Combining ARFIMA models and fuzzy time series for the forecast of long memory time series

Long memory time series are stationary processes in which there is a statistical long range dependency between the current value and values in different times of the series. Therefore, in this class of series, there is a slow decay of the autocorrelation function as the time difference increases. Many practical forecasting problems fall in this class, for instance, in financial time series, hydrology and earth sciences applications. This research introduces a hybrid method combining Auto Regressive Fractional Integrated Moving Average (ARFIMA) models and Fuzzy Time Series (FTS) for the forecast of long memory (long-range) time series. The proposed method is developed as one algorithm consisting of two phases. The first phase is related to the autoregressive part of the model, while the second phase is related to the Moving Average part. Based on these ideas, the combined ARFIMA and FTS model is introduced. For the parameter estimation of the model, Particle Swarm Optimization (PSO) method is selected, based on its performance on similar optimization problems. In order to illustrate the benefit and potential of the proposed ARFIMA-FTS method, it has been applied to the two stock index databases, namely Taiwan Capitalization Weighted Stock Index (TAIEX)and Dow Jones Industrial Average (DJIA), together with exchange rate data of nine main currencies versus USD. Based on the reported results, it is possible to conclude the superiority of the proposed hybrid method, compared with classical ARFIMA models and other methods in the literature.