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-

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

 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

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 ) =

∑

i=n−3

K(iT )

;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

Support/Resistance Indicator.

sl = Support level

rl = Resistance level

sl = Avg(nT)− 2 ∗ σ(nT ),

rl = Avg(nT)+ 2 ∗ σ(nT ),

where

σ(nT ) =

∑

i=n−m

(sp(day

) − Avg(day

))

Avg(nT) = ap

∑

i=n−m

sp(day

)

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):

ROC

(nT ) =

R(nT ) − R((n − 30)T ))

R((n − 30)T )

n ≥ 30,

d(ROC)

(nT ) = Y

ROC

((n − 2)T ) −Y

ROC

(nT ),

n ≥ 2,

(nT ) = D(nT ),

(nT ) = K(nT ),

D−K

(nT ) = Y

(nT ) −Y

(nT ),

Res

(nT ) = Avg(nT ) + 2 ∗ σ(nT ) − R(nT ),

n ≥ 30,

Sup

(nT ) = R(nT ) − (Avg(nT ) + 2 ∗ σ(nT )),

n ≥ 30,

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

0,5

Output

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

Dim =

log

(number − o f − boxes)

log

(

size−o 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

or calculate the Hurst expo-

nent (values in interval [0;1]) as:

H =

log(

)

log(

)

(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 =

∑

i=1

− ¯x)(y

− ¯y)

∑

i=1

− ¯x)

∑

i=1

− ¯y)

(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:

res

∑

i=1

∗ (a

+ a

∗ x

+ ... + a

∗ x

)

∑

i=1

(5)

There are more methods available how to calcu-

late variables E

which denote how much the rule R

matches. We have used c-mean clustering that calcu-

lates E

using

(

||~x−~c

∑

k=1

||~x−~c

)

m−1

(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.

First, we used the following merging formula,

which we denote as absolute merging:

result = rFC +

(rT I − 0.5) ∗ 100

rFR − 50

(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

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

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

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.

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