Analog-Based Forecasting of Solar Cycles Using Dynamic Time Warping and Residual Bootstrapping: Real-Time Application to Cycle 25

Abstract Understanding and predicting the shape and intensity of solar cycles is important for anticipating their effects on Earth. In this study, we present an interpretable analog-based forecasting framework. Using Dynamic Time Warping (DTW), we identify past solar cycles with similar morphology and blend them—after alignment in time and amplitude—to capture key features such as rise–decay asymmetry, cycle duration, and peak variation.To assess forecast uncertainty, we implement a phase-aware residual bootstrapping method based on historical prediction errors. This allows us to construct realistic, nonparametric confidence intervals. The framework is validated on Solar Cycles 20 to 24, showing strong performance in both peak amplitude and timing. All validation and figures are computed on a frozen SILSO monthly-smoothed dataset through 2024-12-01 (1749-07–2024-12; $N=3306$) under a fixed random seed to avoid look-ahead and ensure exact reproducibility. In this frozen setup, the $k=2$ configuration minimizes average RMSE across SC20–24 (Avg RMSE $=15.85$, R² ≈ 0.92, r ≈ 0.97).We then apply the method to forecast Solar Cycles 25 and 26 using only pre-cycle data, ensuring all projections remain within the bounds of historical variability. Baseline forecasts from the frozen snapshot yield SC25 peak $193.86$ at month $47$ (length $133$ months) and SC26 peak $151.66$ at month $42$ (length $129$ months). A real-time extension of the model updates analog selection as new SC25 observations become available. While this enhances short-term accuracy, it may introduce trade-offs in morphological coherence, particularly in capturing long-term shape features due to incomplete phase coverage. In the as-of-2024-12-01 real-time run, DTW selects SC17 and SC13 as analogs (weights $0.506$, $0.494$); on the observed SC25 window we obtain MAE $=11.15$, RMSE $=13.90$, R²=0.93, $r=0.98$, with under-coverage of nominal 98% intervals (~47% zero-mean, ~56% raw).Overall, the core model yields consistent and interpretable long-range forecasts, while the real-time variant offers greater adaptability at the expense of structural consistency.