A Fourier-Based Dynamic Harmonic Regression Framework for Overcoming Computational Barriers in High-Frequency Seasonal ARIMA Modelling

This paper presents a systematic investigation into the computational limitations of Seasonal ARIMA (SARIMA) models when applied to high-frequency time series data and demonstrates Dynamic Harmonic Regression (DHR) with Fourier terms as a principled alternative. Through two controlled case studies - monthly surface temperature data from India (s=12) and daily minimum temperature data from Melbourne (s = 365) - we demonstrate that while SARIMA achieves a Mean Absolute Error (MAE) of 0.547° C for low-frequency seasonal data, its state-space estimation procedure fails to converge entirely for high-frequency daily cycles due to dimensionality scaling proportional to the seasonal period. We formalize this computational boundary and show that a DHR model employing four Fourier harmonics with ARIMA errors reduces the MAE from 3.07° C to 2.04° C (a 33.6% improvement) for daily data while maintaining statistical rigour through comprehensive residual diagnostics. The findings provide practitioners with an empirically grounded decision framework for method selection based on the seasonal period magnitude.