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1、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：Richard A. Johnson and Dean W. Wichern. Applied Multivariate Statistical Analysis (5th Ed). Pearson Education, Inc. 2003。我看的是中国统计出版社(China Statistics Press)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 12 题。第二题用的是第一题的数据。1 习题1
2、.6. The data in Table 1.5 are 42 measurements on air-pollution variables recorded at 12:00 noon in the Los Angeles area on different days.(a) Plot the marginal dot diagrams for all the variables.(b) Construct the x , Sn, and R arrays, and interpret the entries in R.TABLE 1.5 AIR-POLLUTION DATAWind (
3、x1)SolarO3 (X6)HC (x7)radiation (x2) CO (x3)NO (x4)NO2(x5)89872128271074395371034356310885281546914281038905212124984741215557264211447825111113864521394671541033691421273772741810310704211731072418103977419103876417738715316449674213239693395310625314449884276388042131145303352368351102348843276367
4、8421111387921710366243983103731723871411073752411284548658436754110243103541692885419102586316122586721318277974925377952862668621114384043652Source: Data courtesy of Professor G.C. Tiao.8.12. Consider the air-pollution data listed in Table 1.5. Your job is to summarize these data in fewer than p=7
5、dimensions if possible. Conduct a principal component analysis of the data using both the covariance matrix S and the correlation matrix R. What have you learned? Does it make any difference which matrix is chosen for analysis? Can the data be summarized in three or fewer dimensions? Can you interpr
6、et the principal components?2 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Wind Solar radiationCONOno2O3HCAverage7.573.8571434.5476192.190476210.0476199.40476193.0952381Stdev1.581138817.3353881.23372091.08735743.37098375.56583450.6917466Excel 给出的协方差矩阵 SSolarWindradiationCONONO2O3HCradiation-2.714286 293.36054
7、CO-0.369048 3.8163265 1.4858277NO-0.452381 -1.353741 0.6575964 1.154195NO2-0.571429 6.6020408 2.2596372 1.0623583 11.092971O3-2.178571 30.057823 2.7545351 -0.791383 3.0521542 30.24093HC0.1666667 0.6088435 0.138322 0.1723356 1.0192744 0.5804989 0.4671202Excel 给出相关系数矩阵 RWindSolarradiationCONOno2O3HCWi
8、nd1Solar radiation-0.1014421CO-0.1938030.18279341NO-0.269543-0.0735690.50215251NO2-0.1098250.1157320.55658380.29689811O3-0.2535930.31912370.4109288-0.1339520.16664221HC0.15609790.05201040.16603230.23470430.44776780.15445061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
9、个主成分，将03与Solar radiation 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。WindSolar2.44047622.2 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: ExtractionAnalysis 选项中选中 Correlation Matr
10、ix。SPSS给出的相关系数矩阵(Correlation Matrix)，与Excel计算的结果一样。Correlation MatixWINDSolar radiationCONONO2O3HCWIND1.000-.101-.194-.270-.110-.254.156Solar radiation-.1011.000.183-.074.116.319.052CO-.194.1831.000.502.557.411.166NO-.270-.074.5021.000.297-.134.235NO2-.110.116.557.2971.000.167.44803-.254.319.411-.13
11、4.1671.000.154HC1560521662354481541 000公因子方差(Communalities)表如下。公因子方差变化于0.5440.795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%。CommunalitiesInitialExtractio nWIND1.000.737Solar radiation1.000.544CO1.000.725NO1.000.795NO21.000.681O31.000.722HC1 000722Extraction Method: Principal Compone
12、nt Analysis.特征根与方差贡献(Total Variance Explained)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%。Total Variance ExplainedComponentIni tial Eicernali iesExtraction Sums of Squared L oadincsTotal% of VarianceCumulative %Total% of VarianceCumulative %12.33733.38333.3832.33733.38333.38321.38619.80053.1831.38619.80053.18331.20417.20170.3841.20417.20170.3844.72710.38780.7715.6539.33590.1066.5377.66797.77371562 227100 000Extraction Method: Principabmponent Analysis.elvauvne主成分载荷矩阵(Component Matrix )见下表。Component Ma
