Collective Computational Processes of Portfolio Management Using Cellular Automata
The paper presents simulation of selection elements of portfolio applying the classical Markowitz’s portfolio analysis theory using parallel collective computational environment — cellular automata. The basics of classical portfolio analysis and theory of cellular automata and their collective work...
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| Опубліковано в: : | Электронное моделирование |
|---|---|
| Дата: | 2009 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
2009
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/101531 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Collective Computational Processes of Portfolio Management Using Cellular Automata / A. Ulfik // Электронное моделирование. — 2009. — Т. 31, № 6. — С. 99-108. — Бібліогр.: 9 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859533954587033600 |
|---|---|
| author | Ulfik, A. |
| author_facet | Ulfik, A. |
| citation_txt | Collective Computational Processes of Portfolio Management Using Cellular Automata / A. Ulfik // Электронное моделирование. — 2009. — Т. 31, № 6. — С. 99-108. — Бібліогр.: 9 назв. — англ. |
| collection | DSpace DC |
| container_title | Электронное моделирование |
| description | The paper presents simulation of selection elements of portfolio applying the classical Markowitz’s portfolio analysis theory using parallel collective computational environment — cellular automata. The basics of classical portfolio analysis and theory of cellular automata and their collective work are presented. Structure of drawn up simulation and its results are presented. Simulations was performed application in Borland C++ Builder.
Представлено моделирование отдельных элементов портфеля инвестиций с применением классической теории анализа портфеля инвестиций Марковица с использованием парал-лельной коллективной вычислительной среды клеточных автоматов. Представлены основы классического анализа портфеля ценных бумаг и теории клеточных автоматов и их совместного использования, а также структура процедуры моделирования и его результаты. Моделирование выполнено в системе Борлэнд С++ Билдер.
Наведено моделювання окремих елементів портфеля інвестицій з застосуванням теорії аналізу інвестиційного портфеля Марковіца з використанням паралельного обчислювального середовища — кліткових автоматів. Наведено основи класичного аналізу портфеля цінних паперів і теорії кліткових автоматів та їхнього сумісного використання, а також структура процедури моделювання та його результати. Моделювання виконано у системі Борленд С++ Білдер.
|
| first_indexed | 2025-11-25T23:07:30Z |
| format | Article |
| fulltext |
A. Ulfik, PhD
Czestochowa University of Technology
(E-mail: ulfik@zim.pcz.pl)
Collective Computational Processes
of Portfolio Management Using Cellular Automata
The paper presents simulation of selection elements of portfolio applying the classical Markowitz’s
portfolio analysis theory using parallel collective computational environment — cellular automata.
The basics of classical portfolio analysis and theory of cellular automata and their collective work are
presented. Structure of drawn up simulation and its results are presented. Simulations was performed
application in Borland C++ Builder.
Ïðåäñòàâëåíî ìîäåëèðîâàíèå îòäåëüíûõ ýëåìåíòîâ ïîðòôåëÿ èíâåñòèöèé ñ ïðèìåíåíèåì
êëàññè÷åñêîé òåîðèè àíàëèçà ïîðòôåëÿ èíâåñòèöèé Ìàðêîâèöà ñ èñïîëüçîâàíèåì ïàðàë-
ëåëüíîé êîëëåêòèâíîé âû÷èñëèòåëüíîé ñðåäû êëåòî÷íûõ àâòîìàòîâ. Ïðåäñòàâëåíû îñíî-
âû êëàññè÷åñêîãî àíàëèçà ïîðòôåëÿ öåííûõ áóìàã è òåîðèè êëåòî÷íûõ àâòîìàòîâ è èõ
ñîâìåñòíîãî èñïîëüçîâàíèÿ, à òàêæå ñòðóêòóðà ïðîöåäóðû ìîäåëèðîâàíèÿ è åãî ðåçóëü-
òàòû. Ìîäåëèðîâàíèå âûïîëíåíî â ñèñòåìå Áîðëýíä Ñ++ Áèëäåð.
K e y w o r d s: portfolio analysis, portfolio management, cellular automata.
Portfolio analysis. In last few years, Polish stock market has strongly devel-
oped. About 500 stock companies are quoted on Warsaw Stock Exchange every
working day. Stock market in Poland in spite of today’s global capital markets
crisis is a good object for researches on behavior of stock prices. Apart from
stock, investors on Polish capital market can invest money in bonds, futures con-
tracts and other financial instruments. The main aim for investors is to estimate
correctly future value of securities and then choose the biggest profit ones. Prob-
lem is in high level of risk which those securities are characterized by very often.
Investors can reduce that risk by investing in more than one security. Portfolio
analysis is one of the basic techniques of capital market analysis.
According to the theory of portfolio analysis we can predict future values of
stock basing on historical quotation. From this data the rate of return is counted
interpreted as an expected profit and standard deviation which is a measure of
dispersion and is interpreted as a risk connected with this expected profit. Invest-
ments are usually interested in shares with the large profit and low risk level.
Portfolio analysis studies how these values will change if we will invest in more
than one share. This theory also shows how to choose assets during constructing
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 99
the portfolio to diversify a risk which means that the risk of portfolio would be
lower than the risk of shares that are this portfolio’s components.
H. Markowitz published in 1952 his first paper about the portfolio selection
[1]. It has started a real revolution for capital markets and created completely
new technique of making investment decisions today, called portfolio analysis.
Although this theory showed how to choose the best stocks to get the highest in-
come with the lowest level of risk, it was very hard to apply it in practice because
of computational complexity. After few years another scientist W. Sharp has
simplified this model and made the portfolio analysis able to be applied in prac-
tice [2]. He has added to this theory two coefficients giving investors a clear hint
which stocks are giving better results than a general tendency on the considered
market. He has suggested stock exchange index as a factor reflecting the market
tendency. On stock market in Poland investors most commonly use Warsaw
Stock Exchange Index for this purpose.
Rate of return and risk. In classical Markowitz’s model of portfolio selec-
tion the most important profiles of assets are rate of return and standard deviation
[3]. These two values have to be calculated for all shares taken by investor into
consideration. Positive value of rate of return is interpreted as an expected profit
and negative value — as an expected loss. This quantity is counted in a period of
time t with following formula:
R
P P D
P
t
t t t
t
�
� �
�
�
1
1
,
where Rt — the rate of return in period t; Pt — the price of securities in period t;
Pt –1 — the price of securities in period t – 1; Dt — paid-out dividend in period t.
Rate of return is appointed for every period of time t, there for it becomes a
function of time. The real value of income depends on many factors and investor
cannot be sure that he will get calculated profit. This is the reason that we use ex-
pression: expected rate of return. It is counted with the following formula:
R
R
N
t
t
t
N
�
�
�
1
,
where R — expected rate of return from securities; Rt — the rate of return in pe-
riod t; N — the number of all analyzed rates of return.
The level of profit or loss defined in this way is always accompanied by the
investment risk. The risk in portfolio analysis is calculated using statistics. Stan-
A. Ulfik
100 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
dard deviation is interpreted as the amount of risk. The formula to calculate vari-
ation of stock’s rate of return is shown below:
S
n
R Rt
t
n
2 2
1
1
1
�
�
�
�
�( ) ,
where S2
— the variation of the rate of return; R, Rt — as above; n — the number
of all analyzed rates of return; S — standard deviation of the rate of return.
Map of risk and income. Both values: expected income and risk can be
shown on a graph. It is called a map of risk and income. This graph makes pos-
sible retrieval companies with possibly the lowest risk and largest income. The
following figure represents a map of risk and income for counted all companies’
quotation from the Polish stock market in the period from October 1, 2008 to Oc-
tober 1, 2009.
As we can see on the map of risk and income, most of stock companies have
achieved standard deviation much higher than a standard deviation of Wars-
zawski Index Gieldowy (WIG) which is in the center of Fig. 1. There has been
only 2.5 % of stock companies with dispersion of its quotation lower than WIG,
and 97.5 % of stock companies with dispersion of its quotation higher than WIG.
That means that values of quotations of stock companies have been changing
more often than values of index WIG, so they have been more risky for inves-
tors. This situation happens very often on the Polish stock market.
Collective Computational Processes of Portfolio Management Using Cellular Automata
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 101
0.020
0.015
0.010
0.005
0
0.005
0.010
�
�
0 ,10 0 10 0.15 0.20 0.25. S
R
Fig. 1. Map of risk and income for companies on Warsaw Stock Exchange from 01.10.2008
to 01.10.2009
At the same time 45.6 % of stock companies expected the income higher
than index WIG, and 54.4 % of stock companies expected the income lower than
index WIG. This situation does not occur too often on the Polish stock market.
Usually situation is much worse for potential investors.
Two-element portfolios. The expected rate of return and standard deviation
are two basic characteristics of individual securities. When we consider more than
one security there is one more important value. It is the coefficient of correlation be-
tween two securities. It defines connection of rates of return of two stocks. The coef-
ficient of correlation can be calculated by the following formula:
�
12
1 1 2 2
1
1 2
�
� �
�
�( ) ( )R R R R
S S
i i
i
n
,
where �
12
— coefficient of correlation of rates of return of stock 1 and 2; R i1
—
rate of return in period i; R
1
— expected rate of return from the first stock; S1 —
standard deviation of rate of return of the first stock; R i2
, R
2
, S2 — the same for
the second stock; n — the number of all analyzed rates of return.
Investors are seeking for the possibility of investing capital in stocks with
high rate of return. However these securities are very often characterized by high
level of risk. The investor is interested in such investments, in which with
growth of rate of return, the risk will go down. The portfolio analysis gives us
such a possibility [4—6].
A portfolio is a set of stock, which we have or we want to buy. The rate of re-
turn of the portfolio is the sum of rates of return individual values multiplied by
their parts in investment:
R x R x Rp A A B B� � ,
0 1� �x A , 0 1� �xB , x xA B� �1,
where Rp — the rate of return from two component portfolio; xA — the part of
share A in portfolio; RA — the rate of return of stock A; xB, RB — the same for
stock B.
Calculations of risk for two-element investment portfolio are more compli-
cated. Variation of two-component portfolio is defined as:
S x S x S x x S S rp A A B B A B A B AB
2 2 2 2 2
2� � � ,
where S p
2
— variation of investment portfolio; xA, xB, RA, RB — as before; rAB —
the coefficient of correlation between the rate of return of stock A and B; Sp —
standard deviation of two-component portfolio.
A. Ulfik
102 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
Many-element portfolio. For more than two-element portfolio, the formu-
las are as follows:
R x Rp i i
i
N
�
�
�
1
,
S x S x x S S rp i i
i
N
i
N
j
N
i j i j ij
2 2 2
1 1 1
2� �
� � �
� � � .
Formulas above show computational complexity of the problem of selection
of effective portfolios. The portfolio is effective when the rate of return is higher
than any other portfolio with the same risk, or the level of risk is the lowest of the
portfolios with the same rate of return.
The Sharpe measure of portfolio effectiveness value is also called the
Sharpe efficiency of the portfolio. It is quotient of risk premium achieved by the
portfolio and risk of the total portfolio which has been taken. It must be calcu-
lated with the following formula:
SH
E R R
E S
J
J F
J
�
�( )
( )
,
where SHJ — Sharpe measure of portfolio effectiveness value; E RJ( ) — the
value of the expected rate of return on portfolio J; RF — risk-free rate of return;
SJ — standard deviation of portfolio returns J.
The higher is the value of this measure, the better is the quality of portfolio
management. Portfolios lying on the line of the greatest slope in the total system
risk are preferred as those of additional income.
Problem of selection of effective portfolio is not simple in spite of existence
of theoretical solution. With choosing portfolio the investor has first to choose
securities in which he wants to invest, and then establish how the invested capital
will be divided among securities. Very often the best solution we got as a result
of watching nature. The most popular usage of such processes are genetic algo-
rithms and neural networks. Another one is a cellular automaton. Cellular auto-
mata were created in the fortieth of the last century to emulate processes occur-
ring in nature. Soon after this invention, cellular automata proved to be very in-
teresting and useful. There are still some researches done to find new application
for cellular automata. One of them can be a problem of selection of portfolio in
the stock market.
Cellular automata (CA) are discrete models used mainly in physics, mathe-
matics and computability theory. They were created in the 40’s of the last cen-
tury by Stanislaw Ulman and later developed by his colleague John von Neu-
Collective Computational Processes of Portfolio Management Using Cellular Automata
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 103
mann who was also working at Los Alamos National Laboratory. CA are struc-
tures of the same cells put into lattice. It is usually one, two-or three-dimensional
and involves the great number of cells (theoretically this number is infinite but it
is impossible to implement such a model). Every cell has a defined type and
starting value. It also has its algorithm called a function of conversion. This
function defines what will be future value depending on the values of neighbor-
ing cells in current time [7].
There are two types of neighborhoods in CA: von Neuman’s and Moore’s.
Von Neuman’s neighborhood is made up of four cells adjacent vertically and
horizontally. Eight cells surrounding the given cell from right, left, top, bottom
and on a slant make up Moore’s neighborhood. In every iteration a current value
of every cell is calculated on the basis of values of adjacent cells from previous
iteration [8].
The best known example of CA is a game «Life» created by John Conway.
In this game every cell receives starting state: it can be active or unactive. This
game simulates real environment where animals can be born or day when they
do not have enough food. In this game there are settled rules of behavior of every
cell. If it is unactive and three other active cells surround it, it comes to life, and
so becomes active. If a cell is active and in its surrounding has two or three other
active cells, it stays active too. In every different case a cell is unactive — it stays
unactive if such it was earlier or if earlier conditions are not fulfilled. So simple
rules lead to astounding solutions. The final effect of CA can be systematized as
following: CA achieves stable state, in which nothing changes it (all cells stay in
determinate state); state of CA changes cyclically after some iterations; CA
achieves a chaotic state in which it is hard to find any order; in CA we can find
stable local configurations with long life.
The cellular automata are already applied for a long time in many fields of
science. In every issue it is necessary to establish three basic parameters of CA
as: type of cells creating CA, which means to establish what kind of information
they have to contain; starting value of every cell; function of passage which is an
algorithm deciding what will be the state of cells in a current iteration on the base
of values of neighboring cells in the previous iteration.
It has been proved that CA can simulate mechanisms occurring at the stock
market and show how this simulation can be useful for potential capital market
investors using its collective work to choose effective portfolios [3, 5], although
further researches are needed, especially in time of capital markets crisis. Struc-
ture of CA simulating stock market using collective work of all cells can be orga-
nized in many ways, as it was shown before [9].
Simulation results. To find an effective portfolio we have to establish work
of CA and their three basic parameters: type of cells, starting value and function
A. Ulfik
104 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
Collective Computational Processes of Portfolio Management Using Cellular Automata
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 105
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of passage from one iteration to the other. In suggested solution all cells of cellu-
lar automata work collectively exchanging information between each other [3].
All cells of CA have one aim: to choose an effective portfolio, so they have to in-
clude information about composition of this portfolio and during iterations im-
prove it. To be able to choose the portfolio, they should also have access to all
input data. Every cell can be interpreted as an artificial investor choosing the
most efficient portfolio. This artificial investor is working on his own choosing
the best result but at the same time he exchanges all information with other artifi-
cial investors which are in CA arranged in grid [9]. At the beginning of simula-
tion all cells of CA get randomly chosen portfolio. Then they calculate the basic
characteristics of portfolio: rate of return and standard deviation. Then every cell
communicates with its neighbors with Moore’s or von Neumann’s neighbor-
hood (it is that moment when they work collectively). If in another cell there is a
portfolio with better characteristics, then the cell gets this portfolio from its
neighbor. If portfolios in adjoining cells have worse characteristics, then the cell
stays with its original portfolio. At the next step all cells work individually, looking
for a new, better portfolio. They get this portfolio as completely new, randomly cho-
sen one. If the new portfolio is better, it stays as current one if not, then it is passed
over. This is repeated until all cells in CA get the same effective portfolio.
A. Ulfik
106 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
Fig. 2. Map of risk and income for two-element (a), three-element (b), five-element (c), ten-ele-
ment (d) portfolios
Table includes characteristics of portfolios achieved in independent simula-
tions. Table includes characteristics of two-, three-, five-, ten-element portfo-
lios. Additionally characteristics of index WIG for two-element portfolios are
shown in Table. These characteristics are rate of return R, S standard deviation
and Sharpe measure of portfolio effectiveness SHj.
All realizations of two-, three-, five- and ten-elements portfolios are shown
in Fig. 2. Precise data for the maps of risk and income are in Table. On all maps
of risk and income, there is also additionally index WIG as a mark of reference.
Data of index WIG characteristics are in Table.
Conclusions. The conducted simulations confirmed that cellular automata can
select portfolio on stock market, basing on classical Markowitz’s model of portfolio
analysis using collective work of cells. All portfolios obtained in simulation were
different in every simulation. There were shown ten random portfolios gained in
simulations for two-, three-, five- and ten-element portfolios.
On Fig. 2 we can see maps of risk and income for two-, three-, five- and ten-ele-
ment portfolios. All of those portfolios have the rate of return higher than that calcu-
lated for index WIG and almost all of those portfolios have higher standard devia-
tion, so they are more risky for investors. With this two characteristics it is hard to
rate those portfolios that are results of collective work of cellular automata.
The third calculated characteristic of portfolios — which is SHJ — Sharpe mea-
sure of portfolio effectiveness — is helpful when we want to compare results of sim-
ulations. In all cases SHJ is higher than SHJ calculated for index WIG. It is a clear
proof, that portfolios chosen by cellular automata are the effective ones.
Íàâåäåíî ìîäåëþâàííÿ îêðåìèõ åëåìåíò³â ïîðòôåëÿ ³íâåñòèö³é ç çàñòîñóâàííÿì òåîð³¿
àíàë³çó ³íâåñòèö³éíîãî ïîðòôåëÿ Ìàðêîâ³öà ç âèêîðèñòàííÿì ïàðàëåëüíîãî îá÷èñëþâàëü-
íîãî ñåðåäîâèùà — êë³òêîâèõ àâòîìàò³â. Íàâåäåíî îñíîâè êëàñè÷íîãî àíàë³çó ïîðòôåëÿ
ö³ííèõ ïàïåð³â ³ òåî𳿠êë³òêîâèõ àâòîìàò³â òà ¿õíüîãî ñóì³ñíîãî âèêîðèñòàííÿ, à òàêîæ
ñòðóêòóðà ïðîöåäóðè ìîäåëþâàííÿ òà éîãî ðåçóëüòàòè. Ìîäåëþâàííÿ âèêîíàíî ó ñèñòåì³
Áîðëåíä Ñ++ Á³ëäåð.
1. Markowitz H. Portfolio selection// Journal of Finance. — 1952. — P. 77—91.
2. Sharpe W. A Simplified model for portfolio analysis// Management Science. — 1963. — 9,
Iss. 2. — P. 277—293.
3. Katkow A. Ulfik A. Network implementation of investment management// Elektron. model-
ing. — 2007. — 29, ¹ 4. — P. 7—17.
4. Haugen R. Nowa nauka o finansach. — Warszawa : WIG-Press, 1999. — 828 ð.
5. Katkow A. Ulfik A. Simulation of Process of Portfolio Management Using Cellular Autom-
ata. Sbornik trudov konferencii MODELIROVANIE — 2006, Simulation. — Kiev : Insti-
tute of Modelling Problems in Power Engineering, National Academy of Sciences of
Ukraine, 2006. — P. 47—52.
Collective Computational Processes of Portfolio Management Using Cellular Automata
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 107
6. Jajuga K., Jajuga T. Inwestycje.— Warszawa: Wydawnictwo Naukowe PWN, 2005. —
230 ð.
7. Baldwin J. T., Shelah S. On the classifiability of cellular automata// Theoretical Computer
Science. — 2000. — 230, Iss. 1—2. — P. 117—129.
8. Kari Jarkko Theory of cellular automata: A survey// Theoretical Computer Science. —
2005. — 334, Iss. 1— 3. — P. 3—33.
9. Ulfik A. The simulation of parallel chaotic processes in sequential environments// Elektron.
modeling. — 2005. — 27, ¹ 2. — P. 67—81.
Submitted on 04.11.09
A. Ulfik
108 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
|
| id | nasplib_isofts_kiev_ua-123456789-101531 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0204-3572 |
| language | English |
| last_indexed | 2025-11-25T23:07:30Z |
| publishDate | 2009 |
| publisher | Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| record_format | dspace |
| spelling | Ulfik, A. 2016-06-04T13:19:43Z 2016-06-04T13:19:43Z 2009 Collective Computational Processes of Portfolio Management Using Cellular Automata / A. Ulfik // Электронное моделирование. — 2009. — Т. 31, № 6. — С. 99-108. — Бібліогр.: 9 назв. — англ. 0204-3572 https://nasplib.isofts.kiev.ua/handle/123456789/101531 The paper presents simulation of selection elements of portfolio applying the classical Markowitz’s portfolio analysis theory using parallel collective computational environment — cellular automata. The basics of classical portfolio analysis and theory of cellular automata and their collective work are presented. Structure of drawn up simulation and its results are presented. Simulations was performed application in Borland C++ Builder. Представлено моделирование отдельных элементов портфеля инвестиций с применением классической теории анализа портфеля инвестиций Марковица с использованием парал-лельной коллективной вычислительной среды клеточных автоматов. Представлены основы классического анализа портфеля ценных бумаг и теории клеточных автоматов и их совместного использования, а также структура процедуры моделирования и его результаты. Моделирование выполнено в системе Борлэнд С++ Билдер. Наведено моделювання окремих елементів портфеля інвестицій з застосуванням теорії аналізу інвестиційного портфеля Марковіца з використанням паралельного обчислювального середовища — кліткових автоматів. Наведено основи класичного аналізу портфеля цінних паперів і теорії кліткових автоматів та їхнього сумісного використання, а також структура процедури моделювання та його результати. Моделювання виконано у системі Борленд С++ Білдер. en Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України Электронное моделирование Collective Computational Processes of Portfolio Management Using Cellular Automata Коллективные вычислительные процессы в управлении портфелем инвестиций с использованием клеточных автоматов Article published earlier |
| spellingShingle | Collective Computational Processes of Portfolio Management Using Cellular Automata Ulfik, A. |
| title | Collective Computational Processes of Portfolio Management Using Cellular Automata |
| title_alt | Коллективные вычислительные процессы в управлении портфелем инвестиций с использованием клеточных автоматов |
| title_full | Collective Computational Processes of Portfolio Management Using Cellular Automata |
| title_fullStr | Collective Computational Processes of Portfolio Management Using Cellular Automata |
| title_full_unstemmed | Collective Computational Processes of Portfolio Management Using Cellular Automata |
| title_short | Collective Computational Processes of Portfolio Management Using Cellular Automata |
| title_sort | collective computational processes of portfolio management using cellular automata |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/101531 |
| work_keys_str_mv | AT ulfika collectivecomputationalprocessesofportfoliomanagementusingcellularautomata AT ulfika kollektivnyevyčislitelʹnyeprocessyvupravleniiportfeleminvesticiisispolʹzovaniemkletočnyhavtomatov |