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Genetic Programming of Fuzzy Logic Production Rules
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Fuzzy Sets,f i s t is a threshold function of the type. then I 2 I,then 1 a Y,Figure 1 Threshold function, This is used to generate fuzzy sets with linguistic values such as greater than less than much larger than. large small etc The two parameters a and g control the position and steepness of the threshold The second. set type has a range function,Y 525 e Xffi,i l2 x 1 4. Figure 2 Range function, The two parameters a and c control the width and position of the fuzzy set respectively The range function is. used to generate fuzzy sets with linguistic values such as close to fairly close to medium etc. Genetic Programming of the Fuzzy Rule Sets,The initial population.
The size of the initial population may be varied in our application but we have found that Koza s suggested size. of 500 provides a good selection of genetic material without imposing excessive processing overheads. We generate rule sets using Koza s ramped half and half method In this equal numbers of rules are generated. for each depth between 2 and the maximum default 5 Of these half the trees are full i e all branches are. full length and the rest are grown i e branches may be different lengths Our default maximum depth is 9. Inputs to a fuzzy link And Or Not may be other links or fuzzy sets Inputs to fuzzy sets can only be terminals. leaf nodes The inputs to comparative fuzzy sets must be different so if A is greater than A is illegal. However we have not at present disallowed deeper forms of contradiction or tautology e g if A is greater than. B or A is much greater than B,We exclude the expression NotNot. Terminal nodes are selected randomly from the general pool of input values. We normalised the data we supplied to the genetic programming system to the interval 0 l This enabled us to. limit the fuzzy set parameters to the interval 1 1. The fuzzy set functions above were recast to give two parameters that controlled the centre or transition point of. the set and the rate of transition or crispness of the set In the initial generation the centre parameters were. selected randomly with a uniform distribution The crispness parameter was randomly chosen with a gaussian. distribution centred on 0 and the result squared to give a single ended result. Subsequent generations, Subsequent generations are produced by a roulette selection of tree pairs based upon their fitness calculated in. the previous generation see below for details of the fitness algorithm The rules for legal sets are the same as. those described above but the maximum depth of created trees is greater default 9. After selecting a pair of trees we randomly determine whether they will be passed directly to the next generation. probability 0 1 or mated together probability 0 9 Mating is done by randomly selecting crossover points. anywhere on each tree then swapping the cut limbs This may involve a single node or almost the entire tree. When mating produces an illegal tree as defined above an alternate cross point is selected If a pair of legal. trees cannot be generated then the parent trees are transferred to the next generation unchanged. Elitism where the best tree so far is always passed to the next generation was employed for all the examples. Experiments, We are interested in the apparent ability of this technique to generate rules that optimise some external process. that are both effective and easily understood We have chosen the generation of financial trading rules as a test of. this technique partly because of personal and corporate interest but also because the making of money is the. most widely accepted measure of effectiveness, The process in this case to be optimised is the trading of financial instruments We use as input the price history. of the instrument of interest consisting of 500 daily samples The samples contain the open price High achieved. during the day low achieved and closing price, The other vital constituent is a simulation of trading Financial trading is not a homogenous activity in particular.
the time scale over which traders work varies dramatically Because only daily data was initially available we. chose to simulate trades with a minimum duration of 1 day In most of the markets we looked at short trading. was possible i e as well as buying an instrument in the expectation of a price rise one can also borrow an. instrument in the expectation of a price drop and thus make a profit on either direction of market movement. Instruments are traded with bid ask spreads this means that the selling price is higher than the buying price by. some narrow margin which is a source of profit to the dealer and there can be commission charges We. simulated each of these attributes of trading and had our simulation checked by a suitably qualified external. organisation, The inputs to the rules were constructed by moving a time window over the historical data the rules were. evaluated with the input data and a trading decision for each day was generated The trading decisions were to. buy or not to buy not buying was interpreted as a command to go short or to borrow a quantity of the. instrument We settled on an output above 0 5 representing a buy decision In our simulation we kept the capital. employed at a nominal 1 million profits were accumulated but not committed The data window was swept. across the historical data and a trading history generated along with a total final profit or loss figure. The pool of input values used for input to the rules was created from the sampled time series using a time. window If we were determining the rules trading advice for day n and had for example set a window size of 3. the pool from which inputs were initially randomly extracted would consist of Clos l Close z Close 3 Open. l 0pen 2 etc In previous work 8 9 we have made much of the importance of window size in non linear. time series prediction here we select only the maximum size of the window and let natural selection decide on. embedding parameters In the work described here a window size of 20 was used. Fitness Algorithm, The profit earned by a trading rule was calculated as described above and a running total profit loss series was. generated Various measures derived from this series have been tried out as fitness algorithm our current. favourite is a mixture of final profit and linearity of the equity curve It has also proved helpful to pre test the. initial generation for profitability and discard and replace non profit making rules until the entire generation is. profitable,In Sample Performance, Initial training runs on foreign exchange data produced results rapidly as can be seen in Figure 3 which shows that. the best profit evolved after 12 generations, The best result found in our initial trials however was IF Dall2 is small OR Day3 is small OR Day4 is small. then buy DM which yielded a profit of 18 This is simply a summary of the simple rule buy low sell high. Whilst it was encouraging to find that the system spotted the obvious this is clearly not useful information since. the system clearly took advantage of the normalisation to which the data was subjected before use In taking. advantage of the normalisation the system made use of data not available to the trader Trading rules which. make use of level information will not generalise well to subsequent years in financial trading We can obtain. rules that generalise better by making use of the relationships between input values We found that comparative. sets were driven out of the population by absolute sets as training continued. We overcame this by only allowing comparative sets This yielded the results shown in Figure 4 Note that the. profit returned by this algorithm is 62,1 3 5 7 9 I 1 13 15 17 19 21 23 25 27 29 31 33 35.
Generation, Figure 3 Training run and best rule with mixed sets. B e s t Training Profit 62 0 p a,Generations, i h g r e z t e r t h a ElDBOR LOW3isgreaterth L O W O R v l G H i i s d o s e t o O P E N 4 O R 8107isc. PROFIT 62 2,8108 LOW3 HlGHi OPENI,LOW LOW OPEN4 8107. Figure 4 Training run and best rule with comparative sets only. Evolutionary anomalies, We observed an occasional tendency for the population to evolve towards a non productive special case For. example a population produced with the same initial conditions as the example above Figure 4 evolved into a. population containing only Or links and only Range sets To prevent this we added mutation to the parse trees. All nodes were given a probability of 0 005 of mutation If selected for mutation the following transformation. would be performed on the node,If an operator,either a threshold range range 3 threshold.
or b a new pair of parameters is randomly chosen,or c the input terminals are exchanged. with equal probability of each, This made little difference to the general performance of the algorithm but the anomalous evolution described. above has not recurred,Effects of Population Size, We tried training with both very large and very small populations to determine the trade off between training. times and growth in fitness The time for one generation is approximately proportional to the size of population. although smaller populations seem to generate larger mean tree sizes The results are plotted in Figure 5. Average of 10 best rules in,0 1 160 000,250 500 1000 2000. 0 5 10 15 20 25 Population,Thousands of Rule Evaluations.
Figure 5 Mean profit growth and best profit for different population sizes. Out of Sample Performance, In order to find out if this technique produced rules that generalised well with out of sample data we tried 5. different daily time series of over 500 days and left the last 30 trading days out of the training set as an out of. sample test It is our practice to store the best 10 rules in terms of in sample performance of any particular. training run as they are generated The figures below are the average in sample and out of sample scores of those. ten rules annualised so as to make comparison possible. Average in sample Average out of sample,Annualised profit Annualised profit. US T Bond 61 35 11 58,NIKKEI 61 57 7 67,FTSE 73 75 13 11. S P 8 1 93 3 00,DM 39 58 13 73, The system was retrained for each series The random number generator we use is seeded with the time on first. use in a training session Each run therefore contains different sets of random numbers 101 Is a good overview. of the non stationary and non linear nature of financial time series that make them such a stern task for this. Conclusion, We have shown that the combination of Genetic Programming and a Fuzzy Logic Inference engine produces a.
powerful methodology for the generation of Fuzzy production rules that are both effective and intelligible We. have provided a simple example of the use of this methodology drawn from the world of financial trading that. generates trading signals that are profitable both in and out of sample. Acknowledgements, Our thanks to David Vanrenen and Reuters Ltd for funding and provision of data and resources and Walton. Asset Management Ltd for the verification of our trading simulation and checking of results. References, 13 Koza John R Genetic Programming on the programming of computers by means of natural selection MIT. Press 1992, 2 Zadeh L A Fuzzy Sets Information and Control 8 338 353 1965. 3 Feigenbaum E An Informal Processing Theory of Verbal Learning Santa Monica California The Rand. 4 Takagi Sugeno Derivation of fuzzy control rulesfrom human operator s control actions Proc of the. IFAC Symposium on Fuzzy Information Knowledge Representation and Decision Analysis 55 60 July 1983. 5 Tsukamoto Y An approach to fuzzy reasoning method In Mada M Gupta Rammohan K Ragade Ronald. R Yager Editors Advances in Fuzzy Set Theory and Applications 137 149 North Holland Amsterdam 1979. 6 C C Lee Fuzzy Logic in Control Systems Fuzzy Logic Controller IEEE transactions on Systems Man. Cybernetics 20 2 4 19 435 1990, 7 Holland John H Adaptation in Natural and Artijicial Systems Ann Arbor The University of Michigan. Press 1975, 8 Edmonds A N Multivariate prediction offinancial time series using recent developments in chaos theory.
Proceedings of the 1st International Workshop on Neural Networks in the Capital Markets London Business. School Nov 1993, 9 Edmonds A N Burkhardt D Adjei 0 Simultaneous Prediction of Multiple Financial Time Series using. Supervised Learning and Chaos Theory Proceedings of the IEEE World Congress on Computational. Intelligence ICNN section Orlando June 1994, IO Alfred0 Medio Chaotic Dynamics Theory and Applications to Economics Cambridge University Press.


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