回归分析大作业

回归大作业国内旅游消费影响的回归分析一、问题引入我国第三产业发展迅速，在2010年其已占国内生产总值的43.14%，而旅游业在第三产业中占有重要地位，且与餐饮、住宿、休闲、运输等产业联系密切，所以此次分析以探究国内旅游消费的影响为目的，并建立回归模型。二、模型设计运用多元线性模型拟合，若拟合效果不显著，则进行log或平方根变换或使用多项式拟合等其他模型。1、相关性分析，首先确定与因变量有相关性的变量。2、建立全模型多元线性回归，若回归方程F检验未通过，则查找原因、更换模型；若有部分回归系数检验未通过，则进行选元（步骤2），剔除部分变量再继续；若所有检验都良好，则模型初步确立，跳过步骤2。3、运用逐步回归方法筛选变量，并进行t检验，若效果显著，则可初步确立多元线性回归模型；若仍有部分变量未通过检验，则再单独进行变量筛选，综合运用AIC准则等确定剔除变量，直至所有变量都通过t检验。4、回归诊断。进行残差分析，检验残差是否满足正态分布，是否有相关性，也即自变量间是否有自相关性，检验是否存在异常值和强影响值，是否存在异方差性，是否存在多重共线性。若以上问题存在，则需修改模型，或重新筛选变量，或增减样本。5、模型最终确立。三、数据yearincomenumberexpenselevelroadrail199448108.5524195.3320.0111.785.90199559810.5629218.7345.1115.706.24199670142.5640256.2377.6118.586.49199778060.9644328.1394.6122.646.60199883024.3695345.0417.8127.856.64199988479.2719394.0452.3135.176.74200098000.5744426.6491.0140.276.872001108068.2784449.5521.2169.807.012002119095.7878441.8557.6176.527.192003135174.0870395.7596.9180.987.302004159586.81102427.5645.3187.077.442005183618.51212436.1695.2334.527.542006215883.91394446.9761.9345.707.712007266411.01610482.6843.4358.377.802008315274.71712511.0916.8373.027.972009341401.51902535.41001.6386.088.552010403260.02103598.21062.6400.829.12yearairrailtranroadtranshiptranairtrantravel1994104.561087389539402616540391023.51995112.9010274510408102392451171375.71996116.659479711221102289555551638.41997142.509330812045832257356302112.71998150.589508512573322054557552391.21999152.2210016412690041915160942831.92000150.2910507313473921938667223175.52001155.3610515514027981864575243522.42002163.7710560614752571869385943878.42003174.959726014643351714287593442.32004204.94111764162452619040121234710.72005199.85115583169738120227138275285.92006211.35125656186048722047159686229.72007234.30135670205068022835185767770.62008246.18146193268211420334192518749.32009234.511524512779081223142305210183.72010276.511676093052738223922676912579.8数据来源：《中国统计年鉴2011》数据说明：Year：年份。Income:国民总收入，单位亿元。Number:旅游人数。Expense：人均旅游花费，单位元。Level：居民消费水平指数，以1978年为基年。Road:公路里程，单位万公里。Rail:铁路里程，单位万公里。Air：民航里程，单位万公里。Roadtran:公路客运量，单位万人。Railtran:铁路客运量，单位万人。Shiptran:水路客运量，单位万人。Airtran：民航客运量，单位万人。Travel：国内旅游消费总额，单位亿元。四、回归分析1、相关性首先分析相关性，画出散布阵。2000110000O40O500000^00^色00-000000^000^10.000E0001001000000□□BoH00aQo0GhraBsaHHHasHBa叽0TH00HH0HEa5rooo5C可较为直观地看出，travel与各变量间有较强的相关性，除了road,和shiptran两项,做相关性检验，可见,travel与road是线性相关的，相关系数为0.93,p-value=4.563e-08,而travel与shiptran不相关，p-value=0.9983，所以可先排除shiptran，再做回归。2、全回归模型直接建立多元回归模型，得结果:Coefficients:EstimateStd.ErrortvaluePr(〉|t|)(Intercept)-5.972e+033.193e+03-1.8700.110617income2.151e-024.779e-034.5010.004100**number1.039e+001.446e+000.7190.499354expense6.805e+001.124e+006.0520.000922**:level-5.815e+001.261e+00-4.6100.003653**roadT.468e+001.019e+00-1.4410.199608rail6.274e+024.462e+021.4060.209292air-4.155e+002.790e+00-1.4900.186935railtran2.524e-028.492e-032.9720.024903*roadtran-4.093e-044.554e-04-0.8990.403410airtran1.058e—011.272e—010.8320.43732711110.1Signif.codes:0‘***'0.001‘**'0.01‘*'0.050.1Residualstandarderror:84.55on6degreesoffreedomMultipleR-squared:0.9998,AdjustedR-squared:0.9994F-statistic:2462on10and6DF,p-value:5.061e-10其中，R2=0.9998,F检验的p-value:2.632e-08，可见回归模型的检验是成立的，但回归系数并不是全能通过检验，所以应该进行选元。3、选元先进行逐步回归，逐步回归排除了roadtran,number两个变量，以AIC准则为主要判断依据，调整后的AIC值为153.73，达到最小值。再检验一下回归模型：Coefficients:(Intercept)EstimateStd.ErrortvaluePr(>|t|)-4.393e+032.102e+03-2.0900.070022.income1.898e-022.320e-038.1793.72e-05***expense7.038e+009.369e-017.5126.85e-05***level-5.427e+001.057e+00-5.1330.000893***road-1.460e+009.339e-01-1.5640.156518rail3.697e+022.865e+021.2900.232935air-3.589e+002.496e+00-1.4380.188431railtran2.166e-026.843e-033.1650.013295*airtran2.032e-015.464e-023.7190.005879**Signif.codes:0‘***'0.001‘**'0.01‘*'0.05‘.'0.1‘'1Residualstandarderror:78.95on8degreesoffreedomMultipleR-squared:0.9997,AdjustedR-squared:0.9994F-statistic:3529on8and8DF,p-value:2.252e-13可见回归模型改善，自由度调整负相关系数达到了0.9994，有所提高，这与AIC准则的判断相符，而回归系数的检验也有所好转，但仍然有road，rail，air通不过检验。若去掉一个变量回归，可见：DfSumofSqRSSAIC<none>49866153.73income1416943466809189.75expense1351763401629187.19level1164237214103176.50road11524165107156.26rail11038060246154.94air11288662752155.63railtran162438112303165.53airtran186215136081168.79去掉rail,AIC增加最小，同时RSS增加最小，而回归方程系数检验:Coefficients:(Intercept)EstimateStd.ErrortvaluePr(>|t|)-1.773e+035.648e+02-3.1400.011936*income1.935e-022.386e-038.1121.98e-05***expense7.977e+006.116e-0113.0433.77e-07***level-5.126e+001.069e+00-4.7970.000978***road-2.214e+007.550e-01-2.9330.016676*air-5.129e+002.272e+00-2.2570.050398.railtran1.495e-024.613e-033.2410.010144*airtran2.603e-013.323e-027.8322.62e-05***只有air一项在a=0.05的情况下是不能通过检验的，若排除air,则:Coefficients:EstimateStd.ErrortvaluePr(>|t|)(Intercept)-2.450e+035.683e+02-4.3100.00154**income1.834e-022.782e-036.5936.13e-05***expense7.465e+006.742e-0111.0726.21e-07***level-5.389e+001.261e+00-4.2730.00163**road-2.381e+008.921e-01-2.6690.02355*railtran1.933e-024.970e-033.8890.00301**airtran2.451e-013.864e-026.3438.42e-05***所有回归系数通过检验，回归模型初步确立。4、回归诊断计算得出残差，进行W正态性检验，得到p-value=0.9066，不能拒绝正态性假设。而回归值与标准化残差的残差图为:pepulepepule电20004000600080001000012000从图中也可看出，残差分布均匀且无规律，所以线性回归的基本假设满足，且没有自相关性。而再看:「7011「7011「7011「7011Resid7SFittedcfl-rnnpwENormalQ-Q3-l2np-30」P0z-P」fi3pul31a)C15-1-2-1Th&oreticalQuantilesIm(travel-income+expense+■level■+road+railtran+airIran)Scale-LocalionP-I2np-3O」poN-p」Scale-LocalionP-I2np-3O」poN-p」(T3puraFittedvaluesIm(travel-income+expense+level■+road+railtran+airIran)Cooktdistance15UDaoUEfpw-^000upo1011151015Obs.numbIm(travel-income+expense+■level■+road+railtran+airIran)综合看上面四幅图，11和15号观测值可能为强影响值，但产生原因还需要探究，可能是统计过程上的，亦可能是分析方法上的，去掉后回归效果减弱，所以暂不剔除。再检验多重共线性，kappa=1346.411〉1000，所以存在多重共线性，接近零的特征值及其相应特征向量为：0.004087919，[,6][1,]0.74512169[2,]0.07020978[3,]-0.60233849[4,]0.13346499[5,]-0.14256057[6,]-0.197271830.005567391[,5][1,]-0.264478984[2,]0.115775260[3,]-0.550564160[4,]0.004567634[5,]-0.073879174[6,]0.779773728可见，1,3,6之间即income与level,airtran之间可能存在严重的多重共线性关系，更可能的是在income与level之间，这在经济意义上也可以理解，国民收入越高，消费水平越高，而坐飞机的人才越多，前两者关系更直接。所以引起原因可能是有多余的自变量，分别去掉income,level，airtran做回归，并计算kappa值。从结果知，不管去掉哪一个，kappa值均减少一半左右，而只有去掉level时，回归方程几乎无影响，Coefficients:(Intercept)EstimateStd.ErrortvaluePr(〉|t|)-3.824e+037.511e+02-5.0910.000349***income1.217e-023.811e-033.1940.008552**expense5.483e+007.843e-016.9912.3e-05***road-4.247e+001.247e+00-3.4070.005855**railtran2.708e-027.416e-033.6510.003811**airtran1.929e-015.876e-023.2840.007288**Signif.codes:0‘***'0.001‘**'0.01‘*'0.05‘.'0.1‘'1Residualstandarderror:155.7on11degreesoffreedomMultipleR-squared:0.9985,AdjustedR-squared:0.9978F-statistic:1450on5and11DF,p-value:4.078e-15所以可以剔除level。再做一下异方差性的检验，用等级相关系数法，计算残差的绝对值与自变量间的等级相关系数，分别为0.2156863，0.05637255，0.2156863，0，0.2156863发现并无相关的，所以模型拟合良好。5、模型确立

Travel=-3.824e+03+1.217e-02*income+5.483*expense-4.247*road+2.708e-02*railtran+1.929e-01*airtran五、模型评注从模型来看，国内旅游消费量可由国民收入、人均旅游花费、铁路客运量、民航客运量、公路里程来建模模拟预测，这与实际意义相符。前两者可归纳为人民生活水平，后三者是国家交通建设方面，而恰恰包括了公路、铁路、航空三个方面。所以回归方程的建立与其实际意义大致相符，影响因素也基本确定。但是受开始自变量选择的影响，有可能存在重要变量为选入。六、程序代码及输出（编程语言：R）x=read.csv（"数据.csv",head=T）a=x[,2:13]plot（a）^raH0HH00E0EEE柠H0HHs00H0o0a仙细aEHH0a0H0H00sBTaHBlarasaBB顶u」s0H000HEHsashYrfTTmTHInTTTMTtrniT^-0oo^raH0HH00E0EEE柠H0HHs00H0o0a仙细aEHH0a0H0H00sBTaHBlarasaBB顶u」s0H000HEHsashYrfTTmTHInTTTMTtrniT^-0oo启§Ag0OOQ0-000^-D貝1000001&000HHHH0HH0HHTEOOO1001000000/*相关性检验*/>cor.test(road,travel)/*相关性检验*/Pearson'sproduct—momentcorrelationdata:roadandtravelt=10.0692,df=15,p-value=4.563e-08alternativehypothesis:truecorrelationisnotequalto095percentconfidenceinterval:0.82099800.9761007sampleestimates:cor0.9333393>cor.test(shiptran,travel)Pearson'sproduct-momentcorrelationdata:shiptranandtravelt=0.0021,df=15,p-value=0.9983alternativehypothesis:truecorrelationisnotequalto095percentconfidenceinterval:-0.48022170.4810676sampleestimates:cor0.0005500457>model=lm(travel~income+number+expense+level+road+rail+air+railtran+roadtran+airtran)>summary(model)/*建立回归模型*/Call:lm(formula=travel~income+number+expense+level+road+rail+air+railtran+roadtran+airtran)Residuals:Min1QMedian3QMax-72.549-44.8603.56244.80690.603Coefficients:EstimateStd.ErrortvaluePr(>|t|)(Intercept)-5.972e+033.193e+03-1.8700.110617income2.151e-024.779e-034.5010.004100**number1.039e+001.446e+000.7190.499354expense6.805e+001.124e+006.0520.000922**level-5.815e+001.261e+00-4.6100.003653**road-1.468e+001.019e+00-1.4410.199608rail6.274e+024.462e+021.4060.209292air-4.155e+002.790e+00-1.4900.186935railtran2.524e-028.492e-032.9720.024903*roadtran-4.093e-044.554e-04-0.8990.403410airtran1.058e-011.272e-010.8320.43732711110.1Signif.codes:0‘***'0.001‘**'0.01‘*'0.050.1Residualstandarderror:84.55on6degreesoffreedomMultipleR-squared:0.9998,AdjustedR-squared:0.9994F-statistic:2462on10and6DF,p-value:5.061e-10>model1=step(model)/*逐步回归*/Start:AIC=155.17travel~income+number+expense+level+road+rail+air+railtran+roadtran+airtranDfSumofSqRSSAIC-number1369346589154.57-airtran1494847844155.02<none>42897155.17-roadtran1577548671155.31-rail11413757033158.01-road11485057746158.22-air11586258758158.52-railtran163136106033168.55-income1144834187731178.26-level1151949194845178.90-expense1261858304755186.50Step:AIC=154.57travel~income+expense+level+road+rail+air+railtran+roadtran+airtranDfSumofSqRSSAIC-roadtran1327649866153.73<none>46589154.57-rail11173558325156.39-air11565762246157.50-road11700963598157.86-airtran158169104758166.34-railtran164855111444167.40-income1148468195057176.91-level1163524210114178.18-expense1353482400071189.12Step:AIC=153.73travel~income+expense+level+road+rail+air+railtran+airtranDfSumofSqRSSAIC<none>49866153.73-rail11038060246154.94-air11288662752155.63-road11524165107156.26-railtran162438112303165.53-airtran186215136081168.79-level1164237214103176.50-expense1351763401629187.19-income1416943466809189.75>summary(model1)Call:lm(formula=travel~income+expense+level+road+rail+air+railtran+airtran)Residuals:Min1QMedian3QMax-66.673-57.7662.79646.74991.039Coefficients:(Intercept)EstimateStd.ErrortvaluePr(>|t|)-4.393e+032.102e+03-2.0900.070022.income1.898e-022.320e-038.1793.72e-05***expense7.038e+009.369e-017.5126.85e-05***level-5.427e+001.057e+00-5.1330.000893***road-1.460e+009.339e-01-1.5640.156518rail3.697e+022.865e+021.2900.232935air-3.589e+002.496e+00-1.4380.188431railtran2.166e-026.843e-033.1650.013295*airtran2.032e-015.464e-023.7190.005879**Signif.codes:0‘***'0.001‘**'0.01‘*'0.05‘.'0.1‘'1Residualstandarderror:78.95on8degreesoffreedomMultipleR-squared:0.9997,AdjustedR-squared:0.9994F-statistic:3529on8and8DF,p-value:2.252e-13>model2=drop1(model1)/*减少一个变量做回归*/>model2SingletermdeletionsModel:travel~income+expense+level+road+rail+air+railtran+airtranDfSumofSqRSSAIC<none>49866153.73income1416943466809189.75expense1351763401629187.19level1164237214103176.50road11524165107156.26rail11038060246154.94air11288662752155.63railtran162438112303165.53airtran186215136081168.79model3=update(modell,.~.-rail)/*剔除rail*/summary(model3)Call:lm(formula=travel~income+expense+level+road+air+railtran+airtran)Residuals:Min1QMedian3QMax-77.120-62.739-7.68257.07396.157Coefficients:(Intercept)EstimateStd.ErrortvaluePr(>|t|)-1.773e+035.648e+02-3.1400.011936*income1.935e-022.386e-038.1121.98e-05***expense7.977e+006.116e-0113.0433.77e-07***level-5.126e+001.069e+00-4.7970.000978***road-2.214e+007.550e-01-2.9330.016676*air-5.129e+002.272e+00-2.2570.050398.railtran1.495e-024.613e-033.2410.010144*airtran2.603e-013.323e-027.8322.62e-05***Signif.codes:0‘***'0.001‘**'0.01‘*'0.05‘.'0.1‘'1Residualstandarderror:81.82on9degreesoffreedomMultipleR-squared:0.9997,AdjustedR-squared:0.9994F-statistic:3756on7and9DF,p-value:7.348e-15model4=update(model3,.~.-air)summary(model4)Call:lm(formula=travel~income+expense+level+road+railtran+airtran)

Residuals:Min1QMedian3QMax-165.78-44.4312.8649.24123.92Coefficients:EstimateStd.ErrortvaluePr(>|t|)(Intercept)-2.450e+035.683e+02-4.3100.00154**income1.834e-022.782e-036.5936.13e-05***expense7.465e+006.742e-0111.0726.21e-07***level-5.389e+001.261e+00-4.2730.00163**road-2.381e+008.921e-01-2.6690.02355*railtran1.933e-024.970e-033.8890.00301**airtran2.451e-013.864e-026.3438.42e-05***Signif.codes:0‘***'0.001‘**'0.01‘*'0.05‘.'0.1‘'1Residualstandarderror:97.14on10degreesoffreedomMultipleR-squared:0.9995,AdjustedR-squared:0.9991F-statistic:3108on6and10DF,p-value:9.282e-16>resid=resid(model4)>resid61234532.124983-8.71978212.857759-83.08903650.68695747.681664789101112-54.769913-28.532399123.92000880.382480-165.78294633.0082931314151617-28.870183-44.425769-112.19928996.48354949.243624>shapiro.test(resid)/*W正态性检验*/Shapiro-Wilknormalitytestdata:residW=0.9756,p-value=0.9066>y=predict(model4)>rstandard=rstandard(model4)>plot(y,rstandard)FittedvaluesFittedvalueslm(travel~income十expense+level十road+railtran十airtran)pepulepepule电20004000600080001000012000plot(model4,l)plot(model4,2)plot(model4,3)plot(model4,4)ResidualsvsFittedOSL00L090OS—S(unp_s3虫09oogOSL00L090OS—S(unp_s3虫09oog■'C—IS-°1120004000°1120004000600080001000012000NormalQ-Qs_(unpas_(unpa<D」P3NP」(UPU-2S-2-1012TheoreticalQuantileslm(travel-income+expense十level+road+railtran+airtran)S・L0・LSO-S(unp-S3S・L0・LSO-S(unp-S3」P3Z-P」PPU2S-P00Cook'sdistance15101510Obs.numberlm(trave-l-income十expense十level十「o日对+railtran十airtran)attach(x)aa=data.frame(travel,income,expense,level,road,raiItran,airtran)b=aa[,2:7]bb=cor(b)kappa(bb,exact=T)/*计算kappa值*/[1]1366.411eigen(bb)/*求解矩阵特征值及特征向量*/$values[1]5.5857780070.2846304690.0915242580.0284119570.005567391[6]0.004087919$vectors[,1][,2][,3][,4][,5][1,]-0.4203320-0.112070060.15936640.4002601-0.264478984[2,]-0.37471550.861240140.1616264-0.27091930.115775260[3,]-0.42105000.07658249-0.09816320.3758770-0.550564160[4,]-0.4080864-0.16947687-0.7878004-0.40777420.004567634[5,]-0.4033123-0.433640390.5637918-0.5528584-0.073879174[6,]-0.4200367-0.151902770.01877780.39138070.779773728[,6][1,]0.74512169[2,]0.07020978[3,]-0.60233849[4,]0.13346499[5,]-0.14256057[6,]-0.19727183bbb=kappa(cor(b[,colnames⑹！="level"]))/*去掉变量level后求kappa值*/bbb[1]529.9542kappa(cor(b[,colnames(b)！="income"]))[1]537.9962kappa(cor(b[,colnames(b)！="airtran"]))[1]624.6458summary(update(model4,.~.-level))Call:lm(formula=travel~income+expense+road+railtran+airtran)Residuals:Min1QMedian3QMax-322.63-58.04-11.6295.45214.69Coefficients:EstimateStd.ErrortvaluePr(>|t|)(Intercept)-3.824e+037.511e+02-5.0910.000349***income1.217e-023.811e-033.1940.008552**expense5.483e+007.843e-016.9912.3e-05***road-4.247e+001.247e+00-3.4070.005855**railtran2.708e-027.416e-033.6510.003811**airtran1.929e-015.876e-023.2840.007288**Signif.codes:0‘***'0.001‘**'0.01‘*'0.05‘.'0.1‘'1Residualstandarderror:155.7on11degreesoffreedomMultipleR-squared:0.9985,AdjustedR-squared:0.9978F-statistic:1450on5and11DF,p-value:4.078e-15>summary(update(model4,.~.-income))Call:lm(formula=travel~expense+level+road+railtran+airtran)Residuals:Min1QMedian3QMax-466.59-84.23-15.25150.26246.75Coefficients:EstimateStd.ErrortvaluePr(>|t|)(Intercept)-4.844e+039.639e+02-5.0250.000387***expense7.056e+001.480e+004.7670.000583***level-1.075e+002.377e+00-0.4520.659854road-4.336e+001.855e+00-2.3370.039350*railtran3.611e-029.409e-033.8380.002755**airtran3.690e-017.443e-024.9580.000430***Signif.codes:0‘***'0.001‘**'0.01‘*'0.05‘.'0.1‘'1Residualstandarderror:214.1on11degreesoffreedomMultipleR-squared:0.9971,AdjustedR-squared:0.9958F-statistic:765.5on5and11DF,p-value:1.357e-13>summary(update(model4,.~.-airtran))Call:lm(formula=travel~income+expense+level