Adaptive stochastic modeling of nonlinear time series: ARFIMA meets improved fractional Ornstein-Uhlenbeck

Abstract In this paper, a novel hybrid time series forecasting approach under the category of additive mathematical modeling of time series is presented. The algorithm leverages ARFIMA, whose residuals are subsequently modeled by a newly developed improved fractional Ornstein-Uhlenbeck (IfOU) stochastic differential equation. The fractional differencing in ARFIMA modeling captures most of the correlation dependencies in non-stationary data, considering both long and short memory. However, nonlinearities and unknown multi-physics phenomena governing the measurement process lead to ARFIMA nonconformity in the time series. To address this, IfOU learns and models the residual data and compensates for the ARFIMA forecasting error. The OU process is a continuous Gaussian Markov process with constant mean and exponentially decaying autocovariance, capturing short-term mean-reverting fluctuations. To model both short and long-range dependence in volatility, we adopt the fOU extension in our adaptive IfOU algorithm, which augments the classical OU dynamics with fractional noise to capture persistent correlations over longer time scales. IfOU improves on fOU by introducing an affine transformation of fOU, along with adaptively estimating Hurst parameters in overlapped segments of residuals. The affine transformation reduces the measured residual modeling noise and the ARFIMA global modeling noise. The proposed hybrid algorithm is evaluated and compared with existing state-of-the-art methods on three benchmark environmental datasets: Sunspot, Lake Erie level, and CO 2 . Experimental results on out-of-sample data demonstrate that IfOU, in conjunction with ARFIMA, outperforms innovative algorithms for time series prediction in terms of various statistical tests.