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.

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修正灰色预测模型在利用 MLP 和 MLE 预测城市固体废物产生中的应用

灰色预测理论是一种利用有限数据建立预测模型以产生更好预测结果的方法。该预测理论有一个基本模型，表示为 GM(1,1) 模型，其特征在于一个变量的一阶微分方程。它有可能在没有任何统计假设的情况下进行准确和可靠的预测。该研究提出了一种方法来推导改进的 GM(1,1) 模型，并提高了预测精度。残差序列由 GM(1,1) 模型预测，以修改实际预测值。该研究主要解决两个基本问题：预测残差的符号预测和制定灰色模型的过程。准确的符号预测非常复杂，尤其是在模型缺乏数据的情况下。预测残差的符号是使用多层感知器来确定的，以克服这个缺点。通常，基本模型是按惯例制定的，包含无法直接计算的参数。因此，最大似然估计被结合到修改后的模型中来解决这个缺点。评估三个统计指标，相对残差、后验方差检验和灰色指数的绝对度，以确定模型的适应度和有效性。最后，利用沙特阿拉伯的实际城市固体废物产生数据进行了实证研究，并将预测精度与线性回归和原始 GM(1,1) 进行了比较。对所有模型的 MAPE 进行严格检验和比较，得出 GM(1,1) 模型的预测精度，修改后的 GM(1,1) 模型，线性回归分别为 15.97%、8.90% 和 27.90%。实验结果证实，修正后的灰色模型比其他比较模型更适合预测方法。