Today, it can be said that in every field in which TIMEly information is needed, we can use the applications of TIME-SERIES prediction. In this paper, among so many CHAOTIC systems, the Mackey-Glass and Loranz are chosen. To predict them, Multi-Layer Perceptron Neural Network (MLP NN) trained by a variety of heuristic methods are utilized such as genetic, particle swarm, ant colony, evolutionary strategy algorithms, and population-based incremental learning. Also, in addition to expressed methods, we propose two algorithms of Bio-geography-Based Optimization (BBO) and fuzzy system to predict these CHAOTIC systems. Simulation results show that if the MLP NN is trained based on the proposed meta-heuristic algorithm of BBO, training and testing accuracy will be improved by 28.5% and 51%, respectively. Also, if the presented fuzzy system is utilized to predict the CHAOTIC systems, it outperforms approximately by 98.5% and 91.3% in training and testing accuracy, respectively.

‎Artificial neural networks with amazing properties‎, ‎such as universal approximation‎, ‎have been utilized to approximate the nonlinear processes in many fields of applied sciences‎. ‎This work proposes the rough-neural networks (R-NNs) for the one-step ahead prediction of CHAOTIC TIME SERIES‎. ‎We adjust the parameters of R-NNs using a continuous-TIME Lyapunov-based training algorithm‎, ‎and prove its stability using the continuous form of Lyapunov stability theory‎. ‎Then‎, ‎we utilize the R-NNs to predict the well-known Mackey-Glass TIME SERIES‎, ‎and Henon map‎, ‎and compare the simulation results with some well-known neural models‎.

Since the pioneering work of Zadeh, fuzzy set theory has been applied to a myriad of areas. Song and Chissom introduced the concept of fuzzy TIME SERIES and applied some methods to the enrolments of the University of Alabama. Thereafter we apply fuzzy techniques for system identification and apply statistical techniques to modelling system. An automatic methodology framework that combines fuzzy techniques and statistical techniques for nonparametric residual variance estimation is proposed. The methodology framework is creating regression model by using fuzzy techniques. Identification is performed through learning from examples method introduced by Wang and Mendel algorithm. Delta test residual noise estimation is used in order to select the best subset of inputs as well as the number of linguistic labels for the inputs. An experimental result for CHAOTIC TIME SERIES prediction is compared with statistical model and shows the advantages of the proposed methodology in terms of approximation accuracy, generalization capability and linguistic interpretability.

A deep learning method is developed for CHAOTIC TIME SERIES classification. We investigate the CHAOTIC state of a dynamical system, based on the output of the system. One of the main obstacles in TIME SERIES classification is mapping a high-dimensional vector into a scalar value. To reduce the dimensions, it is common to use an average pooling layer block after feature extraction block. This blind process results in models with high computational complexity and potent to overfitting. One alternative is to extract the features manually, then apply shallow learning models to classify the TIME SERIES. In fact, since complexity lies between the chaos and order, it is a sound idea to refer to complex systems characteristics to explore the CHAOTIC region entrance. Therefore, CHAOTIC state of a dynamical system can be recognized solely based on these characteristics. Inspired by this concept, we conclude that there is a feature space in which the output vector can be sparsified. Thus, we propose a deep learning method which the feature space dimensions successively are reduced in the feature extraction process. Specifically, we employ a fully convolutional network and add on two maximum pooling layers to the relevant feature extraction block. To validate the proposed model, the Lorenz system is employed which exhibits CHAOTIC/non-CHAOTIC states. We generate a labeled dataset containing 10000 samples each with 20000 features of the output of Lorenz system. The proposed model achieves 99.45 percent accuracy over 2000 unseen samples, higher than all the other competitor methods.

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.

This paper presents a fuzzy approach to the prediction of highly nonlinear TIME SERIES.The optimized Mamdani-type fuzzy system denoted SQP-FLC is applied for the input-output modeling of measured data. In order to tune fuzzy membership functions, a sequential quadratic programming (SQP) method is employed. The proposed method is evaluated and validated on a highly complex TIME SERIES, daily gold price data. The TIME SERIES is primarily investigated for its CHAOTIC properties.Correlation dimension and autocorrelation function (ACF) for the TIME SERIES are discussed. Accordingly, TIME delay and embedding dimension are computed. Month selection in each stage is based on computed correlation coefficients. Thus, for the proposed fuzzy predictor, 3, 5, and 7 dynamics are selected and the TIME SERIES are verified. The simulation results for one-step-ahead prediction of daily gold price in 2010, compared with methods of ANFIS and GA-FLC, demonstrate comparably better performance of the proposed SQP-FLC until the higher significant dynamics of the CHAOTIC trend is taken into account.