A Recurrent Fractal Forecasting Framework Integrating ARIMA and Neural Networks

Abstract

In this paper, we propose a new recurrent rational cubic fractal model that incorporates scale and
a single shape parameter to capture self-similar structures in the data effectively. The convergence analysis is presented to establish the theoretical foundation of the proposed model. To provide a statistical benchmark for forecasting performance, the autoregressive integrated moving average (ARIMA) model, a widely used method in time series analysis, is applied. In addition, we design an artificial neural network (ANN) framework based on recurrent iterated function systems (RIFs), where the approximation is represented by a rational function with a cubic numerator and a quadratic denominator. This hybrid design enhances the flexibility of the ANN in capturing complex nonlinear patterns. A numerical example shows how well the proposed methods work, and a comparison shows that the recurrent fractal and ANN-based methods are more accurate and efficient than ARIMA.