Grey forecasting theory is an approach to build a prediction model with limited data to produce better forecasting results. This forecasting theory has an elementary model, represented as the GM(1,1) model , characterized by the first-order differential equation of one variable. It has the potential for accurate and reliable forecasting without any statistical assumption. The research proposes a methodology to derive the modified GM(1,1) model with improved forecasting precision. The residual series is forecasted by the GM(1,1) model to modify the actual forecasted values. The study primarily addresses two fundamental issues: sign prediction of forecasted residual and the procedure for formulating the grey model. Accurate sign prediction is very complex, especially when the model lacks in data. The signs of forecasted residuals are determined using a multilayer perceptron to overcome this drawback. Generally, the elementary model is formulated conventionally, containing the parameters that cannot be calculated straightforward. Therefore, maximum likelihood estimation is incorporated in the modified model to resolve this drawback. Three statistical indicators, relative residual, posterior variance test, and absolute degree of grey indices, are evaluated to determine the model fitness and validation. Finally, an empirical study is performed using actual municipal solid waste generation data in Saudi Arabia, and forecasting accuracies are compared with the linear regression and original GM(1,1). The MAPEs of all models are rigorously examined and compared, and then it is obtained that the forecasting precision of GM(1,1) model , modified GM(1,1) model, and linear regression is 15.97%, 8.90%, and 27.90%, respectively. The experimental outcomes substantiate that the modified grey model is a more suitable forecasting approach than the other compared models.
Canonical models for torus canards in elliptic bursters
