Projection of residential annual water consumption in Hengshui City based on dynamic gray model groups
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摘要: 针对居民用水量序列的随机性和周期性以及传统灰色模型由于离散程度大而产生的过拟合问题，依据灰色模型理论构造了由5个GM(1,1)模型组成的灰色动态模型群；基于衡水市2007—2019年居民年用水量数据，采用灰色动态模型群对衡水市2020—2030年居民年用水量变化进行预测研究，并对预测结果进行残差检验以及残差修正；将灰色动态模型群分别与5个GM(1,1)模型进行对比，以检验模型性能。结果表明：灰色动态模型群的预测相对误差整体小于传统GM(1,1)模型，具有更好的准确性和适用性；衡水市2019年居民年用水量为1 795.00万m3，2030年预计增至2 862.21万m3，未来衡水市居民年用水量呈明显增长趋势，这与衡水市未来人口增长和社会经济发展趋势相吻合。衡水市未来居民用水量预测结果可为供水优化调度和水资源优化配置提供参考。Abstract: To address the randomness and periodicity of the residential water consumption (RWC) data along with overfitting problem caused by the large dispersion of traditional gray model, a dynamic gray model group consisting of five GM(1,1) models was proposed based on gray model theory. Based on the annual RWC data of Hengshui City from 2007 to 2019, the dynamic gray model group was used to project the future changes of annual RWC in Hengshui City during 2020-2030, and meanwhile residual tests and corrections were conducted using the projected results; the dynamic gray model group was compared with five GM(1,1) models to test the model accuracy. The results showed that the projected relative error of the dynamic gray model group was smaller than that of the traditional GM(1, 1) model, implying better accuracy and applicability. The annual RWC in Hengshui approached 17.95 million m3 in 2019 and was expected to increase to 28.62 million m3 in 2030, which indicated that the future RWC in Hengshui City would be at an obvious uptrend, and this result was in line with the future population growth and socio-economic development trend. The projected results of RWC in this study was capable of providing reference for optimal water supply and water resources allocation in Hengshui City.
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图 1 2015—2019年衡水市居民年用水量预测值与实际值对比及相对误差
Figure 1. Comparison and relative error between predicted and actual annual RWC of Hengshui City from 2015 to 2019
图 2 2015—2019年衡水市居民用水量灰色动态模型群误差检验
注：纵坐标为单一GM(1,1)模型与灰色动态模型群预测结果的相对误差。
Figure 2. Error test of dynamic gray model group of RWC in Hengshui City from 2015 to 2019
图 3 衡水市居民用水量预测拟合效果及未来用水量预测
Figure 3. Prediction and fitting effect and change trend of RWC in Hengshui City in the future
表 1 检验精度指示表
Table 1. Check accuracy indicator
精度 P C 等级 良好 ≥0.95 ≤0.35 1 合格 0.80~0.95 0.35~0.50 2 勉强接受 0.70~0.80 0.50~0.65 3 不合格 <0.70 >0.65 4 
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表 2 灰色模型精度要求
Table 2. Grey model accuracy requirements
预测时间段 相对误差/% 短期预测（≤1 a） 2~5 中期预测（1~5 a） 10~20 长期预测（5~10 a） 30~40
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表 3 2007—2019年衡水市居民年用水量
Table 3. Annual RWC in Hengshui City from 2007 to 2019
万 m3 年份 年用水量 年份 年用水量 2007 1 264.65 2014 1 184.43 2008 1 293.37 2015 1 254.85 2009 891.51 2016 1 603.20 2010 905.00 2017 1 846.50 2011 993.00 2018 1 785.30 2012 1 086.20 2019 1 795.00 2013 1 104.52
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表 4 GM(1,1)模型群及残差检验
Table 4. GM(1,1) model group and residual test
模型 GM(1,1)预测模型 P C 精度 等级 1 ${\hat x}^{\left(1\right)}\left( {k + 1} \right) = {\text{100\;714} }{\text{.11} }{ {\rm{e} }^{0.010\;2k} } - 99\;449.46$ 1 0.643 6 合格 3 2 ${\hat x}^{\left(1\right)}\left( {k + 1} \right) = 14\;234.23{ {\rm{e} }^{0.060\;0k} } - 12\;940.86$ 1 0.017 6 良好 1 3 ${\hat x}^{\left(1\right)} \left( {k + 1} \right) = 1{\text{4\;253} }{\text{.85} }{ {\rm{e} }^{0.063\;0k} } - 13\;362.34$ 1 0.0294 良好 1 4 ${\hat x}^{\left(1\right)} \left( {k + 1} \right) = {\text{1} }8\;071.92{ {\rm{e} }^{0.054\;1k} } - 17\;166.92$ 1 0.022 1 良好 1 5 ${\hat x}^{\left(1\right)} \left( {k + 1} \right) = 23\;885.56{ {\rm{e} }^{0.044\;1k} } - 22\;892.56$ 1 0.031 8 良好 1
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表 5 残差修正前2015—2019用水量预测值与实际值对比
Table 5. Comparison of projected and actual water consumption data from 2015-2019 before residual correction
年份 预测值/万m³ 实际值/万m³ 相对误差/% 残差/万m³ 模型1 模型2 模型3 模型4 模型5 均值 2015 1 109.22 1 258.26 1 266.94 1 246.49 1 227.02 1 221.59 1 254.85 −2.65 −33.26 2016 1 120.93 1 338.17 1 350.82 1 316.98 1 282.93 1 281.97 1 603.20 −20.04 −321.23 2017 1 132.42 1 420.86 1 438.57 1 390.21 1 340.70 1 344.55 1 846.50 −27.18 −501.95 2018 1 144.04 1 508.65 1 532.02 1 467.50 1 401.08 1 410.66 1 785.30 −20.98 −374.64 2019 1 155.77 1 601.87 1 631.54 1 549.10 1 464.18 1 480.49 1 795.00 −17.52 −314.51
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表 6 GM(1,1)残差修正后预测模型群
Table 6. GM(1,1) residual-corrected prediction model group
模型 残差修正后GM(1,1)预测模型 1 ${\hat Y}^{\left(1\right)} \left( {k + 1} \right)$= ${\text{100\;714} }{\text{.11} }{{\rm{e}}^{0.010\;2k} } - 282.02{{\rm{e}}^{ - 0.762\;5k} } - 99\;167.44$ 2 ${\hat Y}^{\left(1\right)} \left( {k + 1} \right)$= $14\;234.23{{\rm{e}}^{0.060\;0k} } + 154{{\rm{e}}^{00.002\;2k} } - 12\;994.86$ 3 ${\hat Y}^{\left(1\right)} \left( {k + 1} \right)$= $1{\text{4\;253} }{\text{.85} }{{\rm{e}}^{0.063\;0k} } + 85.84{{\rm{e}}^{0.298\;9k} } - 13\;247.50$ 4 ${\hat Y}^{\left(1\right)} \left( {k + 1} \right)$= ${\text{1} }8\;071.92{{\rm{e}}^{0.054\;1k} } - 51.63{{\rm{e}}^{ - 0.004\;5k} } - 17\;115.92$ 5 ${\hat Y}^{\left(1\right)} \left( {k + 1} \right)$= $23\;885.56{{\rm{e}}^{0.044\;1k} } + 54.83{{\rm{e}}^{0.003\;9k} } - 22\;946.56$
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表 7 残差修正后2015—2019用水量预测值与实际值对比
Table 7. Comparison of predicted and actual water consumption data for 2015-2019 after residual correction
年份 预测值/万m³ 实际值/万m³ 平均相对
误差/%残差/万m³ 模型1 模型2 模型3 模型4 模型5 平均值 2015 1 490.22 1 268.26 1 271.94 1 256.49 1 437.02 1 344.79 1 254.85 7.17 89.94 2016 1 501.93 1 348.17 1 355.82 1 326.98 1 522.93 1 411.17 1 603.20 −11.98 −192.03 2017 1 513.42 1 430.86 1 443.57 1 400.21 1 623.70 1 482.35 1 846.50 −19.72 −364.15 2018 1 525.04 1 518.65 1 537.02 1 477.50 1 711.08 1 553.86 1 785.30 −12.96 −231.44 2019 1 536.77 1 611.87 1 636.54 1 559.10 1 784.18 1 625.69 1 795.00 −9.43 −169.31
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表 8 衡水市居民年用水量预测结果
Table 8. Projected annual water consumption of Hengshui residents
万 m3 年份 预测值 实际值 模型1 模型2 模型3 模型4 模型5 平均值 2015 1 490.22 1 268.26 1 271.94 1 256.49 1 437.02 1 344.79 1 254.85 2016 1 501.93 1 348.17 1 355.82 1 326.98 1 522.93 1 411.17 1 603.20 2017 1 513.42 1 430.86 1 443.57 1 400.21 1 623.70 1 482.35 1 846.50 2018 1 525.04 1 518.65 1 537.02 1 477.50 1 711.08 1 553.86 1 785.30 2019 1 536.77 1 611.87 1 636.54 1 559.10 1 784.18 1 625.69 1 795.00 2020 1 543.93 1 917.53 1 950.41 1 735.23 1 830.12 1 795.44 2021 1 556.28 2 036.01 2 070.61 1 826.15 1 899.03 1 877.62 2022 1 568.75 2 161.81 2 198.63 1 922.12 1 961.04 1 962.47 2023 1 581.36 2 295.39 2 334.96 2 023.43 2 024.29 2 051.89 2024 1 594.09 2 437.22 2 480.14 2 130.37 2 107.12 2 149.79 2025 1 616.95 2 587.81 2 634.76 2 243.26 2 193.01 2 255.16 2026 1 629.95 2 747.71 2 799.43 2 362.43 2 282.76 2 364.46 2027 1 663.08 2 917.48 2 974.79 2 488.22 2 376.56 2 484.03 2028 1 678.34 3 097.75 3 161.54 2 621.01 2 474.58 2 606.64 2029 1 689.74 3 289.16 3 360.42 2 761.18 2 577.01 2 735.50 2030 1 700.28 3 492.39 3 572.23 2 909.14 2 637.01 2 862.21
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