epileptic seizures
from standard EEG recordings // Lancet. — 2001. — 357. — P. 183–188.
7. Grassberger P., Proccacia I. Characterization of strange attractors // Phys. Rev. Lett.
— 1983. — 50. — P. 346–349.
8. Abarbanel H. D. I., Gilpin M. E., Rotenberg M. Analysis of observed chaotic data. —
Berlin: Springer-Verlag, 1998. — 284 p.
9. Tetko I. V., Aksenova T. I., Patiokha A. A. at all. Pharmaceutical fingerprinting in
phase space. 2 // Pattern Recognition. Anal. Chem. — 1999. — 71. — P. 2431–
2439.
10. Aksenova T. I., Tetko I. V., Ivakhnenko A. G. at all. Pharmaceutical fingerprinting in
phase space. 1. // Construction of Phase Fingerprints. Anal. Chem. — 1999, 71,
P. 2423–2430.
11. Makarenko A.., Poliarush A.., Krachkovsky V. Pattern recognition and object classi-
fication in cytological investigations of neuronal system cells. First Nat. // Conf.
Of Young Scientist «System Analysis and Informational Technologies». —
Kyiv, 1999. — P. 42–43.
12. Villa A. E. P., Tetko I. V., Donati F. Functional characteristics of the epileptic area
determined by bi-spectral analysis in intractable medial temporal lobe epilepsy //
Proc. secind intern. conf. EMBC – 02. — Veinna, 2002. — P. 712–714.
13. Basseville M., Nikiforov I. V. Detection of abrupt changes: Theory and Application.
— Prentice Hall Inc., 1994. — 350 p.
14. Tetko I. V. Neural network studies. 4. Introduction to associative neural networks // J.
Chem Inf Comput Sci. — 2002. — 42. — P. 717–728.
15. Ivakhnenko A. G., Ivakhnenko G. A., Savchenko E. A., Wunsch D. Problems of fur-
ther development of GMDH Algorithms: Part 2 // Pattern Recognition and Image
Analysis. — 2002. — 12. — P. 6–19.
16. Ivakhnenko A. G., Ivakhnenko G. A., Tetko I. V., Sarychev A. P. Recognition of the
type of neurons' interaction from the histograms of pulse delay of their activity //
Pattern Recognition & Image Analysis. — 2000. — 10. — P. 164–168.
Received 29.10.2002
|