A hybrid neuro-fuzzy-evolutionary framework for multivariate time series forecasting: convergence analysis and error bounds

Propose a hybrid neuro-fuzzy-evolutionary (neuro-fuzzy-evolutionary) framework for forecasting multivariate locally stationary time series arising in complex dynamical systems. The model integrates recurrent neural architectures (Long Short-Term Memory) with interval type-2 fuzzy inference systems and genetic algorithm-based parameter optimization. Establish theoretical convergence guarantees for the evolutionary optimization process under the Polyak–Łojasiewicz condition and derive asymptotic error bounds for the forecasting component. Crucially, introduce an effective dimensionality assumption whereby the Long Short-Term Memory layer induces a low-dimensional latent representation, circumventing the curse of dimensionality inherent in high-dimensional nonparametric regression. Numerical experiments on empirical datasets using rolling-origin cross-validation demonstrate a significant reduction in Root Mean Square Error compared to Autoregressive Integrated Moving Average models and a coefficient of determination equals very high value.