ARIMA模型预测.doc

5 ARIMA模型预测5.1 模型选取目前，学术界较为成熟的预测方法很多，各种不同的预测方法有其所面向的特定对象，不存在一种普遍“最好”的预测方法。GM（1,1）模型预测是以灰色系统理论为基础，通过原始数据的分析处理和建立灰色模型，对系统未来状态作出科学的定量预测的一种方法。我们采用GM（1,1）模型是基于以下两方面的考虑：第一，GM（1,1）模型对数据要求较低，而其他多数预测方法以数理统计为基础，对样本量有较高要求。我们用来做预测的数据时序只有14年，预测使用GM（1,1）模型较好；第二，GM（1,1）模型的计算量相对较小，计算方法相对简单，适用性较好。5.2 模型假设前提1、假设未来重庆地区经济发展基本态势不变；2、假设未来中央政府对重庆实施的政策方向基本不变；3、假设未来不会出现战争、瘟疫及其它不可抗拒的自然或社会因素。5.3 预测数据来源预测样本为19972008年的重庆市农资价格指数、化学肥料价格指数、饲料价格指数。具体预测样本数据如下：表5.1 19972008年重庆部分农资价格指数单位：%19971998199920002001200220032004200520062007农资105.9 90.4 93.2 96.8 100.5 103.5 102.5 109.8 106.9 101.2 110.0 化肥93.6 87.4 96.5 97.3 98.9 99.7 101.0 106.3 108.3 99.9 102.2 饲料96.6 91.0 91.9 95.8 101.1 106.6 105.6 115.7 105.5 101.1 109.2 为提高数据预测的科学性，我们以1996年（直辖前）的农资价格为基期，假设1996年农资产品价格为100元，则以后第年的农资产品价格计算公式如下： 经此换算，得到19972008年的预测样本。其中，NZJG表示换算后的农资，HXFL表示换算后的化肥，SL表示换算后的饲料。具体见下表：表5.2 19972008年转换后的预测样本单位：元19971998199920002001200220032004200520062007NZJG105.9 95.7 89.2 86.4 86.8 89.8 92.1 101.1 108.1 109.4 120.3 HXFL93.6 81.8 78.9 76.8 76.0 75.7 76.5 81.3 88.1 88.0 89.9 SL96.6 87.9 80.8 77.4 78.2 83.4 88.1 101.9 107.5 108.7 118.7 5.4 GM（1,1）模型建立与检验5.4.1 序列的建立设由n个原始数据组成的原始序列为x(0)(k)=x(0)(1),x(0)(2),x(0)(n)。那么可以得到四个样本原始序列：NZJG x(0)(k)= 105.9,95.7,120.3；HXFL x(0)(k)= 93.6,81.8,89.9；SL x(0)(k)= 96.6,87.9,118.7。5.4.2 级比检验级别检验是GM（1,1）建模的数据检验，经计算可得：NZJG级比序列= 0.904,0.932 , 1.198；HXFL地区序列= 0.874, 0.965, 1.200 ；SL地区序列= 0.910, 0.919, 1.170；都落在界区（0.7515，1.3307）内。这表明，以上三个样本序列均可以进行GM（1,1）模型建模。5.4.3 模型的方程通过一次累加生成新序列：x(1)(k)=x(1)(1),x(1)(2),x(1)(n)，则GM(1,1)模型相应的微分方程为：其中，称为发展灰度，为内生控制灰度，它们是方程中重要的参数。通过求解微分方程，即可得到预测模型。由于GM（1,1）预测模型种类较多，我们选取其中较常用的一种如下： 通过测算，我们得到三组、值和相应的四个拟合预测方程如下表表5.3 拟合预测方程和、值拟合预测方程NZJG-0.04691572.875525=(105.9-)e+HXFL-0.02734869.157548=(93.6-)e+SL-0.05742164.105061=(96.6-)e+5.4.4 后验差检验后验差检验包括残差的方差比（C）和小误差概率（P）。首先算得知残差平均值为历史数据方差为其中历史数据平均值为残差方差为 后验差比值为小误差概率为经计算，我们同样得到四组方差比（）和小误差概率（）如下：表5.4 后验差检验结果表NZJG0.273916 1HXFL0.406991SL0.1728 1 根据下列后验差检验结果判别表，我们认为四个模型均通过了后验差检验，有较好的预测精度。表5.5 后验差检验结果判别表0.350.95好0.500.80合格0.650.70勉强合格0.650.70不合格5.4.5 相对误差检验经过对模型进行拟合测算，得到以下四个相对误差表：表5.6 NZJG GM(1,1)模型拟合相对误差表时期实际值拟合值相对误差（%）199895.779.70 -16.72 199989.283.53 -6.36 2008144.2 127.41 -11.64 平均相对误=6.56%表5.7 HXFL GM(1,1)模型拟合相对误差表时期实际值拟合值相对误差（%）199881.872.71 -11.12 199978.974.72 -5.29 2008107.995.58 -11.42 平均相对误差=5.41%表5.8 SL GM(1,1)模型拟合相对误差表时期实际值拟合值相对误差（%）199896.671.69 -18.44 199987.975.93 -6.03 2008138.8127.30 -8.28 平均相对误差=6.26%综上分析，我们发现：以上四个模型的相对误差还是在合理的范围内，模型可靠，拟合精度较高，可以进行预测。5.5 模型预测利用模型预测2009-2020年NZJG、HXFL、SL的具体数值，结果见表5.9。表5.9 GM(1,1)模型预测结果 单位：元年份NZJGHXFLSL2009133.598.2134.82010139.9100.9142.82011146.7103.7151.22012153.7106.6160.22013161.1109.6169.62014168.8112.6179.72015176.9115.7190.32016185.4118.9201.52017194.3122.2213.42018203.7125.6226.12019213.5129.1239.42020223.7132.7253.6经过换算，我们可以得到2009-2020年重庆市农资价格指数、化学肥料价格指数、饲料价格指数的预测值如下：表5.10 GM(1,1)模型预测换算结果 单位：%年份农资价格指数化肥价格指数饲料价格指数200992.6 91.0 97.1 2010104.8 102.8 105.9 2011104.8 102.8 105.9 2012104.8 102.8 105.9 2013104.8 102.8 105.9 2014104.8 102.8 105.9 2015104.8 102.8 105.9 2016104.8 102.8 105.9 2017104.8 102.8 105.9 2018104.8 102.8 105.9 2019104.8 102.8 105.9 2020104.8 102.8 105.9 附录GM(1,1)模型模拟结果NZJGHXFL期数实际值模拟值残差期数实际值模拟值残差1998 79.70 -16.00 -16.72 1998 1998 72.71 -9.09 1999 83.53 -5.67 -6.36 1999 1999 74.72 -4.18 2000 87.54 1.14 1.32 2000 2000 76.79 -0.01 2001 91.74 4.94 5.70 2001 2001 78.92 2.92 2002 96.15 6.35 7.07 2002 2002 81.11 5.41 2003 100.77 8.67 9.41 2003 2003 83.36 6.86 2004 105.61 4.51 4.46 2004 2004 85.67 4.37 2005 110.68 2.58 2.39 2005 2005 88.05 -0.05 2006 116.00 6.60 6.03 2006 2006 90.49 2.49 2007 121.57 1.27 1.06 2007 2007 93.00 3.10 2008 127.41 -16.79 -11.64 2008 2008 95.58 -12.32 SL期数实际值模拟值残差1998 71.69 -16.21 -18.44 1999 75.93 -4.87 -6.03 2000 80.42 3.02 3.90 2001 85.17 6.97 8.91 2002 90.20 6.80 8.16 2003 95.53 7.43 8.44 2004 101.18 -0.72 -0.71 2005 107.16 -0.34 -0.32 2006 113.49 4.79 4.41 2007 120.20 1.50 1.26 2008127.30 -11.50 -8.28 Null Hypothesis: NZJG has a unit rootExogenous: NoneLag Length: 1 (Automatic based on SIC, MAXLAG=11)t-StatisticProb.*Augmented Dickey-Fuller test statistic-0.4843870.5029Test critical values:1% level-2.5953405% level-1.94508110% level-1.614017*MacKinnon (1996) one-sided p-values.Augmented Dickey-Fuller Test EquationDependent Variable: D(NZJG)Method: Least SquaresDate: 08/31/09 Time: 20:34Sample (adjusted): 2003M03 2009M07Included observations: 77 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.NZJG(-1)-0.0006700.001384-0.4843870.6295D(NZJG(-1)0.6556820.0873467.5066840.0000R-squared0.429121Mean dependent var-0.105195Adjusted R-squared0.421509S.D. dependent var1.724201S.E. of regression1.311403Akaike info criterion3.405702Sum squared resid128.9833Schwarz criterion3.466580Log likelihood-129.1195Hannan-Quinn criter.3.430053Durbin-Watson stat1.936552Null Hypothesis: D(NZJG) has a unit rootExogenous: NoneLag Length: 0 (Automatic based on SIC, MAXLAG=11)t-StatisticProb.*Augmented Dickey-Fuller test statistic-3.9485590.0001Test critical values:1% level-2.5953405% level-1.94508110% level-1.614017*MacKinnon (1996) one-sided p-values.Augmented Dickey-Fuller Test EquationDependent Variable: D(NZJG,2)Method: Least SquaresDate: 08/31/09 Time: 20:35Sample (adjusted): 2003M03 2009M07Included observations: 77 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.D(NZJG(-1)-0.3429800.086862-3.9485590.0002R-squared0.169859Mean dependent var-0.029870Adjusted R-squared0.169859S.D. dependent var1.432064S.E. of regression1.304783Akaike info criterion3.382852Sum squared resid129.3868Schwarz criterion3.413291Log likelihood-129.2398Hannan-Quinn criter.3.395027Durbin-Watson stat1.934426Null Hypothesis: D(NZJG,2) has a unit rootExogenous: NoneLag Length: 1 (Automatic based on SIC, MAXLAG=11)t-StatisticProb.*Augmented Dickey-Fuller test statistic-9.5992680.0000Test critical values:1% level-2.5961605% level-1.94519910% level-1.613948*MacKinnon (1996) one-sided p-values.Augmented Dickey-Fuller Test EquationDependent Variable: D(NZJG,3)Method: Least SquaresDate: 08/31/09 Time: 20:36Sample (adjusted): 2003M05 2009M07Included observations: 75 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.D(NZJG(-1),2)-1.5468980.161147-9.5992680.0000D(NZJG(-1),3)0.3704290.1064843.4787210.0009R-squared0.630862Mean dependent var0.041333Adjusted R-squared0.625805S.D. dependent var2.147223S.E. of regression1.313488Akaike info criterion3.409554Sum squared resid125.9432Schwarz criterion3.471353Log likelihood-125.8583Hannan-Quinn criter.3.434229Durbin-Watson stat2.099426Date: 08/31/09 Time: 20:51Sample: 2003M01 2009M07Included observations: 78AutocorrelationPartial CorrelationACPACQ-StatProb. |* |. |* |10.6490.64934.1530.000. |* |. | . |20.389-0.05646.5920.000. |* |. |* |30.3700.24157.9650.000. |* |. | . |40.3680.07169.3540.000. |* |. | . |50.2990.00676.9760.000. |* |. |*. |60.2730.08183.4230.000. |* |. | . |70.218-0.06387.5990.000. |*. |.*| . |80.129-0.06689.0830.000. | . |.*| . |90.051-0.08089.3220.000. | . |. | . |100.024-0.02989.3720.000.*| . |*| . |11-0.147-0.31291.3890.000*| . |*| . |12-0.331-0.243101.730.000*| . |. |* |13-0.2100.221105.960.000.*| . |.*| . |14-0.160-0.114108.460.000*| . |. | . |15-0.247-0.020114.480.000*| . |. | . |16-0.300-0.021123.520.000*| . |. |*. |17-0.2290.124128.900.000*| . |. | . |18-0.2180.006133.860.000*| . |. | . |19-0.243-0.032140.130.000*| . |.*| . |20-0.258-0.073147.300.000*| . |.*| . |21-0.268-0.106155.140.000*| . |.*| . |22-0.295-0.071164.830.000*| . |.*| . |23-0.244-0.150171.590.000.*| . |. | . |24-0.171-0.054174.970.000.*| . |. |*. |25-0.1200.182176.650.000.*| . |. | . |26-0.112-0.050178.160.000.*| . |.*| . |27-0.111-0.082179.680.000.*| . |. | . |28-0.0810.056180.500.000.*| . |. | . |29-0.0900.033181.530.000.*| . |. | . |30-0.083-0.048182.420.000. | . |. | . |31-0.0190.006182.460.000. | . |. | . |320.031-0.019182.590.000偏相关1阶 自相关4阶 拖尾 ARMA(1,4)DNZJG C MA(1) MA(2) MA(3) MA(4) AR(1)Dependent Variable: DNZJGMethod: Least SquaresDate: 08/31/09 Time: 21:15Sample (adjusted): 2003M03 2009M07Included observations: 77 after adjustmentsConvergence achieved after 16 iterationsMA Backcast: 2002M11 2003M02VariableCoefficientStd. Errort-StatisticProb.C-0.2713300.675130-0.4018920.6890AR(1)0.8564480.1282186.6796380.0000MA(1)-0.1972760.163094-1.2095880.2304MA(2)-0.3737700.136868-2.7308860.0080MA(3)0.0127070.125