This paper deals with the control of chaotic dynamics by using the approximated system equations which are obtained by using the Genetic Programming (GP). Well known OGY method utilizes already existing unstable orbits embedded in the chaotic attractor, and use linearlization of system equations and small perturbation for control. However, in the OGY method we need transition time to attain the control, and the noise included in the linealization of equations moves the orbit into unstable region again. In this paper we propose a control method which utilize the estimated system equations obtained by the GP so that the direct nonlinear control is applicable to the unstable orbit at any time. In the GP, the system equations are represented by parse trees and the performance (fitness) of each individual is defined as the inversion of the root mean square error between the observed data and the output of the system equation. By selecting a pair of individuals having higher fitness, the crossover operation is applied to generate new individuals. In the simulation study, the method is applied at first to the artificially generated chaotic dynamics such as the Logistic map and the Henon map. The error of approximation is evaluated based upon the prediction error. The effect of noise included in the time series on the approximation is also discussed. In our control, since the system equations are estimated, we only need to change the input incrementally so that the system moves to the stable region. By assuming the targeted dynamic system f(x(t)) with input u(t)=0 is estimated by using the GP (denoted (x(t))), then we impose the input u(t) so that xf=(t+1)=(x(t))+u(t) where xf is the fixed point. Then, the next state x(t+1) of targeted dynamic system f(x(t)) is replaced by x(t+1)+u(t). The control method is applied to the approximation and control of chaotic dynamics generating various time series and even noisy time series by using one dimensional and higher dimensional system. As a result, if the noise level is relatively large, the method of the paper provides better control compared to conventional OGY method.

In this paper, we show the analysis of price changes in artificial double auction markets consisting of multi-agents who learn from past experiences based on the Genetic Programming (GP) and its applications. For simplicity, we focus on the double auction in an electricity market. Agents in the market are allowed to buy or sell items (electricity) depending on the prediction of situations. Each agent has a pool of individuals (decision functions) represented in tree structures to decide bid price by using the past result of auctions. A fitness of each individual is defined by using successful bids and a capacity utilization rate of production units for a production of items, and agents improve their individuals based on the GP to get higher return in coming auctions. In simulation studies, changes of bid prices and returns of bidders are discussed depending on demand curves of customers and the weight between an average profit obtained by successful bids and the capacity utilization rate of production units. The validation of simulation studies is examined by comparing results with classical models and price changes in real double auction markets. Since bid prices bear relatively large changes, we apply an approximate method for a control by forcing agents stabilize the changes in bid prices. As a result, we see the stabilization scheme of bid prices in double auction markets is not realistic, then it is concluded that the market contains substantial instability.

This paper deals with the approximation of multi-dimensional chaotic dynamics by using the multi-stage fuzzy inference system. The number of rules included in multi-stage fuzzy inference systems is remarkably smaller compared to conventional fuzzy inference systems where the number of rules are proportional to an exponential of the number of input variables. We also propose a method to optimize the shape of membership function and the appropriate selection of input variables based upon the genetic algorithm (GA). The method is applied to the approximation of typical multi-dimensional chaotic dynamics. By dividing the inference system into multiple stages, the total number of rules is sufficiently depressed compared to the single stage system. In each stage of inference only a portion of input variables are used as the input, and output of the stage is treated as an input to the next stage. To give better performance, the shape of the membership function of the inference rules is optimized by using the GA. Each individual corresponds to an inference system, and its fitness is defined by using the prediction error. Experimental results lead us to a relevant selection of the number of input variables and the number of stages by considering the computational cost and the requirement. Besides the GA in the optimization of membership function, we use the GA to determine the input variables and the number of input. The selection of input variable to each stage, and the number of stages are also discussed. The simulation study for multi-dimensional chaotic dynamics shows that the inference system gives better prediction compared to the prediction by the neural network.

This paper deals with the identification of system equation of the chaotic dynamics by using smaller number of data based upon the genetic programming (GP). The problem to estimate the system equation from the chaotic data is important to analyze the structure of dynamics in the fields such as the business and economics. Especially, for the prediction of chaotic dynamics, if the number of data is restricted, we can not use conventional numerical method such as the linear-reconstruction of attractors and the prediction by using the neural networks. In this paper we utilize an efficient method to identify the system equation by using the GP. In the GP, the performance (fitness) of each individual is defined as the inversion of the root mean square error of the spectrum obtained by the original and predicted time series to suppress the effect of the initial value of variables. Conventional GA (Genetic Algorithm) is combined to optimize the constants in equations and to select the primitives in the GP representation. By selecting a pair of individuals having higher fitness, the crossover operation is applied to generate new individuals. The crossover operation used here means the replacement of a part of tree in individual A by a part of tree in individual B. To avoid the meaningless genetic operation, the validity of prefix representation of the subtree to be embedded to the other tree is probed by using the stack count. These newly generated individuals replace old individuals with lower fitness. The mutation operation is also used to avoid the convergence to the local minimum. In the simulation study, the identification method is applied at first to the well known chaotic dynamics such as the Logistic map and the Henon map. Then, the method is applied to the identification of the chaotic data of various time series by using one dimensional and higher dimensional system. The result shows better prediction than conventional ones in cases where the number of data is small.

This paper deals with the use of neural network rule extraction techniques based on the Genetic Programming (GP) to build intelligent and explanatory evaluation systems. Recent development in algorithms that extract rules from trained neural networks enable us to generate classification rules in spite of their intrinsically black-box nature. However, in the original decompositional method looking at the internal structure of the networks, the comprehensive methods combining the output to the inputs using parameters are complicated. Then, in our paper, we utilized the GP to automatize the rule extraction process in the trained neural networks where the statements changed into a binary classification. Even though the production (classification) rule generation based on the GP alone are applicable straightforward to the underlying problems for decision making, but in the original GP method production rules include many statements described by arithmetic expressions as well as basic logical expressions, and it makes the rule generation process very complicated. Therefore, we utilize the neural network and binary classification to obtain simple and relevant classification rules in real applications by avoiding straightforward applications of the GP procedure to the arithmetic expressions. At first, the pruning process of weight among neurons is applied to obtain simple but substantial binary expressions which are used as statements is classification rules. Then, the GP is applied to generate ultimate rules. As applications, we generate rules to prediction of bankruptcy and creditworthiness for binary classifications, and the apply the method to multi-level classification of corporate bonds (rating) by using the financial indicators.

This paper deals with an adaptive credit assignment algorithm to select strategies having higher capabilities in the learning classifier system (LCS) based upon the genetic algorithm (GA). We emulate a kind of prizes and incentives employed in the economies with imperfect information. The compensation scheme provides an automatic adjustment in response to the changes in the environment, and a comfortable guideline to incorporate the constraints. The learning process in the LCS based on the GA is realized by combining a pair of most capable strategies (called classifiers) represented as the production rules to replace another less capable strategy in the similar manner to the genetic operation on chromosomes in organisms. In the conventional scheme of the learning classifier system, the capability s(k, t) (called strength) of a strategy k at time t is measured by only the suitableness to sense and recognize the environment. But, we also define and utilize the prizes and incentives obtained by employing the strategy, so as to increase s(k, t) if the classifier provide good rules, and some amount is subtracted if the classifier k violate the constraints. The new algorithm is applied to the portfolio management. As the simulation result shows, the net return of the portfolio management system surpasses the average return obtained in the American securities market. The result of the illustrative example is compared to the same system composed of the neural networks, and related problems are discussed.

This paper deals with the prediction of stock trends by using the wavelet transform and the multi-stage fuzzy inference system based upon the optimization of membership function by using the GA. The system is expected to recognize the short-term feature which is usually used to estimate the rise/fall of price by human experts. In the prediction of stock prices, the wavelet transform is used to describe the short term feature of the stock trend. The fractal dimension and the variance of the time series are also used as the input variables. By dividing the inference system into multiple stages, the total number of rules is sufficiently depressed compared to the single stage system. In each stage of inference only a portion of input variables are used as the input, and output of the stage is treated as an input to the next stage. To give better performance, the shape of the membership function of the inference rules is optimized by using the GA. Each individual corresponds to an inference system, and its fitness is calculated as the ratio of the correct recognition. In the simulation study, we define the rise and fall of prices by considering the threshold value for the price change, and the interval of prediction. Then, the parameters of the system are adjusted by using the data for learning and the performance is evaluated by comparing the prediction and observation. The simulation study shows that the inference system gives about a 70% correct prediction of the price change of stocks. The result is compared to the prediction by the neural network, and we see better classification of the fuzzy system.

This paper shows a novel automatic classification method for the electroencephalogram (EEG) based on syntactical pattern recognition. The syntactical method is effective to represent the complicated structure of the features of the EEG which contains transient-waves as well as the background-wave. For the extraction of transient-waves an adaptive autoregressive-moving average (ARMA) model fitting is utilized where the input to the model is replaced by the modified input if the prediction error grows more than a given threshold. By the adaptive ARMA model transient-waves and the spectrum of the background-wave are obtained from the prediction error and ARMA parameters, respectively. Since transient-waves may contain noisy patterns or variances, a relaxation scheme is applied. As the second stage all of the features of the EEG including the spectrum are described syntactically according to the generative grammar. Then the syntactical description Ti inherent to the diagnosis is obtained. In order to reduce the ambiguity and to suppress the complexity of syntactical descriptions, numerical values representing the details of EEG are separated from the syntactical description, and are added as the attributes (this method is generally called the attributed grammar). For the input EEG having syntactical description Ti, the final diagnostic decision is made by using the statistical Bayes estimation about the attributes within the group for Ti. As the result of automatic EEG classification for 200 EEG samples correct recognition of about 80 percent is observed.

This paper deals with the decomposition of surface data into several fractal signal based on the parameter estimation by the Mean Likelihood and Importance Sampling (IS) based on the Monte Carlo simulations. The method is applied to the feature extraction of surface data. Assuming the stochastic models for generating the surface, the likelihood function is defined by using wavelet coefficients and the parameter are estimated based on the mean likelihood by using the IS. The approximation of the wavelet coefficients is used for estimation as well as the statistics defined for the variances of wavelet coefficients, and the likelihood function is modified by the approximation. After completing the decomposition of underlying surface data into several fractal surface, the prediction method for the fractal signal is employed based on the scale expansion based on the self-similarity of fractal geometry. After discussing the effect of additive noise, the method is applied to the feature extraction of real distribution of surface data such as the cloud and earthquakes.

This paper shows the design of multi-stage fuzzy inference system with smaller number of rules based upon the optimization of rules by using the genetic algorithm. Since the number of rules of fuzzy inference system increases exponentially in proportion to the number of input variables powered by the number of membership function, it is preferred to divide the inference system into several stages (multi-stage fuzzy inference system) and decrease the number of rules compared to the single stage system. In each stage of inference only a portion of input variables are used as the input, and the output of the stage is treated as an input to the next stage. If we use the simplified inference scheme and assume the shape of membership function is given, the same backpropagation algorithm is available to optimize the weight of each rule as is usually used in the single stage inference system. On the other hand, the shape of the membership function is optimized by using the GA (genetic algorithm) where the characteristics of the membership function is represented as a set of string to which the crossover and mutation operation is applied. By combining the backpropagation algorithm and the GA, we have a comprehensive optimization scheme of learning for the multi-stage fuzzy inference system. The inference system is applied to the automatic bond rating based upon the financial ratios obtained from the financial statement by using the prescribed evaluation of rating published by the rating institution. As a result, we have similar performance of the multi-stage fuzzy inference system as the single stage system with remarkably smaller number of rules.

This paper deals with the chaoticity and fractality analysis of price time series for artificial stock market generated by the multi-agent systems based on the co-evolutionary Genetic Programming (GP). By simulation studies, if the system parameters and the system construction are appropriately chosen, the system shows very monotonic behaviors or sometime chaotic time series. Therefore, it is necessary to show the relationship between the realizability (reproducibility) of the system and the system parameters. This paper describe the relation between the chaoticity of an artificial stock price and system parameters. We also show the condition for the fractality of a stock price. Although the Chaos and the Fractal are the signal which can be obtained from the system which is generally different, we show that those can be obtained from a single system. Cognitive behaviors of agents are modeled by using the GP to introduce social learning as well as individual learning. Assuming five types of agents, in which rational agents prefer forecast models (equations) or production rules to support their decision making, and irrational agents select decisions at random like a speculator. Rational agents usually use their own knowledge base, but some of them utilize their public (common) knowledge base to improve trading decisions. By assuming that agents with random behavior are excluded and each agent uses the forecast model or production rule with most highest fitness, those assumptions are derived a kind of chaoticity from stock price. It is also seen that the stock price becomes fractal time series if we utilize original framework for the multi-agent system and relax the restriction of systems for chaoticity.

The paper deals with the estimation method of system equations of dynamic behavior of an input-pricing mechanism by using the Genetic Programming (GP) and its applications. The scheme is similar to recent noise reduction method in noisy speech which is based on the adaptive digital signal processing for system identification and subtraction estimated noise. We consider the dynamic behavior of an input-pricing mechanism for a service facility in which heterogeneous self-optimizing customers base their future join/balk decisions on their previous experiences of congestion. In the GP, the system equations are represented by parse trees and the performance (fitness) of each individual is defined as the inversion of the root mean square error between the observed data and the output of the system equation. By selecting a pair of individuals having higher fitness, the crossover operation is applied to generate new individuals. The string used for the GP is extended to treat the rational form of system functions. The condition for the Li-Yorke chaos is exploited to ensure the chaoticity of the approximated functions. In our control, since the system equations are estimated, we only need to change the input incrementally so that the system moves to the stable region. By assuming the targeted dynamic system f(x(t)) with input u(t)=0 is estimated by using the GP (denoted (x(t))), then we impose the input u(t) so that xf= (t+1)=(x(t))+u(t) where xf is the fixed point. Then, the next state x(t+1) of targeted dynamic system f(x(t)) is replaced by x(t+1)+u(t). We extend ordinary control method based on the GP by imposing the input u(t) so that the deviation from the targeted level xL becomes small enough after the control. The approximation and control method are applied to the chaotic dynamics generating various time series based on several queuing models and real world data. Using the GP, the control of chaos is straightforward, and we show some example of stabilizing the price expectation in the service queue.

There are several methods for generating multi-fractal time series, but the origin of the multi-fractality is not discussed so far. This paper deals with the multi-fractality analysis of time series in an artificial stock market generated by multi-agent systems based on the Genetic Programming (GP) and its applications to feature extractions. Cognitive behaviors of agents are modeled by using the GP to introduce the co-evolutionary (social) learning as well as the individual learning. We assume five types of agents, in which a part of the agents prefer forecast equations or forecast rules to support their decision making, and another type of the agents select decisions at random like a speculator. The agents using forecast equations and rules usually use their own knowledge base, but some of them utilize their public (common) knowledge base to improve trading decisions. For checking the multi-fractality we use an extended method based on the continuous time wavelet transform. Then, it is shown that the time series of the artificial stock price reveals as a multi-fractal signal. We mainly focus on the proportion of the agents of each type. To examine the role of agents of each type, we classify six cases by changing the composition of agents of types. As a result, in several cases we find strict multi-fractality in artificial stock prices, and we see the relationship between the realizability (reproducibility) of multi-fractality and the system parameters. By applying a prediction method for mono-fractal time series as counterparts, features of the multi-fractal time series are extracted. As a result, we examine and find the origin of multi-fractal processes in artificial stock prices.