Modelling of time series of external-driven events with echo state network and its application to substorm activity analysis

Substorms are a magnetospheric phenomenon which causes high geomagnetic and auroral disturbances. It is widely accepted that substorm activity is controlled by solar wind conditions. It is, however, difficult to predict substorms deterministically because of the complex physical processes underlying substorm occurrences. We propose a framework for modelling time series of event occurrences controlled by external forcing. In this framework, occurrences of external-driven events are represented with a non-stationary Poisson process, and its intensity, which corresponds to the occurrence rates per unit time, is described with a simple machine learning model, the echo state network, which is fed with forcing variables. The echo state network is trained by maxmising the likelihood given the event time series data. 

We apply this approach for analysing time series of substorm onsets identified from Pi2 pulsations, which are irregular geomagnetic oscillations associated with substorm onsets. We train the echo state network to well describe the response of substorm activity to solar-wind conditions. We then examine the characteristics of the substorm activity by feeding synthetic solar-wind data into the echo state network. The results show what solar wind variables effectively contribute to the substorm occurrence. 

Our echo state network model is also useful for examining the statistical properties of the substorm occurrence rate. For example, we can evaluate what mainly controls the seasonal and UT variations of substorm activity. There are two explanations for the seasonal and UT variations. One explanation is that the seasonal and UT variations is controlled by the inner product between the solar-wind magnetic field and the Earth's dipole axis. The other is that the variations are due to the angle between the solar-wind flow and the Earth's dipole axis. Since these two effects are related with different input variables in our echo state network model, we can examine the contribution of each effect to the substorm occurrence frequency. The result shows that the seasonal and UT variations are mostly dependent on the angle between the solar-wind flow and the Earth's dipole axis.