Chaotic time series part II: System identification and prediction

1996

In this work methods for performing time series prediction on complex real world time series are examined. In particular series exhibiting non-linear or chaotic behaviour are selected for analysis. A range of methodologies based on Takens’ embedding theorem are considered and compared with more conventional methods. A novel combination of methods for determining the optimal embedding parameters are employed and tried out with multivariate financial time series data and with a complex series derived from an experiment in biotechnology. The results show that this combination of techniques provide accurate results while improving dramatically the time required to produce predictions and analyses, and eliminating a range of parameters that had hitherto been fixed empirically. The architecture and methodology of the prediction software developed is described along with design decisions and their justification. Sensitivity analyses are employed to justify the use of this combination of methods, and comparisons are made with more conventional predictive techniques and trivial predictors showing the superiority of the results generated by the work detailed in this thesis.

Physica D: Nonlinear Phenomena, 1998

Local linear prediction, based on the ordinary least squares (OLS) approach, is one of several methods that have been applied to prediction of chaotic time series. Apart from potential linearization errors, a drawback of this approach is the high variance of the predictions under certain conditions. Here, a different set of so-called linear regularization techniques, originally derived to solve ill-posed regression problems, are compared to OLS for chaotic time series corrupted by additive measurement noise. These methods reduce the variance compared to OLS, but introduce more bias. A main tool of analysis is the singular value decomposition (SVD), and a key to successful regularization is to damp the higher order SVD components. Several of the methods achieve improved prediction compared to OLS for synthetic noise-corrupted data from well-known chaotic systems. Similar results were found for real-world data from the R-R intervals of ECG signals. Good results are also obtained for real sunspot data, compared to published predictions using nonlinear techniques.

2001

We address two aspects in chaotic time series analysis, namely the definition of embedding parameters and the largest Lyapunov exponent. It is necessary for performing state space reconstruction and identification of chaotic behavior. For the first aspect, we examine the mutual information for determination of time delay and false nearest neighbors method for choosing appropriate embedding dimension. For the second aspect we suggest neural network approach, which is characterized by simplicity and accuracy.

WSEAS Transactions on Computer Research, 2008

The prediction of chaotic time series with neural networks is a traditional practical problem of dynamic systems. This paper is not intended for proposing a new model or a new methodology, but to study carefully and thoroughly several aspects of a model on which there are no enough communicated experimental data, as well as to derive conclusions that would be of interest. The recurrent neural networks (RNN) models are not only important for the forecasting of time series but also generally for the control of the dynamical system. A RNN with a sufficiently large number of neurons is a nonlinear autoregressive and moving average (NARMA) model, with "moving average" referring to the inputs. The prediction can be assimilated to identification of dynamic process. An architectural approach of RNN with embedded memory, "Nonlinear Autoregressive model process with eXogenous input" (NARX), showing promising qualities for dynamic system applications, is analyzed in this paper. The performances of the NARX model are verified for several types of chaotic or fractal time series applied as input for neural network, in relation with the number of neurons, the training algorithms and the dimensions of his embedded memory. In addition, this work has attempted to identify a way to use the classic statistical methodologies (R/S Rescaled Range analysis and Hurst exponent) to obtain new methods of improving the process efficiency of the prediction chaotic time series with NARX.

Expert Systems with Applications, 2011

In this paper, two CI techniques, namely, single multiplicative neuron (SMN) model and adaptive neurofuzzy inference system (ANFIS), have been proposed for time series prediction. A variation of particle swarm optimization (PSO) with cooperative sub-swarms, called COPSO, has been used for estimation of SMN model parameters leading to COPSO-SMN. The prediction effectiveness of COPSO-SMN and ANFIS has been illustrated using commonly used nonlinear, non-stationary and chaotic benchmark datasets of Mackey-Glass, Box-Jenkins and biomedical signals of electroencephalogram (EEG). The training and test performances of both hybrid CI techniques have been compared for these datasets.

Neural Processing …, 2011

The accuracy of a model to forecast a time series diminishes as the prediction horizon increases, in particular when the prediction is carried out recursively. Such decay is faster when the model is built using data generated by highly dynamic or chaotic systems. This paper presents a topology and training scheme for a novel artificial neural network, named "Hybrid-connected Complex Neural Network" (HCNN), which is able to capture the dynamics embedded in chaotic time series and to predict long horizons of such series. HCNN is composed of small recurrent neural networks, inserted in a structure made of feed-forward and recurrent connections and trained in several stages using the algorithm back-propagation through time (BPTT). In experiments using a Mackey-Glass time series and an electrocardiogram (ECG) as training signals, HCNN was able to output stable chaotic signals, oscillating for periods as long as four times the size of the training signals. The largest local Lyapunov Exponent (LE) of predicted signals was positive (an evidence of chaos), and similar to the LE calculated over the training signals. The magnitudes of peaks in the ECG signal were not accurately predicted, but the predicted signal was similar to the ECG in the rest of its structure.

Chaos, Solitons & Fractals, 2008

Based on the genetic algorithm (GA) and steepest descent method (SDM), this paper proposes a hybrid algorithm for the learning of neural networks to identify chaotic systems. The systems in question are the logistic map and the Duffing equation. Different identification schemes are used to identify both the logistic map and the Duffing equation, respectively. Simulation results show that our hybrid algorithm is more efficient than that of other methods.

Journal of Computational Methods in Sciences and Engineering, 2016

In this paper, we used four types of artificial neural network (ANN) to predict the behavior of chaotic time series. Each neural network that used in this paper acts as global model to predict the future behavior of time series. Prediction process is based on embedding theorem and time delay determined by this theorem. This ANN applied to the time series that generated by Mackey-glass equation that has a chaotic behavior. At the end, all neural networks are used to solve this problem and their results are compared and analyzed.

2008

This paper examines how efficient neural networks are relative to linear and polynomial approximations to forecast a time series that is generated by the chaotic Mackey-Glass differential delay equation. The forecasting horizon is one step ahead. A series of regressions with polynomial approximators and a simple neural network with two neurons is taking place and compare the multiple correlation coefficients. The neural network, a very simple neural network, is superior to the polynomial expansions, and delivers a virtually perfect forecasting. Finally, the neural network is much more precise, relative to the other methods, across a wide set of realizations.

Technological Forecasting and Social Change, 1991

It is well known that a nonlinear recursive equation can produce a chaotic sequence at certain values of the parameter. Furthermore, in the chaotic regime, extremely small changes in the initial value or in the value of the parameter produce very large changes in the sequence. It is surmising therefore that a short segment of a chaotic sequence can be used to reconstruct large portions of the sequence and to forecast future values of the sequence over short ranges. Furthermore, while the accuracy of the forecasts is dependent on the precision of the data, the relationship is much less sensitive than might have been expected. This is demonstrated by fitting a two-parameter model to two different types of chaotic equations: one a polynomial and the other a trigonometric function. This leads to the expectation that under certain circumstances, it may be possible to forecast values in a chaotic series over limited ranges, even if initial data are somewhat degraded.