Use of Multivariate Relevance Vector Machines in forecasting multiple geomagnetic indices

The forecasting ability of Multivariate Relevance Vector Machines (MVRVM), used previously to generate forecasts for the Dst index, is extended to forecast the Dst, AL, and PC indices during the years 1975-2007. Such learning machines are used in forecasting because of their robustness, efficiency, and sparseness. The MVRVM model was trained on solar wind and geomagnetic activity data sampled every hour with activity periods of various intensities, durations, and features. It was found that during the training phase, for a given error threshold, 14.60% of the training data was needed to explain the features of the data. The trained model was then tested on 177 different storm intervals, at various levels of geomagnetic activity, to generate simultaneous forecasts of the three indices at a lead time of one hour (1-h). The focus of the modeling was to assess the forecasts during main storm (MS) time periods when the indices show enhanced activity above quiet time values. The forecasts obtained by the MVRVM model reported in this paper returned a MS time average prediction efficiency, (PE) over bar of 82.42%, 84.40%, and 76.00% and (RMSE) over bar of 13.70 nT, 97.00 nT, and 0.77 mV/m, for the Dst, AL, and PC indices, respectively. The qualitative numbers indicated that the model underestimated the peak amplitude of the indices during the geomagnetic activity, but the peaks were forecasted on time by the model, on average. The forecasting results indicate a robust model generalization and the MVRVM's ability to learn the input-output relationship through a sparse model framework. A qualitative comparison with the previous univariate RVM forecast of Dst indicates that the model goodness of fit numbers improved in the present study.