USING FUZZY AND FRACTAL METHODS FOR ANALYZING
MARKET TIME SERIES
P. Kroha and M. Lauschke
Department of Computer Science, University of Technology, Strasse der Nationen 62, 09111 Chemnitz, Germany
Keywords:
Time series, Fuzzy-controller, Fractal analysis, Market data.
Abstract:
In this contribution, we investigate the possibilities of using fuzzy and fractal methods for analyzing time
series of market data. First, we implemented and tested a fuzzy component that provides fuzzyfication by the
Mamdani Larsen inference method with static rules using not only Gauss but also Cauchy and Mandelbrot
distribution. Second, we implemented and tested a fractal component that provides fuzzy clustering by the
Takagi Sugeno method with dynamic fuzzy rules. Looking for an optimum, we simulated many parameter
combinations and compared the results. We present some interesting results of our experiments.
1 INTRODUCTION
Currently, it is not difficult to collect and store very
large data representing time series. However, there
is a question whether it is possible to extract any in-
formation usable for trend forecasting (meteorology,
biology, seismology, finance) and how to do it.
Data about financial markets is very interesting.
There are large time series available and the eventu-
ally obtained forecast can be easily tested. The ques-
tion is whether a forecast exists, of course.
There are different hypotheses about processes in
markets. Under Efficient market hypothesis (Fama,
1970), (Malkiel, 1996), markets were assumed to be
efficient in the sense that prices reflected all current
information that could anticipate future events. There
is a statistical requirement that market returns were
normally distributed as white noise. This traditional
capital market theory has been modeled by probabili-
ties since the first approach in (Bachelier, 1964) (orig-
inally published in 1900 as Ph.D. Thesis).
More recently, Markowitz (Markowitz, 1952),
(Markowitz, 1959) used the standard deviation as a
measure of the risk of investment, and the covari-
ance of returns as a measure of diversification of in-
vestment, where uncorrelated or negative correlated
stocks reduced the risk of portfolio. The next famous
work based on probabilities is Black-Scholes option
pricing model (Black and Scholes, 1973).
Later, some anomalies in market development
have been found. They were explained by the fact that
different investors have different access to informati-
on (e.g. insiders know more), different investment
horizon (short-term investors, long-term investors),
and different interpretation procedures. Based on
these phenomenons, an Inefficient market hypothe-
sis was formulated (Shleifer, 2000) but the anomalies
cannot be modeled easily.
The probability model of markets used in (Fama,
1970), (Markowitz, 1952), (Black and Scholes, 1973)
has one advantage and one disadvantage:
The advantage is that it can be simply described
by tools for Gaussian statistics.
The disadvantage is that the measured data, i.e.
the market returns, are not distributed normally.
Using the current computer technology and the
known time series (e.g. 103-years known daily
prices of Dow Jones Industrial), the difference be-
tween the theoretically supposed distribution and
the distribution found by experiments is very sig-
nificant. This was documented in many works
starting with Mandelbrot (Mandelbrot, 1962),
(Peters, 1994) and others. Compared to normal
distribution, the real distribution of market returns
is characterized by asymmetry, by higher peaks at
the mean and fatter tails that do not converge to
zero.
The Fractal market hypothesis (Peters, 1994), (Pe-
ters, 1996) places no statistical requirements on the
market development process. The goal is to include
the investor behavior and to find a model that fits
to observated time series. The components of in-
vestor behavior are investment horizon and crowd be-
85
Kroha P. and Lauschke M..
USING FUZZY AND FRACTAL METHODS FOR ANALYZING MARKET TIME SERIES.
DOI: 10.5220/0003075200850092
In Proceedings of the International Conference on Fuzzy Computation and 2nd International Conference on Neural Computation (ICFC-2010), pages
85-92
ISBN: 978-989-8425-32-4
Copyright
c
2010 SCITEPRESS (Science and Technology Publications, Lda.)
havior. Investors have different horizons of invest-
ment. Hence, they have different strategies for buying
and selling. Further, there are panics and stampedes
caused by the known crowd behavior of investors. It
has been found that markets have a memory and their
behavior is not characterized by white noise (no mem-
ory) as suggested in (Fama, 1970) but by black noise
(Peters, 1994).
All investors are interested in the prediction of
market movements. Short-term investors follow tech-
nical analysis, long-term investors follow fundamen-
tal analysis. Often a combination of both approaches
will be used.
Theoretically, market movements cannot be pre-
dicted succesfully as both efficient market theory and
fractal market theory say. But there is not only local
noise. Also, non-regular and non-periodic global de-
terministic movements called “trends” are present. It
is very probably not predictable when they start and
how long they take. Investors try to estimate the trend
begin and the trend end and use loss-limiting strate-
gies that control buying and selling stocks.
So, the goal of such a strategy is to indicate that
a trend started, resp. finished and generate a corre-
sponding buy signal, resp. a sell signal, in such a way
that the gain is greater than the loss in the investment
horizon. To simplify the problem, we do not discuss
problems of taxes, problems of money management,
problems of hedging and other more complex strate-
gies that are used by traders and investors.
The motivation of our project is to build con-
trollers based on fuzzy and fractal technology and
test, what can be gained with fuzzy and fractal strate-
gies compared to the often used strategies based on
technical indicators of technical analysis.
Our original approach is that we implemented and
tested not only Gauss but also Cauchy and Mandel-
brot distributions in our fuzzy component. The pub-
lished fuzzy-controllers discussed in Section 2 (Re-
lated work) use Gaussian distribution of price devia-
tions even though it is known that the Gaussian nor-
mal distribution does not fit to the reality very well.
We compared the results and found that the Cauchy
distribution is more efficient than the others.
Further, we implemented and tested a fractal com-
ponent using fuzzy clustering and the Takagi Sugeno
method of dynamic fuzzy rules. This method brings
the best results. All methods were tested on daily
prices of 100 stocks of NASDAQ100 (see Section 9
for more details).
The rest of the paper is organized as follows. In
Section 2, we discuss related work. In Section 3,
we introduce the developed system. In Sections 4, 5,
and 6, its components are described. The synthesis of
the components is discussed in Section 7. The goals
of our investigation are explicitly stated in Section 8.
Section 9 describes the implementation, experiments,
and results. In the last section we conclude.
2 RELATED WORK
The idea that the market returns are not normally dis-
tributed is not new. It has been published in (Man-
delbrot, 1962). In (Dourra and Siy, 2002), a system
using the Mamdani Larsen fuzzy inference method is
described but with the Gaussian distribution in back-
ground. In (Castillo and Melin, 2002), a system us-
ing the Takagi Sugeno inference method was used to
calculate the fractal dimension of a time series. We
extended this approach using the Hurst exponent and
correlation quotient. In (Raimondi et al., 2007), a
system with technical indicators and lately a fuzzy
system (using triangular membership function) with
static fuzzy rules is described. This system was tested
with data from January, 1st, 2000 to July, 7th, 2006
and documented that the fuzzy system performed bet-
ter than the system using technical indicators.
3 THE SYSTEM DEVELOPED
The main components of our system and their fea-
tures:
Technical indicator component
(used MA, TBI, MACD, MAcut, dTD),
fuzzy-control component
(ROC, stochastic indicator, and support/resistence
indicator are used to get input data for fuzzy com-
ponent),
Static fuzzy rules,
Mamdani Larsen fuzzy inference,
Defuzzyfication.
Fractal analysis component
using additional input data (fractal dimension,
Hurst exponent, correlation, trend)
Fuzzy clustering and dynamic fuzzy rules
(Takagi Sugeno inference method),
Interpretation of the result.
Decision strategies,
Synthesis of results.
These parts will be discussed in the next sections.
ICFC 2010 - International Conference on Fuzzy Computation
86
User
System
entry
Dataset
Subsystem :
fuzzy-control
Subsystem :
technical
indicator
Subsystem :
fractal analysis
Merge results Decision
Figure 1: The system architecture and dataflow.
4 TECHNICAL INDICATORS
COMPONENT
Technical analysis is based on a study of patterns on
charts, as well as price trends, as well as support and
resistence levels at which rising or falling trends may
be halted or reversed. The main idea is that all in-
formation necessary to forecast the market is stored
in the existing time series. Some technical indicators
have been defined to indicate a trend’s begin, its end,
and its strength. Most short-term investors use tech-
nical analysis because it reflects the current investors’
behavior. The effectivity of technical indicators has
been investigated by (Hellstroem and Holmstroem,
1997b), (Hellstroem and Holmstroem, 1997a).
We have used it for two purposes:
First, to get input parameters for the fuzzy con-
trol component (technical indicators (Kirkpatrick
and Dahlquist, 2006) used: rate of change indi-
cator, stochastic indicator, and Support/resistance
indicator)—see Section 5.
Second, to get input parameters for the techni-
cal indicator component (technical indicator used:
MA, TBI, MACD, MAcut, dTD)—see Section
9.2. We normalized the used technical parameters
into the interval [0..1].
5 FUZZY COMPONENT
Fuzzyfication provides the transformation of numeric
input data (sharp data) into fuzzy data (unsharp data).
Values of technical indicators such as rate of change
indicator, stochastic indicator, and Support/resistance
indicator have been used as input data.
5.1 Technical Indicators as Input for the
Fuzzy Component
In this component, technical indicators will be com-
puted to be used as basics of input data for the fuzzy
component.
Y j
j m
Z k
k k
Figure 2: Fuzzy component.
Rate of Change (ROC). This indicator describes the
absolute difference between the the current stock
price and the price n days ago:
sp = last closing stock price
ROC = sp(today) sp(today ndays)
Stochastic Indicator. The main idea behind stochas-
tic indicator is that rising price tends to close near
its previous highs, and falling price tends to close
near its previous lows (definitions are given be-
low). K - D stochastic indicatr was introduced by
Lane (Lane, 1984). Usually, indicator K(nT ) (de-
noted often as %K - fast line) und D(nT )) (de-
noted often as %D - slow line) are used. We com-
puted two values for time interval n, where
sp = last closing stock price
l p = the lowest price
hp = the highest price
ap = average price of m days
K(nT ) =
sp(today) l p
hp l p
100
D(nT ) =
n
i=n3
K(iT )
3
;n 3
Low resp. high price means here the lowest resp.
the highest stock price in the given time interval.
The lag of 3 days used in D(nT) is a value recom-
mended by traders. Very probably, it represents
an experience that price changes older that 3 days
have a very small influence.
USING FUZZY AND FRACTAL METHODS FOR ANALYZING MARKET TIME SERIES
87
Support/Resistance Indicator.
sl = Support level
rl = Resistance level
sl = Avg(nT) 2 σ(nT ),
rl = Avg(nT)+ 2 σ(nT ),
where
σ(nT ) =
s
n
i=nm
(sp(day
i
) Avg(day
i
))
2
m
,
Avg(nT) = ap
=
n
i=nm
sp(day
i
)
m
5.2 Convergence Module:
More Input Parameters
In this component, the indicators mentioned above are
used to generate more parameters. We used the fol-
lowing equations from (Dourra and Siy, 2002):
Y
ROC
(nT ) =
R(nT ) R((n 30)T ))
R((n 30)T )
,
n 30,
Y
d(ROC)
(nT ) = Y
ROC
((n 2)T ) Y
ROC
(nT ),
n 2,
Y
D
(nT ) = D(nT ),
Y
K
(nT ) = K(nT ),
Y
DK
(nT ) = Y
D
(nT ) Y
K
(nT ),
Y
Res
(nT ) = Avg(nT ) + 2 σ(nT ) R(nT ),
n 30,
Y
Sup
(nT ) = R(nT ) (Avg(nT ) + 2 σ(nT )),
n 30,
Y
Avg
(nT ) = R(nT ) Avg(nT ), n 30,
R(nT ) is the stock price on the n-th day, D(nT ) und
K(nT ) are indicators defined above and Avg(nT ) is
the average stock price during the observation time
interval.
5.3 Fuzzyfication, Fuzzy Processing,
and Defuzzyfication
This part of our system generates forecast using the
Mamdani Larsen inference method. The membership
functions and rules are implemented as being static.
The indicators described above have been used as in-
put parameters in a similar way as in (Dourra and Siy,
2002). We used 11 fuzzy rules for fuzzyfication and
the Gaussian bell function (in the first approach - the
imp rovement is given in Section 9.1) as the member-
ship function (SUP = 100 und INF = 0). The output
membership function is shown in Fig. 3.
0 10 20 30 40 50 60 70 80 90 100
0
0,5
1
Output
x
µ(x)
Figure 3: Output membership function.
The output membership function is sent to the
defuzzyfication module. We used the center-of-area
method producing a value in the interval [0..100].
6 FRACTAL ANALYSIS
COMPONENT
This component uses stock prices as input data. In the
part Fractal data calculation (see Fig. 4), the follow-
ing data will be computed:
Box dimension and fractal dimension,
Hurst exponent,
Correlation,
Trend in interval.
Fractal data
calculation
Stockdata Fuzzyfication Fuzzy processing
Forecast
data
Fuzzy clustering
Figure 4: Fractal architecture.
The fractal dimension of the time series is a real
number of the interval [1;2] which can be seen as a
metric measuring how much the time series is jagged.
A value near 1 means that there is a trend similar to
a line, a value near 2 means that there are very many
positive or negative changes in the interval. To mea-
sure the fractal dimension, a box covering method is
used. The graph of the time series has to be covered
by a set of smallest quadratic nonoverlapping boxes of
the same size. The fractal dimension (exact: the frac-
tal capacity dimension) is the number of such boxes
that contain at least one point of the object (Castillo
and Melin, 2002). Then, we can calculate the fractal
dimension using the following formula:
ICFC 2010 - International Conference on Fuzzy Computation
88
Dim =
log
10
(number o f boxes)
log
10
(
1
sizeo f boxes
)
(1)
The box size is normalized in relation to the size of
the interval.
To obtain the Hurst exponent, we first have to
eliminate all linear trends from the data. Then, we
define a time interval N and find the range R and the
standard deviation S in this interval. We can derive
the Hurst quotient as
R
S
or calculate the Hurst expo-
nent (values in interval [0;1]) as:
H =
log(
R
S
)
log(
N
2
)
(2)
Time series with H near to 1 have trends, time se-
ries with H near to 0 are near to a white noise and no
trend can be found and forecasted. The Hurst expo-
nent has a close relation to the fractal dimension:
Dim = 2 H (3)
The correlation quotient (value of interval [-1;1])
specifies the linear correlation between elements of
the time series. A value near to 1 indicates a positive
trend, values near to 1 indicate a negative trend. The
following formula (Bravais-Pearson) will be used ( ¯x is
a mean value):
r =
n
i=1
(x
i
¯x)(y
i
¯y)
p
n
i=1
(x
i
¯x)
2
·
p
n
i=1
(y
i
¯y)
2
(4)
The trend in an interval is calculated as the differ-
ence between the stock value at the begin and at the
end of the interval.
The fuzzy-component used for the fractal analy-
sis subsystem (Fig. 1) uses fractal data (fractal di-
mension, Hurst exponent, correlation coefficient, and
trend) as input parameters. They are processed by a
fuzzy control system using the Takagi Sugeno infer-
ence method. The membership function and the rules
are not defined as being static. They will be found
dynamically using fuzzy clustering of time series in a
similar way as in (Castillo and Melin, 2002).
We assigned the cluster number to 4, being in-
spired by (low, medium, big, large). The clustering
is analysed by the c-mean clustering method. It gives
us the weights a
i j
we need for the output function cal-
culation. In our system, this process will be repeated
any 150 days with the last 364 data sets.
Having the membership functions given by clus-
tering the fuzzyfication can start to calculate the out-
put function directly according to:
y
res
=
n
i=1
E
i
(a
i0
+ a
i1
x
1
+ ... + a
im
x
m
)
n
i=1
E
i
(5)
There are more methods available how to calcu-
late variables E
i
which denote how much the rule R
i
matches. We have used c-mean clustering that calcu-
lates E
i
using
E
i
=
1
(
||~x~c
i
||
c
k=1
||~x~c
k
||
)
2
m1
(6)
The value of the output function has to be qualified
in some way, i.e. we have to specify the threshold for
buy and sell signals. To find some empirical values we
analysed 100 stocks in the time interval 2003 2009.
We found using simulation that is was most effective
(at least in the analysed time interval) to have +6% as
a threshold that should be crossed to generate a buy
signal and 6% as a threshold for sell signal.
7 SYNTHESIS OF COMPONENT
RESULTS
In the previous parts, all three components have been
described. The obtained results, i.e. the buy and sell
signals, have to be merged together as shown in Fig.
1.
First, we used the following merging formula,
which we denote as absolute merging:
result = rFC +
(rT I 0.5) 100
2
+
rFR 50
2
(7)
where rFC is the result of fuzzy-control, rT I is the
result of the technical indicators, rFR is the result of
the fractal component. The variable result is in the
interval [50...150].
Second, we used the following formula, which can
be denoted as mean merging:
FC = doFC rFC (8)
FI = doT I (rT I 100) (9)
FR = doFR rFR (10)
result =
FC +T I + FR
doFC + doT I + doFR
(11)
where variables doFC, doTI, and doFR have val-
ues 0 or 1 and indicate whether the results of the fuzzy
control (FC), the technical indicators (TI), and/or the
fractal control (FR) are present.
USING FUZZY AND FRACTAL METHODS FOR ANALYZING MARKET TIME SERIES
89
7.1 Decision Strategies
The resulting value of the whole system is a numeric
one, but we need a qualitative value for the decision.
This means, we need thresholds for the definition of
signals for buy and sell. We defined two thresholds
UTL (upper limit) and LT L (lower limit) and imple-
mented the following strategies:
Low risk strategy - LT L = 49 and UT L = 51 -
very careful but the frequency of buy and sell can
be very high.
High risk strategy - LTL = 40 and UTL = 60 -
more risk but the frequency of buying and selling
is not as high.
Variable border - the last values (0 means all past
data, 256 means the last 256 days) have been anal-
ysed and the best combination of LTL and UTL
will be used.
In Section 9, we show which decision strategy brings
the best results.
8 GOALS OF OUR
INVESTIGATION
As we stated in Section 1, market returns (exactly, in-
crements of market returns) are not distributed nor-
mally in real markets, even though normal distribu-
tion will be used in routine business. We stated the
following questions as objectives of our investigation:
which parameter combination used in the tech-
nical indicator component will bring the highest
gain,
which distribution function will bring the highest
gain when used in the fuzzy-controller,
which tuple of fractal analysis parameters will
bring the highest gain,
which component of our system will generate the
highest gain,
which combination of components can generate
more gain than each of them separately.
9 IMPLEMENTATION,
EXPERIMENTS, AND RESULTS
The presented system has been implemented in Java
as a multi-threaded component of our information
system (Kroha and Gemeinhardt, 2001), (Kroha and
Baeza-Yates, 2005), (Kroha et al., 2006), (Kroha and
Reichel, 2007), (Kroha et al., 2007), (Kroha and
Nienhold, 2010). A detailed description of the de-
sign and implementation of the presented system is
out of the scope of this paper and is given completely
in (Lauschke, 2010).
For our experiments with the implemented system
we used time series of all stocks from NASDAQ100
(daily prices) in the time interval from January, 1st,
2003 to October, 1st, 2009. When looking for the pa-
rameters’ value by simulation, we used an investment
of $ 10.000 and transaction costs of $ 10. The imple-
mented system runs on Apple MacPro with 8 cores, 8
GB of memory, and 2.26 GHz.
9.1 Non-Gaussian Distribution used in
the Fuzzy Component Contribution
In the following Table 1, we can see the results, i.e.,
the value (in thousands) of the invested $ 10,000)
when changing the distribution in the fuzzy-controller
and the strategy. The elapsed time was about 15 min-
utes.
Table 1: Results of Fuzzy-Control component.
Strategy Gauss Cauchy Mandelbrot
Low Risk $ 15.3 $ 16.8 $ 14.5
High Risk $ 17.8 $ 19.9 $ 19.3
Var. Border 0 $ 17.7 $ 19.7 $ 18.3
Var. Border 256 $ 15.2 $ 14.7 $ 15.1
We found that in most cases (Table 1) the Cauchy
distribution used in the fuzzy-controller gives the best
results (about 99 %) and the Gauss distribution the
worst results (about 78 %). Considering strategies,
the high risk strategy results were the best.
9.2 Technical Indicators Component
and its Contribution
We used the technical indicators with the following
parameters: Moving Average (17), TBI (9,17), TBI-
line = 100, MACD(12,26) - always calculated any 3
days. The highest gain (about 60%) was achieved for
the most simple merging (for variable border 0), but it
was rather small compared to fuzzy-control. Hence,
we do not describe the details about the used methods
of technical indicators merging here.
9.3 Fractal Component and its
Contribution
In this experiment, we used the 6 % threshold as ex-
plained above and used the rules recomputation, i.e.
ICFC 2010 - International Conference on Fuzzy Computation
90
the new clustering, every 150 days. The highest gain
(about 100 %) was achieved when using the tuple
[box-dimension;correlation] for fuzzy clustering. The
gain was higher than that of the other components. To
get the results, we needed about 32 minutes.
Table 2: Results of fractal component.
Input Result
Box-corr $ 20,024.19
Hurst-corr $ 17,797.11
Box-Hurst-corr $ 16,038.22
Box-Hurst-corr-Trend $ 16,951.81
9.4 Synthesis of Components and their
Contributions
We just described the individual behavior of compo-
nents. The next question was whether we can get
more profit when using some specific combination of
the components’ results. Because of the very large
number of possible combinations, the simulation all-
together took about 71 hours. The best gain of 88.4
% was obtained with the following parameters of our
system:
Synthesis of the components’ results = mean,
Distribution used in fuzzy-controller fuzzyfica-
tion = Cauchy,
Clustering in fractal analysis according to [Hurst
exponent; correlation],
Decision strategy = high risk.
We do not discuss the details of the synthesis be-
cause we can see that the best solution is to use only
the fractal component (higher gain than the combina-
tion with other components).
9.5 Comparison with the Strategy Buy
& Hold
One of the often used strategies is Buy & Hold. A
stock will be bought at the begin of the interval and
sold at the end of interval. In the next experiment, we
choose an interval (21.11.2000 - 26.9.2008) in which
the value of the German market index DAX was more
or less equal at the begin and at the end.
Then, we simulated how much an investor would
have earned when using our fuzzy or fractal controller
for his/her transactions. The Tables 3, 4 contains quo-
tients that denote how many times the investement
would have been increased. We can see that the frac-
tal method brings slightly better results than the Buy
& Hold strategy but in this case technical indicators
bring good results.
The number of transactions is important, too. The
Buy & Hold used only 2 transactions, fuzzy approach
used 7 transactions, technical indicators 70 transac-
tions, and fractal approach 117 transactions.
Table 3: Comparison with Buy & Hold - Fuzzy and Fractal
methods.
Index Buy & Hold Fuzzy Fractal
DAX 0.9365 1.2236 1.0326
Nasdaq100 0.6636 0.5206 0.8755
Table 4: Comparison with Buy & Hold - Technical Indica-
tors.
Index Techn. Ind. The best combination
DAX 1.5593 0.6884
Nasdaq100 0.6306 0.7256
10 CONCLUSIONS
As we have discussed, market returns are not dis-
tributed normally and the Gauss distribution used
in our fuzzy-controller delivered the worst results.
According to the Fractal market hypothesis, non-
periodical trends exist that correspond to the global
determinism component of the process and black
noise that corresponds to the local randomness com-
ponent.
The result is that it is not possible to forecast mar-
ket changes but it seems to be possible to achieve
some gain in the long-term investment when usig
fuzzy or fractal technology. Of course, we did not
consider taxes but they are different in various coun-
tries and changes in time. Further, the data process-
ing, especially in the case of fractal analysis, is very
time consuming as given above.
In the light of the recent financial crisis we have
to mention that we can only use public data for our
processing and forecasting. May be that there is a
currently hidden accounting fraud by a company like
by Enron in 2001 which was named by the magazine
Fortune as the America’s Most Innovative Company
for six consecutive years or by Lehman & Brothers in
2008. May be that there is a currently hidden account-
ing fraud by a state caused by years of unrestrained
spending like in Argentina in 1999 or in Greece in
2008. We cannot expect too much from public data
processing.
USING FUZZY AND FRACTAL METHODS FOR ANALYZING MARKET TIME SERIES
91
In further work, we will try to combine the fuzzy
and fractal methods with our text classification of
market news (Kroha et al., 2006), (Kroha and Re-
ichel, 2007), (Kroha et al., 2007), (Kroha and Nien-
hold, 2010). The goal would be to investigate whether
the result could be improved.
REFERENCES
Bachelier, L. (1964). Theory of speculation. In Coot-
ner, P.(Ed.): The Random Character of Stock MAr-
ket Prices. Cambridge, M.I.T. Press, 1964 (Originally
published in 1900 as Ph.D. Thesis).
Black, F. and Scholes, M. (1973). The pricing of options
and corporate liabilities. In Journal of Political Econ-
omy, May/June.
Castillo, O. and Melin, P. (2002). Hybrid intelligent sys-
tems for time series prediction using neural networks,
fuzzy logic, and fractal theory. In IEEE Transactions
on Neural Networks Vol 13. No. 6.
Dourra, H. and Siy, P. (2002). Investment using technical
analysis and fuzzy logic. In Fuzzy Sets and Systems,
Volume 127, pp. 221 - 240, Issue 2 (April 2002). Else-
vier North-Holland.
Fama, E. (1970). Efficient capital markets: A review of
theory and empirical work. In Journal of Finance, 25,
pp. 383-417.
Hellstroem, T. and Holmstroem, K. (1997a). Predictable
Patterns in Stock Returns. Technical Report Series
IMa-TOM-1997-09. Maelarden University.
Hellstroem, T. and Holmstroem, K. (1997b). Predicting
the Stock Market. Technical Report Series IMa-TOM-
1997-07. Maelarden University.
Kirkpatrick, C. and Dahlquist, J. (2006). Technical Anal-
ysis: The Complete Resource for Financial Market
Technicians. Financial Times Prent. Int.
Kroha, P. and Baeza-Yates, R. (2005). A case study: News
classification based on term frequency. In Proceedings
of 16th International Conference DEXA’2005, Work-
shop on Theory and Applications of Knowledge Man-
agement TAKMA’2005, pp. 428-432. IEEE Computer
Society.
Kroha, P., Baeza-Yates, R., and Krellner, B. (2006). Text
mining of business news for forecasting. In Proceed-
ings of 17th International Conference DEXA’2006,
Workshop on Theory and Applications of Knowledge
Management TAKMA’2006, pp. 171-175. IEEE Com-
puter Society.
Kroha, P. and Gemeinhardt, L. (2001). Using xml in
a web-oriented information system. In A.M. Tjoa,
R.R. Wagner (Eds.). Proceedings DEXA’2001, Work-
shop Network-Based Information Systems, 12th Inter-
national Workshop on Database and Expert Systems
Applications, pp. 217-221. IEEE Computer Society.
Kroha, P. and Nienhold, R. (2010). Classification of mar-
ket news and prediction of market trends. In Proceed-
ings of ICEIS’2010 - 12th International Conference of
Enterprise Information systems - Volume 2 (Artificial
Intelligence and Decision Support Systems), pp. 187 -
192, Madeira. INSTICC.
Kroha, P. and Reichel, T. (2007). Using grammars for text
classification. In In: Cardoso, J., Cordeiro, J., Filipe,
J.(Eds.): Proceedings of the 9th International Confer-
ence on Enterprise Information Systems ICEIS’2007,
Volume Artificial Intelligence and Decision Support
Systems, pp. 259-264. INSTICC with ACM SIGMIS
and AAAI.
Kroha, P., Reichel, T., and Krellner, B. (2007). Text mining
for indication of changes in long-term market trends.
In Tochtermann, K., Maurer, H. (Eds.): Proceed-
ings of I-KNOW’07 7th International Conference on
Knowledge Management as part of TRIPLE-I 2007,
Journal of Universal Computer Science, pp. 424-431.
Lane, G. (May/June 1984). Lane’s stochastics. In Techni-
cal Analysis od Stocks and Commodities magazine -
second issue, pp. 87 - 90.
Lauschke, M. (2010). Time series analysis of stock prices
using fuzzy logic, technical indicators, fractal analy-
sis. University of Technology Chemnitz, M.Sc. The-
sis,(In German).
Malkiel, B. (1996). A Random Walk Down Wall Street.
W.W. Norton, New York.
Mandelbrot, B. (1962). The variation of certain speculation
prices. In IBM Research Report. IBM.
Markowitz, H. (1952). Portfolio selection. In Journal of
Finance 7.
Markowitz, H. (1959). Portfolio Selection: Efficient Diver-
sification of Investments. John Wiley.
Peters, E. (1994). Fractal market analysis. John Wiley.
Peters, E. (1996). Chaos and Order in the Capitel Markets,
Second Edition. John Wiley.
Raimondi, F., Via, P., and Mul
`
e, M. (2007). A new fuzzy
logic controller for trading on the stock market. In
Proceedings of Conference ICEIS, pp. 322-329.
Shleifer, A. (2000). Inefficient Markets – An Introduction to
Behavioral Finance. Oxford University Press.
ICFC 2010 - International Conference on Fuzzy Computation
92