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In this paper, we introduce ProNet, an novel deep learning approach designed for multi-horizon time series forecasting, adaptively blending autoregressive (AR) and non-autoregressive (NAR) strategies. Our method involves dividing the forecasting horizon into segments, predicting the most crucial steps in each segment non-autoregressively, and the remaining steps autoregressively. The segmentation process relies on latent variables, which effectively capture the significance of individual time steps through variational inference. In comparison to AR models, ProNet showcases remarkable advantages, requiring fewer AR iterations, resulting in faster prediction speed, and mitigating error accumulation. On the other hand, when compared to NAR models, ProNet takes into account the interdependency of predictions in the output space, leading to improved forecasting accuracy. Our comprehensive evaluation, encompassing four large datasets, and an ablation study, demonstrate the effectiveness of ProNet, highlighting its superior performance in terms of accuracy and prediction speed, outperforming state-of-the-art AR and NAR forecasting models. • Forecasting of electricity energy consumption and solar energy generation. • Informer-based model marrying forecasting horizon segmentation and variational inference techniques. • Comparing SARIMAX, DeepAR, DeepSSM, LogTrans, N-BEATS, and Informer models forecasting ability. • Performance evaluation based on four real-world datasets. [ABSTRACT FROM AUTHOR]

• Optimized deep learning, statistical and hybrid models are proposed for deterministic (point) and probabilistic (interval) forecasting of crude oil price (COP). • A DE-DL method employing differential evolution algorithm is proposed to optimize the architecture and other hyper-parameters of deep learning models for a problem. • Recurrence plot is drawn, recurrence quantification analysis is carried out and uncertainty analysis of prediction error is performed using Gaussian and t-Location Scale distributions. • Extensive non-parametric statistical analysis on the obtained results are carried out to rank the optimized models and draw decisive conclusions relating to deterministic and probabilistic forecasting of COP. • Eight research questions related to deterministic and probabilistic forecasting of COP are addressed. In this paper, individual and hybrid methods are proposed employing optimized statistical and deep learning (DL) models for deterministic (point) and probabilistic (interval) forecasting of crude oil price time series. The statistical models are optimized using the Forecast package of R. To enhance the performance of DL models, a novel pruning DE-DL method is proposed, which employs the differential evolution (DE) algorithm to optimize architecture and continuous and discrete-valued hyper-parameters. The proposed DE-DL method is so generic that it can be applied to optimize different DL models for any supervised learning problem. Five DL models (LSTM, BiLSTM, GRU, CNN, and ConvLSTM) are optimized for forecasting monthly crude oil prices and hybridized with an optimized ARIMA model for developing optimized additive and multiplicative hybrid forecasting models. The effectiveness of the proposed methods is evaluated through deterministic and probabilistic forecasting measures, comparing the results with six optimized statistical models, thirteen machine learning models, five optimized DL models, and ten optimized hybrid models. It is observed from the simulation results that the proposed optimized Additive-ARIMA-GRU hybrid model provides statistically superior forecasts, and the t Location Scale distribution is more suitable than the Gaussian distribution for computing reliable prediction intervals with different significance levels. [ABSTRACT FROM AUTHOR]