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<p>The number of sunspots is a key indicator of solar magnetic activity and strongly influences space weather, affecting technological systems and Earth’s environment. This study develops a long-memory statistical framework based on the Auto-Regressive Fractionally Integrated Moving Average (ARFIMA) model to forecast monthly mean sunspot numbers (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\hbox {SN}_m\)</EquationSource> </InlineEquation>) for Solar Cycles 25 and 26 using historical data from January 1749 to October 2025. The model parameters are selected using the Bayesian Information Criterion (BIC), and the fractional integration parameter <i>d</i> is estimated via maximum likelihood (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\hat{d} \approx 0.27599\)</EquationSource> </InlineEquation>), indicating significant long-memory behavior in the series. The selected ARFIMA (3,d,2) model captures the persistent dynamics of solar activity and provides accurate in-sample fitting, with a high correlation coefficient (0.989) between observed and fitted values. Forecast results predict a maximum <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\hbox {SN}_m\)</EquationSource> </InlineEquation> of 224.7 for Solar Cycle 25 (observed peak: approximately 216 in August 2024) and 179.3 for Solar Cycle 26 around March 2035, suggesting a slightly weaker upcoming cycle. Model performance is evaluated using standard accuracy measures, including RMSE, MAE, and relative error metrics, computed against observed data within a validation framework. The proposed model achieves an RMSE of 3.37 and a SMAPE of 9.25%, indicating improved forecasting accuracy.</p>

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Long-memory modeling and forecasting of monthly mean sunspot numbers for cycles 25 & 26 using ARFIMA model

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月22日

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  • H. I. Abdel Rahman,
  • Doaa Eid

摘要

The number of sunspots is a key indicator of solar magnetic activity and strongly influences space weather, affecting technological systems and Earth’s environment. This study develops a long-memory statistical framework based on the Auto-Regressive Fractionally Integrated Moving Average (ARFIMA) model to forecast monthly mean sunspot numbers ( \(\hbox {SN}_m\) ) for Solar Cycles 25 and 26 using historical data from January 1749 to October 2025. The model parameters are selected using the Bayesian Information Criterion (BIC), and the fractional integration parameter d is estimated via maximum likelihood ( \(\hat{d} \approx 0.27599\) ), indicating significant long-memory behavior in the series. The selected ARFIMA (3,d,2) model captures the persistent dynamics of solar activity and provides accurate in-sample fitting, with a high correlation coefficient (0.989) between observed and fitted values. Forecast results predict a maximum \(\hbox {SN}_m\) of 224.7 for Solar Cycle 25 (observed peak: approximately 216 in August 2024) and 179.3 for Solar Cycle 26 around March 2035, suggesting a slightly weaker upcoming cycle. Model performance is evaluated using standard accuracy measures, including RMSE, MAE, and relative error metrics, computed against observed data within a validation framework. The proposed model achieves an RMSE of 3.37 and a SMAPE of 9.25%, indicating improved forecasting accuracy.