一组空气污染大数据地主成分分析报告

实用文档一组空气污染数据的主成分分析【说明】下面的多元统计分析练习题摘自 R.A.Johnson 等编写的《应用多元统计分析（第五版）》，原书为：RichardA.JohnsonandDeanW.Wichern. AppliedMultivariateStatisticalAnalysis (5thEd).PearsonEducation,Inc.2003 。我看的是中国统计出版社（ChinaStatisticsPress ）2003年发行的影印本。第一题为原书第 1.6题，即第 1章的第6题，第二题为原书第 8.12题，即第 8章的第题。第二题用的是第一题的数据。习题1.6.ThedatainTable1.5are42measurementsonair-pollutionvariablesrecordedat12:00noonintheLosAngelesareaondifferentdays.(a)Plotthemarginaldotdiagramsforallthevariables.(b)Constructthex,Sn,andRarrays,andinterprettheentriesinR.TABLE1.5AIR-POLLUTIONDATASolarradiationWind(x1)(x2)CO(x3)NO(x4)NO2(x5)O3(x6)HC(7)x898721282710743953710343563108852815469142810389052121249847412155572642114478251111138645213946715410336914212737727418103107042117310724181039774191038764177387153164496742132396933953文案大全实用文档106253144498842763880421311453033523683511023488432763678421111387921710366243983103731723871411073752411284548658436754110243103541692885419102586316122586721318277974925377952862668621114384043652Source:DatacourtesyofProfessorG.C.Tiao.8.12.Considertheair-pollutiondatalistedinTable1.5.Yourjobistosummarizethesedatainfewerthanp=7dimensionsifpossible.ConductaprincipalcomponentanalysisofthedatausingboththecovariancematrixSandthecorrelationmatrixR.Whathaveyoulearned?Doesitmakeanydifferencewhichmatrixischosenforanalysis?Canthedatabesummarizedinthreeorfewerdimensions?Canyouinterprettheprincipalcomponents?部分解答2.1 部分统计参数利用Excel计算的平均值（ x）和标准差SolarWind radiation CO NO NO2 O3 HCAverage 7.5 73.857143 4.5476192.190476210.0476199.40476193.0952381Stdev1.5811388 17.3353881.23372091.08735743.37098375.56583450.6917466文案大全实用文档Excel给出的协方差矩阵 SSolarWindradiationCONONOOHC23Wind2.4404762Solarradiation-2.714286293.36054CO-0.3690483.81632651.4858277NO-0.452381-1.3537410.65759641.154195NO-0.5714296.60204082.25963721.062358311.0929712O-2.17857130.0578232.7545351-0.7913833.052154230.240933HC0.16666670.60884350.1383220.17233561.01927440.58049890.4671202Excel给出相关系数矩阵 RSolarWindradiationCONONO2O3HCWind1Solarradiation-0.1014421CO-0.1938030.18279341NO-0.269543-0.0735690.50215251NO-0.1098250.1157320.55658380.296898112O3-0.2535930.31912370.4109288-0.1339520.16664221HC0.15609790.05201040.16603230.23470430.44776780.15445061从相关系数矩阵可以看出， CO与NO、NO2相关性明显， O3与Solar radiation 、CO相关性明显。后面的主成分分析将 CO与NO、NO2归并到一个主成分，将 O3与Solarradiation归并到一个主成分，将 HC、Wind归并到一个主成分。 HC与Wind的相关系数并不高，但从正相关的角度看，二者的数值倒是最高的。方差极大正交旋转之后， HC与CO、NO、NO2归并到一个因子，因为 HC与NO2的相关系数较高，与 CO、NO的相关系数高于其他变量。2.2 主成分分析之一——数据未经标准化下面是从相关矩阵 R出发，SPSS给出的结果。原始数据未经标准化。所谓从 R出发，就是在SPSS的Factor Analysis: Extraction —Analysis 选项中选中 Correlation Matrix。SPSS给出的相关系数矩阵（ CorrelationMatrix ），与Excel计算的结果一样。文案大全实用文档CorrelationMatrixWINDSolarradiationCONONO2O3HCWIND1.000-.101-.194-.270-.110-.254.156Solarradiation-.1011.000.183-.074.116.319.052CO-.194.1831.000.502.557.411.166NO-.270-.074.5021.000.297-.134.235NO2-.110.116.557.2971.000.167.448O3-.254.319.411-.134.1671.000.154HC.156.048.1541.000公因子方差（Communalities）表如下。公因子方差变化于0.544～0.795之间，相差不是很大。但是，公因子方差值没有达到0.8以上的，可见每一个变量体现在三个主成分中的信息都不超过80%。CommunalitiesInitialExtractionWIND1.000.737Solarradiation1.000.544CO1.000.725NO1.000.795NO21.000.681O31.000.722HC1.000.722ExtractionMethod:PrincipalComponentAnalysis.特征根与方差贡献（TotalVarianceExplained）如下表。可见提取三个主成分可以解释原来7格变量的70.384%。TotalVarianceExplainedInitialEigenvaluesExtractionSumsofSquaredLoadingsComponentTotal%ofVarianceCumulative%Total%ofVarianceCumulative%12.33733.38333.3832.33733.38333.38321.38619.80053.1831.38619.80053.18331.20417.20170.3841.20417.20170.3844.72710.38780.7715.6539.33590.1066.5377.66797.7737.1562.227100.000ExtractionMethod:PrincipalComponentAnalysis.文案大全eulavnegiE

实用文档ScreePlot2.52.00.01 2 3 4 5 6 7ComponentNumber主成分载荷矩阵（ ComponentMatrix ）见下表。ComponentMat rixaWINDSolarradiationCONONO2O3HC

Component123-.362.328.706.314-.620.246.842-8.03E-03-.125.577.512-.447.796-.667.175.488.362.594ExtractionMethod:PrincipalComponentAnalysis.a.3componentsextracted.将上表从SPSS中复制到 Excel中，进行涂色分类，结果如下表所示。Component123WIND-0.362020.3278090.706084Solarradiation0.31424-0.619970.24631CO0.842417-0.00803-0.12466NO0.5772430.511736-0.44671NO0.7612940.2351830.2156822O30.496126-0.667490.175399HC0.4882570.3624660.593692主成分分类如下：文案大全实用文档第一主成分的主要相关变量： CO、NO、NO2。第二主成分的主要相关变量： Solarradiation 、O3。第三主成分的主要相关变量： Wind、HC。在主成分载荷图（ ComponentPlot）中，三个变量分别落入三个不同的主成分代表的区域。主成分得分表如下。最后一栏对几个典型的样本给出了简单的解释。注意解释的时候看清主成分载荷矩阵中载荷值的正负号。Casesf1f2f3典型的说明S10.61591-0.8186-0.38418S20.03194-0.36015-0.26343S3-0.34752-0.54481-0.49701S40.2425-0.302931.80367样本4代表的区域Wind、HC污染严重S5-0.12729-0.91941-0.4042S60.72612-0.192781.21954S72.036860.899821.4607样本7和8代表的区域与CO、NO、NO污染有明显2S82.573090.77732-0.34124的关系S90.09802-0.817360.30334S100.506640.788030.88735S110.39040.97744-1.48345S120.14485-0.45848-0.27016S131.924770.88883-0.66029S14-0.506620.631390.91242S15-0.89378-0.170361.19632S16-0.66037-0.398620.93758文案大全实用文档S17-0.87787-0.36350.3701S180.887331.53060.65731S19-0.429351.092530.48155S20-0.7510.924240.11384S21样本21代表的区域Solarradiation、O污染较30.428261.961331.18659小S22-0.69373-0.097470.51522S230.414840.206811.21242S24-1.162631.39047-2.12097S250.86691-1.703350.91799S26-0.91899-0.139150.18106S270.09994-0.51948-0.37202S28-1.32458-0.69110.65186S29-0.104720.39184-1.08681S30-1.85931.379330.6047S31-0.62672-0.083470.47051S32-0.142640.649410.72066S330.674211.56899-2.63096样本33代表的区域Wind、HC污染较小S340.24874-1.956810.22088S35-1.714290.39216-0.08554S36-0.80238-1.13269-0.0517S37-1.00653-1.92662-1.17569样本37和38代表的区域Solarradiation、O3S381.29486-1.77265-1.32357污染严重S391.68145-1.04272-0.66334S40-0.48079-0.49683-1.07633S410.72122-0.53042-0.57934S42-1.177760.98919-1.555382.3 主成分分析之二——数据未经标准化下面是从协方差矩阵 S出发，SPSS给出的结果。原始数据未经标准化。所谓从 S出发，就是在SPSS的FactorAnalysis:Extraction —Analysis 选项中选中 CovarianceMatrix 。公因子方差（Communalities）表如下。在未经处理的（Raw）公因子方差一栏，其Initial数值都是原始数据的方差。不过与前面 Excel给出的协方差矩阵有所不同， Excel给出的是总体方差，SPSS给出的是抽样方差。例如以Wind的Initial 值为例，2.4404762×42/41=2.5，或者2.5×41/42=2.4404762（对照前面的协方差矩阵） 。重标的（Rescaled）结果是 Extraction 值与Initial 值之比。文案大全实用文档CommunalitiesRawRescaledInitialExtractionInitialExtractionWIND2.5003.067E-021.0001.227E-02Solarradiation300.516300.1341.000.999CO1.5226.017E-021.0003.953E-02NO1.1826.750E-031.0005.709E-03NO211.364.1791.0001.575E-02O330.9793.8461.000.124HC.4791.667E-031.0003.484E-03ExtractionMethod:PrincipalComponentAnalysis.公因子方差的合计结果如下：RawRescaledInitialExtractionInitialExtractionWIND2.50.030665110.012266Solarradiation300.51568300.1336710.9987288CO1.52206740.060166610.0395295NO1.18234610.006750210.0057091NO211.3635310.179005910.0157527O330.9785133.845942810.1241487HC0.47851340.001667110.0034839合计348.54065304.2578671.1996188特征根与方差贡献（TotalVarianceExplained）如下表。在Raw一栏中显示，提取一个主成分似乎可以解释原来7格变量的87.295%。但重标之后显示的数值却是17.137%。根据公因子方差表和合计结果，重标之前，全部的方差解释为304.25786/348.54065*100=87.295% ；重标之后，全部的方差解释为1.1996188/7*100 ＝17.137%。文案大全实用文档TotalVarianceExplainedaExtractionSumsofSquaredLoadingsInitialEigenvaluesComponentTotal%ofVarianceCumulative%Total%ofVarianceCumulative%Raw1304.25887.29587.295304.25887.29587.295228.2768.11395.408311.4643.28998.69742.524.72499.42151.280.36799.7886.529.15299.9407.2106.014E-02100.000Rescaled1304.25887.29587.2951.20017.13717.137228.2768.11395.408311.4643.28998.69742.524.72499.42151.280.36799.7886.529.15299.9407.2106.014E-02100.000ExtractionMethod:PrincipalComponentAnalysis.a.Whenanalyzingacovariancematrix,theinitialeigenvaluesarethesameacrosstherawandrescaledsolution.ScreePloteulavnegiE

40030020010001 2 3 4 5 6 7ComponentNumber主成分载荷矩阵（ ComponentMatrix ）见下表。可以看来，由于变量 Solarradiation的方差很大，它绝对地控制了第一主成分。文案大全实用文档ComponentMatrix

aWINDSolarradiationCONONO2O3HC

RawRescaledComponeComponentnt11-.175-.11117.324.999.245.199-.082-.076.423.1261.961.352.041.059ExtractionMethod:PrincipalComponentAnalysis.a.1componentsextracted.2.4 主成分分析之三——数据经过标准化下面是从协方差矩阵 S出发，SPSS给出的结果。原始数据经过标准化。可以看到所有的结果重标前后一样，并且与从相关矩阵 R出发计算的结果一样。公因子方差（ Communalities）表如下，重标前后的结果一样。CommunalitiesRawRescaledInitialExtractionInitialExtractionWIND1.000.7371.000.737Solarradiation1.000.5441.000.544CO1.000.7251.000.725NO1.000.7951.000.795NO21.000.6811.000.681O31.000.7221.000.722HC1.000.7221.000.722ExtractionMethod:PrincipalComponentAnalysis.特征根与方差贡献（ TotalVarianceExplained ）如下表。重标前后结果一样。文案大全eulavnegiE

实用文档TotalVarianceExplainedaExtractionSumsofSquaredLoadingsInitialEigenvaluesComponentTotal%ofVarianceCumulative%Total%ofVarianceCumulative%Raw12.33733.38333.3832.33733.38333.38321.38619.80053.1831.38619.80053.18331.20417.20170.3841.20417.20170.3844.72710.38780.7715.6539.33590.1066.5377.66797.7737.1562.227100.000Rescaled12.33733.38333.3832.33733.38333.38321.38619.80053.1831.38619.80053.18331.20417.20170.3841.20417.20170.3844.72710.38780.7715.6539.33590.1066.5377.66797.7737.1562.227100.000ExtractionMethod:PrincipalComponentAnalysis.a.Whenanalyzingacovariancematrix,theinitialeigenvaluesarethesameacrosstherawandrescaledsolution.ScreePlot2.52.00.01 2 3 4 5 6 7ComponentNumber主成分载荷矩阵（ ComponentMatrix）见下表，重标前后一样。可以看到，第一主成分的相对重要性受到标准化的极大影响。 结论自然是：如果在极其不同的范围内测量变量， 或者测量单位的量纲不同，变量必须经过标准化。否则，应该从相关系数矩阵出发开展主成分分析。文案大全实用文档ComponentMatrixaRawRescaledComponentComponent123123WIND-.362.328.706-.362.328.706Solarradiation.314-.620.246.314-.620.246CO.842-.008-.125.842-.008-.125NO.577.512-.447.577.512-.447NO2.761.235.216O3.496-.667.175.496-.667.175HC.488.362.594.488.362.594ExtractionMethod:PrincipalComponentAnalysis.a.3componentsextracted.ComponentPlot1.0no hcno2 wind.5coComponent20.0solaro3radiation-.51.00.00.0-.5-.5Component1Component32