主成分分析
叶田
Monday, March 16, 2015
1.使用princomp函数，对比将参数设置为cor=T OR F的结果
setwd("C:/Users/lenovo/Desktop")
X<-read.table("d7.2.txt",header=T)
attach(X)
cor(X)#变量之间相关性较强，适合做主成分分析## X1 X2 X3 X4 X5 X6 X7
## X1 1.0000000 0.2569697 0.7252526 0.3853672 0.8990457 0.8284572 0.7145260
## X2 0.2569697 1.0000000 0.4537807 0.5765121 0.3575064 0.5420120 0.4045314
## X3 0.7252526 0.4537807 1.0000000 0.5831419 0.7823418 0.8924742 0.7744004
## X4 0.3853672 0.5765121 0.5831419 1.0000000 0.4665789 0.6291140 0.6911234
## X5 0.8990457 0.3575064 0.7823418 0.4665789 1.0000000 0.8795439 0.7853531
## X6 0.8284572 0.5420120 0.8924742 0.6291140 0.8795439 1.0000000 0.8133081
## X7 0.7145260 0.4045314 0.7744004 0.6911234 0.7853531 0.8133081 1.0000000
## X8 0.7218909 0.6277509 0.7220538 0.6254195 0.7517683 0.8435436 0.7183218
## X8
## X1 0.7218909
## X2 0.6277509
## X3 0.7220538
## X4 0.6254195
## X5 0.7517683
## X6 0.8435436
## X7 0.7183218
## X8 1.0000000PCA1=princomp(X,cor=T)#主成分分析,用相关系数阵，标准化数据
PCA1#标准差代表yi的特征值开根号## Call:
## princomp(x = X, cor = T)
##
## Standard deviations:
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7
## 2.3877119 1.0142326 0.7101294 0.5222697 0.4314432 0.4015967 0.2955459
## Comp.8
## 0.2415456
##
## 8 variables and 31 observations.PCA2=princomp(X,cor=F)#主成分分析，用协方差矩阵，未标准化数据
PCA2#系数较大## Call:
## princomp(x = X, cor = F)
##
## Standard deviations:
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6
## 1129.24848 323.03218 199.26641 153.00821 138.80794 92.76297
## Comp.7 Comp.8
## 68.40697 53.71349
##
## 8 variables and 31 observations.summary(PCA1)## Importance of components:
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
## Standard deviation 2.387712 1.0142326 0.71012939 0.52226971 0.43144321
## Proportion of Variance 0.712646 0.1285835 0.06303547 0.03409571 0.02326791
## Cumulative Proportion 0.712646 0.8412295 0.90426494 0.93836065 0.96162855
## Comp.6 Comp.7 Comp.8
## Standard deviation 0.40159675 0.29554587 0.241545582
## Proportion of Variance 0.02015999 0.01091842 0.007293034
## Cumulative Proportion 0.98178855 0.99270697 1.000000000summary(PCA2)#第一主成分解释度较好## Importance of components:
## Comp.1 Comp.2 Comp.3 Comp.4
## Standard deviation 1129.2484798 323.03217820 199.26641417 153.00821309
## Proportion of Variance 0.8627261 0.07059687 0.02686347 0.01583884
## Cumulative Proportion 0.8627261 0.93332294 0.96018641 0.97602525
## Comp.5 Comp.6 Comp.7 Comp.8
## Standard deviation 138.80794361 92.762973040 68.406974154 53.713492273
## Proportion of Variance 0.01303535 0.005821611 0.003165881 0.001951914
## Cumulative Proportion 0.98906059 0.994882205 0.998048086 1.000000000PCA1$loadings##
## Loadings:
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8
## X1 -0.353 -0.429 0.175 0.299 -0.377 0.651
## X2 -0.249 0.677 0.521 -0.399 -0.129 0.134
## X3 -0.374 -0.789 0.261 0.116 0.372
## X4 -0.302 0.472 -0.628 0.225 0.249 -0.416
## X5 -0.376 -0.324 0.123 0.127 -0.281 -0.267 -0.695 0.298
## X6 -0.404 -0.200 0.132 -0.156 -0.857
## X7 -0.371 -0.442 -0.584 0.535 0.166
## X8 -0.374 0.118 0.282 0.409 0.522 0.546 0.141
##
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8
## SS loadings 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
## Proportion Var 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125
## Cumulative Var 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000PCA2$loadings#主成分载荷很多没显示##
## Loadings:
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8
## X1 -0.710 0.539 -0.435 -0.117
## X2 -0.445 -0.276 0.615 -0.486 -0.261 -0.118 -0.155
## X3 -0.119 -0.169 0.301 -0.119 -0.889 0.236
## X4 -0.407 -0.247 -0.511 -0.399 0.579
## X5 -0.530 -0.101 0.798 -0.217 0.117
## X6 -0.394 -0.491 -0.160 0.114 0.658 0.111 0.308 -0.159
## X7 -0.162 -0.225 -0.571 -0.122 -0.746 0.154
## X8 -0.124 0.259 0.943
##
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8
## SS loadings 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
## Proportion Var 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125
## Cumulative Var 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000library(mvstats)
princomp.rank(PCA1,m=2,plot=T)
## Comp.1 Comp.2 PC rank
## 北京 -6.12230233 1.5225207 -4.95377713 2
## 天津 -3.01010636 0.5367841 -2.46795766 5
## 河北 0.88750148 0.6923451 0.85767147 16
## 山西 1.10374781 0.6013737 1.02695901 19
## 内蒙古 -0.53334901 1.8477269 -0.16939720 10
## 辽宁 -0.09437659 0.6551539 0.02019052 11
## 吉林 0.32707448 1.4246843 0.49484615 13
## 黑龙江 1.68861172 0.9958838 1.58272699 28
## 上海 -7.08467075 -1.0693203 -6.16521342 1
## 江苏 -1.14131407 -0.4536945 -1.03621016 7
## 浙江 -3.82110679 0.1721339 -3.21073265 4
## 安徽 1.12338014 -0.3518062 0.89789520 17
## 福建 -1.17171809 -1.3776085 -1.20318881 6
## 江西 1.66938246 -0.5484835 1.33037759 26
## 山东 -0.48111168 0.8084609 -0.28399815 8
## 河南 1.27722788 0.6479033 1.18103447 21
## 湖北 1.00945390 -0.1165043 0.83734914 15
## 湖南 0.36506902 0.2007324 0.33994986 12
## 广东 -4.03195586 -2.4804874 -3.79481107 3
## 广西 1.62739176 -1.2305704 1.19054702 22
## 海南 1.87311537 -2.3528197 1.22717346 24
## 重庆 -0.39403198 0.4623324 -0.26313511 9
## 四川 1.15376310 -0.5180723 0.89822000 18
## 贵州 2.01403098 -0.6594701 1.60538145 29
## 云南 2.42950002 -0.4177647 1.99429039 30
## 西藏 2.72036533 -1.0105476 2.15008847 31
## 陕西 0.88797732 0.1169535 0.77012492 14
## 甘肃 1.32452136 0.1444992 1.14415281 20
## 青海 1.76845611 0.2089337 1.53008025 27
## 宁夏 1.31733038 0.4960324 1.19179348 23
## 新疆 1.31814288 1.0526951 1.27756871 25princomp.rank(PCA2,m=1,plot=T)
## PC rank
## 北京 -2260.23573 -2260.23573 3
## 天津 -881.18567 -881.18567 5
## 河北 763.03936 763.03936 24
## 山西 835.11384 835.11384 26
## 内蒙古 498.54214 498.54214 19
## 辽宁 115.55714 115.55714 9
## 吉林 727.24553 727.24553 23
## 黑龙江 1044.15077 1044.15077 31
## 上海 -3629.69562 -3629.69562 1
## 江苏 -549.47579 -549.47579 7
## 浙江 -2105.72900 -2105.72900 4
## 安徽 343.49144 343.49144 15
## 福建 -874.55179 -874.55179 6
## 江西 662.25145 662.25145 22
## 山东 154.59504 154.59504 11
## 河南 916.00562 916.00562 29
## 湖北 293.37528 293.37528 13
## 湖南 308.83977 308.83977 14
## 广东 -2433.28370 -2433.28370 2
## 广西 385.01137 385.01137 17
## 海南 132.09115 132.09115 10
## 重庆 -89.77927 -89.77927 8
## 四川 178.58360 178.58360 12
## 贵州 624.15707 624.15707 21
## 云南 382.36226 382.36226 16
## 西藏 407.63299 407.63299 18
## 陕西 543.91420 543.91420 20
## 甘肃 802.07876 802.07876 25
## 青海 921.71498 921.71498 30
## 宁夏 898.28168 898.28168 28
## 新疆 885.90113 885.90113 27参数设置为cor=T，表示使用相关系数矩阵结果进行主成分分析，即将数据进行标准化。此时，按照累计方差贡献率需大于等于80%的原则，需要选出两个主成分；参数设置为cor=F，表示使用协方差矩阵结果进行主成分分析，不难看出，此时只需选定一个主成分即可。在综合得分排序时使用协方差矩阵的结果也与使用相关系数矩阵结果的主成分分析结果有一些不同。
2.用谱分解方法进行分析
未标准化数据
cov<-cov(d7.2)
e<-eigen(cov,symmetric=T)
e$value## [1] 1317708.867 107828.114 41030.674 24191.897 19909.900 8891.801
## [7] 4835.498 2981.311sqrt(e$value)## [1] 1147.91501 328.37191 202.56030 155.53745 141.10245 94.29635
## [7] 69.53774 54.60138e$vectors## [,1] [,2] [,3] [,4] [,5]
## [1,] -0.70954291 0.5390663 -0.43521118 -0.001876291 0.11702489
## [2,] -0.07029417 -0.4450158 -0.27598131 -0.615031769 0.48628388
## [3,] -0.11888696 -0.1689435 -0.03245231 0.073167828 -0.30116231
## [4,] -0.09219859 -0.4071370 -0.24702945 0.511323006 0.39909910
## [5,] -0.52980695 -0.1010012 0.79830426 -0.097734231 0.21745160
## [6,] -0.39382287 -0.4910485 -0.15965404 -0.113735783 -0.65780625
## [7,] -0.16166948 -0.2245413 0.01756632 0.570593282 0.12190802
## [8,] -0.08392865 -0.1237633 -0.09623489 -0.082844149 0.07070014
## [,6] [,7] [,8]
## [1,] -0.00664234 0.03982262 -0.034918618
## [2,] -0.26100921 0.11761063 -0.155434379
## [3,] -0.11872592 0.88918196 0.235538025
## [4,] 0.57937677 0.08219183 -0.039383367
## [5,] 0.11718826 0.03442519 0.007635471
## [6,] 0.11058302 -0.30823184 -0.159389887
## [7,] -0.74554493 -0.15397686 -0.056135935
## [8,] -0.01605792 -0.25938582 0.942855744as.matrix(d7.2) %*%e$vectors#Y矩阵## [,1] [,2] [,3] [,4] [,5] [,6]
## 北京 -6271.737 -472.14218 -1478.7720 -47.721642 670.0846 -197.4040
## 天津 -4892.687 -80.69336 -1680.0065 468.804807 642.5301 -440.8240
## 河北 -3248.461 -142.58568 -1052.8677 161.459560 730.6497 -325.9395
## 山西 -3176.387 -287.80663 -952.9766 27.908060 602.0532 -487.2531
## 内蒙古 -3512.959 -480.82319 -1140.9499 -208.459846 700.9511 -473.6798
## 辽宁 -3895.944 128.15536 -1424.9970 192.599398 818.0546 -381.6057
## 吉林 -3284.255 -251.59105 -1253.5557 143.034021 790.3206 -451.8751
## 黑龙江 -2967.350 -142.87870 -1189.9996 8.172715 651.5781 -302.3414
## 上海 -7641.196 165.85432 -1230.5422 -188.637492 583.0494 -407.1886
## 江苏 -4560.977 35.31018 -1425.7811 18.149052 319.9015 -395.0486
## 浙江 -6117.230 -80.43122 -1145.7491 -245.495957 692.1764 -330.7203
## 安徽 -3668.009 222.85883 -1365.1952 -6.957738 467.6136 -398.1544
## 福建 -4886.053 391.30589 -1233.3928 86.680723 557.8603 -659.8796
## 江西 -3349.249 305.33829 -1342.2211 -119.420965 424.4819 -460.7248
## 山东 -3856.906 -227.33267 -1061.3222 -63.375078 712.8425 -495.6177
## 河南 -3095.495 -119.36938 -1120.7817 -53.671806 602.3980 -388.7157
## 湖北 -3718.126 226.61657 -1406.8402 -88.134677 538.4257 -470.1937
## 湖南 -3702.661 -43.40191 -1309.8025 -18.580840 457.1074 -367.0418
## 广东 -6444.785 253.07907 -608.0827 206.794612 572.2519 -427.8501
## 广西 -3626.489 411.41245 -1245.4843 127.464766 415.1013 -315.9280
## 海南 -3879.410 654.40143 -830.3841 228.024851 548.1847 -268.6813
## 重庆 -4101.280 64.21307 -1443.1986 -11.589613 651.8433 -438.0735
## 四川 -3832.917 397.68453 -1287.4281 -140.710939 575.5561 -321.1221
## 贵州 -3387.344 263.03119 -1175.8815 -167.166245 428.9832 -428.4575
## 云南 -3629.139 609.53498 -1244.3362 -3.613962 857.7951 -220.6917
## 西藏 -3603.868 1032.87564 -1388.8594 -193.249168 902.7327 -452.9071
## 陕西 -3467.587 -36.18118 -1290.7885 40.858539 421.1680 -313.2432
## 甘肃 -3209.422 -33.92919 -1146.2330 -53.411235 470.2348 -357.5851
## 青海 -3089.786 46.95751 -1181.3101 -55.270695 508.7677 -241.8020
## 宁夏 -3113.219 -51.16864 -1116.2861 61.118194 670.9278 -448.3887
## 新疆 -3125.600 -99.47641 -1142.9285 -188.221320 711.5164 -388.0790
## [,7] [,8]
## 北京 338.1326 -46.93540207
## 天津 262.7464 -68.06335480
## 河北 329.1848 -88.64297392
## 山西 200.4013 -153.73040440
## 内蒙古 223.8099 -13.10436469
## 辽宁 170.3246 -55.07060619
## 吉林 134.9042 -52.41184623
## 黑龙江 136.3281 -90.64088177
## 上海 199.0403 13.60840001
## 江苏 224.6134 -113.97505321
## 浙江 141.7991 -216.26974579
## 安徽 159.9425 -107.00704251
## 福建 220.8554 -95.38599704
## 江西 325.3652 -43.25605156
## 山东 354.3797 -106.08449797
## 河南 311.9963 -57.90190734
## 湖北 283.9291 -165.29453809
## 湖南 262.1497 -104.05884628
## 广东 264.6169 -80.14799721
## 广西 206.5708 -70.16690022
## 海南 307.4828 -61.60926534
## 重庆 413.6344 -167.50485784
## 四川 333.1896 -80.74317703
## 贵州 206.5979 -110.47488954
## 云南 213.7375 -194.85291383
## 西藏 230.0083 -0.08183429
## 陕西 177.1975 -76.73062051
## 甘肃 211.8872 -19.80288724
## 青海 246.9396 -17.17355476
## 宁夏 252.4040 -60.43164552
## 新疆 275.6191 -56.70631902PCA2$scores## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
## 北京 -2260.23573 -557.91050 -255.644483 45.0564829 -66.9510339
## 天津 -881.18567 -166.46168 -456.878957 -471.4699657 -39.3964785
## 河北 763.03936 -228.35400 170.259894 -164.1247187 -127.5160847
## 山西 835.11384 -373.57495 270.150955 -30.5732191 1.0803799
## 内蒙古 498.54214 -566.59151 82.177656 205.7946873 -97.8174958
## 辽宁 115.55714 42.38704 -201.869472 -195.2645563 -214.9210304
## 吉林 727.24553 -337.35937 -30.428196 -145.6991797 -187.1870151
## 黑龙江 1044.15077 -228.64702 33.127990 -10.8378738 -48.4444953
## 上海 -3629.69562 80.08600 -7.414618 185.9723329 20.0841998
## 江苏 -549.47579 -50.45814 -202.653513 -20.8142110 283.2320983
## 浙江 -2105.72900 -166.19954 77.378498 242.8307979 -89.0428120
## 安徽 343.49144 137.09051 -142.067683 4.2925793 135.5200262
## 福建 -874.55179 305.53757 -10.265277 -89.3458819 45.2733419
## 江西 662.25145 219.56997 -119.093518 116.7558060 178.6516688
## 山东 154.59504 -313.10099 161.805374 60.7099195 -109.7088962
## 河南 916.00562 -205.13770 102.345863 51.0066471 0.7356478
## 湖北 293.37528 140.84825 -183.712673 85.4695181 64.7079136
## 湖南 308.83977 -129.17023 -86.674968 15.9156818 146.0262432
## 广东 -2433.28370 167.31075 615.044824 -209.4597710 30.8817316
## 广西 385.01137 325.64413 -22.356707 -130.1299250 188.0322680
## 海南 132.09115 568.63311 392.743464 -230.6900093 54.9488917
## 重庆 -89.77927 -21.55525 -220.071029 8.9244546 -48.7097217
## 四川 178.58360 311.91621 -64.300519 138.0457799 27.5775302
## 贵州 624.15707 177.26287 47.246060 164.5010868 174.1504466
## 云南 382.36226 523.76666 -21.208660 0.9488038 -254.6615081
## 西藏 407.63299 947.10732 -165.731842 190.5840098 -299.5991385
## 陕西 543.91420 -121.94950 -67.660963 -43.5236976 181.9655865
## 甘肃 802.07876 -119.69751 76.894550 50.7460765 132.8988383
## 青海 921.71498 -38.81081 41.817440 52.6055358 94.3659203
## 宁夏 898.28168 -136.93696 106.841432 -63.7833522 -67.7942401
## 新疆 885.90113 -185.24473 80.199078 185.5561613 -108.3827824
## Comp.6 Comp.7 Comp.8
## 北京 191.5320286 -92.333008 35.666275
## 天津 -51.8879304 -16.946794 14.538322
## 河北 62.9965279 -83.385211 -6.041297
## 山西 -98.3170619 45.398326 -71.128728
## 内蒙古 -84.7437593 21.989712 69.497312
## 辽宁 7.3303443 75.475026 27.531070
## 吉林 -62.9390772 110.895380 30.189830
## 黑龙江 86.5946640 109.471482 -8.039205
## 上海 -18.2525273 46.759317 96.210077
## 江苏 -6.1125618 21.186198 -31.373377
## 浙江 58.2157431 104.000569 -133.668069
## 安徽 -9.2183518 85.857123 -24.405366
## 福建 -270.9435385 24.944182 -12.784320
## 江西 -71.7888071 -79.565585 39.345625
## 山东 -106.6817020 -108.580089 -23.482821
## 河南 0.2203520 -66.196634 24.699769
## 湖北 -81.2576930 -38.129509 -82.692861
## 湖南 21.8941870 -16.350061 -21.457170
## 广东 -38.9140659 -18.817239 2.453679
## 广西 73.0080625 39.228844 12.434776
## 海南 120.2547641 -61.683173 20.992411
## 重庆 -49.1374375 -167.834790 -84.903181
## 四川 67.8139494 -87.390018 1.858500
## 贵州 -39.5214378 39.201724 -27.873213
## 云南 168.2442936 32.062078 -112.251237
## 西藏 -63.9710810 15.791352 82.519842
## 陕西 75.6928007 68.602119 5.871056
## 甘肃 31.3509370 33.912467 62.798789
## 青海 147.1340420 -1.139974 65.428122
## 宁夏 -59.4526500 -6.604354 22.170031
## 新疆 0.8569861 -29.819461 25.895358数据标准化之后
sd7.2<-scale(d7.2)#
covv<-cov(sd7.2)
ee<-eigen(covv,symmetric=T)
ee$value## [1] 5.70116807 1.02866772 0.50428374 0.27276565 0.18614324 0.16127995
## [7] 0.08734736 0.05834427diag(sqrt(ee$value))## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## [1,] 2.387712 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [2,] 0.000000 1.014233 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [3,] 0.000000 0.000000 0.7101294 0.0000000 0.0000000 0.0000000 0.0000000
## [4,] 0.000000 0.000000 0.0000000 0.5222697 0.0000000 0.0000000 0.0000000
## [5,] 0.000000 0.000000 0.0000000 0.0000000 0.4314432 0.0000000 0.0000000
## [6,] 0.000000 0.000000 0.0000000 0.0000000 0.0000000 0.4015967 0.0000000
## [7,] 0.000000 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.2955459
## [8,] 0.000000 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [,8]
## [1,] 0.0000000
## [2,] 0.0000000
## [3,] 0.0000000
## [4,] 0.0000000
## [5,] 0.0000000
## [6,] 0.0000000
## [7,] 0.0000000
## [8,] 0.2415456ee$vectors## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] -0.3530160 -0.42913372 0.17506584 0.29900682 0.0016050 -0.37676548
## [2,] -0.2494594 0.67707366 0.52110127 -0.09613622 -0.3991862 -0.12923592
## [3,] -0.3738401 -0.08881424 -0.07173444 -0.78926796 0.2609431 0.06375756
## [4,] -0.3016294 0.47157654 -0.62799220 0.22489321 0.2492672 -0.41577639
## [5,] -0.3760539 -0.32423582 0.12307893 0.12696200 -0.2808157 -0.26739703
## [6,] -0.4040134 -0.06946256 0.09017054 -0.19986045 0.1315060 -0.08849495
## [7,] -0.3709635 -0.05611496 -0.44195504 0.06921165 -0.5843723 0.53528642
## [8,] -0.3743738 0.11849177 0.28227996 0.40862952 0.5222835 0.54639431
## [,7] [,8]
## [1,] 0.65135285 0.07039520
## [2,] 0.13386770 0.06767919
## [3,] 0.11648213 0.37221164
## [4,] -0.03626539 0.07324460
## [5,] -0.69498247 0.29803953
## [6,] -0.15611394 -0.85695769
## [7,] 0.16637806 -0.05846782
## [8,] -0.08877945 0.14119220sd7.2 %*%ee$vectors#Y矩阵## [,1] [,2] [,3] [,4] [,5]
## 北京 -6.02274607 1.4977627 -0.0006871739 -0.16391109 0.79502001
## 天津 -2.96115828 0.5280553 -1.9513295787 0.23628153 0.22209825
## 河北 0.87306960 0.6810867 -0.9914908511 -0.18032321 -0.07035523
## 山西 1.08579949 0.5915946 -0.3666618862 -0.27837354 -0.81949926
## 内蒙古 -0.52467609 1.8176806 0.8905192709 0.02118982 -0.11409306
## 辽宁 -0.09284191 0.6445003 -0.7208770816 1.06639086 0.02201026
## 吉林 0.32175584 1.4015172 -0.6018210432 0.86149578 -0.27109466
## 黑龙江 1.66115278 0.9796895 -0.2051386719 0.57671770 0.08006449
## 上海 -6.96946517 -1.0519318 1.2773562523 0.47039265 0.36513503
## 江苏 -1.12275488 -0.4463169 -0.0896687938 -0.55598134 0.29151819
## 浙江 -3.75897083 0.1693347 0.8772435556 0.37591228 -0.73052101
## 安徽 1.10511258 -0.3460854 0.0891491147 0.13804480 0.08563301
## 福建 -1.15266449 -1.3552068 -0.0039062405 -0.06991405 -0.75996923
## 江西 1.64223622 -0.5395645 0.6532926862 -0.72904723 0.32177245
## 山东 -0.47328821 0.7953144 0.0998177113 -0.65486444 -0.61640125
## 河南 1.25645857 0.6373676 0.0886737948 -0.44624987 0.07817857
## 湖北 0.99303892 -0.1146098 0.3215625962 -0.53120541 -0.40191131
## 湖南 0.35913254 0.1974682 -0.0514180753 -0.49067727 0.19581667
## 广东 -3.96639124 -2.4401516 -0.7904133404 -0.09605024 -0.48064261
## 广西 1.60092834 -1.2105598 -0.4894787764 0.05720286 0.50872627
## 海南 1.84265618 -2.3145599 -0.9877987233 -0.01609807 0.29572783
## 重庆 -0.38762452 0.4548143 -0.1673779595 -0.90974825 -0.37859619
## 四川 1.13500148 -0.5096478 0.5968162283 -0.32914890 0.31433573
## 贵州 1.98128033 -0.6487463 0.7514554478 -0.31841016 -0.08773246
## 云南 2.38999332 -0.4109714 -0.1730464102 0.88871466 -0.34815123
## 西藏 2.67612880 -0.9941148 1.3934198622 1.24423286 -0.01023678
## 陕西 0.87353770 0.1150517 -0.2617342408 -0.02537647 0.47640866
## 甘肃 1.30298299 0.1421495 0.2420791861 -0.08821778 0.47877789
## 青海 1.73969881 0.2055362 0.1551351630 -0.01382864 0.81194277
## 宁夏 1.29590895 0.4879663 -0.3017998447 0.01458779 -0.19953635
## 新疆 1.29670824 1.0355770 0.7181278224 -0.05373765 -0.05442545
## [,6] [,7] [,8]
## 北京 -0.607016797 0.030533312 0.194482609
## 天津 0.290678906 0.679090704 0.001188019
## 河北 -0.246650021 -0.226660578 0.325268532
## 山西 0.182777166 -0.333579876 -0.196914532
## 内蒙古 0.575893319 -0.272300579 0.127756689
## 辽宁 0.120810958 0.234838381 0.038193070
## 吉林 0.499647897 -0.063569439 -0.056866191
## 黑龙江 -0.073313761 -0.286449976 -0.250789466
## 上海 0.329072083 -0.056652481 0.046159677
## 江苏 0.053493226 0.218739240 -0.428148819
## 浙江 -0.809968015 -0.223735812 -0.429083237
## 安徽 0.090533718 0.146935766 -0.393879327
## 福建 0.738723928 0.389656932 -0.059220257
## 江西 0.305596948 0.277557669 0.087747544
## 山东 0.045424193 -0.071122651 0.330066469
## 河南 0.082593962 -0.163483049 0.202399769
## 湖北 -0.217913070 0.411249415 -0.152242142
## 湖南 -0.073658171 0.051161084 -0.177050148
## 广东 0.159700082 -0.598755027 0.209905025
## 广西 0.019841346 0.002684683 -0.215956347
## 海南 -0.261516365 -0.371217541 0.293149328
## 重庆 -0.533041258 0.519716499 0.227454984
## 四川 -0.408091243 0.140470293 0.183892081
## 贵州 0.086599315 -0.039920012 -0.279832631
## 云南 -1.184964470 0.074604613 -0.014614089
## 西藏 0.132667658 0.474353950 0.451341524
## 陕西 0.067108289 -0.111347823 -0.355857206
## 甘肃 0.360890497 -0.264751563 -0.085831174
## 青海 -0.052583663 -0.287309010 0.023669653
## 宁夏 0.319152039 -0.105666399 0.143746587
## 新疆 0.007511306 -0.175070723 0.209864008PCA1$scores## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
## 北京 -6.12230233 1.5225207 -0.000698533 -0.16662054 0.80816173
## 天津 -3.01010636 0.5367841 -1.983585145 0.24018727 0.22576954
## 河北 0.88750148 0.6923451 -1.007880240 -0.18330396 -0.07151821
## 山西 1.10374781 0.6013737 -0.372722824 -0.28297507 -0.83304562
## 内蒙古 -0.53334901 1.8477269 0.905239595 0.02154009 -0.11597902
## 辽宁 -0.09437659 0.6551539 -0.732793213 1.08401835 0.02237409
## 吉林 0.32707448 1.4246843 -0.611769172 0.87573634 -0.27557587
## 黑龙江 1.68861172 0.9958838 -0.208529623 0.58625087 0.08138796
## 上海 -7.08467075 -1.0693203 1.298471009 0.47816826 0.37117073
## 江苏 -1.14131407 -0.4536945 -0.091151023 -0.56517173 0.29633699
## 浙江 -3.82110679 0.1721339 0.891744431 0.38212613 -0.74259655
## 安徽 1.12338014 -0.3518062 0.090622754 0.14032668 0.08704853
## 福建 -1.17171809 -1.3776085 -0.003970811 -0.07106973 -0.77253155
## 江西 1.66938246 -0.5484835 0.664091644 -0.74109841 0.32709137
## 山东 -0.48111168 0.8084609 0.101467703 -0.66568938 -0.62659039
## 河南 1.27722788 0.6479033 0.090139577 -0.45362641 0.07947087
## 湖北 1.00945390 -0.1165043 0.326878041 -0.53998626 -0.40855492
## 湖南 0.36506902 0.2007324 -0.052268018 -0.49878819 0.19905353
## 广东 -4.03195586 -2.4804874 -0.803478909 -0.09763795 -0.48858766
## 广西 1.62739176 -1.2305704 -0.497569883 0.05814843 0.51713554
## 海南 1.87311537 -2.3528197 -1.004127081 -0.01636417 0.30061623
## 重庆 -0.39403198 0.4623324 -0.170144725 -0.92478643 -0.38485441
## 四川 1.15376310 -0.5180723 0.606681628 -0.33458975 0.31953171
## 贵州 2.01403098 -0.6594701 0.763877040 -0.32367349 -0.08918268
## 云南 2.42950002 -0.4177647 -0.175906875 0.90340516 -0.35390619
## 西藏 2.72036533 -1.0105476 1.416453156 1.26480009 -0.01040599
## 陕西 0.88797732 0.1169535 -0.266060720 -0.02579594 0.48428371
## 甘肃 1.32452136 0.1444992 0.246080766 -0.08967602 0.48669211
## 青海 1.76845611 0.2089337 0.157699554 -0.01405723 0.82536422
## 宁夏 1.31733038 0.4960324 -0.306788610 0.01482892 -0.20283469
## 新疆 1.31814288 1.0526951 0.729998508 -0.05462593 -0.05532510
## Comp.6 Comp.7 Comp.8
## 北京 -0.617050813 0.031038029 0.197697415
## 天津 0.295483841 0.690316105 0.001207657
## 河北 -0.250727157 -0.230407287 0.330645235
## 山西 0.185798481 -0.339093966 -0.200169538
## 内蒙古 0.585412862 -0.276801720 0.129868512
## 辽宁 0.122807969 0.238720270 0.038824403
## 吉林 0.507907099 -0.064620245 -0.057806192
## 黑龙江 -0.074525640 -0.291185007 -0.254935028
## 上海 0.334511659 -0.057588949 0.046922699
## 江苏 0.054377471 0.222355009 -0.435226138
## 浙江 -0.823356823 -0.227434176 -0.436176003
## 安徽 0.092030244 0.149364621 -0.400390171
## 福建 0.750935068 0.396097979 -0.060199171
## 江西 0.310648480 0.282145710 0.089198015
## 山东 0.046175057 -0.072298312 0.335522483
## 河南 0.083959244 -0.166185432 0.205745446
## 湖北 -0.221515183 0.418047386 -0.154758712
## 湖南 -0.074875744 0.052006779 -0.179976795
## 广东 0.162339932 -0.608652475 0.213374764
## 广西 0.020169324 0.002729061 -0.219526115
## 海南 -0.265839242 -0.377353784 0.297995100
## 重庆 -0.541852454 0.528307436 0.231214825
## 四川 -0.414837010 0.142792274 0.186931825
## 贵州 0.088030805 -0.040579891 -0.284458277
## 云南 -1.204551986 0.075837831 -0.014855661
## 西藏 0.134860660 0.482195042 0.458802220
## 陕西 0.068217592 -0.113188407 -0.361739542
## 甘肃 0.366856033 -0.269127919 -0.087249967
## 青海 -0.053452873 -0.292058241 0.024060914
## 宁夏 0.324427636 -0.107413069 0.146122725
## 新疆 0.007635468 -0.177964650 0.2133330703.验证yi与yj之间的协方差=0
var(as.matrix(d7.2)%*%as.matrix(e$vectors))#未标准化数据## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.317709e+06 5.423734e-10 -3.632969e-10 2.897404e-11 2.159107e-10
## [2,] 5.423734e-10 1.078281e+05 -1.956206e-10 4.326672e-11 -9.722534e-12
## [3,] -3.632969e-10 -1.956206e-10 4.103067e+04 -5.564269e-11 1.301448e-11
## [4,] 2.897404e-11 4.326672e-11 -5.564269e-11 2.419190e+04 2.702135e-12
## [5,] 2.159107e-10 -9.722534e-12 1.301448e-11 2.702135e-12 1.990990e+04
## [6,] -1.833271e-11 -1.048812e-11 -1.572549e-12 -3.339366e-11 1.682555e-11
## [7,] 5.396128e-11 3.576091e-11 -1.490642e-11 1.234060e-11 -1.784717e-11
## [8,] -7.006780e-12 1.660581e-11 6.830936e-12 -1.093494e-11 4.108343e-12
## [,6] [,7] [,8]
## [1,] -1.833271e-11 5.396128e-11 -7.006780e-12
## [2,] -1.048812e-11 3.576091e-11 1.660581e-11
## [3,] -1.572549e-12 -1.490642e-11 6.830936e-12
## [4,] -3.339366e-11 1.234060e-11 -1.093494e-11
## [5,] 1.682555e-11 -1.784717e-11 4.108343e-12
## [6,] 8.891801e+03 1.953106e-11 -2.481203e-12
## [7,] 1.953106e-11 4.835498e+03 3.445022e-13
## [8,] -2.481203e-12 3.445022e-13 2.981311e+03var(as.matrix(sd7.2)%*%as.matrix(ee$vectors))#标准化数据## [,1] [,2] [,3] [,4] [,5]
## [1,] 5.701168e+00 -1.204032e-15 3.446010e-16 -1.918412e-16 -4.413871e-16
## [2,] -1.204032e-15 1.028668e+00 -2.466180e-16 -1.012089e-16 1.683099e-16
## [3,] 3.446010e-16 -2.466180e-16 5.042837e-01 -7.534110e-17 2.229315e-16
## [4,] -1.918412e-16 -1.012089e-16 -7.534110e-17 2.727656e-01 -3.087761e-16
## [5,] -4.413871e-16 1.683099e-16 2.229315e-16 -3.087761e-16 1.861432e-01
## [6,] -5.658161e-16 3.318076e-16 9.573573e-17 2.366053e-16 1.456191e-17
## [7,] 6.643431e-16 -1.454385e-16 -1.643267e-16 7.478050e-17 1.127247e-16
## [8,] 1.202019e-16 1.956750e-16 -2.409459e-17 -4.391750e-17 -8.071103e-17
## [,6] [,7] [,8]
## [1,] -5.658161e-16 6.643431e-16 1.202019e-16
## [2,] 3.318076e-16 -1.454385e-16 1.956750e-16
## [3,] 9.573573e-17 -1.643267e-16 -2.409459e-17
## [4,] 2.366053e-16 7.478050e-17 -4.391750e-17
## [5,] 1.456191e-17 1.127247e-16 -8.071103e-17
## [6,] 1.612799e-01 -8.299693e-17 -1.304569e-16
## [7,] -8.299693e-17 8.734736e-02 -2.945077e-17
## [8,] -1.304569e-16 -2.945077e-17 5.834427e-02无论是未标准化的数据还是标准化后的数据，由协方差矩阵均可知，对角线上的值等于X谱分解后的特征值，而其他元素近似于0，即命题得证。
4.验证Y与X的方差和相等
sum(diag(cov))## [1] 1527378sum(var(as.matrix(d7.2)%*%as.matrix(e$vectors)))## [1] 1527378sum(diag(covv))## [1] 8sum(var(as.matrix(sd7.2)%*%as.matrix(ee$vectors)))## [1] 8由此可得，可证明Ｙ矩阵与Ｘ矩阵的方差和相等。
5.验证相关系数公式
y<-as.matrix(d7.2)%*%as.matrix(e$vectors)
cor(y[,1],d7.2[,1]) ## [1] -0.971572cov.eigen=eigen(cov(d7.2))
sqrt(cov.eigen$values[1])*cov.eigen$vector[1,1]/sqrt(var(d7.2[,1]))## [1] -0.971572由此可得，相关系数公式成立。