Bayesian network based probabilistic weighted high-order fuzzy time series forecasting

The present article proposes a probabilistic weighted high-order fuzzy time series (FTS) forecasting model employing Bayesian network (BN) to address complex relationships and uncertainty hidden in time series. As considerable FTS forecasting models considers the fuzzy relationships between precedent moments and the consequent moment as the complex relationships in time series, BN structure learning is utilized to discover and model dependence relationships between each moment in time series. The combination of the fuzzy relationships modeled by fuzzy logical relationships and the dependence relationships modeled by the BN provides a comprehensive establishment and representation of the complex relationships inherent in time series data. The proposed FTS forecasting method calculates the fuzzy-probabilistic weights of each fuzzy logical relationship group using the improved fuzzy empirical probabilities to model both aleatoric and epistemic uncertainty in time series. To this end, the improved fuzzy empirical probabilities are formulated by integrating fuzzy empirical probabilities with the BN to incorporate dependence relationships from the original time series into the FTS forecasting procedure. The efficiency of the proposed forecasting model is validated on fourteen publicly available time series. Experimental results confirm the better performance of the proposed method comparing with nine existing FTS models and six numeric models. Hypothesis tests also validate the robustness and reliability of the proposed method.