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Symmetry 2018, 10(8), 324;

A Fuzzy Set-Valued Autoregressive Moving Average Model and Its Applications

Department of Statistics, Guangzhou University, No. 230 Waihuanxi Road, Higher Education Mega Center, Guangzhou 510006, China
School of Applied Mathematics, Guangdong University of Technology, No. 161 Yinglong Road, Tianhe District, Guangzhou 510520, China
Author to whom correspondence should be addressed.
Received: 23 June 2018 / Revised: 19 July 2018 / Accepted: 2 August 2018 / Published: 7 August 2018
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Autoregressive moving average (ARMA) models are important in many fields and applications, although they are most widely applied in time series analysis. Expanding the ARMA models to the case of various complex data is arguably one of the more challenging problems in time series analysis and mathematical statistics. In this study, we extended the ARMA model to the case of linguistic data that can be modeled by some symmetric fuzzy sets, and where the relations between the linguistic data of the time series can be considered as the ordinary stochastic correlation rather than fuzzy logical relations. Therefore, the concepts of set-valued or interval-valued random variables can be employed, and the notions of Aumann expectation, Fréchet variance, and covariance, as well as standardized process, were used to construct the ARMA model. We firstly determined that the estimators from the least square estimation of the ARMA (1,1) model under some L2 distance between two sets are weakly consistent. Moreover, the justified linguistic data-valued ARMA model was applied to forecast the linguistic monthly Hang Seng Index (HSI) as an empirical analysis. The obtained results from the empirical analysis indicate that the accuracy of the prediction produced from the proposed model is better than that produced from the classical one-order, two-order, three-order autoregressive (AR(1), AR(2), AR(3)) models, as well as the (1,1)-order autoregressive moving average (ARMA(1,1)) model. View Full-Text
Keywords: stochastic process; fuzzy sets; autoregressive model; forecasting stochastic process; fuzzy sets; autoregressive model; forecasting

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Wang, D.; Zhang, L. A Fuzzy Set-Valued Autoregressive Moving Average Model and Its Applications. Symmetry 2018, 10, 324.

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