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In this paper, we propose hesitant fuzzy sets-based hybrid time series forecasting method using particle swarm optimization and support vector machine. Length of unequal intervals, weights of intervals and process of defuzzification are major factors that affect the forecasting accuracy of hesitant fuzzy sets-based time series models. The proposed hybrid fuzzy time series forecasting method uses hesitant fuzzy sets which are constructed using fuzzy sets with equal and unequal length intervals. Particle swarm optimization and linear programming are used to optimize length of unequal intervals and weights of equal and unequal intervals. The proposed hybrid method of time series forecasting uses support vector machine for setting input-target pattern for defuzzification. Outperformance of proposed hybrid method of time series forecasting method is revealed by applying it on widely used time series data of enrollments of the University of Alabama, market share price of State Bank of India share at Bombay stock exchange and car sell in Quebec City of Canada. Validity of the proposed hybrid fuzzy time series forecasting method is verified using values of Willmott index and tracking signal.

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Computational methods for time series forecasting have always an edge over conventional methods of forecasting due to their easy implementation and prominent characteristics of coping with large amount of time series data. Many computational methods for fuzzy time series (FTS) forecasting have been developed in past using fuzzy set, intuitionistic fuzzy set (IFS), and hesitant fuzzy set (HFS) for incorporating uncertainty, non-determinism, and hesitation in time series forecasting. Since probabilistic fuzzy set (PFS) incorporates both probabilistic and non-probabilistic uncertainties simultaneously, we have proposed PFS and particle swarm optimization (PSO) based computational method for FTS forecasting. First, we have developed a PFS based computational method for FTS forecasting and then it is integrated with PSO to enhance the accuracy in forecasted outputs. Unlike other PSO based for FTS forecasting method, PSO is used to optimize both number of partitions and length of intervals. Three diversified time series data of enrolments of the University of Alabama, market price of State Bank of India (SBI) share at Bombay stock exchange (BSE) India, and death cases due to COVID-19 in India are used to compare the performance of PFS based computational method of FTS forecasting before and after its integration with PSO in terms of root mean square error (RMSE). After integration of PFS based computational method with PSO, accuracy in the forecasted outputs is increased significantly and its performance is found better than many other existing FTS forecasting methods. Goodness of the proposed FTS forecasting method is also tested using tracking signal and Willmott index.