基于改进灰色预测与最小二乘修正的城市用水需求预测模型研究

Research on Urban Water Demand Forecasting Model Based on Improved Grey Prediction and Least Squares Correction

摘要: 城市年度供水计划通常要在样本年限较短、影响因素又不断变化的条件下完成，单纯按历史均值或线性趋势外推，容易忽略产业结构调整、节水措施和气候波动带来的扰动。针对这一现实场景，文章以年度城市用水需求为对象，建立GM(1, 1)灰色预测与最小二乘残差修正相结合的预测方法。建模时先通过一次累加生成削弱原始序列中的随机起伏，再由白化微分方程得到参数估计、时间响应函数和还原公式；随后把基础预测留下的残差视作可继续识别的信息，构造线性补偿项。与直接叠加复杂算法不同，该思路把“预测后剩下的误差”转化为二次分析对象，用趋势残差修正GM(1, 1)指数型还原序列的刚性。以南京市2018~2024年城市供水总量为样本进行检验，其中2018~2022年为建模样本、2023~2024年为预测检验样本，组合模型的MAPE、RMSE和MAE均低于原始GM(1, 1)，对应误差降幅分别为11.25%、11.62%和11.21%。

Abstract: Urban annual water supply plans are typically completed under conditions of short sample periods and constantly changing influencing factors. Simply extrapolating based on historical averages or linear trends can easily overlook disturbances caused by industrial restructuring, water conservation measures, and climate fluctuations. To address this real-world scenario, this paper establishes a prediction method combining GM(1, 1) grey prediction with least squares residual correction, focusing on annual urban water demand. In the modeling process, a single accumulation is used to weaken the random fluctuations in the original sequence. Then, parameter estimates, time response functions, and restoration formulas are obtained from the whitening differential equation. Subsequently, the residuals left by the basic prediction are treated as information that can be further identified, and a linear compensation term is constructed. Unlike complex algorithms that directly superimpose data, this approach transforms the “remaining error after prediction” into a secondary analysis object, using trend residuals to correct the rigidity of the GM(1, 1) exponential restoration sequence. The total urban water supply in Nanjing from 2018 to 2024 was used as a sample for testing. The sample from 2018 to 2022 was used for modeling, and the sample from 2023 to 2024 was used for prediction testing. The combined model’s MAPE, RMSE, and MAE were all lower than the original GM(1, 1), with corresponding error reductions of 11.25%, 11.62%, and 11.21%, respectively.