Finite Iterative Forecasting Model Based on Fractional Generalized Pareto Motion
Résumé
In this paper, an efficient prediction model based on the fractional generalized Pareto motion (fGPm) with Long-Range Dependent (LRD) and infinite variance characteristics is proposed. Firstly, we discuss the meaning of each parameter of the generalized Pareto distribution (GPD), and the LRD characteristics of the generalized Pareto motion are analyzed by taking into account the heavy-tailed characteristics of its distribution. Then, the mathematical relationship H=1⁄α between the self-similar parameter H and the tail parameter α is obtained. Also, the generalized Pareto increment distribution is obtained using statistical methods, which offers the subsequent derivation of the iterative forecasting model based on the increment form. Secondly, the tail parameter α is introduced to generalize the integral expression of the fractional Brownian motion, and the integral expression of fGPm is obtained. Then, by discretizing the integral expression of fGPm, the statistical characteristics of infinite variance is shown. In addition, in order to study the LRD prediction characteristic of fGPm, LRD and self-similarity analysis are performed on fGPm, and the LRD prediction conditions H>1⁄α is obtained. Compared to the fractional Brownian motion describing LRD by a self-similar parameter H, fGPm introduces the tail parameter α, which increases the flexibility of the LRD description. However, the two parameters are not independent, because of the LRD condition H>1⁄α. An iterative prediction model is obtained from the Langevin-type stochastic differential equation driven by fGPm. The prediction model inherits the LRD condition H>1⁄α of fGPm and the time series, simulated by the Monte Carlo method, shows the superiority of the prediction model to predict data with high jumps. Finally, this paper uses power load data in two different situations (weekdays and weekends), used to verify the validity and general applicability of the forecasting model, which is compared with the fractional Brown prediction model, highlighting the “high jump data prediction advantage” of the fGPm prediction model.