women<-read.table(file="Data-HW4-track-women.dat", header=FALSE, quote="", sep="\t")
rforw<-data.matrix(women[,2:8])center = function(v){v - mean(v)}
x = apply(rforw, 2, center)y = women$V1; y## [1] ARG AUS AUT BEL BER BRA CAN CHI CHN COL
## [11] COK CRC CZE DEN DOM FIN FRA GER GBR GRE
## [21] GUA HUN INA IND IRL ISR ITA JPN KEN KOR, S
## [31] KOR, N LUX MAS MRI MEX MYA NED NZL NOR PNG
## [41] PHI POL POR ROM RUS SAM SIN ESP SWE SUI
## [51] TPE THA TUR USA
## 54 Levels: ARG AUS AUT BEL BER BRA CAN CHI CHN COK COL CRC CZE DEN ... USAcor(rforw)## V2 V3 V4 V5 V6 V7 V8
## V2 1.0000000 0.9410886 0.8707802 0.8091758 0.7815510 0.7278784 0.6689597
## V3 0.9410886 1.0000000 0.9088096 0.8198258 0.8013282 0.7318546 0.6799537
## V4 0.8707802 0.9088096 1.0000000 0.8057904 0.7197996 0.6737991 0.6769384
## V5 0.8091758 0.8198258 0.8057904 1.0000000 0.9050509 0.8665732 0.8539900
## V6 0.7815510 0.8013282 0.7197996 0.9050509 1.0000000 0.9733801 0.7905565
## V7 0.7278784 0.7318546 0.6737991 0.8665732 0.9733801 1.0000000 0.7987302
## V8 0.6689597 0.6799537 0.6769384 0.8539900 0.7905565 0.7987302 1.0000000eigen(cor(rforw))## eigen() decomposition
## $values
## [1] 5.80762446 0.62869342 0.27933457 0.12455472 0.09097174 0.05451882
## [7] 0.01430226
##
## $vectors
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] -0.3777657 -0.4071756 -0.1405803 0.58706293 -0.16706891 0.53969730
## [2,] -0.3832103 -0.4136291 -0.1007833 0.19407501 0.09350016 -0.74493139
## [3,] -0.3680361 -0.4593531 0.2370255 -0.64543118 0.32727328 0.24009405
## [4,] -0.3947810 0.1612459 0.1475424 -0.29520804 -0.81905467 -0.01650651
## [5,] -0.3892610 0.3090877 -0.4219855 -0.06669044 0.02613100 -0.18898771
## [6,] -0.3760945 0.4231899 -0.4060627 -0.08015699 0.35169796 0.24049968
## [7,] -0.3552031 0.3892153 0.7410610 0.32107640 0.24700821 -0.04826992
## [,7]
## [1,] 0.08893934
## [2,] -0.26565662
## [3,] 0.12660435
## [4,] -0.19521315
## [5,] 0.73076817
## [6,] -0.57150644
## [7,] 0.08208401women.pca = prcomp(rforw, center=TRUE, scale.=TRUE)
women.pca$rotation[,1:2] ## PC1 PC2
## V2 -0.3777657 0.4071756
## V3 -0.3832103 0.4136291
## V4 -0.3680361 0.4593531
## V5 -0.3947810 -0.1612459
## V6 -0.3892610 -0.3090877
## V7 -0.3760945 -0.4231899
## V8 -0.3552031 -0.3892153summary(women.pca)## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6
## Standard deviation 2.4099 0.79290 0.5285 0.35292 0.3016 0.23349
## Proportion of Variance 0.8297 0.08981 0.0399 0.01779 0.0130 0.00779
## Cumulative Proportion 0.8297 0.91947 0.9594 0.97717 0.9902 0.99796
## PC7
## Standard deviation 0.11959
## Proportion of Variance 0.00204
## Cumulative Proportion 1.00000pca2<- svd(x); pca2## $d
## [1] 120.5869891 14.5905502 3.7706286 2.4639297 0.8953146 0.3708287
## [7] 0.1808009
##
## $u
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.026681074 0.058604086 0.0160996392 0.1176296017 0.2858444484
## [2,] 0.086684247 -0.156200588 0.0678554520 -0.0605078062 0.1258865787
## [3,] -0.004511426 -0.106016357 -0.0536623468 -0.0372799730 -0.0366487883
## [4,] 0.087797994 0.035221002 -0.0243959315 0.1687955331 0.0369066426
## [5,] -0.170626202 -0.079397016 -0.0768413253 0.1968639016 0.1168377763
## [6,] 0.052533645 -0.049622456 0.0590101849 0.0734581731 0.0128180870
## [7,] 0.045551129 -0.107240610 -0.0354972319 -0.0831523595 -0.1740961750
## [8,] 0.009590292 0.137750213 0.0390375236 -0.0633315612 0.0117992387
## [9,] 0.119845874 -0.057093597 -0.1379086112 0.0735775676 -0.1144074052
## [10,] -0.010868733 -0.166252257 0.1802048561 -0.1040704997 0.0183994250
## [11,] -0.493604209 0.220825513 -0.2153879593 0.0654359145 0.0261872148
## [12,] -0.089259474 -0.026037615 0.1703296497 -0.1334850145 0.0277725979
## [13,] 0.073441302 -0.215215218 0.1175412448 0.0041451489 0.1936709173
## [14,] 0.034428655 0.097833810 -0.0806739674 -0.0509857932 -0.0230765927
## [15,] -0.107194174 -0.015031543 0.1335368333 -0.0745387120 -0.0649324984
## [16,] 0.048300493 -0.091818590 -0.0223868838 0.0284936924 0.0645130804
## [17,] 0.047976209 -0.227197023 0.0123737997 0.0094275018 -0.1864448552
## [18,] 0.104814586 -0.219320106 0.0470273466 0.0278069224 0.0150761800
## [19,] 0.154083835 -0.041810440 0.0105880627 0.0428668220 0.1612381942
## [20,] 0.003347425 -0.104131997 -0.0170239727 0.0440673899 -0.3617845890
## [21,] -0.149780138 0.126797305 -0.0319172643 -0.1012968217 0.0069172066
## [22,] 0.042772909 0.006150549 -0.0629993255 -0.1203914516 0.0959999254
## [23,] -0.008706477 0.212450673 0.0006592577 0.1180808719 -0.1032274783
## [24,] -0.035942567 -0.102729048 -0.0094755354 -0.0119927261 0.1485681808
## [25,] 0.094834210 0.027792179 -0.0348195690 -0.1637164451 0.1387734040
## [26,] -0.022730502 -0.014883484 0.0404121481 0.0240382007 0.0778907485
## [27,] 0.084514672 0.023686687 -0.0651564887 0.0681189823 -0.0007182163
## [28,] 0.117039584 0.114180378 0.0617737085 -0.0833903159 -0.0754084278
## [29,] 0.125196551 0.098455055 0.0155063579 -0.2317099766 0.1980060738
## [30,] 0.059993977 0.183576164 0.0241942094 -0.0549068265 -0.1361949206
## [31,] 0.063933305 0.382784812 0.0136475922 -0.2699901613 -0.3339393465
## [32,] 0.032089810 0.319266862 -0.0636353269 0.1221919436 0.0673847629
## [33,] -0.129595520 -0.082821009 -0.0615853732 -0.0556052065 0.0317052085
## [34,] -0.113558871 0.077808936 -0.1302642671 -0.0374781021 0.0799074570
## [35,] 0.081535862 -0.124416356 0.2087816837 -0.2696980615 -0.2766266101
## [36,] -0.040601946 0.037148665 -0.0234281538 -0.1452974864 0.0863845594
## [37,] 0.084747805 0.030356021 -0.0474734833 0.0098227907 -0.1526558890
## [38,] 0.059385256 0.031188200 0.0023657619 -0.0721997824 -0.0340316077
## [39,] 0.103079554 0.132701237 -0.0392404327 -0.0834341992 -0.0025409953
## [40,] -0.559541583 -0.301261838 -0.2623896301 -0.3077521988 -0.0373154934
## [41,] -0.100409912 0.089589388 -0.0622731866 0.2998593951 -0.1994262143
## [42,] 0.080683810 -0.123975550 -0.0199373127 0.0343475119 -0.0306890029
## [43,] 0.085370493 0.070701786 -0.0852591863 0.0008402203 0.0569766249
## [44,] 0.093920108 -0.065585095 -0.0470451769 -0.0508736132 0.2747242615
## [45,] 0.104623720 -0.119124214 -0.0628541616 0.1000515327 -0.0843091150
## [46,] -0.318631602 0.049073485 0.7058290292 0.2009564677 0.0055255176
## [47,] -0.010150428 0.232284965 0.2045382779 -0.0806577932 0.2052251021
## [48,] 0.061076011 -0.112071295 -0.0398825410 -0.0422698095 0.0043298671
## [49,] 0.027066351 -0.002183049 -0.0631694884 0.0490897531 -0.0776689004
## [50,] 0.067597459 0.014917039 -0.0425139045 -0.0513640391 0.0977089412
## [51,] -0.049320772 -0.008530645 -0.0203961507 0.3558819134 0.0573359338
## [52,] -0.073128174 -0.022869000 0.1430998265 0.1632642758 -0.1257213944
## [53,] 0.017334836 0.079859480 -0.2398123514 0.1526693467 0.0732902322
## [54,] 0.106289666 -0.148168495 -0.0951059057 0.2895953612 -0.1717398730
## [,6] [,7]
## [1,] 0.159727266 -0.003108278
## [2,] 0.024384038 0.132806560
## [3,] -0.141403656 -0.213422823
## [4,] 0.009530014 -0.083486030
## [5,] -0.221484457 0.034487063
## [6,] 0.022860490 -0.162806753
## [7,] -0.053009375 0.058860471
## [8,] -0.194766548 -0.125328078
## [9,] -0.023636503 0.034877310
## [10,] 0.173930303 -0.014975720
## [11,] -0.174151693 -0.011357393
## [12,] 0.159260583 -0.051276333
## [13,] -0.170224479 -0.271401391
## [14,] 0.066769215 0.001661770
## [15,] 0.161446536 -0.154368187
## [16,] 0.183468834 0.045762244
## [17,] 0.017868424 -0.082002258
## [18,] -0.008484552 -0.051589157
## [19,] 0.102306739 -0.056694193
## [20,] -0.126125182 0.141859090
## [21,] 0.125704918 0.062288659
## [22,] -0.040989600 0.031819909
## [23,] 0.076939721 0.114850089
## [24,] -0.208935574 0.041038745
## [25,] 0.032591823 0.213039462
## [26,] -0.013619339 0.196886324
## [27,] -0.104165644 0.004459760
## [28,] 0.132976674 0.028543536
## [29,] -0.131306458 0.090174249
## [30,] 0.130813705 0.273936758
## [31,] -0.164867343 -0.362395224
## [32,] 0.241795359 -0.157015163
## [33,] 0.351328482 0.021022913
## [34,] 0.141735462 -0.248911741
## [35,] 0.083425778 0.215804659
## [36,] -0.068777889 -0.040803930
## [37,] 0.020047891 -0.279546379
## [38,] -0.037560158 -0.114872966
## [39,] -0.040995173 0.302556282
## [40,] 0.076650305 0.030586273
## [41,] 0.071154937 0.119660155
## [42,] 0.013517768 -0.023102329
## [43,] -0.110836950 0.098087070
## [44,] -0.125220256 -0.103525963
## [45,] -0.155631808 -0.018438188
## [46,] -0.133413854 0.017334539
## [47,] 0.046037573 0.067933846
## [48,] 0.061363484 -0.057149295
## [49,] -0.021283547 -0.034364279
## [50,] -0.168238355 0.122504190
## [51,] 0.239743422 -0.042582908
## [52,] -0.260611108 0.103583011
## [53,] -0.155223022 0.186648955
## [54,] 0.127582782 -0.028548934
##
## $v
## [,1] [,2] [,3] [,4] [,5]
## [1,] -0.016123307 0.11485619 0.17348957 -0.292124494 9.325622e-01
## [2,] -0.038657909 0.29039299 0.38670581 -0.794578949 -3.543286e-01
## [3,] -0.107793074 0.93844399 -0.22555878 0.238107150 -1.355429e-03
## [4,] -0.004504024 0.01340703 0.03608601 0.009212813 1.547251e-02
## [5,] -0.013072642 0.03631915 0.26812163 0.068410358 -6.671958e-02
## [6,] -0.039484872 0.07871002 0.83389231 0.470904246 -9.287424e-03
## [7,] -0.992409201 -0.11878027 -0.03025618 -0.009843642 -2.323022e-05
## [,6] [,7]
## [1,] 0.0226814395 -0.030430700
## [2,] -0.0832625101 0.028266709
## [3,] 0.0055727214 -0.009943020
## [4,] 0.3768162267 0.925301584
## [5,] 0.8833633010 -0.370349317
## [6,] -0.2649535334 0.069511587
## [7,] -0.0005351329 -0.001613339#ranking
women[order(women.pca$x[,1], decreasing = TRUE),1]## [1] USA GER RUS CHN FRA GBR CZE POL ROM AUS
## [11] ESP CAN ITA NED BEL FIN AUT GRE POR SUI
## [21] IRL BRA MEX KEN TUR SWE HUN NZL NOR JPN
## [31] IND DEN COL ARG ISR TPE CHI MYA KOR, S THA
## [41] BER KOR, N MAS LUX INA MRI PHI CRC DOM SIN
## [51] GUA PNG COK SAM
## 54 Levels: ARG AUS AUT BEL BER BRA CAN CHI CHN COK COL CRC CZE DEN ... USA#My intuition tells me that the larger more developed countries with plenty of resources are leading the ranks. Such is the case for the row [1] and row [8], these are highly developed countries. It starts to go down for second tier countries, still developed but less so that the first two rows.
women2<-women
women2[,5:8]<-(women2[,5:8])*60
women2[,2]<-100/(women2[,2])
women2[,3]<-200/(women2[,3])
women2[,4]<-400/(women2[,4])
women2[,5]<-800/(women2[,5])
women2[,6]<-1500/(women2[,6])
women2[,7]<-3000/(women2[,7])
women2[,8]<-42195/(women2[,8])#PCA Analysis
women.pca2 = prcomp(rforw, center=TRUE)
women.pca2$rotation[,1:2]## PC1 PC2
## V2 -0.016123307 0.11485619
## V3 -0.038657909 0.29039299
## V4 -0.107793074 0.93844399
## V5 -0.004504024 0.01340703
## V6 -0.013072642 0.03631915
## V7 -0.039484872 0.07871002
## V8 -0.992409201 -0.11878027summary(women.pca2)## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6
## Standard deviation 16.5639 2.00417 0.51794 0.33845 0.12298 0.05094
## Proportion of Variance 0.9841 0.01441 0.00096 0.00041 0.00005 0.00001
## Cumulative Proportion 0.9841 0.99856 0.99952 0.99993 0.99999 1.00000
## PC7
## Standard deviation 0.02483
## Proportion of Variance 0.00000
## Cumulative Proportion 1.00000rforw2<-data.matrix(women2[,2:8])
women2.pca = prcomp(rforw2, center=TRUE)
women2.pca$rotation[,1:2]## PC1 PC2
## V2 0.3102442 -0.37596510
## V3 0.3573948 -0.43376925
## V4 0.3787367 -0.51873227
## V5 0.2993405 0.05313551
## V6 0.3912131 0.21084397
## V7 0.4595909 0.39557338
## V8 0.4227291 0.44458346summary(women2.pca)## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6
## Standard deviation 0.8557 0.2934 0.18270 0.12238 0.09408 0.07853
## Proportion of Variance 0.8285 0.0974 0.03777 0.01695 0.01002 0.00698
## Cumulative Proportion 0.8285 0.9259 0.96372 0.98067 0.99068 0.99766
## PC7
## Standard deviation 0.04545
## Proportion of Variance 0.00234
## Cumulative Proportion 1.00000women2[order(women2.pca$x[,1], decreasing = TRUE),1]## [1] USA CHN RUS GER GBR FRA ROM POL CZE AUS
## [11] ESP CAN ITA NED IRL POR KEN FIN BEL SUI
## [21] MEX AUT GRE TUR HUN NOR BRA NZL SWE JPN
## [31] DEN IND COL ARG KOR, S ISR MYA CHI TPE KOR, N
## [41] LUX MAS THA INA BER MRI PHI CRC DOM SIN
## [51] GUA PNG COK SAM
## 54 Levels: ARG AUS AUT BEL BER BRA CAN CHI CHN COK COL CRC CZE DEN ... USA#The components between the standardized predictors and the converted measures are very similar. The numbers for PC1 in the standardized predictors (problem b) do not change that much (.8297) in comparison to PC1 in the converted speed measures (.8258). I prefer the converted measures anaylsis because it captures 93% of the variation after the second principal component compared to the standardized principal component at 92%.
men<-read.table(file="Data-HW4-track-men.dat", header=FALSE, quote="", sep="")
rform<-data.matrix(men[,2:9])centermen = function(v){v - mean(v)}
x = apply(rform, 2, center)cor(rform)## V2 V3 V4 V5 V6 V7 V8
## V2 1.0000000 0.9147554 0.8041147 0.7119388 0.7657919 0.7398803 0.7147921
## V3 0.9147554 1.0000000 0.8449159 0.7969162 0.7950871 0.7613028 0.7479519
## V4 0.8041147 0.8449159 1.0000000 0.7677488 0.7715522 0.7796929 0.7657481
## V5 0.7119388 0.7969162 0.7677488 1.0000000 0.8957609 0.8606959 0.8431074
## V6 0.7657919 0.7950871 0.7715522 0.8957609 1.0000000 0.9165224 0.9013380
## V7 0.7398803 0.7613028 0.7796929 0.8606959 0.9165224 1.0000000 0.9882324
## V8 0.7147921 0.7479519 0.7657481 0.8431074 0.9013380 0.9882324 1.0000000
## V9 0.6764873 0.7211157 0.7126823 0.8069657 0.8777788 0.9441466 0.9541630
## V9
## V2 0.6764873
## V3 0.7211157
## V4 0.7126823
## V5 0.8069657
## V6 0.8777788
## V7 0.9441466
## V8 0.9541630
## V9 1.0000000eigen(cor(rform))## eigen() decomposition
## $values
## [1] 6.703289951 0.638410110 0.227524494 0.205849181 0.097577441 0.070687912
## [7] 0.046942050 0.009718862
##
## $vectors
## [,1] [,2] [,3] [,4] [,5]
## [1,] -0.3323877 -0.52939911 -0.343859303 0.38074525 0.29967117
## [2,] -0.3460511 -0.47039050 0.003786104 0.21702322 -0.54143422
## [3,] -0.3391240 -0.34532929 0.067060507 -0.85129980 0.13298631
## [4,] -0.3530134 0.08945523 0.782711152 0.13427911 -0.22728254
## [5,] -0.3659849 0.15365241 0.244270040 0.23302034 0.65162403
## [6,] -0.3698204 0.29475985 -0.182863147 -0.05462441 0.07181636
## [7,] -0.3659489 0.33360619 -0.243980694 -0.08706927 -0.06133263
## [8,] -0.3542779 0.38656085 -0.334632969 0.01812115 -0.33789097
## [,6] [,7] [,8]
## [1,] -0.36203713 0.3476470 -0.065701445
## [2,] 0.34859224 -0.4398969 0.060755403
## [3,] 0.07708385 0.1135553 -0.003469726
## [4,] -0.34130845 0.2588830 -0.039274027
## [5,] 0.52977961 -0.1470362 -0.039745509
## [6,] -0.35914382 -0.3283202 0.705684585
## [7,] -0.27308617 -0.3511133 -0.697181715
## [8,] 0.37516986 0.5941571 0.069316891men.pca = prcomp(rform, center=TRUE, scale.=TRUE)
men.pca$rotation[,1:2]## PC1 PC2
## V2 -0.3323877 -0.52939911
## V3 -0.3460511 -0.47039050
## V4 -0.3391240 -0.34532929
## V5 -0.3530134 0.08945523
## V6 -0.3659849 0.15365241
## V7 -0.3698204 0.29475985
## V8 -0.3659489 0.33360619
## V9 -0.3542779 0.38656085summary(men.pca)## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6
## Standard deviation 2.5891 0.7990 0.47700 0.45371 0.3124 0.26587
## Proportion of Variance 0.8379 0.0798 0.02844 0.02573 0.0122 0.00884
## Cumulative Proportion 0.8379 0.9177 0.94615 0.97188 0.9841 0.99292
## PC7 PC8
## Standard deviation 0.21666 0.09858
## Proportion of Variance 0.00587 0.00121
## Cumulative Proportion 0.99879 1.00000men[order(men.pca$x[,1], decreasing = TRUE),1]## [1] U.S.A. GreatBritain Kenya France
## [5] Australia Italy Brazil Germany
## [9] Portugal Canada Belgium Poland
## [13] Russia Spain Japan Switzerland
## [17] Norway Netherlands Mexico NewZealand
## [21] Denmark Greece Hungary Finland
## [25] Ireland Sweden Austria Chile
## [29] China CzechRepublic Romania Argentina
## [33] Korea,South India Columbia Turkey
## [37] Israel Mauritius Luxembourg Taiwan
## [41] DominicanRepub Bermuda Thailand Indonesia
## [45] CostaRica Korea,North Malaysia Guatemala
## [49] Philippines Myanmar(Burma) PapuaNewGuinea Singapore
## [53] Samoa CookIslands
## 54 Levels: Argentina Australia Austria Belgium Bermuda Brazil ... U.S.A.#My intiution tells me that the highly developed countries lead the pack. I also happen to know that Kenyan always win marathons. So, even though my intuition leans to wealthy countries, Kenya has a long history of producing fast runners. So, this does not surprise me and it makes sense to me. ### (f)-(e)
men2<-men
men2[,5:9]<-(men2[,5:9])*60
men2[,2]<-100/(men2[,2])
men2[,3]<-200/(men2[,3])
men2[,4]<-400/(men2[,4])
men2[,5]<-800/(men2[,5])
men2[,6]<-1500/(men2[,6])
men2[,7]<-5000/(men2[,7])
men2[,8]<-10000/(men2[,8])
men2[,9]<-42195/(men2[,9])rform2<-data.matrix(men2[,2:9])
men2.pca = prcomp(rform2, center=TRUE)
men2.pca$rotation[,1:2]## PC1 PC2
## V2 0.2439701 0.43237108
## V3 0.3113827 0.52345617
## V4 0.3168151 0.46905827
## V5 0.2775048 0.03280175
## V6 0.3642621 -0.06284374
## V7 0.4276861 -0.26134677
## V8 0.4209180 -0.30988613
## V9 0.4163706 -0.38688033summary(men2.pca)## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6
## Standard deviation 0.7029 0.2150 0.11795 0.11542 0.08673 0.07582
## Proportion of Variance 0.8444 0.0790 0.02378 0.02277 0.01286 0.00983
## Cumulative Proportion 0.8444 0.9234 0.94713 0.96990 0.98276 0.99258
## PC7 PC8
## Standard deviation 0.05675 0.03348
## Proportion of Variance 0.00550 0.00192
## Cumulative Proportion 0.99808 1.00000men2[order(men2.pca$x[,1], decreasing = TRUE),1]## [1] U.S.A. Kenya GreatBritain France
## [5] Australia Italy Belgium Germany
## [9] Portugal Brazil Spain Canada
## [13] Russia Poland Japan Switzerland
## [17] Netherlands Mexico Norway NewZealand
## [21] Denmark Ireland Finland Greece
## [25] Hungary Sweden Austria China
## [29] Romania Chile CzechRepublic Argentina
## [33] Korea,South India Turkey Columbia
## [37] Israel Luxembourg Mauritius Taiwan
## [41] Korea,North CostaRica Thailand Guatemala
## [45] DominicanRepub Philippines Indonesia Bermuda
## [49] Malaysia Myanmar(Burma) Singapore PapuaNewGuinea
## [53] Samoa CookIslands
## 54 Levels: Argentina Australia Austria Belgium Bermuda Brazil ... U.S.A.pollution<-read.table(file="Data-HW4-pollution.dat", header=FALSE, quote="", sep="")
names(pollution) = c("wind","solar","co","no","no2","o3","hc")cov(pollution)## wind solar co no no2 o3
## wind 2.5000000 -2.7804878 -0.3780488 -0.4634146 -0.5853659 -2.2317073
## solar -2.7804878 300.5156794 3.9094077 -1.3867596 6.7630662 30.7909408
## co -0.3780488 3.9094077 1.5220674 0.6736353 2.3147503 2.8217189
## no -0.4634146 -1.3867596 0.6736353 1.1823461 1.0882695 -0.8106852
## no2 -0.5853659 6.7630662 2.3147503 1.0882695 11.3635308 3.1265970
## o3 -2.2317073 30.7909408 2.8217189 -0.8106852 3.1265970 30.9785134
## hc 0.1707317 0.6236934 0.1416957 0.1765389 1.0441347 0.5946574
## hc
## wind 0.1707317
## solar 0.6236934
## co 0.1416957
## no 0.1765389
## no2 1.0441347
## o3 0.5946574
## hc 0.4785134# center columns
center = function(v){v - mean(v)}
Xc = apply(pollution, 2, center)
# factor loading by principal component solution
fit.pca_pollution = eigen(cor(Xc))
#eigenvectors
v = fit.pca_pollution$vectors
rownames(v) = colnames(pollution)
#factor loadings
L1 = v[,1] * sqrt(fit.pca_pollution$values[1])
L2 = v[,2] * sqrt(fit.pca_pollution$values[2])
round(L1,3)## wind solar co no no2 o3 hc
## 0.362 -0.314 -0.842 -0.577 -0.761 -0.496 -0.488round(L2,3)## wind solar co no no2 o3 hc
## 0.328 -0.620 -0.008 0.512 0.235 -0.667 0.362# commonalities
# m = 1
round(L1^2, 3)## wind solar co no no2 o3 hc
## 0.131 0.099 0.710 0.333 0.580 0.246 0.238# m = 2
round(L1^2 + L2^2, 3)## wind solar co no no2 o3 hc
## 0.239 0.483 0.710 0.595 0.635 0.692 0.370sum(L1^2) / length(L1)## [1] 0.3338261sum(L1^2) / length(L1) + sum(L2^2) / length(L1)## [1] 0.5318262varimax(cbind(L1, L2), normalize = FALSE)## $loadings
##
## Loadings:
## L1 L2
## wind 0.160 0.461
## solar -0.695
## co -0.735 -0.412
## no -0.752 0.171
## no2 -0.781 -0.160
## o3 -0.114 -0.824
## hc -0.602
##
## L1 L2
## SS loadings 2.117 1.606
## Proportion Var 0.302 0.229
## Cumulative Var 0.302 0.532
##
## $rotmat
## [,1] [,2]
## [1,] 0.8768458 0.4807718
## [2,] -0.4807718 0.8768458##Interpretion of varimax ##L1 #co, no, and no2 items load relatively high to component L1 (absolute values: .735, .752 & .781, respectively). I would say that hc also loads relatively high to component L1 all of which are in contrast to wind which loads to .160 to component L1. ##L2 #item o3 loads the highest on the component L2. Yet, wind (.461), solar (.695) and co(.412) also load high on the component L1, all of which in contrast to no(.171) and no2(-.160)