The hybrid model of autoregressive integrated moving average and fuzzy time series Markov chain on long-memory data

Abstract

IntroductionThe price of crude oil as an essential commodity in the world economy shows a pattern and identifies the component factors that influence it in the short and long term. The long pattern of the price movement of crude oil is identified by a fractionally time series model where the accuracy can still be improved by making a hybrid residual model using a fuzzy time series approach.MethodsTime series data containing long-memory elements can be modified into a stationary model through the autoregressive fractional integrated moving average (ARFIMA). This fractional model can provide better accuracy on long-memory data than the classic autoregressive integrated moving average (ARIMA) model. The long-memory data are indicated by a high level of fluctuation and the autocorrelation value between lags that decreases slowly. However, a more accurate model is proposed as a hybridization time series model with fuzzy time series Markov chain (FTSMC).ResultsThe time series data collected from the monthly period of West Texas Intermediate (WTI) oil price as the standard for world oil prices for the 2003–2021 time period. The data of WTI oil price has a long-memory data pattern to be modeled fractionally, and subsequently their hybrids. The times series model of crude oil price is obtained as the new target model of hybrid ARIMA and ARFIMA with FTSMC, denoted as ARIMA-FTSMC and ARFIMA-FTSMC, respectively.DiscussionThe accuracy model measured by MAE, RMSE, and MAPE shows that the hybrid model of ARIMA-FTSMC has better performance than ARIMA and ARFIMA, but the hybrid model of ARFIMA-FTSMC provides the best accuracy compared to all models. The superiority of the hybrid time series model of ARFIMA-FTSMC on long-memory data provides an opportunity for the hybrid model as the best and more precise forecasting method.

Hybridization of long short-term memory neural network in fractional time series modeling of inflation

“…For example, assuming data X t is non-stationary regarding variance, it can be transformed with the formula , where λ is the transformation parameter. Stationarity is achieved when λ = 1 yields a rounded value process (Devianto et al, 2022 ).…”

“…Step 4 . Estimating the differentiating parameters d using the Geweke and Porter-Hudak model, denoted as with the following formula (Devianto et al, 2022 ):…”

“…Since the data are not stationary, data transformation was performed to obtain a rounded value (λ) using Box-Cox transformation T. For example, assuming data X t is non-stationary regarding variance, it can be transformed with the formula T(X t ) = (X λ−1 t )/λ, where λ is the transformation parameter. Stationarity is achieved when λ = 1 yields a rounded value process (Devianto et al, 2022).…”

“…In 1980, Granger and Joyeux introduced ARFIMA model, an advancement of ARIMA model capable of predicting both short and long-memorypatterned time series data. This flexibility extends to the ability of ARFIMA to predict data exhibiting short-term as well as longterm memory patterns (Devianto et al, 2022). Subsequently, in 1981, Hosking examined the properties of long memory patterns in both stationary and non-stationary ARFIMA models.…”