Approximation of Chaotic Dynamics by Using Smaller Number of Data Based upon the Genetic Programming and Its Applications

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  author =       "Yoshikazu Ikeda and Shozo Tokinaga",
  title =        "Approximation of Chaotic Dynamics by Using Smaller
                 Number of Data Based upon the Genetic Programming and
                 Its Applications",
  journal =      "IEICE Transactions on fundamentals of electronics,
                 communications and computer sciences",
  volume =       "E83A",
  number =       "8",
  pages =        "1599--1607",
  year =         "2000",
  keywords =     "genetic algorithms, genetic programming, nonlinear
                 dynamics, system identification, Nonlinear Signal
                 Processing, chaotic dynamics,
  organisation = "The Institute of Electronics, Information and
                 Communication Engineers. JAPAN",
  publisher =    "Oxford University Press",
  ISSN =         "0916-8524",
  ISSN =         "0916-8508",
  URL =          "",
  URL =          "",
  broken =       "",
  abstract =     "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 use 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

Genetic Programming entries for Yoshikazu Ikeda Shozo Tokinaga