Ensemble of stochastic models, machine learning, deep learning and wavelet based techniques: A Whale optimization approach

High variability in datasets leads to less accurate forecasts. The application of different models, including advanced statistical and machine learning techniques, yields varying forecast performances, making it challenging to choose the appropriate model for the final forecast. In the present investigation, an algorithm based on Whale Optimization is proposed to combine fifteen models, including stochastic models, machine learning, deep learning techniques and wavelet-based hybrid models, for data with different patterns, i.e., nonlinearity, nonnormality, high volatility, etc. Two datasets, the daily close log return price of the S&P 500 (Standard & Poor's 500) and the monthly log return price of potatoes were considered to examine the potential of the proposed algorithm. An empirical comparison of the forecasting accuracy of all the models was made using the criteria of the Mean Absolute Error (MAE), Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE). It was found that the deep learning techniques performed better in terms of forecasting than the usual machine learning techniques. The proposed WOA-based ensemble reduced RMSE by up to 98.6% for the S&P 500 and 97.0% for potato prices compared to individual models and the lowest MAPE values of 0.01 and 0.26 for S&P 500 and potato prices. Finally, when Whale Optimization was used to combine the predictions of different models, it outperformed any individual model or technique.