Pythagorean fuzzy time series approach based on hybrid high-order artificial neural network for nonlinear time series...

Fuzzy time series forecasting methods rely on the fuzzification of time-series observations through membership functions and the subsequent use of fuzzy-valued variables in the forecasting process. Although Pythagorean fuzzy time series models provide a richer representation of uncertainty than classical and intuitionistic fuzzy approaches, existing methods still face important challenges, including high-dimensional input structures generated by membership and non-membership values, a lack of systematic lag selection mechanisms, and limited capability to model complex nonlinear relationships. To address these limitations, this study employs an artificial neural network with a hybrid architecture combining a modified Pi-Sigma artificial neural network and simple exponential smoothing to capture both the linear and nonlinear dynamics and model the relationship in Pythagorean fuzzy time series. To mitigate the high-dimensional input structure arising from membership and non-membership values, principal component analysis was used for input dimension reduction and orthogonalization. Particle swarm optimization was used to estimate all the parameters in the Pythagorean fuzzy time series method. To evaluate its performance, the proposed forecasting algorithm was assessed against existing approaches in the literature using 10 distinct sub-time series generated via a rolling window approach from the S&P 500 and Dow-Jones 30 stock market composite indices, and the proposed method's superior performance was demonstrated experimentally. From an artificial intelligence perspective, these contributions jointly define a novel Pythagorean fuzzy time series forecasting framework that advances hybrid neural network modelling and optimization, while from an engineering perspective, they provide a systematic and implementable methodology for constructing accurate forecasting models for complex nonlinear time series, particularly in financial and other engineering-oriented applications.