A Hybrid Minkowski-Log-Cosh Loss Function for Robust LSTM-Based Time Series Forecasting

Designing robust loss functions is essential for improving deep learning performance on noisy and sparse time series. We propose the Minkowski–Log–Cosh (MLC) loss, a hybrid formulation that integrates the smooth curvature of Log-Cosh with a tunable exponent p to control robustness against outliers. We evaluate MLC on monthly malaria case forecasting in Ghana’s Oti Region using an LSTM-based model across both clean and outlier-injected datasets. On clean data, we assess performance using MAE, MAPE, and RMSE, respectively. MLC with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> = 2.25 achieved the lowest MAE (2515.38), MAPE (18.23%), and RMSE (3671.74), respectively, outperforming MSE (2892.85, 20.96%, 3677.12), MAE (3120.22, 24.69%, 3812.34), and Log-Cosh (2930.70, 20.53%, 3701.01), respectively, while also recording the fastest training time (502.65s). For noisy data, robustness was evaluated using MedAE, SMAPE, MAE, and MAPE, respectively. MLC with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> = 1.5 achieved the lowest MedAE (2192.06), SMAPE (24.62%), MAE (3676.36), and MAPE (22.73%), respectively. It showed substantial improvements over MSE (2492.13, 30.58%, 4089.78, 29.94%), MAE (2416.78, 29.94%, 3838.88, 26.48%), and Log-Cosh (2370.88, 29.19%, 4027.66, 27.41%), respectively, and again trained fastest (403.36s). These results demonstrate that MLC’s performance varies with the exponent <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i>: values around <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> = 2.25 improve forecasting precision on clean data, while values near <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> = 1.5 enhance robustness under noise. MLC thus provides a tunable, data-adaptive loss function for noise-resilient time series forecasting. Code is available at GitHub.