Kooths/Ringhut - INFRATELLIGENCE
Contents
1. 10 lt gt 4 gt wl wr The genetic operators need to be adapted to real coded individuals We apply non uniform mutation and Max Min arithmetical crossover to create individuals for the next generation t 1 If an individual C is randomly selected for mutation one of its elements c one of the two borders or a center is identified to be mutated A random binary number determines whether to increase or decrease the value of c Assume that the selected element c to be mutated to c is a center fc then Cn A t fe c ifo 1 d5 ch Cn A t cCn f amp a if amp 0 The result of the function A t y is a value in the range 0 y and the probability of the result s being close to 0 increases in t according to 16 At y y 1 r If a pair of individuals Cp Cp2 in the current population is selected for crossover four offspring are created according to Ci a Cp 1 a Cp2 7 Cor a Cp 1 a Cpi Cau1 with cs min cpi cps V cpr Cp and cp Cpr Cast with of max cp Cp Vv Cpl E Cp and Cp E Cp The parameter a is a constant in the range 0 1 and has to be set by the user To keep the size of the population unchanged only the two best of the four offspring are copied in the next population We use the same fitness function as in the previous simplification step for 18 evaluating the population Since each individual C represents a complete FB it is linked to the2. The neural learning operations described so far reflect the standard error backpropagation algorithm EBP whose results do not necessarily account for the specific needs of a fuzzy rule base Sometimes it is suitable to exclude the adaptation of certain parameters or to restrict their adjustment ranges to keep the whole fuzzy system in a sound state Otherwise it could happen that the EBP algorithm ruffles the FB leaving us with a degenerate system that lacks economic interpretability e g negative or extremely small fuzzy set widths or fuzzy set centers that are difficult to distinguish In these situations special MEBP filters intervene to get a differentiated fuzzification of the relevant crisp data intervals e g by ensuring a minimum overlap of adjacent fuzzy sets The underlying idea of the adjustment filters is to form the various criteria for a sound FRB and to inhibit the learning process whenever one or more of these criteria is running the risk to be violated The MEBP algorithm is especially designed for applying a given FRB to observations that might differ considerably from the training data set In this case GENEFER finds itself on Virgin soil which might be due to a radical change within the economic system Virgin soil appears whenever the fuzzy sets do not cover the relevant crisp interval in an adequately 1999 ch 3 For details in neural technologies see Hecht Nielsen 1991 21 differentiated way an
3. Systems vol 82 pp 65 71 Lin C T Lee C S G 1991 Neural Network Based Fuzzy Logic Control and Decision System in IEEE Transactions on Computers Vol 40 No 12 Dec 1991 Mamdani E H Assilian S 1975 An Experiment in Linguistic Synthesis with a Fuzzy Logic Controller in International Journal of Man Machines Studies Vol 7 pp 1 13 Marengo L Tordjman H 1996 Speculation Heterogeneity and Learning A Simulation Model of Exchange Rate Dynamics in Kyklos vol 49 p 407 438 McFadden D 1999 Rationality for Economists in Journal of Risk and Uncertainty vol 19 1 3 pp 73 105 Nauck D Klawonn F Kruse R 1994 Neuronale Netze und Fuzzy Systeme Grundlagen des Konnektionismus Neuronaler Fuzzy Systeme und der Kopplung mit wissensbasierten Methoden Braunschweig Wiesbaden Olsen R B Dacrogna M M Miiller U A Pictet O V 1992 Going Back to the Basics Rethinking Market Efficiency O amp A Working Paper 1992 09 07 http www olsen ch library research oa_working html Pedrycz W Gomide F 1998 An Introduction to Fuzzy Sets Analysis and Design MIT Press Cambridge Massachusetts London Vriend Nicolaas J 2000 An illustration of the essential difference between individual and social learning and its consequences for computational analyses in Journal of Economic Dynamics amp Control vol 24 pp 1 19 Zimmermann H J 1991 Fuz
4. and fuzzy technologies is given by Nauck Klawonn Kruse 1994 pp 231 ff A very concise description of the basic concepts of neural networks can be found in Buckley Feuring 20 the network not only the output node are responsible for the network error on account of their influence on the signal propagated through the network During the training phase the signal s direction is reversed so that each learning round starts at the output node with the network error being fed into the network This error signal is then backpropagated layer by layer until it reaches the basic input nodes The global network error is thereby distributed over all relevant nodes Since each node s signal can only be changed by adjusting its link weights these are the object of the learning process The adjustment of each weight w is proportional to its marginal influence on the network error This method implies a linear approximation of the error function in the environment of the current weight values OMSE 19 Aw D We la If we plot the network error as a function of all link weights we get a mountain like error surface in which 19 describes the steepest descent with the exogenous learning rate u determining the step size We omit further details and formulas of the parameter adjustment procedures here KOOTHS 1998 section 2 5 3 4 but it should be noticed that they depend on the fuzzy types the fuzzy AND operator and the rule weighting option
5. comes into play Some cognitive scientists believe that agents form mental models of the world in order to deal with complex environments see Johnson Laird 1983 They look for patterns hypothesize try make mistakes learn and adapt MARENGO TORDJMAN 1996 p 410 In doing so they inductively form expectations ARTHUR 1994 The attempt to model their underlying mental processes explicitly is challenging but it helps to ensure that theorists are not taking too much for granted and that their theories are not vague incoherent or like mystical insights only properly understood by their proponents JOHNSON LAIRD SHAFIR 1993 In recent years there is a growing literature about the use of Artificial Intelligence AI methods for modeling these mental models and their adaptation to a constantly evolving economic environment Several papers are about John Holland s Classifier Systems see HOLLAND 1975 and their applications to various economic problems see VRIEND 2000 ARTHUR ET AL 1996 MARENGO TORDJMAN 1996 and BELTRAMETTI ET AL 1997 As discussed in section 2 this technique does not meet all the demands placed on an Al based approach for modeling expectations Therefore we propose an alternative way to model the formation of expectations in a complex environment using Fuzzy Rule Bases FRB as an operational representation of mental models We divide the design process of such an FRB into four ma
6. created or simplified RB to determine the FRB s output for MSE calculation 4 3 4 Neural Tuning Error Backpropagation The neural FRB tuning option makes use of the fact that both fuzzy and neural systems are based on a distributed knowledge representation This allows to transform a fuzzy system into an equivalent neural network and hence to apply neural learning procedures to fuzzy systems For this purpose we interpret the FRB as a hybrid neural fuzzy system connectionist fuzzy control system according to the basic technology presented by Lin Lee 1991 and Lin 1994 This approach uses a layered feed forward neural network with a total of five layers see Figure 4 Each layer enacts a specific function in the fuzzy inference process Figure 4 Neural fuzzy system with two input variables layer 1 layer 2 layer 3 layer 4 layer 5 basic input input term antecedent consequent nodes defuzzifying node nodes nodes nodes output term nodes output node rule base inference engine crisp value of input 1 crisp forecast value crisp value of input 2 xy conclusion defuzzification of the accumulation output value signal fuzzification of the distribution input values aggregation The nodes in layer one basic input nodes are sensors to the outside world Their task is to receive the crisp values of input variables and transmit them to the appropriate nodes in layer two input ter
7. offered whose settings can be found in This algorithm runs over all periods in the fuzzification interval and guarantees a minimum membership value equal to the overlap in for each observation of the selected variable within this interval Maximum and minimum values of the chosen variable within this interval are displayed in It is important to point to the S Shape Border Sets option If a new observation is outside the support of all fuzzy sets this option helps to avoid an inability to generate an inference result For a detailed description of the FC FS algorithm see Lin Cunningham Coggeshall 1996 pp 65 71 In addition to their proposal all input time series are scaled within the unit interval in order to avoid distorted results due to significant differences in standard deviations Note that this failure cannot occur during the FRB design process since the interval for inductive learning can only be equal or part of the interval in the previous step 10 Figure 2 Fuzzification Dialog Fuzzification Dialog C fal p as P AGE T 8 Shae J S Shape Border Sets Maximum Value 99 7051 uzzification ol i Se bos T Width Symmetry Minimum Value 0 8047 JaGDP Differential U JaGDP Differential U Uy G by sets C gaussian E 2 Clustering of Sets Number of Fuzzy Sets 2 F 1 7 50 Overlap 90 Learning Rate E aa a NA F 10 Stop L
8. on a population of encoded RBs An RB is encoded as a binary string C of length M which can be regarded as a sequence of switches that either turn a rule on 1 or off 0 A string only containing 1 s represents the created RB of the previous step All other individuals in the population are initialized randomly The population is of constant size K with k lagkh The user may choose between three selection procedures i Stochastic Universal Sampling SUS rang based ii SUS fitness proportional and iii Tournament Selection to determine the individuals for the mating pool The activation of the elitist selection option guarantees the survival of the fittest individual in the next population The offspring population is created by the classical binary multipoint crossover and uniform mutation operators The fitness of one individual Cx is determined by its MSE over the TDS The lower this error the better the individual Since there is a need to fulfill the completeness requirement the fitness value has to be modified in case of a completeness violation We ensure this by requiring that each observation in the TDS has a CV equation 3 greater or equal to t The completeness property for Cx over the complete TDS is defined as training set completeness degree 13 TSCD Ck TDS CV d die TDS The following fitness function penalizes the fitness value if the training set completeness degree is violated MSE Cx if TSC
9. relevant inputs he can choose that option in the complete DESIGN NAVIGATOR and select the desired time series Otherwise he can apply GENEFER s identification algorithm If the user is completely or partially ignorant about the set of independent inputs he can proceed along the tree items none and partial respectively In the latter case preselected time series are definitely included in the final set of inputs before the user runs the FC FS identification algorithm This algorithm eliminates unimportant and related inputs iteratively In the first step it calculates J fuzzy curves FC for each input output combination and determines the MSE between these curves and the observed output data FC Ranking in ascending order yields the most important input lowest MSE Each step is completed by eliminating a percentage B of insignificant inputs highest MSE The next step combines the previously identified input with each remaining possible input and output to calculate 1 B N 1 fuzzy surfaces FS Sorting all FS according to MSE indicates the second most important input Computation of these surfaces proceeds until the set of possible inputs is empty The FC FS algorithm is a computationally fast means for isolating an independent set of significant input variables of a complex poorly defined nonlinear system 3 2 Fuzzification Once the previous procedure has been accomplished a variable s fuzzification can
10. D Ck TDS T 41 ee ERs gt Sout otherwise die TDS It is important to note that GENEFER does not delete any redundant rules The simplifying procedure attempts to improve the performance of a previously created FRB by switching single rules on and off As a result we obtain a binary code of switches that is linked to the generated FRB This linkage might possibly yield a reduced FRB Nevertheless the user is 16 always able to work with the generated one by re activating all rules turning on all switches 3 4 Fuzzy Rule Base Tuning RB simplifying and FRB tuning both aim at improving the system s forecast performance Whereas RB simplifying uses a given FB in order to refine the RB the FRB tuning focuses on adjusting the fuzzy sets parameters The current Rule Base Tuning FCB fuzzification of input s and output might not be as E Genetic Tunin Selection suitable as possible with respect to the goodness of fit igi The search space is limited to the fuzzy sets centres and un El Neural Tuning Error B ackpropagation widths without changing their a priori defined type and General Settings Additional Constraints the degrees of granularity GENEFER offers two popular R m tuning algorithms for fuzzy systems genetic and neural tuning error backpropagation With the latter the FRB is transformed into a Neural FRB with a given topology We do not apply explicit structural learning chan
11. Modeling Expectations with GENEFER an Artificial Intelligence approach by Eric Ringhut and Stefan Kooths O09erri wiwi uni muenster de kooths uni muenster de Institut fiir industriewirtschaftliche Forschung Westf lische Wilhelms Universitat M nster Universit tsstr 14 16 D 48143 M nster Germany http www wiwi uni muenster de iif June 2000 Abstract Economic modeling of financial markets attempts to model highly complex systems in which expectations can be among the dominant driving forces It is necessary then to focus on how agents form expectations We believe that they look for patterns hypothesize try make mistakes learn and adapt Agents bounded rationality leads us to a rule based approach which we model using Fuzzy Rule Bases For example if a single agent believes the exchange rate is determined by a set of possible inputs and is asked to state his relationship his answer will probably reveal a fuzzy nature like IF the inflation rate in the EURO Zone is low and the GDP growth rate is larger than in the US THEN the EURO will rise against the USD Low and larger are fuzzy terms which give a gradual linguistic meaning to crisp intervalls in the respective universes of discourse In order to learn a Fuzzy Fuzzy Rule base from examples we introduce Genetic Algorithms and Artificial Neural Networks as learning operators These examples can either be empirical data or originate from an economic simu
12. agents who instantaneously discount new information into prices so that no technical trading can offer any consistent speculative profits and cause markets to appear to be perfectly efficient Homogeneity assumes mutually consistent perceptions about the environment one commonly shared true model and allows for the representative agent framework There is accumulating behavioral evidence against this rational view as well as theoretical objections including the costs of gathering and processing information the costs of transacting bounded rationality of agents and indeterminacy of deductively formed expectations ARTHUR 1995 p 8 ALBIN 1998 Real financial markets are characterized by heterogeneous agents who have different motives to trade different planning horizons OLSEN ET AL 1992 and different beliefs about the functioning or driving forces of the market and future events that affect their action today and influence asset prices tomorrow Financial decision makers face theoretical competing theories empirical spurious correlation and operational non measurability of inputs difficulties in identifying relevant input data Therefore they do not base their decisions on a uniform input data set or on a single accepted theory but rather on a mixture of theories technical analyses chart analyses what their competitors do and the like Despite all these problems agents have to form expectations when trans acting
13. be accessed directly via the DESIGN NAVIGATOR Fuzzification FCB Clicking on a variable opens the Fuzzification Dialog in Output USD E Figure 2 Starting with a default FB this dialog allows Inputl Budget Deficit EU iro rag EL ae the user to fuzzify each variable separately Neither the Input3 Inflation Rate Differential US EU degree of granularity number of fuzzy sets nor the Input4 gGDP Differential US EU Input Current Account US type of fuzzy set triangular or gaussian is applied Input6 Interest Rate Differential US EU Input Budget Deficit US uniformly but may differ across inputs and output We know from cognitive science that human beings are typically capable of distinguishing up to 7 significant classes so we restrict the number of fuzzy sets per variable to be either 3 5 or 7 ALTROCK 1995 p 153 and PEDRYCZ GOMIDE 1998 p 67 The fuzzification dialog assists the user in fuzzifying the output and the identified inputs easily GENEFER displays the default fuzzification for each variable if the user has not specified a FB yet The user may modify this by changing the settings in the displayed form in figure 2 The selected variable is in The user determines the number of fuzzy sets in and their type in Grid shows the centers and widths but also allows for manual editing if this option is selected deactivation of Additionally a clustering algorithm is
14. d as a user manual but as means of highlighting the most crucial features for model builders to raise interest in applying GENEFER to economic simulations The modular design is strongly influenced by the work of Cord6n Herrera 1997 For better readability and easier access to the mathematical notation we list our frequently used indexes and variables 1 thins Input index out Output in t te 1 T Period index Aj Antecedent of Rule i ie 1 M Rule index Ai Fuzzy set of input j in Rule i TDS Training data set B Output fuzzy set in Rule i pe 1 P Period index only in 3 1 1 a B Bj By Set of all output fuzzy sets ne 1 N Fuzzy set index Axinj Membership value of in inj Inputj int B out Membership value of out in in inj Input vector in t outFRB in Crisp inference result 3 1 Input Identification GENEFER s data interface uses an Excel 5 compatible file format with the following worksheet organization horizontally the worksheet is divided into the output range first J Design Navigator x Input Identification FCB Import Data File rows of numerical data equal in number of periods Pre Knowledge none column and the input range following columns The first row is used for time series labels Then come After importing the data GENEFER offers three ways E partial to proceed If the user has complete pre knowledge of Preselection FC FS Parameters the
15. d or whenever the relevant crisp values lie in the border regions of the fuzzified interval 3 5 Adaptability of the system in a changing environment This section is about GENEFER s application in an economic simulation As we have mentioned in section 2 there is a need to guarantee the system s adaptability to a changing environment due to the learning ability of agents The major steps in the design process reveal some useful procedures that can be used for our purposes here as well Nevertheless GENEFER s spectrum must be extended The software must allow for exploitation of agents existing knowledge as well as for exploration of new knowledge if the observed data conflict with the agents expectations The two tuning procedures in the section above could serve as means for exploitation of knowledge For example an agent who is used to an inflation rate between 1 and 2 will sooner or later change his opinion about the meaning of high if the inflation rate exceeds his experienced top level The tuning routines adapt the fuzzification of input s and output according to a changing economic environment They check for the existence of a better FB one that leads to an improved forecast performance The user specifies a time interval in which no tuning occurs If he wants to prohibit tuning he will set this interval equal to the simulation interval if he wants agents to learn continuously the interval is set to one I
16. d to the agent s current FRB E a 4 3 If the observed output at the end of this simulation period matches the newly appended rule s consequent it will be kept unchanged If this is not the case GENEFER replaces the consequent fuzzy set by the one yielding the highest membership value for the observed outcome Therefore an agent tends to expect what he has learned from observations so far 23 3 6 Key Facts of GENEFER for Economic Model Builders GENEFER is not limited to a specific type of economic model It can be implemented in e macro level simulations using a single FRB for modeling the expectations of the representative agent or the entire market e g Dornbusch s exchange rate model or Laidler s monetarist business cycle model e micro level simulations for multiple agent modeling with multiple FRBs allowing also for analysis of the interaction of heterogeneous expectations The key AlJ related features for expectations design are e three ways to set up a Knowledge Base that consists of fuzzy rules and thus accounts for the agents bounded rationality This allows for interpretability of the FRB inference result e to modify this FRB to improve its performance with the training data set offline or to model the learning processes during a simulation online The modification includes the introduction of new fuzzy rules exploration of knowledge as well as the tuning of existing rules exploi
17. dicators is on our research agenda for the near future GENEFER s modular architecture gives room for further combinations of AI techniques We plan to apply GA in order to evolve a population of Fuzzy Rule Bases and therefore reduce the amount of exogenous parameters e g the learning rate For the download of the software GENEFER as well as additional information including illustrative examples we refer to www genefer de 25 References Altrock C v 1995 Fuzzy Logic Band 1 Technologie 2nd edition Munich Arthur W B 1994 Inductive Reasoning and Bounded Rationality SFI Paper 94 03 014 http www santafe edu arthur Papers Papers html Arthur W B 1995 Complexity in Economic and Financial Markets http www santafe edu arthur Papers Papers html Arthur W B Holland J H LeBaron B Palmer R Taylor P 1996 Asset Pricing under endogenous expectations in an Artificial Stock Market SFI Paper 96 12 093 http www santafe edu sfi publications 96wplist html Beltrametti L Fiorentini R Marengo L Tamborini R 1997 A learning to forecast experiment on the foreign exchange market with Classifier System in Journal of Economic Dynamics amp Control vol 21 pp 1543 1575 Buckley J J Feuring T 1999 Fuzzy and Neural Interactions and Applications Heidelberg New York Cordon O Herrera F 1997 A three stage evolutionary process for learning descriptive and approximate F
18. e consistent with more than one fuzzy rule which may of course differ in their consequents But this does not necessarily imply contradictions just like there is no contradiction in stating that a person is tall to a degree of 0 9 and very tall to a degree of 0 25 To avoid contradictions in FRBs we apply the concepts of positive and negative observations An observation is regarded as positive for a fuzzy rule if it matches its antecedent and consequent with a compatibility degree R d greater than or equal to 5 TDSpo R dp TDS R dp 2 If the antecedent matches but not the consequent the observation is considered negative for the rule 6 TDS Ri dp TDS R dp 0 and A n gt 0 With respect to this relaxation we call an FRB consistent if it provides a sufficiently small number of negative observations measured as the percentage k of positive observations GENEFER offers three options to create an FRB as shown in the DESIGN NAVIGATOR above The two first create an FRB inductively by using the information inside the TDS The third allows for the manual creation of fuzzy rules to make use of expert knowledge a evolutionary The evolutionary method is an iterative process that generates a complete and consistent RB for all observations in the TDS It is therefore necessary to select these observations carefully so that they cover all possible input combinations One step encompasses the four
19. efault consequent This basic RB can then be used as a starting point for further individual modeling By setting influence factors inff for each input variable j the antecedents of an initialized RB can be connected automatically to the respective consequent sets Influence factors reflect ceteris paribus reasoning and are represented by integers going from 3 to 3 3 means strong negative influence zero means no influence and 3 means strong positive influence In the automatic RB connection procedure each input and output fuzzy term is identified by its relative position RP towards the respective mid term For example the linguistic term set very low low medium high very high is represented as 2 1 0 1 2 Given the user defined influence factors for all inputs the relative position RP B of the consequent set in any rule i is found as follows J 1 12 RP B trunc 3 Yinff RP A A RP B gt RP B min RP B lt RP B max j i 15 3 3 2 Rule Base Simplifying Although the previous step already provides a valid FRB the user can try to improve its performance The RB creating process might lead to a larger number of rules than is necessary Redundant rules occur because of overlearning when some observations in the TDS have a higher covering degree than the desired one GORDON HERRERA 1997 p 391 The purpose of simplification is to remove these redundant rules by applying genetic operations
20. ers three and four A link weight of zero means that the respective rule has been deactivated The third and fourth layers constitute the connectionist inference engine that embodies the complete RB of the equivalent fuzzy system The single node in the fifth layer output node defuzzifies the fuzzy inference result and delivers the crisp forecast value The link weights between layers four and five represent the centers and widths of the fuzzy sets that represent the linguistic terms of the output variable The efficiency of a neural network with given topology and node functionality depends only upon the values of the link weights that determine how the node output of layer s is propagated to the subsequent nodes in layer s 1 The knowledge of a neural network is therefore embodied in the values of the link weights The fact that the complete functionality of the fuzzy inference process is represented equivalently by the neural network allows the application of neural learning methods for FRB tuning GENEFER s neural learning method is a modified error backpropagation procedure MEBP The starting point of the learning process is the mean squared error over the TDS T 11 2 18 MSE 5 F2 0utFRB in out t 1 The learning procedure aims at minimizing the error function 18 by finding the weight vector that minimizes MSE The underlying idea of error backpropagation is that all nodes of 10 A good overview of hybridizing neural
21. evel 10 Cool Factor 08d s 07 06 05 fi f38 0775 17 3072 38 0775 _ 04 2 38 0775 55 3847 23 1939 4 03 B 234939 78 5786 42 2529 024 014 A 05 o 05 1 15 2 GDP Differential US EU in 100 oK Cancel 3 3 Rule Base Generating The user s decision about the granularity of input variables determines the maximum number J of possible rules Max N which is equal to every 7 j 1 Rule Base Generating FCB i E Creating possible combination of input fuzzy sets An RB E descriptive er j inductive consisting of Mma rules inevitably becomes T intractable if the number of inputs increases Therefore neura manual it is advantageous to remove or better not to create approximative Simplifying all redundant rules For example if an agent who z Selection distinguishes among low medium and high Parameters Run inflation rates and weak medium and strong growths of GDP has never experienced a combination of low inflation and strong growth there is no reason to have a rule for this case LEGRENZI GIROTTO JOHNSON LAIRD 1993 pp 38 ff The DESIGN NAVIGATOR above shows the two major steps of the RB generating process which are now described For those who might ask what to do if a new observation begets an unexperienced input value combination we refer to section 3 5 11 3 3 1 Rule Base Creating Rule Base Creating has a desc
22. following substeps e Creation of a candidate RB matching all observations within the TDS e Evaluation of all candidate rules according to a fitness function e Copying the best fittest rule to the generated RB and clearing the candidate RB 13 e Removal of all observations in the TDS for which CV d gt A candidate rule is created by linking those fuzzy sets that yield the highest membership value for the values of the variables of the current observation A TDS of P observations therefore leads to a candidate RB of P or less fuzzy rules since doubles are excluded The consecutive evaluation uses a multicriterion fitness function that considers the following criterions 7 High Frequency Value P Ri dp Wrps Ri ae 8 High Average Covering Degree Over Positive Observations Ri dp Go Ri TTDS RDT dpe TDS Ri 9 Small Negative Example Set 1 if TDS R lt k1 TDSp o R a R 1 TIDS R KITDS Ryl expay terse 10 Fitness function F Ri Wrps Ri Go Ri a R The best rule the one with the highest fitness value is copied to the generated RB This generated RB is then used to compute the Covering Value for all d TDS see 4 and all observations whose CV is greater than or equal to are removed The candidate RB is cleared for the next step which runs over the reduced TDS This process terminates when TDS b neural As an alternative t
23. ging the network s topology but focus on parametric learning 4 3 3 Genetic tuning The genetic approach requires a suitable representation of the input and output fuzzification We encode the complete FB as a sequence of real 3 tuples such as fl fc fr with fl fr as a fuzzy set s left right border and fc as its centre The complete encoding C of all fuzzified variables FB is the object to be genetically modified The encoded fuzzification of one input variable j contains 3 N elements c so that the encoded FB is a sequence of 3 Nj NourJ elements The encoded current fuzzification as well as randomly initialized individuals form the starting population for the first generation t Initialization and genetic operations take place in a specified search space The structure of C as well as the search space for fuzzy parameter adaptation is shown in figure 3 17 Figure 3 Genetic Tuning of FB _ Fuzzification A Fuzzification _ Fuzzification Fuzzification Fuzzification C Input 1 ues Input j ee Input J 1 P InputJ P Output Aj in Ain Cy Cy C3 5 5 C3n 2 gt C3n 1 gt Can 5 5 C3N 2 gt C3N 1 gt Can A A L Pes Se eres sees JN J fl fc fr pie fe fl _ fc fl i fle fl fl 1 f 2 2 fr f fee Re fel t te 5 7 fr fc f fre fry fr fe fr 75 a a a gt fl fl fl fe fe fce fr fr fr
24. h AN A IF Input AND Input2 THEN Output pr p P ra low medium medium i PA z low very high high Input i medium very low low very low low medium high very high l 3 f A A A A high very high high a Sf y 3 low low low X V Fs r Ve AN high medium medium _ Input2 3 high very low medium low medium high medium low low i f amp medium medium medium a ge high medium medium Output gt This separation is also realised in GENEFER s data handling which allows for a greater flexibility of linking different FBs to different RBs Both FB and RB can be modified during an agent s learning process The GA approach requires a suitable encoding of these objects in order to apply its genetic operators selection mutation crossover Transforming the FRB into an equivalent Artificial Neural Network opens it up to neural learning techniques such as error backpropagation Whereas fuzzy systems account for D1 and D2 their combination with GA and ANN training techniques meets all three demands The use of Genetic and Neural Fuzzy Rule Bases and their implementation in GENEFER is sketched in the next section 3 Managing Fuzzy Rule Bases with GENEFER GENEFER is software for designing an FRB of the Mamdani multiple input single output type The primarily technical literature about fuzzy control proposes numerous ways of setting up an FRB GENEFER does not foc
25. in financial markets Since their expectations help strongly to determine the aggregate market outcome they try to predict expectations become self referential and a rational deduction of the true model See McFadden 1999 p 75 onwards George Soros one of the most successful and therefore most famous traders expressed his opinion about the standard academic theory as follows this efficient market theory interpretation of the way financial markets operate is severely distorted It may seem strange that a patently false theory should gain such widespread acceptance becomes impossible An agent cannot deductively form his expectations since he needs to know others privately held expectations to form his own consequently these are indeterminate FOLEY 1998 p 53 ff ARTHUR 1995 p 3 ff Even without informational problems an agent might not be able to understand fully the complexity of a market or if the costs of discovering the exact input output relation in finite time exceed the benefits from it Both limited information processing capability stemming from the intrinsic data complexity and a lack of information bound the agents rationality When others expectations are non measurable there is an absolute barrier to rationality But of course agents form expectations everyday How do they do it As mentioned one theory holds that agents look for patterns or rules within the market In d
26. jor steps and apply Genetic Algorithms as well as Artificial Neural Networks to learn and to tune FRBs from observed data In section three these steps of modeling and training FRBs are described We do so by following a typical FRB design session with the software tool GENEFER GEnetic N Eural Fuzzy ExploreR which was developed for handling Neural and Genetic Fuzzy Rule Bases and whose key features for model builders are highlighted in section 3 6 The paper closes with a glance at possible applications of GENEFER and some suggestions for further research 2 A Genetic and Neural Fuzzy Rule Base approach towards modeling expectations The true model of the world economy or national economies or even their single subsystems e g markets industries or firms has yet to be found The massive interaction between millions of heterogeneous agents produces this complexity which is continuously challenging our understanding of the world we live and trans act in Although we have recorded stock and bond prices interest rates exchange rates and many more prices for more than a century we still lack a commonly accepted theory explaining the formation of these prices This is especially so in financial market research where we see several competing theories that seem appropriate to explain specific market phenomena Recent research breaks away from the widespread idea of neoclassical market foundations and the assumption of homogenous rational
27. lation model The software GENEFER GEnetic NEural Fuzzy ExploreR has been developed for designing such a Fuzzy Rule Base The design process is modular and comprises Input Identification Fuzzification Rule Base Generating and Rule Base Tuning The two latter steps make use of genetic and neural learning algorithms for optimizing the Fuzzy Rule Base 1 Introduction Modeling expectations is a major endeavor for economists as well as for psychologists In economics there are a variety of theories requiring explicit expectation modeling Here we deal with financial markets because of the predominant influence of forecasts on asset market transactions Since financial markets display phenomena such as bubbles crashes herd behavior contagion GARCH effects or focal points we ask What moves the asset prices FARMER 1999 gives a first hint by stating to have a good theory of how prices behave we will need to explain the behavior of the agents on whom they depend The standard academic literature is still not very convincing about modeling expectation formation Although agents face a pool of publicly available information which consists of past prices trading volumes economic indicators political events rumors news etc there may be many different perfectly defensible statistical ways based on different assumptions and different error criteria to use them ARTHUR 1995 That is the point where the psychologists view
28. m nodes The input term nodes carry out the fuzzification function for each input Every basic input node is connected with all input term nodes of the respective input variable which represent the different linguistic terms The parameters used for characterizing 19 the membership functions centers and widths can be interpreted as link weights between layer one and two After calculating the degrees of membership for all linguistic input terms the layer two nodes propagate this result to the next layer whose nodes represent the antecedents of the RB Each of them computes the degree of activation of the respective rule by means of the fuzzy AND operator All cross term combinations between all inputs are represented in the aggregation layer so the number of links of each conditional node to the anterior nodes equals the number of inputs Since the aggregation procedure works with unweighted input data i e degrees of membership of the concerned terms the link weights between layer two and three are constant and equal to one Each node at layer four consequent nodes in the conclusion accumulation layer corresponds to one linguistic term of the output variable output term nodes Each of these nodes receives the degrees of application of those conditional nodes that point at the respective consecutive term represented by the node considered at layer four Optionally the degrees of activation can be weighted by means of link weights between lay
29. mplete if it is able to generate an inference result for any input proposition As mentioned above this may lead to an unacceptably large number of rules if certain regions in the input space are not covered or can be excluded Therefore we relax the RB completeness property by requiring each observation d in out in the TDS to be covered to a degree of at least CV d 2 gt 0 The Covering Value CV is calculated as follows 1 Ai Gin min Aj in Aiy Ging 2 Ri d min Aj in Bi out M 3 CV d U R d i M 4 CVd Ri d i l 8 We plan to test GENEFER s forecast abilities and will of course include the approximate approach to see if it shows better results than the descriptive one 12 Equation 1 is the degree of compatibility between the i rule s antecedent and the observation in whereas equation 2 holds for the entire rule i and the observation d Equation 3 is the generic Covering Value However the iterative nature of the evolutionary procedure creating an RB see below requires the modification in equation 4 The second property of an FRB is consistency A Rule Base is called consistent if it does not contain contradictions A contradiction arises if two or more rules have the same antecedents but not the same consequent In FRBs there is a need to relax the consistency requirement due to fuzzy modeling It is the essence of fuzzy modeling that a crisp observation may b
30. n contrast to this exploration procedures search for new rules without adjusting the FB As mentioned above the generation process creates an FRB which covers all observations in the TDS but cannot guarantee to provide a fuzzy rule for all possible values of input output combinations In the case of a missing rule or poor covering of existing rules GENEFER offers two ways to explore the rule search space The first one is continuously to replace an agent s FRB after a user specified number of simulation periods GENEFER would therefore generate a new FRB on i an enlarged TDS including all new observations or ii a TDS of constant size including a user specified number of observations moving window The technical realization might exceed computational capacity when the number of agents increases or when they are forced to replace their FRB quite frequently Additionally one cannot exclude the case of false inference if an uncovered observation occurs during the interval of unchanged FRB For that reason we introduce a second procedure that allows one to generate a new rule in case of a new observation for which no forecast value can be 1l A poor covering may occur with gaussian fuzzy sets which always yield positive membership values but not necessarily significantly different from zero 22 inferred If for example there has never been an observation of low inflation rate differential and high unemployment rate differential there
31. o the evolutionary procedure GENEFER offers an unsupervised competitive neural learning algorithm for detecting rules in a given set of observations The For the underlying feature map algorithm see Kohonen 1988 ch 5 particularly p 132 14 algorithm works as follows At the beginning of the learning process all antecedents 1 are virtually connected with all consequent terms Virtually connected means that there is a potential connection between each antecedent i i 1 Mmax and each consequent set q q 1 Nout with an initial connection weight of zero Wig 0 V i Presenting a training pattern d allows computation of the degree of activation for each antecedent A in and the membership value for each fuzzy set q of the output variable B out These values are used to adapt the connection weights by means of the following learning rule 11 Awig B out Wig A in At the end of the learning procedure after a given fixed number of training patterns the connection with the maximum connection weight of each antecedent is kept and all others are removed c manual GENEFER s expert mode allows one to re design the complete RB or parts of it manually including the rule weights vector setting a rule weight to zero deactivates the rule If no RB has yet been created the program delivers an initialized normalized RB by generating all Mmax antecedents with the output s mid term as the d
32. oing so they mentally form a rule based representation of the market s functioning These rules reflect the agents experiences and may be influenced by economic theory This rule based approach includes the expression of explicit e g technical trading rules as well as tacit knowledge Due to agents bounded rationality however these rules should not be interpreted as exact mathematical functions but rather as a relationship between the agents interpretation of the input and output variables states We contend that agents interpret these crisp states vaguely by associating with them a limited number of linguistic terms e g low medium high An evolving economic environment continuously generates new observations that extend an agent s experience New observations can conflict with current knowledge so that it becomes necessary to learn by changing rules or the interpretation of crisp data From the preceding we are led to the following list of demands on a realistic approach for modeling expectations D1 explicit knowledge representation theory driven rules model building D2 vague formulation of forecasts bounded rationality D3 dependency upon experiences ability to learn To meet these demands we choose Fuzzy Rule Bases as an operational representation of mental models and apply Genetic Algorithms GA and Artificial Neural Networks ANN as learning operators This is oppo
33. riptive and an approximate approach They differ in the linguistic meaning of each fuzzy set when interpreting fuzzy rules The descriptive approach is characterized by a uniform meaning for all fuzzy sets in all rules If the second set of the third input appears in more than one rule it always has the same meaning e g medium and is represented by the same membership function Changing this function consequently affects all rules that contain the respective fuzzy set Hence there are no restrictions on the economic interpretability of a descriptive FRB By contrast the approximate approach allows fuzzy sets to differ from rule to rule free semantics Whereas this may have advantages concerning the goodness of fit it is not amenable to an economic interpretation We do not consider the approximate approach in the following but concentrate on the interpretably descriptive one The designer of an FRB has to pay attention to two properties to obtain good results PEDRYCZ GOMIDE 1998 ch 10 6 and especially GORD N HERRERA 1997 pp 377 380 The completeness property guarantees that the FRB is able to generate an inference result for each observation of input values in the TDS The completeness property can be assigned to the FB and the RB An FB is complete if the union of all fuzzy sets for each input variable covers the related universe of discourse If it covers to a level of 6 0 1 the FB is called complete An RB is called co
34. sed to classifier systems which usually interpret crisp data by means of crisp intervals A single fuzzy rule expresses the vague relation between input s and output like IF the US current account deficit is very high AND the GDP growth rate is lower than in the Euro Zone THEN the EURO is expected to rise strongly against the USD The IF elements of the fuzzy rule are called antecedents whereas the THEN elements are consequents Very high lower and strongly are fuzzy terms The set of all fuzzy rules represents an agents knowledge base KB and is called Fuzzy Rule Base FRB An FRB can be divided into two parts the rule base RB and the fuzzification base FB see Figure 1 The former captures all rules as IF THEN statements whereas the latter provides the fuzzy sets which express the terms linguistic meaning as membership functions These functions map the elements in the universes of discourse to the unit interval and allow for partial quantification of belongingness e g a temperature of 20 C belongs to the fuzzy term warm to a degree of 0 8 We additionally assign a weight to each rule indicating its relative importance within the RB and therefore allow for a simple hierarchy Figure 1 Structure of Fuzzy Rule Bases KB FRB Knowledge Base Fuzzy Rule Base FB Fuzzification Base RB Rule Base low medium hig
35. tation of knowledge e obtaining flexibility in adapting the system s behavior to specific purposes The user can set the agents inputs by default or let GENEFER learn which inputs are the most significant He can define each rule in an FRB or let GENEFER learn an RB or even a combination of both As far as simulation is concerned the frequency to tune simplify or even replace the FRB can be set individually for each agent GENEFER comes with COM server functionality giving access to those FRB design related methods needed for the implementation of all features mentioned above The user is therefore free to choose any preferred programming language for his computational economic simulations 4 Summary and directions for further research In this paper we propose a new AJI based technique for modeling expectations This technique combines fuzzy systems as a representation of knowledge bases with Genetic Algorithms GA and Artificial Neural Networks as learning operators We describe their synthesis and present the software GENEFER GEnetic NEural Fuzzy ExploreR Economic model builders can implement GENEFER in their simulations via a COM interface and make use of its fuzzy 24 inference and learning routines Nevertheless it may also be used for pure forecasting purposes on empirical data We are currently testing GENEFER s performance in forecasting financial time series The development of self documenting business cycle in
36. us on one of the many ways but separates the design process in four major steps and thus allows the user to combine alternative methods i identifying inputs ii fuzzifying crisp input and output data FB iii generating a rule base RB and finally iv tuning the FRB The design methods of each step can be run either in an inductive or in a manual expert mode The former applies combinations of GA or ANN with Fuzzy Logic Controllers LIN LEE 1991 LIN 1994 and CORDON HERRERA 1997 and requires a training data set of observations TDS for learning GENEFER is equipped with a DESIGN NAVIGATOR that allows quick and easy navigation through the process of FRB design The description of each step below shows the respective DESIGN NAVIGATOR s appearance We will not explain the basics of fuzzy inference except to mention that we use t norm operators for calculating the degree of activation AD of a single rule i and accumulate these using the sum procedure min 1 X AD to obtain a compositional degree of activation The fuzzy inference result can be defuzzified by the 1 Center of Maximum weighted by surface ii Center of Maximum weighted by significance iii Center of Gravity weighted by Surface or iv Center of Gravity weighted by Significance Since it is beyond our needs here we also avoid giving details of GA and ANN and refer instead to the standard literature e g PEDRYCZ GOMIDE 1998 This paper is not intende
37. uzzy Logic Controller Knowledge Bases from examples in International Journal of Approximate Reasoning vol 17 pp 369 407 Farmer J Doyne 1999 Physicists attempt to scale the ivory towers of finance in Computing in Science amp Engineering Foley Duncan K 1998 Introduction in Barriers and Bounds to Rationality Essays on Economic Complexity and Dynamics in Interactive Systems Albin Peter S Princeton University Press Princeton New Jersey Hecht Nielsen R 1991 Neurocomputing Reading Mass Holland J H 1975 Adaptation in natural and artificial systems University of Michigan Press Ann Johnson Laird P N 1993 Mental Models Cambridge University Press Cambridge MA Johnson Laird P N Shafir E 1993 The interaction between reasoning and decision making an introduction in Cognition vol 49 pp 1 9 Kohonen T 1988 Self Organisation and Associative Memory 3 ed Berlin 26 Kooths S 1998 Erfahrungsregeln und Konjunkturdynamik Makromodelle mit Neuro Fuzzy generierten Erwartungen Frankfurt Main u a O Legrenzi P Girotto V Johnson Laird P N 1993 Focussing in reasoning and decision making in Cognition vol 49 pp 37 66 Lin C T 1994 Neural Fuzzy Control Systems with Structure and Parameter Learning Singapore Lin Y Cunningham G Coggeshall S 1996 Input variable identification Fuzzy curves and fuzzy surfaces in Fuzzy Sets and
38. will probably be no rule covering such an observation What should an agent expect about the exchange rate assuming that to be the variable Will he throw dice to find out Not likely We assume that the agent tries to derive a new rule and therefore a new forecast result by using his current knowledge Let us consider an agent with the FRB in Figure 1 who experiences a low inflation rate differential and a high unemployment differential Let us also assume that none of his rules covers the observed data so that no result can be inferred The agent is assumed to compare the existing rules antecedents with the observed input data to identify the most similar ones If we index the fuzzy sets according to their position we can encode the FRB as IF Inflation rate differential AND Unemployment rate differential THEN USD 1 NIN QTR OO NO WIN GI ND yr Gr Go NIR INN RR Qtr Go bh Oo Oo N We compute the sum of the squared distances between each index of the new antecedent 1 4 and the ones for all antecedents in the FRB above for each rule The lowest value yields the rule s that we regard as most similar These are the first two in our example 0 1 The agent has to decide whether to expect the first rule s consequent or the second one s We calculate the mean value and round the result which yields high Thus the following rule is adde
39. zy Set Theory and Its Applications gnd ed Boston et al
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