Model Based Method for Determining the Minimum Embedding Dimension from Solar Activity Chaotic Time Series

Predicting future behavior of chaotic time series system is a challenging area in the literature of nonlinear systems. The prediction's accuracy of chaotic time series is extremely dependent on the model and the learning algorithm. On the other hand the cyclic solar activity as one of the natural chaotic systems has significant effects on earth, climate, satellites and space missions. Several methods have been introduced for prediction of solar activity indices especially the sunspot number, which is a common measure of solar activity. In this paper, the problem of embedding dimension estimation for solar activity chaotic time series based on polynomial models is considered. The optimality of embedding dimension has an important role in computational efforts, Lyapunov exponents' analysis and efficiency of prediction. The method of this paper is based on the fact that the reconstructed dynamics of an attractor should be a smooth map, i.e. with no self intersection in the reconstructed attractor. To check this property, a local general polynomial autoregressive model is fitted to the given data and a canonical state space realization is considered. Then, the normalized one-step forward prediction error for different orders and various degrees of nonlinearity in polynomials is evaluated. Besides the estimation of the embedding dimension, a predictive model is obtained which can be used for prediction and estimation of the Lyapunov exponents. This algorithm is applied to indicate the minimum embedding dimension of sunspot numbers (SSN), Disturbance Storm Time or Dst. and Proton Flux indices are some of the most important among solar activity indices and results depict the power of the proposed method in embedding dimension estimation.