A hybrid model for the time series of complex structure (HMTS) was proposed. It is based on the combination of function expansions in a wavelet series with ARIMA models. HMTS has regular and anomalous components. The time series components, obtained after expansion, have a simpler structure that makes it possible to identify the ARIMA model if the components are stationary. This allows us to obtain a more accurate ARIMA model for a time series of complicated structure and to extend the area for application. To identify the HMTS anomalous component, threshold functions are applied. This paper describes a technique to identify HMTS and proposes operations to detect anomalies. With the example of an ionospheric parameter time series, we show the HMTS efficiency, describe the results and their application in detecting ionospheric anomalies. The HMTS was compared with the nonlinear autoregression neural network NARX, which confirmed HMTS efficiency.

In this study, a new multicomponent model (MCM) to determine the time variation of ionospheric parameters is suggested. The model was based on the combination of wavelets with autoregressive-integrated moving average model classes and allowed the study of the seasonal and diurnal variations of ionospheric parameters and the determination of anomalies occurring during ionospheric disturbances. To investigate in detail anomalous changes in the ionosphere, new computational solutions to detect anomalies of different scales and estimate their parameters (e.g., time of occurrence, duration, scale, and intensity) were developed based on a continuous wavelet transform. The MCM construction for different seasons and periods of solar activity was described using ionosphere critical frequency f o F2 data from Kamchatka (Paratunka Station, 52° 58′ N, 158° 15′ E, Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS). A comparison of the MCM with the empiric International Reference Ionosphere (IRI) model and the moving median method for the analyzed region showed that the suggested method was promising for future research, since it had the advantage of providing quantitative estimates for the occurrence time, duration, and intensity of the anomalies, characterizing the ionospheric state and disturbance degree with a higher accuracy. Geomagnetic storms from 17 March and 2 October 2013 were analyzed using the suggested method, and it was shown that the ionospheric disturbances were at maximum during the strongest geomagnetic disturbances. An increase in the electron concentration in comparison with the background level, under calm or weakly disturbed geomagnetic field conditions, was identified before the analyzed magnetic storms.

Geomagnetic data analysis is an important basis for the investigation of the processes in the near-Earth space, Earth magnetosphere, and ionosphere. The negative impact of geomagnetic anomalies on modern technical objects and human health determine the applied significance of the investigation and requires the creation of effective methods for timely detection of the anomalies. Priory complicated structure of geomagnetic data makes their formalization and analysis difficult. This paper proposes a wavelet model for geomagnetic field variations. It describes characteristic changes and anomalies of different amplitude and duration. Numerical realization of the model provides the possibility to apply it in online analysis. We describe the process of model identification and show its efficiency in the detection of sudden, short-period geomagnetic anomalies occurring before and during magnetic storms. Raw second data of the Paratunka and Magadan observatories and post-processed minute data were used in the paper. The question of noise effect on the proposed model results was under consideration.

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.

This work is directed at creation of methods of study of the processes in the ionospheric---magnetospheric system during increased solar and geomagnetic activity. Method of modeling and analysis of the parameters of the ionosphere, which allows prediction of the data and identification of the anomalies during the ionospheric disturbances, are given. Computational solutions for determination and estimation of the geomagnetic disturbances are described. Method of determination of the anomalous changes in the time course of cosmic rays, which allows qualitative estimations of the moments of their origination, duration, and intensity, is suggested.On the basis of the methods elaborated, the data on the periods of strong and moderate magnetic storms are complexly analyzed. Sharp oscillations in the electron density of the ionosphere with positive and negative phases, which originate in the regions analyzed during an increase in geomagnetic activity, are distinguished. Positive phases of the ionospheric disturbances from several hours to one and a half days long were formed before the beginning of the magnetic storms. At the moments of the increase in the electron concentration, a local increase is observed in the level of cosmic rays (several hours before the magnetic storms) that supported the solar nature of these effects. During the strongest geomagnetic disturbances, the electron concentration in the ionosphere decreased significantly and led to prolonged negative phases of ionospheric storms, which coincided with the decrease in the level of cosmic rays (a Forbush decrease).