Fractional Levy Stable Motion (FLSM) Enhanced by Long-Short Term Memory (LSTM): A Generalized Framework for Forecasting Non-Stationary Time Series with Limited Data

Abstract Time series are often characterised by their non-stationarity and volatility, which poses a significant challenge for conventional predictive methods. These series represent data collected over an extended period across various fields like economics, health and energy, necessitating accurate forecasts to inform strategic decisions. The dynamic nature of these series, combined with frequently complex distributions, underscores the need to adopt advanced methodological approaches to overcome the inherent limitations of traditional statistical models in forecasting. The present work proposes a hybrid model that combines Fractional Levy Stable Motion (FLSM) and Long Short-Term Memory (LSTM) to optimise prediction accuracy. The FLSM effectively captures the time series's complex stochastic dynamics and long-term dependencies. In contrast, the LSTM model excels in identifying nonlinear relationships and managing unmodeled residuals. The adopted methodology revolves around a multi-step strategy: initially, the FLSM model is employed to model the series's stochastic dynamics, followed by applying the LSTM model to the residuals to refine forecasts by detecting underlying nonlinear relationships. The model is applied to data related to malaria collected in the Adamawa Region of Cameroon. The evaluation utilises metrics like Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and the coefficient of determination (R²) to estimate the model's performance. The results indicate that the FLSM-LSTM hybrid model significantly outperforms traditional methods like the ARIMA and isolated FLSM and LSTM models, achieving an RMSE of 432.18 and a high R² of 0.92. These performances highlight the relevance of hybridisation for modelling complex time series. This study opens promising prospects for future applications, with recommendations to explore other datasets and optimise the parameters of the stochastic model.