A fuzzy quantile method for AR time series model based on triangular fuzzy random variables

A time series model is a collection of quantitative observations with each element in this sequence representing a value recorded at a moment. In this regard, a quantile model not only provides a method for estimating the conditional quantiles of conventional time series models but also substantially expands the modeling options for time series analysis by allowing for local and quantile-specific time series dynamics. In such methods, the data are often reported by imprecise data instead of exact values. In this paper, a quantile AR time series model is proposed for the cases where the observations are reported by fuzzy triangular numbers. For this purpose, a notion of conditional quantile was introduced for a fuzzy random variable. These notions were illustrated via some numerical evaluations. Then, a quantile-based time series model was developed for the AR model based on a sign distance measure for triangular fuzzy numbers. A hybrid algorithm was also proposed to evaluate optimal quantile level and smoothing parameters based on an absolute error distance measure. Some common goodness-of-fit criteria were employed to examine the performance of the proposed fuzzy time series model compared to others based on different values of the autoregressive parameters. The potential applications of the proposed method were further illustrated based on a simulation study and three applied examples.