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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 l 2 -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. [ABSTRACT FROM AUTHOR]

Time series modelling is gaining spectacular popularity in the prediction process of decision making, with applications including real-world management and engineering. However, for short time series, prediction has to face unavoidable limitation for modelling extremely complex systems. It has to apply inadequate and incomplete data from short time to predict unknown observations. With such limited data source, existing techniques, such as statistical modelling or machine learning methods, fail to predict short time series effectively. To address this problem, this paper provides a global framework for short time series modelling predictions, integrating the rolling mechanism, grey model, and meta-heuristic optimization algorithms. In addition, dragonfly algorithm and whale optimization algorithm are investigated and deployed to optimize the framework by enhancing its predicting accuracy with less computational costs. To verify the performance of the proposed framework, three industrial cases are introduced as simulation experiments in this paper. Experimental results confirm that the framework solves corresponding short time series modelling predictions with greater accuracy and speed than the standard GM(1,1) models. The source codes of this framework are available at: https://github.com/zhesencui/HybridRollingGreyFramework.git. [ABSTRACT FROM AUTHOR]