ARIMA模型預(yù)測(cè).doc

5 ARIMA模型預(yù)測(cè)5.1 模型選取目前，學(xué)術(shù)界較為成熟的預(yù)測(cè)方法很多，各種不同的預(yù)測(cè)方法有其所面向的特定對(duì)象，不存在一種普遍“最好”的預(yù)測(cè)方法。GM（1,1）模型預(yù)測(cè)是以灰色系統(tǒng)理論為基礎(chǔ)，通過(guò)原始數(shù)據(jù)的分析處理和建立灰色模型，對(duì)系統(tǒng)未來(lái)狀態(tài)作出科學(xué)的定量預(yù)測(cè)的一種方法。我們采用GM（1,1）模型是基于以下兩方面的考慮：第一，GM（1,1）模型對(duì)數(shù)據(jù)要求較低，而其他多數(shù)預(yù)測(cè)方法以數(shù)理統(tǒng)計(jì)為基礎(chǔ)，對(duì)樣本量有較高要求。我們用來(lái)做預(yù)測(cè)的數(shù)據(jù)時(shí)序只有14年，預(yù)測(cè)使用GM（1,1）模型較好；第二，GM（1,1）模型的計(jì)算量相對(duì)較小，計(jì)算方法相對(duì)簡(jiǎn)單，適用性較好。5.2 模型假設(shè)前提1、假設(shè)未來(lái)重慶地區(qū)經(jīng)濟(jì)發(fā)展基本態(tài)勢(shì)不變；2、假設(shè)未來(lái)中央政府對(duì)重慶實(shí)施的政策方向基本不變；3、假設(shè)未來(lái)不會(huì)出現(xiàn)戰(zhàn)爭(zhēng)、瘟疫及其它不可抗拒的自然或社會(huì)因素。5.3 預(yù)測(cè)數(shù)據(jù)來(lái)源預(yù)測(cè)樣本為19972008年的重慶市農(nóng)資價(jià)格指數(shù)、化學(xué)肥料價(jià)格指數(shù)、飼料價(jià)格指數(shù)。具體預(yù)測(cè)樣本數(shù)據(jù)如下：表5.1 19972008年重慶部分農(nóng)資價(jià)格指數(shù)單位：%19971998199920002001200220032004200520062007農(nóng)資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 飼料96.6 91.0 91.9 95.8 101.1 106.6 105.6 115.7 105.5 101.1 109.2 為提高數(shù)據(jù)預(yù)測(cè)的科學(xué)性，我們以1996年（直轄前）的農(nóng)資價(jià)格為基期，假設(shè)1996年農(nóng)資產(chǎn)品價(jià)格為100元，則以后第年的農(nóng)資產(chǎn)品價(jià)格計(jì)算公式如下： 經(jīng)此換算，得到19972008年的預(yù)測(cè)樣本。其中，NZJG表示換算后的農(nóng)資，HXFL表示換算后的化肥，SL表示換算后的飼料。具體見(jiàn)下表：表5.2 19972008年轉(zhuǎn)換后的預(yù)測(cè)樣本單位：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）模型建立與檢驗(yàn)5.4.1 序列的建立設(shè)由n個(gè)原始數(shù)據(jù)組成的原始序列為x(0)(k)=x(0)(1),x(0)(2),x(0)(n)。那么可以得到四個(gè)樣本原始序列：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 級(jí)比檢驗(yàn)級(jí)別檢驗(yàn)是GM（1,1）建模的數(shù)據(jù)檢驗(yàn)，經(jīng)計(jì)算可得：NZJG級(jí)比序列= 0.904,0.932 , 1.198；HXFL地區(qū)序列= 0.874, 0.965, 1.200 ；SL地區(qū)序列= 0.910, 0.919, 1.170；都落在界區(qū)（0.7515，1.3307）內(nèi)。這表明，以上三個(gè)樣本序列均可以進(jìn)行GM（1,1）模型建模。5.4.3 模型的方程通過(guò)一次累加生成新序列：x(1)(k)=x(1)(1),x(1)(2),x(1)(n)，則GM(1,1)模型相應(yīng)的微分方程為：其中，稱(chēng)為發(fā)展灰度，為內(nèi)生控制灰度，它們是方程中重要的參數(shù)。通過(guò)求解微分方程，即可得到預(yù)測(cè)模型。由于GM（1,1）預(yù)測(cè)模型種類(lèi)較多，我們選取其中較常用的一種如下： 通過(guò)測(cè)算，我們得到三組、值和相應(yīng)的四個(gè)擬合預(yù)測(cè)方程如下表表5.3 擬合預(yù)測(cè)方程和、值擬合預(yù)測(cè)方程N(yùn)ZJG-0.04691572.875525=(105.9-)e+HXFL-0.02734869.157548=(93.6-)e+SL-0.05742164.105061=(96.6-)e+5.4.4 后驗(yàn)差檢驗(yàn)后驗(yàn)差檢驗(yàn)包括殘差的方差比（C）和小誤差概率（P）。首先算得知?dú)埐钇骄禐闅v史數(shù)據(jù)方差為其中歷史數(shù)據(jù)平均值為殘差方差為 后驗(yàn)差比值為小誤差概率為經(jīng)計(jì)算，我們同樣得到四組方差比（）和小誤差概率（）如下：表5.4 后驗(yàn)差檢驗(yàn)結(jié)果表NZJG0.273916 1HXFL0.406991SL0.1728 1 根據(jù)下列后驗(yàn)差檢驗(yàn)結(jié)果判別表，我們認(rèn)為四個(gè)模型均通過(guò)了后驗(yàn)差檢驗(yàn)，有較好的預(yù)測(cè)精度。表5.5 后驗(yàn)差檢驗(yàn)結(jié)果判別表0.350.95好0.500.80合格0.650.70勉強(qiáng)合格0.650.70不合格5.4.5 相對(duì)誤差檢驗(yàn)經(jīng)過(guò)對(duì)模型進(jìn)行擬合測(cè)算，得到以下四個(gè)相對(duì)誤差表：表5.6 NZJG GM(1,1)模型擬合相對(duì)誤差表時(shí)期實(shí)際值擬合值相對(duì)誤差（%）199895.779.70 -16.72 199989.283.53 -6.36 2008144.2 127.41 -11.64 平均相對(duì)誤=6.56%表5.7 HXFL GM(1,1)模型擬合相對(duì)誤差表時(shí)期實(shí)際值擬合值相對(duì)誤差（%）199881.872.71 -11.12 199978.974.72 -5.29 2008107.995.58 -11.42 平均相對(duì)誤差=5.41%表5.8 SL GM(1,1)模型擬合相對(duì)誤差表時(shí)期實(shí)際值擬合值相對(duì)誤差（%）199896.671.69 -18.44 199987.975.93 -6.03 2008138.8127.30 -8.28 平均相對(duì)誤差=6.26%綜上分析，我們發(fā)現(xiàn)：以上四個(gè)模型的相對(duì)誤差還是在合理的范圍內(nèi)，模型可靠，擬合精度較高，可以進(jìn)行預(yù)測(cè)。5.5 模型預(yù)測(cè)利用模型預(yù)測(cè)2009-2020年NZJG、HXFL、SL的具體數(shù)值，結(jié)果見(jiàn)表5.9。表5.9 GM(1,1)模型預(yù)測(cè)結(jié)果 單位：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經(jīng)過(guò)換算，我們可以得到2009-2020年重慶市農(nóng)資價(jià)格指數(shù)、化學(xué)肥料價(jià)格指數(shù)、飼料價(jià)格指數(shù)的預(yù)測(cè)值如下：表5.10 GM(1,1)模型預(yù)測(cè)換算結(jié)果 單位：%年份農(nóng)資價(jià)格指數(shù)化肥價(jià)格指數(shù)飼料價(jià)格指數(shù)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 附錄GM(1,1)模型模擬結(jié)果NZJGHXFL期數(shù)實(shí)際值模擬值殘差期數(shù)實(shí)際值模擬值殘差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期數(shù)實(shí)際值模擬值殘差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偏相關(guān)1階 自相關(guān)4階 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