Probabilistic sunspot predictions with a gated recurrent units-based combined model guided by pinball loss

Sunspots play a crucial role in both weather forecasting and the monitoring of solar storms. In this work, we propose a novel combined model for sunspot prediction using improved gated recurrent units (GRU) guided by pinball loss for probabilistic forecasts. Specifically, we optimize the GRU parameters using the slime mould algorithm and employ a seasonal-trend decomposition procedure based on loess to tackle challenges related to sequence prediction, such as self-correlations and non-stationarity. To address prediction uncertainty, we replace the traditional $$ -norm loss with pinball loss. This modification extends the conventional GRU-based point forecasting to a probabilistic framework expressed as quantiles. We apply our proposed model to analyze a well-established historical sunspot dataset for both single- and multi-step ahead forecasting. The results demonstrate the effectiveness of our combined model in predicting sunspot values, surpassing the performance of other existing methods.