An additive seasonal decomposition time series model based on triangular fuzzy random variables

In this paper, a common additive seasonally decomposed time series model is developed for the case where the observed data are triangular fuzzy numbers instead of exact values. For this purpose, the concept of autocorrelation and the corresponding estimator for triangular fuzzy random variables are first defined. The most important properties, such as consistency, are then discussed. The unknown fuzzy trend, fuzzy seasonality and fuzzy remainder terms are estimated in three stages. In stage 1, the unknown fuzzy trend term is estimated by a local polynomial regression model. In stage 2, a methodology for estimating the fuzzy seasonality term is extended. A general harmonic regression model determines the center of the fuzzy seasonality term, and we estimate the left and right dispersion using a commonly used nonparametric time series model. We then evaluate the unknown remainder term using a generalized difference operation of fuzzy numbers. An extended autocorrelation function representation ensures that the fuzzy remainder term is approximately noisy. Six well-established criteria for assessing goodness-of-fit are used to measure the effectiveness of the proposed time series model in comparison to other time series models when dealing with triangular fuzzy numbers. The practical applicability and superiority of the proposed time series model are demonstrated by a comparative analysis using a simulation study and two applications with real-life fuzzy data.