Approximation and Analysis of Natural Data Based on NARX Neural Networks Involving Wavelet Filtering

Recurrent neural network (RNN) models continue the theory of the autoregression integrated moving average (ARIMA) model class. In this paper, we consider the architecture of the RNN with embedded memory—«Process of Nonlinear Autoregressive Exogenous Model» (NARX). Though it is known that NN is a universal approximator, certain difficulties and restrictions in different NN applications are still topical and call for new approaches and methods. In particular, it is difficult for an NN to model noisy and significantly nonstationary time series. The paper suggests optimizing the modeling process for a complicated-structure time series by NARX networks involving wavelet filtering. The developed procedure of wavelet filtering includes the application of the construction of wavelet packets and stochastic thresholds. A method to estimate the thresholds to obtain a solution with a defined confidence level is also developed. We introduce the algorithm of wavelet filtering. It is shown that the proposed wavelet filtering makes it possible to obtain a more accurate NARX model and improves the efficiency of the forecasting process for a natural time series of a complicated structure. Compared to ARIMA, the suggested method allows us to obtain a more adequate model of a nonstationary time series of complex nonlinear structure. The advantage of the method, compared to RNN, is the higher quality of data approximation for smaller computation efforts at the stages of network training and functioning that provides the solution to the problem of long-term dependencies. Moreover, we develop a scheme of approach realization for the task of data modeling based on NARX and anomaly detection. The necessity of anomaly detection arises in different application areas. Anomaly detection is of particular relevance in the problems of geophysical monitoring and requires method accuracy and efficiency. The effectiveness of the suggested method is illustrated in the example of processing of ionospheric parameter time series. We also present the results for the problem of ionospheric anomaly detection. The approach can be applied in space weather forecasting to predict ionospheric parameters and to detect ionospheric anomalies.