Computational Intelligence and Neuroscience

Volume 2015, Article ID 341031, 10 pages

http://dx.doi.org/10.1155/2015/341031

## Forecasting Nonlinear Chaotic Time Series with Function Expression Method Based on an Improved Genetic-Simulated Annealing Algorithm

^{1}National Key Laboratory on Electromagnetic Environmental Effects and Electro-Optical Engineering, PLA University of Science and Technology, Nanjing 210007, China^{2}College of Meteorology and Oceanography, PLA University of Science and Technology, Nanjing 211101, China

Received 15 October 2014; Revised 11 March 2015; Accepted 11 March 2015

Academic Editor: Francois B. Vialatte

Copyright © 2015 Jun Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper proposes a novel function expression method to forecast chaotic time series, using an improved genetic-simulated annealing (IGSA) algorithm to establish the optimum function expression that describes the behavior of time series. In order to deal with the weakness associated with the genetic algorithm, the proposed algorithm incorporates the simulated annealing operation which has the strong local search ability into the genetic algorithm to enhance the performance of optimization; besides, the fitness function and genetic operators are also improved. Finally, the method is applied to the chaotic time series of Quadratic and Rossler maps for validation. The effect of noise in the chaotic time series is also studied numerically. The numerical results verify that the method can forecast chaotic time series with high precision and effectiveness, and the forecasting precision with certain noise is also satisfactory. It can be concluded that the IGSA algorithm is energy-efficient and superior.

#### 1. Introduction

Chaos is a universal complex dynamical phenomenon that exists in various natural and social systems, such as communication, atmosphere, economics, and biology. Chaos phenomenon is generated by determinate equations but the appearance follows an apparently unpredictable nonperiodic stochastic pattern, and it is aperiodic, bounded, deterministic, and sensitive to the initial state. So the prediction of chaotic time series is very useful to evaluate characteristics of dynamical systems and is important to the research of chaos. The prediction of chaotic time series has been widely studied over the years. It is proved that short-term prediction of chaotic time series is possible by exploiting the deterministic dynamics in chaotic systems [1–6]. In general, the forecast of chaotic series implies two processes. The first step is to use the immediate past behavior of the time series to reconstruct state space [7]. Estimating the proper embedding dimensions and delaying time is the main work of state space reconstruction. The self-correlation method [8], mutual information method [9], false nearest neighbors (FNN) algorithm [10], and C-C algorithm [11] have been introduced to reconstruct the state space. The second step is to build the forecasting model. Lots of techniques are proposed to build various models in many literatures, such as artificial neural networks [12–16] and polynomial fitting [17, 18]. Each of them has drawback and advantage; no method is superior to all other methods under every evaluating criterion. The paper proposes a novel and simple predictive model named the function expression method to forecast chaotic time series. In fact, there are many methods to establish the proper function expression. Zhang and Xiao [17] proposed a continued fractions method to give explicit expression. Zhou et al. [18] presented a multivariate local polynomial kernel estimator to approximate polynomial. Although those mathematical methods can predict accurately, they are too complex to predict the complex high dimensional chaotic systems, and the convergence speed is not high enough. The paper proposes an improved genetic-simulated annealing algorithm to establish the best approximation to the real dynamic equation. The genetic algorithm has a strong capability of global optimization and has been used in many forecasting problems [19–23]. However, it is easily trapped into the local-best solution, and the quality of solutions is decreased a lot. We find that the simulated annealing algorithm [24, 25] has strong ability to jump out of the local-best solution and search for the best solution, but the local search ability is relatively poor. Therefore, incorporating the simulated annealing algorithm into the genetic algorithm is an ideal way that combines the global optimization ability of GA with the local search ability of SA; GA is developed to rapidly search for an optimum or near-optimum among the solution space, and then SA is utilized to seek a better one on the basis of that solution. In addition, the fitness function and the genetic operators are also improved to further improve the performance of optimization. The performance of the proposed method is verified by some simulations, and the results demonstrate the outstanding optimization capability and higher forecasting precision compared with other methods such as traditional genetic algorithm (GA) [1], continued fractions (CF) method [17], and neural network (NN) [16].

The remaining sections of this paper are organized as follows. In Section 2, the general forecasting principle of chaotic time series is presented. Section 3 elaborates the IGSA algorithm and the detailed forecasting procedure. The simulated numerical results are given in Section 4. The effect of noise in the chaotic time series is presented in Section 5. In Section 6, the discussions upon the proposed method are given. The paper ends with conclusions in Section 7.

#### 2. General Forecasting Principle

The system state of a chaotic system and its delayed versions can be described in (1), where is a scalar index for the time series, is the time delay, and is the embedding dimension. Consider

We want to determine the dependence of the state value on its previous state values. Takens Embedding Theorem [7] guarantees that the system’s state information can be recovered from a sufficiently long observation of the output time series. According to the theorem, the system state follows the existence of a smooth map satisfying

Thus, once the state space has been reconstructed from the time series, the chaotic time series can be forecasted by establishing the functional relation with the proposed IGSA algorithm.

Measurements from a chaotic system are not restricted to a unique variable, but situations in which several variables are observed from the same system are common. In this case, we need to deal with multivariate time series . The model of connection between the different variables can be written in (3), where is the model to be determined. Consider

The general principle of using IGSA method to establish the optimum function expression to forecast chaotic series time is shown in Figure 1.