Decomposition of logarithm mean square error of weighted geometric mean combined forecasting method
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摘要:
针对非线性的加权几何平均组合预测模型，引入对数均方误差，并提出了对数均值偏差、模型解释力系数、对数离差误差方差及预测方法差异性测度等概念。在此基础上，将组合预测对数均方误差分解为对数均值偏差、因模型解释力不足而产生的对数方差以及基于模型解释力系数的对数离差方差3个组成部分，从理论上探讨了对数均方误差的来源。同时，将组合预测对数均方误差分解为单项预测方法对数均方误差的加权平均和预测方法差异性测度的加权平均两部分，获得了有益的结论：提高单项预测方法的精度和预测方法差异性测度水平有利于减少加权几何平均组合预测方法的对数均方误差，为组合预测单项方法的遴选提供了理论支持。通过实际案例，分析了各组成部分以及单项预测方法之间的差异性测度对组合预测对数均方误差的影响。
Abstract:In addition to proposing the ideas of logarithm mean deviation, logarithm explanatory power coefficient, logarithm deviation error variance, and diversity measure of prediction methods, the nonlinear weighted geometric mean combination prediction model was introduced to the logarithm mean square error. On this basis, the logarithm mean square error of the combined prediction is decomposed into three components: the logarithm mean error, the logarithm variance due to the insufficient explanatory power of the model and the logarithm deviation variance based on the logarithm explanatory power coefficient. The source of the logarithm mean square error is discussed theoretically. Simultaneously, the logarithm mean square error of combination forecasting is broken down into the weighted average of the logarithm mean square error of single forecasting methods and the weighted average of diversity measure of forecasting methods. This yields the helpful conclusion that increasing the accuracy of single forecasting methods and the diversity measure level of forecasting methods helps to lower the logarithm mean square error of weighted geometric mean combined forecasting methods. A case study analyzed the influence of the diversity measure of each component and single forecasting method on the logarithm mean square error of combined forecasting.
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图 3 因模型解释力不足而产生的对数方差R
Figure 3. Logarithmic variance due to insufficient explanatory power of the model R
图 4 基于解释力系数的对数离差误差D
Figure 4. Logarithmic deviation error based on explanatory power coefficient D
表 1 实际值及单项方法预测值
Table 1. Actual value and predicted value of single method
实际值 预测值 F1 F2 F3 F4 6 278 6 335 6 280 6 325 6 256 6 325 6 379 6 321 6 372 6 311 6 369 6 370 6 454 6 419 6 362 6 410 6 415 6 522 6 466 6 415 6 461 6 660 6 459 6 512 6 467 6 516 6 663 6 514 6 559 6 519 6 593 6 736 6 590 6 605 6 571 6 676 6 683 6 792 6 651 6 624 6 741 6 858 6 725 6 696 6 676 6 795 6 822 6 904 6 742 6 728 6 827 6 963 6 835 6 787 6 780 6 876 6 996 6 844 6 832 6 833 6 902 6 914 7 051 6 877 6 885 6 929 7 067 6 926 6 922 6 937 6 936 6 931 7 122 6 966 6 989 6 949 7 117 6 948 7 010 7 041 
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表 2 对数误差
Table 2. Logarithmic error
e1 e2 e3 e4 −0.009 0 −0.000 3 −0.007 5 0.003 5 −0.008 5 0.000 6 −0.007 4 0.002 2 −0.000 2 −0.013 3 −0.007 8 0.001 1 −0.000 8 0.017 3 −0.008 7 −0.000 8 −0.030 3 0.000 3 −0.007 9 −0.000 9 −0.022 3 0.000 3 −0.006 6 −0.000 5 −0.021 5 0.000 5 −0.001 8 0.003 3 −0.001 0 −0.017 2 0.003 8 0.007 8 −0.017 2 0.002 4 0.006 7 0.009 7 −0.004 −0.015 9 0.007 8 0.009 9 −0.019 7 −0.001 2 0.005 9 0.006 9 −0.017 3 0.004 7 0.006 4 0.006 3 −0.001 7 −0.021 4 0.003 6 0.002 5 −0.019 7 0.000 4 0.001 0 −0.001 2 0.000 7 −0.026 5 −0.004 3 −0.007 6 −0.023 9 0.000 1 −0.008 7 −0.013 2
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表 3 对数均方误差及其分解指标
Table 3. Log-mean-square error and its decomposition index
方法 L B R D F1 24.983 8×10−5 15.075 5×10−5 0.115 7×10−5 9.792 6×10−5 F2 13.826 1×10−5 4.201 4×10−5 0.228 4×10−5 9.396 4×10−5 F3 4.130 3×10−5 0.253 6×10−5 1.428 4×10−5 2.448 3×10−5 F4 3.812 1×10−5 0.331 9×10−5 0.063 0×10−5 3.417 2×10−5 F12 8.831 1×10−5 7.317 6×10−5 0.183 7×10−5 1.329 7×10−5 F13 3.991 1×10−5 0.588 6×10−5 1.156 4×10−5 2.246 0×10−5 F14 2.967 3×10−5 0.025 0×10−5 0.070 6×10−5 2.871 7×10−5 F23 3.461 1×10−5 0.670 3×10−5 0.729 5×10−5 2.061 2×10−5 F24 2.766 4×10−5 0.001 9×10−5 0.092 8×10−5 2.671 7×10−5 F34 3.128 4×10−5 0.007 6×10−5 0.163 3×10−5 2.957 5×10−5 F123 3.346 9×10−5 1.097 8×10−5 0.568 7×10−5 1.680 4×10−5 F124 2.274 0×10−5 0.268 7×10−5 0.094 5×10−5 1.910 8×10−5 F134 2.788 0×10−5 0.063 3×10−5 0.016 4×10−5 2.708 4×10−5 F234 2.536 9×10−5 0.054 3×10−5 0.013 8×10−5 2.468 8×10−5 F1234 2.249 2×10−5 0.286 2×10−5 0.015 1×10−5 1.941 6×10−5
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表 4 差异性测度的指标值
Table 4. The index value for measures of variability
差异测度 C F1,F2 3.911 30×10−4 F1,F3 2.455 35×10−4 F1,F4 3.148 69×10−4 F2,F3 1.630 16×10−4 F2,F4 1.890 72×10−4 F3,F4 0.334 08×10−4
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