Research On Adaptive Prediction Algorithm For Chaotic Time Series

New adaptive predict algorithms are studied in this paper, which is consisted of the following contents. (1) Many chaotic signals show non-Gaussian distribution, which is employed to separate chaotic signals and Gaussian noise using the fast independent component analysis method. (2)According to the theory of independent component analysis, negative entropy can be approximated using non-polynomial functions, which are included to an adaptive prediction algorithm for enhanced performances. (3) To simulate the dynamic property of a chaotic system, a nonlinear feedback is introduced to the adaptive algorithms for an improved long-term predictability. (4) The approximation expression about error accumulation of a long-term prediction is derived. Basing on the formula, we improve the non-polynomial adaptive algorithm to enlarge the maximum attempting time.New ideas of the paper are listed as following:1. Non-Gaussianity of chaotic signals is used to reduce Gaussion noise in chaotic signals:(1) Non-Gaussianity is the native property of chaotic signals. In communication systems, the most important noise is Gaussian-distributed, and its influence must be considered in chaotic communications. Because of the difference in higher-order cumulants between chaotic signals and Gaussian noise, independent component analysis method is used to separate chaotic signals and Gaussian noise.(2) Basing on the principle of independent component analysis, non-Gaussinity can be characterized not only by higher-order cumulants, but also by negative entropy. From this point, these prediction algorithms that include polynomial functions and Volterra series can be thought as using higher-order cumulants of chaotic signals directly. But higher-order cumulants are of less robust and sensitive to noise. Using negative entropy is another way to design new prediction functions. Since non-polynomial functions can be used to approximation negative entropy, they constitute alternative basis function set that are of robust and an enhanced performance.2. Introducing dynamic character using feedback loop In modeling and predicting chaotic time series, static functions are often used to fit the waves. A drawback of the methods is they can't model the dynamic actions, and may show limited performance not only on the prediction accuracy, but also on the maximum prediction time. Acording to the definition of a dynamic system, the nonlinear feedback is the source of the chaos. By using a nonlinear feedback to a prediction algorithm can introduce some dynamic actions. In this paper, a sigmoid function is used as non-linear feedback to an improved method.3. The approximation expression about error accumulation in a long-term prediction is derived to enlarge the maximum attempting time.To predict the long term trajectory of a chaotic system is very difficulty, for a chaotic system is very sensitive to the initial congditions, and long-term prediction error will become larger and larger with time to cause the predicted trajectory escape the actual one in an exponent speed. Many presented works ascribe the limitation of the long term prediction to the sensitivity property of a chaotic map, which amplify the prediction error and make the algorithm fail to capture the actual trajectory. But there is not a detail study on the affection of the prediction error to the long-term predictability. In this paper, the error accumulation can be obtained analytically by approximating. By analyzing this formula, we find that the factors that can affect the long-term predictability include the model parameters, prediction errors and the derivates of the used basis functions. The ideal case to enlarge the maximum attempting time is to find a model coefficient vector that is orthogonal with the error accumulation vector. But the ideal case is very difficult to arrive at, since prediction errors are random. Then we present a moderated method to reduce the speed of error accumulation, which means using these basis functions with smaller derivative functions and a fast attenuation where out of the time series range to construct an adaptive method.