Hurst Exponent and Trading Signals
Derived from Market Time Series
Petr Kroha and Miroslav
ˇ
Skoula
Czech Technical University, Faculty of Information Technology, Prague, Czech Republic
Keywords:
Time Series, Trading Signals, Fractal Dimension, Hurst Exponent, Technical Analysis Indicators, Decision
Support, Trading, Investment.
Abstract:
In this contribution, we investigate whether it is possible to use chaotic properties of time series in forecasting.
Time series of market data have components of white noise without any trend, and they have components of
brown noise containing trends. We constructed a new technical indicator MH (Moving Hurst) based on Hurst
exponent that describes chaotic properties of time series. Further, we stated and proved a hypothesis that this
indicator can bring more profit than the very well known indicator MACD (Moving Averages Convergence Di-
vergence) that is based on moving averages of time series values. In our experiments, we tested and evaluated
our proposal using hypothesis testing. We argue that Hurst exponent can be used as an indicator of technical
analysis under considerations discussed in our paper.
1 INTRODUCTION
The economists dispute over the problem how much
randomness influences markets and stock prices.
There are the following competing hypothesis trying
to explain it.
Efficient Market Hypothesis (Fama, 1970) states
that markets are efficient in the sense that the current
stock prices reflect completely all currently known
information that could anticipate future market, i.e.,
there is no information hidden that could be used to
predict future market development. This model is
based on the assumption that market changes can be
represented by a normal distribution.
Inefficient Market Hypothesis (Shleifer, 2000)
was formulated later because some anomalies in mar-
ket development have been found that cannot be ex-
plained as being caused by efficient markets. In (Pe-
ters, 1996) and (Lo and MacKinlay, 1999), a strong
statistical evidence is provided that the market does
not follow a normal, Gaussian random walk.
Fractal Market Hypothesis (Peters, 1994) repre-
sents a new framework for modeling the conflicting
randomness and deterministic characteristic of capi-
tal markets. We follow ideas of this hypothesis in our
paper.
Our motivation was to answer the question
whether it is possible to construct a new technical
indicator based on chaos theory which would bring
more profit than some of standard technical indica-
tors, e.g., MACD, that are often used.
We developed a new technical indicator MH (abr.
Moving Hurst) based on fractal dimension of time se-
ries, i.e., on Hurst exponent, and we evaluated the hy-
pothesis that Hurst exponent of market time series can
be successfully used as a technical indicator in a trad-
ing strategy. Exactly, it was used on existing data, and
it brought more profit than using of MACD.
The main idea of our approach is that changes in
fractal dimension of a time series, which describe the
history of prices, invoke changes in behavior of in-
vestors and traders. They buy or sell, and the feedback
can be either negative, i.e., the fluctuation of prices
decreases (a trend appears or continues), or positive,
i.e., the fluctuation of prices increases.
The Hurst exponent derived from fractal dimen-
sion describes chaotic properties of time series, and
it tells us whether there is a long memory process in
a time series or not. We discuss the values of Hurst
exponent in Section 3.6.
In contrast to the works cited in Section 2, we
do not agree with the commonly accepted conclusion
that Hurst exponent has no practical value in trading
forecast. Our original contribution is that we found,
implemented, and tested a construction supporting the
idea of Hurst exponent applicability to trading.
Our paper is structured as follows. In Section 2,
we discuss related works. We present the problems of
Kroha, P. and Škoula, M.
Hurst Exponent and Trading Signals Derived from Market Time Series.
DOI: 10.5220/0006667003710378
In Proceedings of the 20th International Conference on Enterprise Information Systems (ICEIS 2018), pages 371-378
ISBN: 978-989-758-298-1
Copyright
c
2019 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
371
chaotic markets in Section 3. Then we briefly explain
fractal dimension and methods how Hurst exponent
can be estimated in Sections 3.4, 3.6. In Section 4,
we describe the trading strategy we used in our exper-
iments. Our contribution is presented in Sections 4.1.
Our implementation, used data, experiments, and re-
sults are described in Sections 5. Section 5.1 contains
testing of our hypothesis. In Section 6, we conclude.
2 RELATED WORKS
The topic Fractals and Markets is covered by many in-
teresting and famous publications. Mandelbrot (Man-
delbrot, 1963) has shown that financial data have a
fractal nature, i.e., that time series of prices in 15 min-
utes interval have a very similar shape like time series
of daily close prices.
In his book (Mandelbrot, 1997), Mandelbrot col-
lected his papers on the application of the Hurst ex-
ponent to financial time series. Unfortunately, he does
not describe how fractals might be applied to financial
data to achieve more profit. In (Kroha and Lauschke,
2012), fractal dimension was used in fuzzy approach
to market forecasting.
Both Peters’s books (Peters, 1994), (Peters, 1996)
explain and discuss the Hurst exponent and its calcu-
lation using the rescaled range analysis (R/S analy-
sis). Conclusions of Peters support the idea that there
is indeed some local randomness and a global struc-
ture in the financial market. Unfortunately, Peters
only applies Hurst exponent estimation to a few time
series and does not discuss the accuracy of Hurst ex-
ponent calculation for sets of stock prices.
In the book (Lo and MacKinlay, 1999), long-
memory processes in stock market prices are dis-
cussed. But the authors do not find long-term memory
in stock market return data sets they examined. Meth-
ods that compute fractal dimension or Hurst exponent
are described in overview in (Gneiting et al., 2012).
The problem is that all of them are estimators and de-
liver values that differ.
Another promising approach has been presented
in (Selvaratnam and Kirley, 2006), (Qian and
Rasheed, 2004) where Hurst exponent is used as an
input parameter in neuronal networks applied to pre-
dict time series.
We investigated the topic of fractal dimension in
markets in our paper (Kroha and Lauschke, 2012) and
compared the fuzzy and fractal technology.
In paper (Mitra, 2012), the correlation between the
Hurst exponent of a time series and 1-day profit has
been measured, but because the 1-day profit has its
Hurst exponent near 0.5, only weak positive correla-
tion coefficients have been found.
Our goal is to investigate the impact of changes of
the Hurst exponent on trading strategies. This topic is
discussed in (Vantuch, 2014), where Hurst exponent
is used for supposed improvement of performance of
trading strategies based on technical indicators RSI
and CCI. However, no improvement has been found.
3 PROBLEMS OF CHAOTIC
MARKETS
3.1 Deterministic Chaotic Systems and
Non-deterministic Random Systems
The behavior of many physical systems is strongly
given by physical laws and initial conditions, i.e., an
initial state given by values of input parameters at the
start time. Often, we suppose we can restart such sys-
tems using the same initial state. In simple systems
(e.g., some computer programs), we really can do it.
In complex systems (e.g., computer programs coop-
erating with Web), we practically cannot restart the
system twice under the same initial conditions. We
do not know the state of the global network exactly
enough, because of ever-present changes and unex-
pected events. This phenomenon is not a new one.
Old Greeks knew the saying ”No man ever steps in
the same river twice” (Heraclitus of Ephesus).
There are two kinds of such systems:
Deterministic chaotic systems have the property
that their final behavior is extremely dependent on
any imprecision in the initial conditions (Lorenz,
1963). Determinism means that rules of be-
havior do not involve probabilities. These sys-
tems are described by nonlinear differential equa-
tions, whose solutions behave irregularly (Cas-
dagli, 1991). Their properties are discussed in
many publications, e.g., in (Gleick, 1987).
Stochastic nonlinear systems are affected not only
by small differences of input parameters but also
by unpredictable, random, external events having
unpredictable impact on system behavior. They
are indeterministic because their rules of behavior
involve probabilities.
Financial markets can be seen as a complex mix-
ture of deterministic chaotic systems and stochas-
tic non-linear systems, even though fractal market
hypothesis stresses the chaotic part. Processes be-
hind markets have their weak deterministic compo-
nent (some deterministic rules exist, e.g., 1-day re-
ICEIS 2018 - 20th International Conference on Enterprise Information Systems
372
turns have Gaussian distribution), but they have a
strong built-in randomness component, because the
main changes are reactions on unpredictable, random
events in the world, e.g., volcano eruption, terrorist at-
tack on World Trade Center, floods in Thailand, some
political decisions.
Additionally, compared with deterministic chaotic
systems in physics (e.g., in meteorology, etc.), mar-
kets are nonlinear feedback systems, because they
contain a component including psychology of human
investors called behavior finance. This component
brings reflexivity into the system, i.e., circular re-
lationships between cause and effect. For example,
when we would predict weather very exactly, weather
were not change because of it. On the other hand, a
well-known, precise market prediction would change
markets completely.
Before trading, investors try to analyze the market.
There are two main methods of stock analysis.
Fundamental analysis assumes that the markets
prices in the future can be derived from the eco-
nomic results of companies today.
Technical analysis assumes that market prices are
affected mainly by behavior and sentiment of in-
vestors, because investors interpret the economic
results. Components of investor behavior, like
greed, are regarded as being stable, and they spec-
ify some patterns usable for price prediction.
Often, methods of fundamental and technical analysis
are combined.
3.2 The Problem of Volatility
To measure the chaotic behavior of markets is not a
new idea, because the chaotic behavior is very obvi-
ous in every observation. There was an instrument
used by traders before any knowledge about fractal
dimension. It has been denoted as volatility, and it
is the standard deviation of prices. It is defined as
a measure for variation of price of a financial instru-
ment over time, i.e., volatility is simply the range in
which a day trader operates. Volatility is investigated
mainly for purpose of risk management (Brooks and
Persand, 2003). Volatility forecasting is described in
(Northington, 2009), and all aspects of using volatil-
ity are critically discussed in (Goldstein and Taleb,
2007).
3.3 The Problem of Stationarity
Both kinds of analysis use the hypothesis that we can
apply our experience coming from the past to predict
the future. The hypothesis is based on the assumption
that the same set and structure of patterns, which oc-
curred in the past, will occur again in the future, i.e.,
the statistical characteristics of data in the past are the
same as they will be in the future. This property is
called stationarity.
We cannot exactly answer the question whether
time series representing market data stay stationary
in the future. We can only investigate whether time
series were stationary in the past. However, if they
were, it does not mean that they stay stationary for
ever. More or less, we cannot see any reason for that,
because there are unique events that never occurred in
the past.
In (Taleb, 2007), the story of a turkey is presented
as an excellent example of expected (but not well-
founded) stationarity. A turkey is regularly fed by
a farmer for 1,000 days. It derived a simple pattern
saying that it will be fed at the Day 1001, too. But
it was two days before Thanksgiving, and the turkey
was served in the next days as a dinner. We can see
that we can never know whether our patterns from
the past will be valid in the future. In (Nava et al.,
2016), the problem of stationarity of patterns in high
frequency financial data is discussed.
3.4 Fractal Dimension and Its
Estimators
The concept of fractal dimension started with Haus-
dorff dimension. It was introduced in 1919 (Haus-
dorff, 1919) as a measure of smoothness of spatial
data. The Hausdorff method completely covers the
given set X by N(r) balls (circles for E2) of radius at
most r. The Hausdorff dimension is the unique num-
ber d such that N(r) grows as 1/r
d
as r approaches
zero (Falconer, 1990), (Gneiting et al., 2012). The
similar idea in used in Minkowski’s box-counting di-
mension D. It uses an evenly spaced grid (building
boxes) to cover the set. Then, it counts how many
boxes are required to cover all set elements. This
number changes as we make the grid finer by apply-
ing a box-counting algorithm.
Fortunately, we can use a simple relation
H = 2 D (Peters, 1994) between fractal dimension
D (estimation of d) and Hurst exponent H for our
purpose. However, it can be used only for so-called
self-affine processes, because the local properties are
reflected in the global ones in them, and this is the
case of markets. More generally, fractal dimension is
a local property, while the long-memory dependence
characterized by the Hurst exponent is a global char-
acteristic (Gneiting and Schlather, 2004).
Hurst Exponent and Trading Signals Derived from Market Time Series
373
3.5 The Rescaled Range Analysis - R/S
Analysis Method
The rescaled range analysis (R/S analysis) is the origi-
nal method invented and used by Hurst (Hurst, 1951).
Briefly, the R/S analysis is the range of partial sums
of deviations of time series parts from their means,
rescaled by their standard deviations. We use here
the definition following (Peters, 1994), (Kri
ˇ
stoufek,
2010).
First, we start with processing of the complete
time series of length N and divide it into 2
k
(k =
0,1,...) adjacent sub-periods of the same length n, so
that 2
k
n = N. This means, we obtain 1 sub-period of
length N, 2 sub-periods of length N/2, 4 sub-periods
of length N/4, etc. For each sub-period, we calcu-
late the arithmetic mean x
mean
and construct a new se-
ries Z
r
= x
r
x
mean
, r = 1, . . . , n, and in the next step
we create a next new series Y
m
,m = 1, . . . , n of cumu-
lated deviations from the arithmetic mean values, i.e.,
we create a sum of all deviations from the mean in
the given sub-period. Then, we calculate an adjusted
range R
n
, which is defined as a difference between a
maximum and a minimum value of the cumulated de-
viations Y
r
, i.e., R
n
= max(Y
1
,...Y
n
) min(Y
1
,...Y
n
),
and finally, we compute a standard deviation S
n
of
original elements of each sub-period. Each adjusted
range R
n
is then standardized by the corresponding
standard deviation S
n
and forms a rescaled range as
R
n
/S
n
. Then, we calculate an average rescaled range
(R/S)
n
for all sub-periods of fixed length n (i.e., aver-
age of all sub-periods having the same length).
Second, we repeat the process iteratively using
k = 0,1,2,... for each length n = N/2
k
of sub-
periods. They scale as described more formal in the
formula for the estimation of the Hurst exponent (Pe-
ters, 1994):
E
R
n
S
n
= Cn
H
as n = 1, . . . , as n (1)
where:
R
n
is the adjusted range as explained above
S
n
is its standard deviation
E [x] is the expected value - in our case, we used
arithmetic average
n corresponds to the time span of the observation
C is a constant
H is the Hurst exponent
The Hurst exponent H is the slope of the plot of
each ranges log((R/S)
n
) versus each ranges log(n).
Analysis of Hurst exponent estimation and its accu-
racy is given in (Resta, 2012).
3.6 Hurst Exponent and Market Time
Series
In the sections above, we resumed briefly some basics
about fractal dimension and Hurst exponent of time
series. Now, we explain what is a relationship be-
tween Hurst exponent and market trends. This prob-
lem has been investigated since Mandelbrots first pa-
pers in 60-ties, and it is presented in Peters books (Pe-
ters, 1994) and (Peters, 1996).
The Hausdorff definition of fractal dimension and
its relation to Hurst exponent specify that values of
the Hurst exponent range between 0 and 1 because
our objects have fractal dimension between 1 (a line)
and 2 (a line covering an E2-surface completely).
It is known (Peters, 1994) that a value of 0.5 indi-
cates a true random process (a Brownian time series).
A Hurst exponent value H, 0.5 < H < 1 indicates per-
sistent behavior (e.g., a positive autocorrelation - a
trend). A Hurst exponent value 0 < H < 0.5 indicates
anti-persistent behavior (or negative autocorrelation -
change of a trend). An H closer to one indicates a
high risk of large and abrupt changes.
It is also known (Peters, 1994) that the value of
Hurst exponent changes with the length of time pe-
riod used. For example, we measured for DAX in-
dex 0.54 for 1-day return time series and 0.82 for 50-
days return time series. This means that 1-day returns
correspond practically to white noise (Hurst exponent
of white noise = 0.5), but the 50-days returns contain
some trends.
Another known fact is (Mandelbrot, 1963) that
time series gains a long memory character when the
return period increases. Moreover, their distributions
move away from the Gaussian normal to so-called
fat-tailed distributions. Mandelbrot (Mandelbrot and
Hudson, 2004) has shown that prices change in finan-
cial markets did not follow a Gaussian distribution,
but rather more general L
´
evy stable distributions.
4 TRADING STRATEGY
Trading strategy is a set of rules used by traders to
buy or sell their investments. An important part of it
are BUY-signals and SELL-signals denoting the cor-
responding time points.
Buy&Hold is a passive investment strategy. An in-
vestor selects stocks at the beginning, buys them and
holds them for a long period of time, regardless of
fluctuations in the market.
MACD (Moving average convergence diver-
gence) is a basic trend indicator that represents the
relationship between two moving averages of prices.
ICEIS 2018 - 20th International Conference on Enterprise Information Systems
374
Moving averages smooth the price data. MACD is
usually calculated by subtracting the 26-day exponen-
tial moving average (EMA) of the price data from the
12-day EMA of the price data. The result is called
a MACD line. A 9-day EMA of the MACD is then
used as the ”signal line”. In a graph of MACD and
9-day EMA of MACD, we observe the crossings as
triggers for buy and sell signals (Murphy, 1999). This
construction is denoted as MACD(26,12, 9).
4.1 Our Non-linear Method for
Generating Signals - Indicator MH
We correlated fractal dimension with investment risk.
A simple model is given by the following situation.
Experienced traders being in market, i.e., having an
opened position after a BUY-signal of their trading
strategy signalized them a starting trend, see or feel
the growing risk, expect the end of trend, and start
selling. They start to sell earlier than the others (not
so experienced), who continue to buy because they
believe that the market will carry on growing. Prices
can continue growing. But after a specific time pe-
riod, more and more traders start selling their posi-
tions, and also the unexperienced traders begin to un-
derstand the situation and start to sell, too. At this
moment, prices start to fall down.
Our hypothesis is that we can estimate the time
point at which the clever traders start to sell. There is
the possibility that their feeling correlate with fractal
dimension changes. Standard technical indicators re-
act on changes of prices, but it is likely that the fact
that clever traders start to sell does not impact prices
strongly enough to influence the standard technical in-
dicators. We suppose that chaotic properties can re-
act before prices change. Because of that, we expect
more profit when using a strategy based on fractal di-
mension changes.
The idea of our non-linear method is similar to the
mechanism of MACD. However, instead of moving
averages computed from daily values of time series
in a given box size, we used moving Hurst exponents
computed from fractal dimension of daily returns in a
given box size.
We experimented with different moving box sizes
for moving Hurst exponents and with different re-
turn periods R, i.e., for each time series of values, we
built a time series of returns for a given period R and
searched a maximum of a profit-function of three vari-
ables - H-fast, H-slow, and R.
Finally, after time consuming brute force compu-
tations, we obtained the best profit results for the fast
moving Hurst exponent H-fast = H16 (i.e., Hurst ex-
ponent is computed for a time window of the last 16
days), the slow moving Hurst exponent H-slow = H32
(i.e., Hurst exponent is computed for a time window
of the last 32 days), and for the return period of one
day R1. So, we generate BUY- or SELL-signals at
each crossing of H16 and H32 as shown in Fig. 1. In
the following formulas, n denotes the index of a time
series member.
(H16 H32)
n
> 0 and (H16 H32)
n+1
< 0 =
signal BUY
(H16 H32)
n
< 0 and (H16 H32)
n+1
> 0 =
signal SELL
When H16 is crossing down the H32 then we gen-
erate a BUY-signal. We suppose that chaotic prop-
erties will decrease. When H16 is crossing up the
H32 then we generate a SELL-signal. We suppose
that chaotic properties will increase.
Of course, we can imagine a very large number of
other constructions of this kind covering every think-
able combination of Hurst exponent and any of some
hundreds existing technical indicators. However, ex-
periments are time consuming because of the enor-
mous number of combinations and because of input
data volume (about 3,000,000 data elements).
5 IMPLEMENTATION, DATA,
EXPERIMENTS, AND RESULTS
Our prototype was implemented in MATLAB. How-
ever, the Hurst function had to be reimplemented from
C++ according to the version written by Kaplan (Ka-
plan, 2003). We decided to use it because of its per-
formance. We also tried to use the Hurst function
written by Weron (Weron, 2011) that was more ac-
curate, but it was slower by grade and unusable for
bunch of tests we needed.
For simulations, we used all daily close prices
of all stocks (about 31 stocks and DAX value) from
DAX index between 1995-2013 and all daily close
prices of all stocks (about 100 stocks) from NASDAQ
index between 1996-2014. Derived from these close
prices, we used returns R
i
(i.e., profits) of the last i
days (i = 1,...,10) as time series data elements. Ex-
actly, it was more complex because the membership
of a company in an index is not fixed forever. It de-
pends on capitalization of companies. Some compa-
nies are growing and replace companies with decreas-
ing capitalization. Our data are exactly described in
(Skoula, 2017).
Altogether, we used 728,190 data elements for
DAX and 2,349,000 data elements for NASDAQ.
The processing of these data was very time consum-
ing. We found out that the indicator MH is able to
Hurst Exponent and Trading Signals Derived from Market Time Series
375
Figure 1: The non-linear MH indicator used for DAX.
produce about three to four time better results than
Buy & Hold strategy in our experiments.
5.1 Hypothesis Testing
Since the differences between profits from the usage
of MACD and MH were fluctuating, we used paired
t-test statistics to test our hypothesis that MH brings
more profit than MACD. We used the time series of
profits generated by MH and the corresponding time
series of profits generated by MACD for all stocks of
DAX and for all stocks of NASDAQ as described in
Section 5. We stated the following two hypotheses:
H
0
: µ
MH
= µ
MACD
and
H
A
: µ
MH
> µ
MACD
The function used in MatLab was:
alpha = 0.01
test = t.test(macd, y=mh,
alternative=’less’,
paired=TRUE,
conf.level=1-alpha)
The t.test procedure answered for DAX:
t = -6.9046, df = 27, p-value = 1.014e-07
alternative hypothesis:
true difference in means is less than 0
99 percent confidence interval:
-Inf -102.9352
sample estimates:
mean of the differences:
-160.3643
The t.test procedure answered for NASDAQ (data
omitted here):
t = -8.8001, df = 87, p-value = 5.764e-14
alternative hypothesis:
true difference in means is less than 0
99 percent confidence interval:
-Inf -86.42111
sample estimates:
mean of the differences:
-118.2739
So, we reject the theory H
0
with possible error α =
1% for both data collection (DAX and NASDAQ).
The winning theory is H
A
, i.e., MH has a greater in-
come rate than MACD.
Similarly, we tested the hypothesis that the using
of MH indicator brings more profit than the using of
Buy & Hold strategy.
5.2 Practical Aspects
However, it must be said that this profit is purely the-
oretical.
First, there is the problem of stationarity explained
in detail in Section 3.3, i.e., the patterns that brought
profit in the past can bring loss in the future, because
the stationarity of market time series is not guaran-
teed, and it is very probably that it is an oversimplifi-
cation.
Second, we have shown that the using of the MH
indicator generated more profit than the using of the
MACD or Buy & Hold strategy, but it can also mean
that the using of both indicators generates a loss, even
though MH generates a smaller loss.
Third, there is the problem of transaction fre-
quency and fees. Usually, at least for small investors,
transactions on stock exchange are charged by fees
ICEIS 2018 - 20th International Conference on Enterprise Information Systems
376
and taxes. Unfortunately, the using of the MH indica-
tor generates a large number of transactions.
We simulated a real investment of 2000 into NAS-
DAQ for a start, and we used the fee of 1% for each
transaction (BUY or SELL) because it corresponds
with the reality. After the whole period (i.e.,1996-
2014), we finished with 7819.8, and we spent 34497.3
on fees. To make a picture clear, using Buy & Hold
on NASDAQ, we would finish with 16648.1, and we
would pay only 187.7 for fees.
Tax rules are different in different countries, e.g.,
between 15% 25%, but it is evident that they re-
duce the profit, too. Usually, the tax has to be paid
immediately after the transaction is finished. So, the
reinvestment of profit is reduced.
However, the transaction fees depend on the stock
exchange provider. Big investors can use the strategy
of scalping that represents many thousands transac-
tions in a day. Such investors are classified as market
makers, and they are not charged by transactions fees.
The more detailed explanation is out of the scope of
our paper.
6 CONCLUSIONS
The goal of our investigation was to develop and test
a new indicator MH for technical analysis based on
chaos measure represented by Hurst exponent of the
underlying time series of prices.
We found a construction described in Section 4.1,
and we evaluated it in comparison to the strategy us-
ing MACD or Buy & Hold for data described in Sec-
tion 5.
Using hypothesis testing, we proved our hypothe-
sis that the new MH indicator developed in this work,
i.e., our non-linear method described in Section 4.1,
generates more profit compared to the MACD techni-
cal indicator and to the Buy & Hold investment strat-
egy.
On DAX, MH was 4.5 times better than
Buy & Hold and 7.2 times better than MACD. On
NASDAQ, it was 2.9 times better than Buy & Hold
and 16.8 times better than MACD.
In Subsection 5.2, we explained why the indicator
MH cannot be used as a money generating machine.
However, we believe that complex, non-linear sys-
tems with problematic stationarity are an important
research topic. More research has to be done to an-
swer questions about filtering of BUY- and SELL-
signals. In our future research, we will apply meth-
ods of genetic programming to improve it like in our
previous work (Kroha and Friedrich, 2014).
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