Genetic Programming for Financial Trading

1
Genetic Programming for Financial Trading

• Nicolas NAVET
• INRIA, France
• AIECON NCCU, Taiwan
• http//www.loria.fr/nnavet

Tutorial at CIEF 2006, Kaohsiung, Taiwan,
08/10/2006
2
Outline of the talk (1/2)

• PART 1 Genetic programming (GP) ?
• GP among machine learning techniques
• GP on the symbolic regression problem
• Pitfalls GP
• PART 2 GP for financial trading
• Various schemes
• How to implement it ?
• Experimentations GP at work

3
Outline of the talk (2/2)

• PART 3 Analyzing GP results
• Why GP results are usually inconclusive?
• Benchmarking with
• Zero-intelligence trading strategies
• Lottery Trading
• Answering the questions
• is there anything to learn on the data at hand
• is GP effective at this task
• PART 4 Perspectives

4
GP is a Machine Learning technique

• Ultimate goal of machine learning is the
  automatic programming, that is computers
  programming themselves ..
• More achievable goal Build computer-based
  systems that can adapt and learn from their
  experience
• ML algorithms originate from many fields
  mathematics (logic, statistics), bio-inspired
  techniques (neural networks), evolutionary
  computing (Genetic Algorithm, Genetic
  Programming), swarm intelligence (ant, bees)

5
Evolutionary Computing

• Algorithms that make use of mechanisms inspired
  by natural evolution, such as
• Survival of the fittest among an evolving
  population of solutions
• Reproduction and mutation
• Prominent representatives
• Genetic Algorithm (GA)
• Genetic Programming (GP) GP is a branch of GA
  where the genetic code of a solution is of
  variable length
• Over the last 50 years, evolutionary algorithms
  have proved to be very efficient for finding
  approximate solutions to algorithmically complex
  problems

6
Two main problems in Machine Learning

• Classification model output is a prediction
  whether the input belongs to some particular
  class
• Examples Human being recognition in image
  analysis, spam detection, credit scoring, market
  timing decisions
• Regression prediction of the systems output
  for a specific input
• Example predict tomorrow's opening price for a
  stock given closing price, market trend, other
  stock exchanges,

7
Functioning scheme of ML
Learning on a training interval
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GP basics
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Genetic programming

• GP is the process of evolving a population of
  computer programs, that are candidate solutions,
  according to the evolutionary principles (e.g.
  survival of the fittest)

Generate a population of random programs
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In GP, programs are represented by trees (1/3)

• Trees are a very general representation form

• Formula

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In GP, programs are represented by trees (2/3)

• Logical formula

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In GP, programs are represented by trees (3/3)

• Trading rule formula BUY IF (VOLgt10) AND
  (Moving Average(25) gt Moving Average(45))

Picture from BhPiZu02
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Preliminary steps of GP

• The user has to define
• the set of terminals
• the set of functions
• how to evaluate the quality of an individual
  the fitness measure
• parameters of the run e.g. number of
  individuals of the population
• the termination criterion

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Symbolic regression a problem GP is good at

• Symbolic regression find a function that fits
  well a set of experimental data points

• Symbolic means that one looks for both
• the functional form
• - the value of the parameters, e.g.

• Differs from other regressions where one solely
  looks for the best coefficient values for a
  pre-fixed model. Usually the choice of the model
  is the most difficult issue !

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Symbolic regression

• Given a set of points

• Find the function s.t. as far as
  possible

• Possible fitness function

• GP functions

• GP terminals

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GP Operators biologically inspired

• Recombination (aka crossover) 2 individuals
  share genetic material and create one or several
  offsprings
• Mutation introduce genetic diversity by random
  changes in the genetic code
• Reproduction individual survives as is in the
  next generation

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Selection Operators for Crossover/reproduction

• General principles in GP the fittest
  individuals should have more chance to survive
  and transmit their genetic code

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Standard Recombination (aka crossover)

• Standard recombination exchange two randomly
  chosen sub-trees among the parents

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Mutation Operator 1 standard mutation

• Standard mutation replacement of a sub-tree
  with a randomly generated one

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Mutation Operator 2 swap sub-tree mutation

• Swap sub-tree Mutation swap two sub-trees of
  an individual

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Mutation Operator 3 shrink mutation

• Shrink Mutation replacing a branch (a node
  with one or more arguments) with one of his child
  node

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Other Mutation Operators

• Swap mutation (? swap sub-tree mutation)
  exchanging the function associated to a node by
  one having the same number of arguments
• Headless Chicken crossover mutation
  implemented as a crossover between a program and
  a newly generated random program
• .

23
Reproduction / Elitism Operators

• Reproduction an individual is reproduced in
  the next generation without any modification

• Elitism the best n individuals are kept in the
  next generation

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GP is no silver bullet
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GP Issue 1 how to choose the function set ?

• The problem cannot be solved if the set of
  functions is not sufficient
• But Non-relevant functions increases uselessly
  the search space

• Problem there is no automatic way to decide a
  priori the relevant functions and to build a
  sufficient function sets

26
Problem cannot be solved if the set of functions
is not sufficient illustration

• Generating function

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Results with sin(x) in the function set ?
Typical outcome
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Results without sin(x) in the function set ?
Typical outcome
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Yes, sin(x) can be approximated by its Taylors
series ..
Sin(x) and taylor approximation of degree 1, 3 ,
5, 7, 9, 11, 13 image Wikipedia

• Problem 1 there is little hope to discover
  that ..

• Problem 2 what happens outside the training
  interval ?

30
Composition of the function set is crucial
illustration

• GP functions

• Subset is
  extraneous in this context

• Same experimental setup as before

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Function set containing redundant functions ?
(1/2)
Typical outcome
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Function set containing redundant functions ?
(2/2)

• On average, with the extraneous functions the
  best solution is 10 farther from the curve in
  the training interval (much more outside!)

• With the extraneous functions, the average
  solution is better .. because the tree is more
  likely to contain a trigonometric function

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GP Issue 2 code bloat

• Solutions increase in size over generations

Same experimental setup as before
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GP Issue 2 code bloat

• Much of the genetic code has no influence on the
  fitness .. but may constitute a useful reserve
  of genetic material

non-effective code !! aka introns
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Code bloat why is it a problem ?

• Solutions are hard to understand
• learning something from huge solutions is almost
  impossible ..
• One has no confidence using programs one does not
  understand !
• Much of the computing power is spent manipulating
  non-contributing code, which may slow down the
  search

36
Countermeasures .. (1/2)

• Static limit of the tree depth
• Dynamic maximum tree depth SiAl03 the limit
  is increased each time an outstanding individual
  deeper than the current limit is found
• Limit the probability of longer-than-average
  individuals to be chosen by reducing their
  fitness
• Apply operators than ensure limited code growth
• Discard newly created individuals whose
  behavior is too close to the ones of their
  parents (e.g. behavior for regression pb could
  be position of the points Str03)

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Countermeasures .. (2/2)

• Possible symbolic simplification of the tree

can be simplified into

• Needs to be further investigated ! preliminary
  experiments TeHe04 show that simplification
  does not necessarily help (introns may
  constitute a useful reserve of genetic materials)

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GP Issue 3 GP can be disappointing outside the
training set
and such a behavior can hardly be predicted
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GP Issue 3 explanation (1/2)

• Usually GP functions are implemented to have the
  closure property each function must be able to
  handle every possible value
• What to do with
• division by 0 ?
• sqrt(x) with x lt 0 ?

• Solution protected operators, eg. the
  division
• if (abs(denominator) lt value-near-0) return 1

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GP Issue 3 explanation (2/2)

• in our case, fragment of the best GP tree

• Why did it not occur on the training interval ?
• not training points chosen such that

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GP Issue 4 standard GP is not good at finding
numerical constants (1/3)

• Where do numerical values come from ?
• Ephemeral random constants random values
  inserted at the leafs of the GP trees during the
  creation of initial population
• Use of arithmetic operators on existing
  numerical constants
• Generation by combination of variables/functions
  

• Lately, many studies show that standard GP is
  not good at finding constants

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GP Issue 4 standard GP is not good at finding
numerical constants (2/2)

• Experiment find a constant function equal to
  the numeric constant 3.141592

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GP Issue 4 standard GP is not good at finding
numerical constants (3/3)

• There are several more efficient schemes for
  constants generation in GP Dem95
• - local optimization ZuPiMa01,
• numeric mutation EvFe98,
• One of them should be implemented otherwise 1)
  computation time is lost searching for constants
  2) solutions may tend to be bigger

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Some (personal) conclusions on GP (1/3)

• GP is undoubtedly a powerful technique
• Efficient for predicting / classifying .. but
  not more than other techniques
• Symbolic representation of the created solutions
  may help to give good insight into the system
  under study .. not only the best solutions are
  interesting but also how the population has
  evolved over time
• GP is a tool to learn knowledge

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Some (personal) conclusions on GP (2/3)

• Powerful tool but ...
• a good knowledge of the application field is
  required for choosing the right functions set
• prior experience with GP is mandatory to avoid
  common mistakes there is no theory to tell us
  what to do !
• it tend to create solutions too big to be
  analyzable -gt countermeasures should be
  implemented
• fine-tuning the GP parameters is very
  time-consuming

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Some (personal) conclusions on GP (3/3)

• How to analyze the results of GP ?
• efficiency can hardly be predicted, it varies
• from problem to problem
• and from GP run to GP run
• if results are not very positive
• is it because there is no good solution ?
• or GP is not effective and further work is
  needed ?

• There are solutions part 3 of the talk

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Part 2 GP for financial trading
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Why GP is an appealing technique for financial
trading ?

• Easy to implement / robust evolutionary technique
• Trading rules (TR) should adapt to a changing
  environment GP may simulate this evolution
• Solutions are produced under a symbolic form that
  can be understood and analyzed
• GP may serve as a knowledge discovery tool (e.g.
  evolution of the market)

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GP for financial trading

• GP for composing portfolio (not discussed here,
  see Lag03 )
• GP for evolving the structure of neural networks
  used for prediction (not discussed here, see
  GoFe99 )
• GP for predicting price evolution (briefly
  discussed here, see Kab02 )
• Most common GP for inducing technical trading
  rules

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Predicting price evolution general comments ..

• Long term forecast of stock prices remain a
  fantasy Kab02

• Swing trading or intraday trading

CIEF Tutorial 1 by Prof. Fyfe today 1h30 pm !

• 2 excellent starting points
• Kab02 single-day-trading-strategy based on
  the forecasted spread
• SaTe01 winner of the CEC2000 Dow-Jones
  Prediction Prediction t1, t2, t3,, th - a
  solution has one tree per forecast horizon

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Predicting price evolution fitness function

• Definition of the fitness function has been shown
  to be crucial e.g. SaTe01, there are many
  possible

• (Normalized) Mean square error
• Mean Absolute Percentage Error
• (1-?) statistic 1 - MAPE / MAPE-Randow-Walk
• Directional symmetry index (DS)
• DS weighted by the direction and amplitude of
  the error

• Issue a meaningful fitness function is not
  always GP friendly

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Inducing technical trading rules
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Steps of the algorithm (1/3)
1. Extracting training time series from the
database
2. Preprocessing cleaning, sampling, averaging,
normalizing,
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Steps of the algorithm (2/3)
3. GP on the training set 3.1 Creation of the
individuals 3.2 Evaluation
Trading Rules Interpreter
Trading Sequence Simulator
3.3 Selection of the individuals
4. Analysis of the evolution statistics, html
files
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Steps of the algorithm (3/3)
5. Evaluate selected individuals on the
validation set
6. Evaluate best individual out-of sample
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GP at work Demo on the Taiwan Capitalization
Weighted Stock Index
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Part 3 Analyzing GP results
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One may cast doubts on GP efficiency ..

• Highly heuristic no theory ! Problems on which
  GP has been shown not to be significantly better
  than random search
• Few clear-cut successes reported in the financial
  literature
• GP embeds little domain specific knowledge yet ..
• Doubts on the efficiency of GP to use the
  available computing time
• code bloat
• bad at finding numerical constants
• best solutions are sometimes found very early in
  the run ..
• Variability of the results ! e.g. returns

-0.160993, 0.0526153, 0.0526153, 0.0526153,
0.0526153, -0.0794787, 0.0526153, -0.0794787,
0.132354, 0.364311, -0.0990995, -0.0794787,
-0.0855786, -0.094433, 0.0464288, -0.140719,
0.0526153, 0.0526153, -0.0746189, 0.418075, .
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Possible pretest measure of predictability of
the financial time-series

• Actual question how predictable for a given
  horizon with a given cost function?

• Serial correlation
• Kolmogorov complexity
• Lyapunov exponent
• Unit root analysis
• Comparison with results on surrogate data
  shuffled series (e.g. Kaboudan statistics)
• ...

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In practice, some predictability does not imply
profitability ..

• Prediction horizon must be large enough!

• Volatility may not be sufficient to cover
  round-trip transactions costs!

• Not the right trading instrument at hand ..
  typically short selling not available

61
Pretest methodology

• Compare GP with several variants of
• Random search algorithms
• Zero-Intelligence Strategies - ZIS
• Random trading behaviors
• Lottery trading - LT

Issue how to best constrain randomness ?

• Statistical hypotheses testing
• Null GP does not outperform ZIS
• Null GP does not outperform LT

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Pretest 1 GP versus Zero-Intelligence
strategies(Equivalent search intensity Random
Search (ERS) with validation stage)

• Null hypothesis H1,0 GP does not outperform
  equivalent random search - Alternative
  hypothesis is H1,1

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Pretest 1 GP vs zero-intelligence strategies
ERS

• H1,0 cannot be rejected interpretation
• There is nothing to learn or GP is not very
  effective

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Pretest 4 GP vs lottery trading

• Lottery trading (LT) random trading behavior
  according the outcome of a r.v. (e.g. Bernoulli
  law)
• Issue 1 if LT tends to hold positions (short,
  long) for less time that GP, transactions costs
  may advantage GP ..
• Issue 2 it might be an advantage or an
  disadvantage for LT to trade much less or much
  more than GP.
• ex downward oriented market with no short-sell

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Frequency and intensity of a trading strategy

• Frequency average number of transactions per
  unit of time
• Intensity proportion of time where a position
  is held
• For pretest 4
• We impose that average frequency and intensity
  of LT is equal to the ones of GP
• Implementation generate random trading
  sequences having the right characteristics

0,0,1,1,1,0,0,0,0,0,1,1,1,1,1,1,0,0,1,1,0,1,0,0,0,
0,0,0,1,1,1,1,1,1,
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Pretest 4 implementation
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Answering question 1 is there anything to learn
on the training data at hand ?
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Question 1 pretests involved

• Starting point if a set of search algorithms do
  not outperform LT, it gives evidence that there
  is nothing to learn ..
• Pretest 4 GP vs Lottery Trading
• Null hypothesis H4,0 GP does not outperform LT
• Pretest 5 Equivalent Random Search (ZIS) vs
  Lottery Trading
• Null hypothesis H5,0 ERS does not outperform LT

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Question 1 some answers ...

• ?R means that the null hypothesis Hi,0 cannot be
  rejected R means we should favor Hi,1

H4,0 H5,0 Interpretation
Case 1 ?R ?R
Case 2 R R
Case 3 R ?R
Case 4 ?R R

• there is nothing to learn

there is something to learn
there may be something to learn -ERS might not be
powerful enough
there may be something to learn GP evolution
process is detrimental
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Answering question 2 is GP effective ?
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Question 2 some answers ...

• Question 2 cannot be answered if there is nothing
  to learn (case 1)
• Case 4 provides us with a negative answer ..
• In case 2 and 3, run pretest 1 GP vs Equivalent
  random search
• Null hypothesis H1,0 GP does not outperform ERS
• If one cannot reject H1,0 GP shows no evidence
  of efficiency

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Pretests at work Methodology Draw conclusions
from pretests using our own programs and compare
with results in the literature ChKuHo06 on the
same time series
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Setup GP control parameters - same as in
ChKuHo06
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Setup statistics, data, trading scheme

• Hypothesis testing with student t-test with a 95
  confidence level
• Pretests with samples made of 50 GP runs, 50 ERS
  runs and 100 LT runs
• Data indexes of 3 stock exchanges Canada,
  Taiwan and Japan
• Daily trading with short selling
• Training of 3 years Validation of 2 years
• Out-of-sample periods 1999-2000, 2001-2002,
  2003-2004
• Data normalized with a 250 days moving average

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Results on actual data (1/2)

• Evidence that there is something to learn 4
  markets out of 9 (C3,J2,T1,T3)
• Experiments in ChKuHo06, with another GP
  implementation, show that GP performs very well
  on these 4 markets
• Evidence that there is nothing to learn 3
  (C1,J3,T2)
• In ChKuHo06, there is only one (C1) where GP
  has positive return (but less than BH)

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Results on actual data (2/2)

• GP effective 3 markets out of 6
• In these 3 markets, GP outperforms Buy and Hold
  same outcome as in ChKuHo06
• Preliminary conclusion one can rely on pretests
  ..
• When there is nothing to learn, no GP
  implementation did good (except in one case)
• When there is something to learn, at least one
  implementation did good (always)
• When our GP is effective, GP in ChKuHo06 is
  effective too (always)

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Further conclusion

• Our GP implementation is
• is more efficient than random search no case
  where ERS outperform LT and GP did not
• But only slightly more efficient one would
  expect much more cases where GP does better than
  LT and not ERS
• Our GP is actually able to take advantage of
  regularities in data but only of simple ones

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Part 4 Perspectives in the field of GP for
financial trading
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Rethinking fitness functions
From LaPo02

• Fitness functions accumulated return,
  risk-adjusted return,
• Issue on some problems LaPo02, GP is only
  marginally better than random search because
  fitness function induces a difficult" landscape
  
• Come up with GP-friendly fitness functions

80
Preprocessing of the data still an open issue

• Studies in forecasting show the importance of
  preprocessing for GP, often, normalization with
  MA(250) is used - with benefits ChKuHo06
• Length of MA should change according to markets
  volatility, regime changes, etc ?
• Why not consider MACD, Exponential MA,
  differencing, rate of change, log value, FFT,
  wavelet,

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Data division scheme

• There is evidence that GP performs poorly when
  the characteristics of the training interval are
  very different from the out-of-sample interval
• Characterization of the current market condition
  mean reverting, trend following ...
• Relearning on a smaller interval if needed ?

82
More extensive tests are needed .. automating the
test

• A comprehensive test for daily indexes done in
  ChKuHo06, none exists for individual stocks and
  intraday data
• Automated testing on several hundred of stocks
  is fully feasible but require a software
  infrastructure and much computing power

83
Ensemble methods combining trading rules

• In ML, ensemble methods have proven to be very
  effective
• Majority rule tested in ChKuHo06 with some
  success
• Efficiency requirement accuracy (better than
  random) and diversity (uncorrelated errors)
  what does it mean for trading rules?
• More fine grained selection / weighting scheme
  may lead to better results

84
Embed more domain specific knowledge

• Black-box algorithms are usually outperformed by
  domain-specific algorithms
• Domain-specific language is limited as yet
• Enrich primitive set with volume, indexes,
  bid/ask spread,
• Enrich function set with cross-correlation,
  predictability measure,

85
References (1/2)

• ChKuHo06 S.-H. Chen and T.-W. Kuo and K.-M.
  Hoi. Genetic Programming and Financial Trading
  How Much about "What we Know. In 4th NTU
  International Conference on Economics, Finance
  and Accounting, April 2006.
• ChNa06 S.-H. Chen and N. Navet. Pretests for
  genetic-programming evolved trading programs
  zero-intelligence strategies and lottery
  trading, Proc. ICONIP2006.
• SiAl03 S. Silva and J. Almeida, Dynamic
  Maximum Tree Depth - A Simple Technique for
  Avoiding Bloat in Tree-Based GP, GECCO 2003,
  LNCS 2724, pp. 17761787, 2003.
• Str03 M.J. Streeter, The Root Causes of Code
  Growth in Genetic Programming, EuroGP 2003, pp.
  443 - 454, 2003.
• TeHe04 M.D. Terrio, M. I. Heywood, On Naïve
  Crossover Biases with Reproduction for Simple
  Solutions to Classification Problems, GECCO
  2004, 2004.
• ZuPiMa01 G. Zumbach, O.V. Pictet, and O.
  Masutti, Genetic Programming with Syntactic
  Restrictions applied to Financial Volatility
  Forecasting, Olsen Associates, Research
  Report, 2001.
• EvFe98 M. Evett, T. Fernandez, Numeric
  Mutation Improves the Discovery of Numeric
  Constants in Genetic Programming, Genetic
  Programming 1998 Proceedings of the Third Annual
  Conference, 1998.

86
References (2/2)

• Kab02 M. Kaboudan, GP Forecasts of Stock
  Prices for Profitable Trading, Evolutionary
  computation in economics and finance, 2002.
• SaTe02 M. Santini, A. Tettamanzi, Genetic
  Programming for Financial Series Prediction,
  Proceedings of EuroGP'2001, 2001.
• BhPiZu02 S. Bhattacharyya, O. V. Pictet, G.
  Zumbach, Knowledge-Intensive Genetic Discovery
  in Foreign Exchange Markets, IEEE Transactions
  on Evolutionary Computation, vol 6, n 2, April
  2002.
• LaPo02 W.B. Langdon, R. Poli, Fondations of
  Genetic Programming, Springer Verlag, 2002.
• Kab00 M. Kaboudan, Genetic Programming
  Prediction of Stock Prices, Computational
  Economics, vol16, 2000.
• Wag03 L. Wagman, Stock Portfolio Evaluation
  An Application of Genetic-Programming-Based
  Technical Analysis, Genetic Algorithms and
  Genetic Programming at Stanford 2003, 2003.
• GoFe99 W. Golubski and T. Feuring, Evolving
  Neural Network Structures by Means of Genetic
  Programming, Proceedings of EuroGP'99, 1999.
• Dem05 I. Dempsey, Constant Generation for the
  Financial Domain using Grammatical Evolution,
  Proceedings of the 2005 workshops on Genetic and
  evolutionary computation 2005, pp 350 353,
  Washington, June 25 - 26, 2005.

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