Fuzzy-Probabilistic Time Series Forecasting Combining Bayesian Network and Fuzzy Time Series Model

Despite many fuzzy time series forecasting (FTSF) models addressing complex temporal patterns and uncertainties in time series data, two limitations persist: they do not treat fuzzy and crisp time series as a unified whole for analyzing nonlinear relationships between different moments, and they fail to effectively capture how uncertainty in temporal patterns affects predictions. In this paper, we propose an FTSF model integrating Bayesian networks to overcome the limitations. Bayesian network (BN) structure learning is employed to extract fuzzy–crisp dependencies between historical fuzzified data and predicted crisp data alongside temporal crisp dependencies within crisp data. Integrating fuzzy logical relationship groups (FLRGs) and the two BNs representing the fuzzy–crisp and crisp relationships identifies temporal patterns efficiently. BN parameter learning models the occurrence uncertainties of dependencies through conditional probability distributions in BNs, while fuzzy empirical conditional probabilities quantify the occurrence uncertainties of the elements in FLRGs. The defuzzification stage infers the crisp predicted value using the fuzzy-empirical-probability weighted FLRGs and the two BN. We validate the forecasting performance of the proposed model on sixteen diverse time series. Experimental results demonstrate the competitive forecasting performance of the proposed model compared to state-of-the-art methods.