Places
Darwin Mangubat
3/16/2022
Import the Data Needed
places <- read.table("places.txt")
View(places)Note that the data for many variables are strongly skewed to the right. Hence, log transform is used to normalize the data.
placespc <- log10(places[,1:9])
placespc## V1 V2 V3 V4 V5 V6 V7 V8
## 1 2.716838 3.792392 2.374748 2.965202 3.605413 3.440437 2.998259 3.147676
## 2 2.759668 3.910518 3.219060 2.947434 3.688687 3.387034 3.745387 3.420286
## 3 2.670246 3.865637 2.790988 2.986772 3.403292 3.408240 2.374748 2.933993
## 4 2.677607 3.898067 3.155640 2.785330 3.837778 3.531351 3.667920 3.208710
## 5 2.818885 3.923917 3.267875 3.171141 3.816771 3.480869 3.652826 3.416973
## 6 2.716003 3.764848 2.806180 2.861534 3.388101 3.473049 2.523746 3.007748
## 7 2.747412 3.918450 2.793092 2.710963 3.459543 3.497483 3.367915 3.048053
## 8 2.729974 3.812044 2.984527 2.848805 3.696793 3.469085 3.172311 3.107210
## 9 2.748963 3.791761 2.635484 2.600973 3.627980 3.443732 2.408240 3.082785
## 10 2.784617 3.815976 2.825426 3.030600 3.690373 3.455150 3.091667 3.044932
## 11 2.946943 4.205394 3.306425 2.992554 3.597037 3.453777 3.750663 3.499137
## 12 2.290035 4.085469 2.778874 3.087426 3.706803 3.382737 3.370328 3.477121
## 13 2.724276 3.756180 2.763428 2.943495 3.457125 3.392521 2.633468 2.923244
## 14 2.771587 3.757775 2.913814 2.989005 3.432488 3.442793 2.227887 2.787460
## 15 2.737193 4.041945 3.399328 3.028164 3.535674 3.524526 3.878464 3.359456
## 16 2.748188 3.742725 2.776701 3.051153 3.484442 3.340246 2.428135 3.066326
## 17 2.597695 3.896361 2.920645 2.720159 3.518251 3.453930 3.066699 3.364551
## 18 2.841359 3.827499 3.080626 2.752816 3.706376 3.475671 3.143327 3.188084
## 19 2.778874 3.825491 2.781755 2.969882 3.270912 3.422590 3.549739 3.000434
## 20 2.842609 3.919914 3.504471 3.116608 3.924744 3.485295 3.878464 3.134177
## 21 2.788875 4.044305 2.804139 3.273696 3.550962 3.466719 2.793092 3.433130
## 22 2.727541 3.798789 3.254790 2.940516 3.401917 3.464639 3.019947 2.960471
## 23 2.675778 4.016365 3.080266 2.914343 3.595827 3.343999 3.586250 3.255273
## 24 2.638489 3.946010 2.893207 3.020775 3.564666 3.486147 3.728759 3.026533
## 25 2.748188 3.906766 2.623249 3.193403 3.571126 3.408918 3.087071 3.195346
## 26 2.753583 3.961326 3.551694 3.238046 3.869525 3.540455 3.990694 3.466126
## 27 2.654177 3.861952 2.892095 2.813581 3.749195 3.467460 3.299943 3.332034
## 28 2.630428 3.907573 2.534026 3.194514 3.522314 3.420781 3.627058 3.150142
## 29 2.721811 3.802226 2.954243 3.013259 3.667640 3.394977 2.549003 3.216957
## 30 2.626340 3.798513 2.789581 3.118265 3.444357 3.438542 3.254064 3.258398
## 31 2.767898 3.895754 2.935003 2.491362 3.471292 3.403978 3.108565 3.170262
## 32 2.887617 3.920593 2.380211 2.916454 3.577032 3.443732 3.114611 3.505150
## 33 2.752816 3.830011 2.755875 3.075547 3.475526 3.405688 1.897627 3.169380
## 34 2.747412 4.164561 3.425045 2.932981 3.545431 3.562650 3.968670 3.282849
## 35 2.655138 3.919862 2.680336 2.908485 3.798305 3.478278 2.890980 3.310906
## 36 2.766413 3.810098 2.644439 2.908485 3.400711 3.413635 2.831870 3.323458
## 37 2.740363 3.916822 3.003029 2.618048 3.656002 3.484585 3.203848 3.236033
## 38 2.786751 3.833211 3.228400 3.050380 3.714078 3.454997 3.597476 3.091315
## 39 2.173186 3.922466 2.905256 2.615950 3.633771 3.429106 3.083144 3.212188
## 40 2.746634 3.848559 2.863917 2.817565 3.242044 3.458336 3.332842 3.298853
## 41 2.687529 3.937217 2.911158 2.828015 3.770042 3.455454 3.167317 3.205475
## 42 2.772322 3.914925 2.656098 2.944483 3.817896 3.378580 3.377488 3.223236
## 43 2.794488 4.064795 3.724358 3.084576 3.832573 3.541454 4.323087 3.486572
## 44 2.661813 4.076058 2.983175 3.036629 3.851747 3.412796 3.563837 3.603361
## 45 2.643453 3.916033 2.522444 3.038620 3.580355 3.433290 2.187521 3.130012
## 46 2.626340 3.923969 2.641474 2.885361 3.378580 3.434249 3.177825 3.179552
## 47 2.907411 3.957128 2.491362 2.813581 3.222716 3.405517 2.582063 3.295127
## 48 2.811575 4.128044 3.406540 2.974512 3.504743 3.481299 3.922622 3.281715
## 49 2.712650 4.001777 2.989005 2.736397 3.652730 3.419625 2.710963 2.890421
## 50 2.759668 3.917138 2.961895 3.125806 3.580925 3.436004 3.301247 3.085291
## 51 2.643453 3.730459 1.959041 2.988559 3.494015 3.382557 2.209515 3.477121
## 52 2.583199 3.915294 2.806180 3.006894 3.403121 3.473195 3.301464 3.150142
## 53 2.756636 3.906551 3.391817 2.987219 3.731105 3.494294 3.932829 3.387568
## 54 2.804139 3.790918 2.997386 2.849419 3.281033 3.401228 2.117271 2.845718
## 55 2.583199 3.985561 3.257439 2.693727 3.854063 3.521530 3.240799 3.525951
## 56 2.759668 3.865222 2.646404 2.812913 3.631342 3.439964 2.995196 3.063333
## 57 2.603144 3.992951 2.537819 2.995196 3.644439 3.389698 2.481443 3.156852
## 58 2.637490 3.890645 2.922725 2.853698 3.721811 3.418135 2.956168 3.176381
## 59 2.720159 3.935860 2.827369 3.009451 3.871981 3.497897 3.343014 3.230449
## 60 2.755112 3.869349 3.165244 3.174641 3.623973 3.500236 3.476107 3.408410
## 61 2.797268 3.891482 2.850033 2.857935 3.737987 3.461499 3.415808 2.926342
## 62 2.808886 3.855459 2.999565 3.104828 3.785259 3.481586 3.634779 3.092018
## 63 2.790988 3.979138 3.129690 2.878522 3.781109 3.542701 3.152900 3.231470
## 64 2.760422 3.791620 2.751279 2.975891 3.531607 3.382917 3.394977 3.092721
## 65 2.710963 4.037944 3.760875 3.014521 3.888853 3.542327 4.395256 3.455758
## 66 2.780317 3.933841 2.385606 2.976350 3.609274 3.494989 3.216694 3.188366
## 67 2.766413 3.910784 3.330008 2.990339 3.759517 3.465085 3.986234 3.389343
## 68 2.735599 3.778658 2.649335 2.866878 3.347525 3.423901 2.045323 3.346157
## 69 2.762679 3.962275 3.500648 3.056142 3.865282 3.473049 4.103085 3.518514
## 70 2.720986 3.929879 2.857935 3.035830 3.530072 3.439964 3.242790 3.375664
## 71 2.733197 3.886604 3.290257 3.027350 3.590284 3.376029 3.459694 3.124178
## 72 2.720986 3.876160 3.152594 3.182985 3.767823 3.463594 3.396025 3.171434
## 73 2.713491 3.764699 2.920645 2.913814 3.476397 3.425697 3.269746 3.084219
## 74 2.746634 3.908110 3.264109 3.038223 3.639885 3.466571 3.822691 3.305351
## 75 2.558709 3.840671 2.660865 3.125481 3.559428 3.453318 3.299289 3.308991
## 76 2.771587 3.782042 2.880814 2.527630 3.569257 3.526727 2.571709 3.015360
## 77 2.735599 3.969323 3.451018 3.184407 3.793301 3.514415 4.018617 3.363612
## 78 2.755112 4.158965 3.371068 2.738781 3.433770 3.481299 3.870111 3.196453
## 79 2.736397 3.756560 2.773055 2.578639 3.499824 3.468790 1.929419 2.699838
## 80 2.643453 3.907573 3.046495 2.921166 3.591843 3.462548 3.007321 3.283301
## 81 2.735599 3.882809 3.352761 3.061075 3.678973 3.442793 3.841046 3.049993
## 82 2.748963 3.857513 2.859138 3.129368 3.614581 3.416973 2.907949 3.598462
## 83 2.681241 3.868938 2.864511 2.952792 3.587374 3.428621 2.474216 3.087071
## 84 2.716838 4.032981 3.403635 3.135133 3.910891 3.497621 3.928242 3.366236
## 85 2.647383 3.904607 3.098990 3.018700 3.742018 3.417139 3.268812 3.255755
## 86 2.729165 3.930694 3.617210 3.200577 3.681964 3.486289 4.016574 3.394977
## 87 2.526339 3.756484 2.773055 2.968483 3.348694 3.348305 2.068186 2.853698
## 88 2.622214 3.902710 2.806180 2.756636 3.564429 3.431525 2.531479 3.200577
## 89 2.285557 3.781037 3.064083 2.688420 3.716421 3.418135 3.376029 3.492341
## 90 2.729974 3.812980 2.647383 3.039811 3.815511 3.419956 2.956168 3.206826
## 91 2.409933 3.849911 2.902003 2.636488 3.504743 3.471292 3.256958 3.145196
## 92 2.772322 3.865874 2.722634 3.121560 3.568788 3.394277 3.579784 3.041787
## 93 2.716838 3.817764 2.775246 2.719331 3.619928 3.404320 2.547775 3.009876
## 94 2.669317 3.849911 2.749736 2.764923 3.521661 3.477121 3.020361 3.204120
## 95 2.663701 3.834357 2.796574 2.926857 3.363988 3.441538 2.332438 3.079181
## 96 2.781755 3.887336 2.723456 2.802774 3.759970 3.421768 3.307924 3.127105
## 97 2.869818 3.971740 2.731589 2.941511 3.723702 3.493876 3.420121 3.531479
## 98 2.740363 3.828853 2.893762 2.936514 3.543571 3.446692 3.273233 3.210051
## 99 2.808211 3.955062 2.954243 2.935003 3.662947 3.387212 2.874482 3.302114
## 100 2.170262 3.912116 2.963788 2.701568 3.801061 3.398981 3.324488 3.150449
## 101 2.748963 3.797545 2.940516 3.060698 3.643650 3.484442 2.926342 2.850646
## 102 2.739572 3.825166 2.773786 2.736397 3.554004 3.368101 3.282169 3.229170
## 103 2.705008 3.916559 2.816241 2.816241 3.351023 3.447003 2.431364 2.897627
## 104 2.732394 3.857574 2.859739 3.222976 3.691258 3.399847 3.335057 3.131939
## 105 2.737193 3.775392 2.783189 2.712650 3.162564 3.385070 3.009026 2.997386
## 106 2.741939 3.813448 2.912753 3.125156 3.695744 3.492621 2.797960 2.903090
## 107 2.690196 3.997867 2.863917 2.871573 3.421110 3.382557 3.206556 3.623249
## 108 2.757396 4.033826 3.097604 3.186391 3.621799 3.436799 3.306854 3.390051
## 109 2.534026 3.968390 2.737193 2.895975 3.661150 3.436004 2.579784 3.602603
## 110 2.779596 3.946551 2.721811 3.152900 3.331022 3.498862 2.565848 3.313445
## 111 2.683047 3.762228 2.668386 2.821514 3.490239 3.466423 2.161368 3.239550
## 112 2.729165 3.878177 2.684845 2.735599 3.460296 3.448552 1.939519 3.320562
## 113 2.706718 3.828209 3.025306 2.851258 3.733679 3.442793 3.454235 3.233250
## 114 2.722634 3.900695 3.016197 3.129690 3.650502 3.419460 3.810636 3.374015
## 115 2.747412 3.968062 2.567026 3.171141 3.642267 3.381476 3.555820 3.474799
## 116 2.720986 3.730944 2.793790 2.874482 3.337260 3.361539 2.184691 2.477121
## 117 2.604226 3.868527 3.238297 3.219585 3.547405 3.490520 3.523096 3.092370
## 118 2.861534 3.890253 3.157457 3.083861 3.534407 3.448552 3.244525 3.477121
## 119 2.683947 3.883150 3.134814 2.998259 3.767527 3.402433 3.614370 3.287802
## 120 2.677607 3.852480 1.633468 2.754348 3.350442 3.427161 2.780317 3.274850
## 121 2.021189 3.897517 3.044932 2.603144 3.747179 3.434729 3.283527 3.115278
## 122 2.710117 3.890980 3.105169 2.978637 3.538322 3.432167 3.512551 3.280806
## 123 2.612784 3.853881 2.824126 2.898725 3.573684 3.437275 2.603144 3.337659
## 124 2.690196 3.914766 2.848805 2.997386 3.215109 3.455454 3.098298 2.868644
## 125 2.564666 3.924331 2.961895 2.765669 3.578983 3.418633 3.406029 3.284431
## 126 2.796574 3.849051 3.228913 2.985426 3.648653 3.489958 3.622007 3.217747
## 127 2.816241 3.801815 3.100371 3.073718 3.596597 3.510009 3.409764 3.149219
## 128 2.754348 3.890030 2.912753 2.797268 3.535421 3.475671 2.916454 3.173478
## 129 2.733999 3.915241 3.054996 2.950365 3.523486 3.438859 3.364739 3.205204
## 130 2.745075 3.897132 3.319522 2.798651 3.789863 3.508395 3.488974 3.185259
## 131 2.712650 4.066400 3.401573 3.106871 3.852480 3.559667 3.558228 3.252853
## 132 2.794488 3.829947 3.002598 2.883661 3.431846 3.435526 2.274158 2.901458
## 133 2.855519 4.230985 3.113275 2.949878 3.771661 3.398287 3.855398 3.568554
## 134 2.630428 3.850891 2.765669 2.602060 3.058805 3.299943 2.860338 3.431364
## 135 2.627366 3.989450 3.392169 3.175802 3.665206 3.514681 4.044265 3.261263
## 136 2.803457 3.821645 2.942008 2.822822 3.602169 3.402261 3.341435 2.924279
## 137 2.778151 3.798167 2.835691 2.965672 3.425045 3.353532 3.283527 3.031408
## 138 2.745855 3.845842 3.350829 3.000000 3.763727 3.429752 3.802637 3.280123
## 139 2.637490 3.974466 3.386856 2.919078 3.442480 3.453624 3.353147 3.177825
## 140 2.714330 3.832126 2.831870 3.009026 3.681241 3.423901 2.509203 3.286232
## 141 2.614897 3.860038 3.253338 3.037825 3.691700 3.495544 3.506370 3.154424
## 142 2.659916 3.821251 3.072250 3.083144 3.749040 3.484015 3.334856 3.459995
## 143 2.751279 3.786112 2.869232 2.985426 3.250420 3.422590 2.753583 3.070776
## 144 2.668386 3.871981 2.845098 2.933487 3.490239 3.403464 3.038223 3.208173
## 145 2.778874 3.944976 3.245266 3.156549 3.697404 3.410609 3.870404 3.000434
## 146 2.821514 3.786680 3.061452 2.627366 3.403464 3.466126 3.284431 3.333447
## 147 2.737987 3.814514 2.863917 2.547775 3.637790 3.429914 2.823474 2.955688
## 148 2.680336 3.969742 3.024486 2.922725 3.666986 3.457579 3.502017 3.213783
## 149 2.763428 3.712566 2.698970 2.797960 3.368287 3.458033 1.903090 2.903633
## 150 2.721811 3.898670 3.018284 3.049218 3.733919 3.461799 3.316180 3.335057
## 151 2.683947 3.859138 2.784617 2.989450 3.537063 3.455606 1.875061 3.048830
## 152 2.739572 3.909877 3.233250 3.057666 3.699491 3.481156 3.067071 3.019116
## 153 2.739572 3.849788 3.287578 3.166726 3.768564 3.469675 3.744528 3.310268
## 154 2.695482 3.930236 3.028164 2.959518 3.650599 3.465085 3.584783 3.347135
## 155 2.562293 3.810434 2.599883 2.865104 3.456670 3.500648 2.963788 2.968950
## 156 2.826075 3.825556 2.982271 2.793790 3.630733 3.441066 3.519697 3.400365
## 157 2.709270 3.820595 2.775246 2.615950 3.259355 3.462997 2.454845 3.000000
## 158 2.546543 3.919601 2.836324 2.829947 3.785045 3.481012 3.166134 3.290702
## 159 2.693727 3.890868 2.816241 2.667453 3.695131 3.469085 3.349278 3.258637
## 160 2.632457 3.933082 2.738781 3.012837 3.721646 3.362671 3.248464 3.239049
## 161 2.671173 3.840169 2.496930 3.038620 3.550106 3.368473 3.163161 3.268344
## 162 2.710963 4.123263 3.092370 2.914872 3.534280 3.416141 3.573568 3.386499
## 163 2.487138 3.824776 2.509203 3.137671 3.533009 3.476832 2.489958 3.400192
## 164 2.747412 3.936061 3.045714 2.615950 3.591955 3.490941 3.006466 3.059563
## 165 2.681241 3.898012 3.137037 2.951338 3.744840 3.461048 3.623869 3.206556
## 166 2.627366 3.789016 2.667453 3.021189 3.521400 3.451326 2.176091 2.846337
## 167 2.741939 3.842734 2.769377 3.163460 3.475526 3.437116 3.447778 3.206556
## 168 2.745075 3.995898 2.614897 3.281715 3.770852 3.350442 3.200303 3.601625
## 169 2.710117 3.874888 2.793092 3.007748 3.467016 3.431364 3.498311 3.243534
## 170 2.738781 4.017618 3.079904 2.958564 3.553276 3.394277 3.324488 3.273927
## 171 2.680336 3.767156 2.678518 3.062958 3.374015 3.375664 3.107210 3.244772
## 172 2.690196 3.837336 2.880242 2.883093 3.468495 3.430398 2.866878 3.267875
## 173 2.802774 3.921166 3.269513 3.023252 3.610660 3.456518 3.555820 3.147058
## 174 2.717671 3.844229 2.869818 2.931966 3.611086 3.419791 3.130977 3.154728
## 175 2.599883 3.916770 2.889302 2.897077 3.749582 3.459091 3.655427 3.256237
## 176 2.696356 3.861534 3.269746 3.123198 3.621799 3.411788 3.338456 3.164947
## 177 2.698970 3.820070 2.706718 2.989450 3.428135 3.449633 2.523746 2.921166
## 178 2.762679 3.919549 3.043362 2.784617 3.559787 3.411956 3.409087 3.204663
## 179 2.946943 4.142014 3.712060 3.292256 3.637990 3.504471 4.372304 3.596377
## 180 2.789581 3.833275 3.324488 2.971740 3.733999 3.481156 3.691612 3.288249
## 181 2.720986 3.984077 3.034628 2.913284 3.582063 3.394277 3.485295 3.052694
## 182 2.781037 3.844477 2.954243 3.206286 3.618884 3.405688 3.531734 3.230960
## 183 2.807535 3.840984 2.864511 2.808211 3.690993 3.447623 3.033021 3.158061
## 184 2.650308 3.794836 2.773055 2.893762 3.497483 3.423410 3.156852 3.080626
## 185 2.577492 3.995504 3.336059 2.891537 3.784189 3.483872 3.709524 3.288696
## 186 2.606381 3.993877 2.867467 2.801404 3.662286 3.435844 2.676694 2.922725
## 187 2.746634 3.837652 2.481443 3.030195 3.458789 3.458033 2.743510 3.054230
## 188 2.376577 3.727948 2.570543 2.922206 3.325721 3.422261 3.090258 3.024896
## 189 2.786041 3.954628 2.408240 2.862131 3.545555 3.446692 3.268578 3.151063
## 190 2.764923 3.940566 2.713491 3.016616 3.408240 3.449324 2.640481 3.579784
## 191 2.710963 3.846028 3.310268 3.172603 3.795672 3.447778 3.651859 3.299725
## 192 2.802089 4.011444 3.364363 3.390759 3.716170 3.459242 3.684576 3.633468
## 193 2.747412 4.084040 3.413132 2.839478 3.623042 3.548881 3.906227 3.203033
## 194 2.773055 4.066400 2.946452 2.810233 3.666143 3.495267 2.863323 3.225826
## 195 2.780317 3.938119 1.986772 3.066699 3.725095 3.383097 2.641474 3.176670
## 196 2.662758 4.007577 3.484727 2.916980 3.694166 3.483445 3.942801 3.462697
## 197 2.466868 3.980412 3.594834 2.957128 3.748653 3.478999 4.068705 3.334051
## 198 2.645422 3.826334 3.167022 3.179264 3.524396 3.443889 3.246499 3.335257
## 199 2.805501 3.936011 2.540329 3.062206 3.301030 3.417638 2.800029 2.920645
## 200 2.788875 4.066699 3.394802 2.913284 3.429752 3.445137 3.878694 3.549494
## 201 2.658011 3.791691 2.519828 2.980912 3.557026 3.389698 3.184123 3.187803
## 202 2.683947 3.829561 2.920123 2.911158 3.545183 3.378034 3.137987 3.046105
## 203 2.724276 3.763428 2.977266 2.893762 3.635986 3.472025 3.397592 3.154728
## 204 2.763428 3.805569 2.844477 3.186674 3.525434 3.419956 2.723456 3.425860
## 205 2.730782 4.031691 2.930949 2.655138 3.521138 3.435844 3.326745 3.182700
## 206 2.778151 3.892095 3.267172 2.992995 3.701568 3.441381 3.637690 3.266937
## 207 2.816904 4.046807 3.593175 2.752816 3.326131 3.509740 3.936514 3.568788
## 208 2.808211 3.907787 2.715167 3.005181 3.507721 3.387212 2.884229 3.161368
## 209 2.712650 4.021561 3.095169 2.955688 3.690196 3.495267 3.133539 3.085291
## 210 2.765669 4.059185 3.315551 2.950851 3.773640 3.543447 3.894980 3.205204
## 211 2.765669 4.009366 2.745075 2.801404 3.653695 3.511081 3.065953 3.358125
## 212 2.697229 3.930185 3.412629 3.205204 3.660771 3.476397 3.901894 3.544068
## 213 2.804821 4.125741 3.894870 3.397592 3.935759 3.474799 4.753928 3.553762
## 214 2.778874 4.152900 3.613419 3.164650 3.545802 3.526598 4.153022 3.259594
## 215 2.743510 3.885700 2.705008 2.889302 3.745153 3.404492 2.984977 3.272538
## 216 2.800717 3.932879 3.286007 2.998695 3.507181 3.479143 3.850462 3.471878
## 217 2.811575 4.304297 3.403121 2.795880 3.548512 3.481299 3.861714 3.355643
## 218 2.959041 4.118430 3.373280 3.185542 3.808211 3.422590 3.789722 3.379124
## 219 2.522444 3.829304 2.689309 3.122871 3.579555 3.456973 2.424882 3.490661
## 220 2.780317 3.825361 2.584331 3.229938 3.288696 3.434729 2.413300 2.954243
## 221 2.743510 3.856487 3.210319 3.112940 3.649237 3.463594 3.685114 3.241048
## 222 2.860937 3.917138 2.528917 2.876218 3.610979 3.419129 2.850033 3.161667
## 223 2.643453 3.852968 3.408070 3.003461 3.763877 3.486997 3.578295 3.296007
## 224 2.706718 4.007449 3.197005 2.905256 3.518382 3.439964 3.844291 3.270912
## 225 2.659916 3.913602 2.883661 3.222976 3.769894 3.473633 3.428297 3.459543
## 226 2.719331 3.829947 2.909556 2.754348 3.356599 3.396374 2.905256 2.930440
## 227 2.949390 4.146128 3.043755 2.898176 3.349860 3.333447 3.442323 3.329398
## 228 2.729165 3.804344 2.303196 3.128399 3.443732 3.397940 2.877947 3.319938
## 229 2.790285 3.823279 2.822822 2.688420 3.643354 3.398461 1.959041 3.059942
## 230 2.766413 3.795741 2.773055 2.771587 3.243038 3.230704 2.190332 3.291369
## 231 2.767898 3.975983 3.048053 2.871573 3.675595 3.485437 2.681241 3.179839
## 232 2.729165 3.811508 2.750508 3.167908 3.465085 3.464490 3.290925 3.334454
## 233 2.691081 3.923658 3.073352 2.964260 3.472318 3.389520 3.360593 3.227372
## 234 2.799341 3.919601 3.712481 3.024896 3.771073 3.577607 4.237292 3.296446
## 235 2.729165 3.950414 3.199755 3.103119 3.674769 3.468643 3.660201 3.393048
## 236 2.665581 3.753889 2.790285 3.067815 3.222976 3.407221 2.571709 2.899273
## 237 2.767898 3.908431 3.533136 2.836957 3.749427 3.549494 4.044108 3.331427
## 238 2.683047 3.892484 2.841359 2.804821 3.575072 3.513750 2.357935 3.152288
## 239 2.683947 3.908485 2.921166 2.915400 3.714749 3.473195 3.371253 3.527114
## 240 2.885361 3.996161 3.201397 3.177248 3.774298 3.524136 3.712650 3.403464
## 241 2.671173 3.998521 2.775246 2.676694 3.365675 3.480869 2.217484 3.143015
## 242 2.688420 3.999174 2.550228 2.801404 3.619719 3.462098 2.894870 3.222716
## 243 2.767898 3.967267 3.392169 2.999131 3.738305 3.551206 3.789016 3.354685
## 244 2.698970 3.969463 2.296665 2.685742 3.657629 3.417970 3.297761 3.518514
## 245 2.696356 3.821972 2.670246 3.072250 3.544192 3.423737 3.116276 3.209247
## 246 2.695482 3.951483 2.968950 3.023252 3.551206 3.436481 3.068557 3.304491
## 247 2.810904 3.915400 3.541080 2.991669 3.815843 3.538448 3.758155 3.205746
## 248 2.788168 3.881613 3.062206 2.717671 3.494155 3.481156 3.044540 3.190051
## 249 2.822168 3.933690 2.437751 2.950365 3.757927 3.392873 2.926857 3.384533
## 250 2.728354 4.095134 2.788875 3.047664 3.830396 3.402949 3.344392 3.377670
## 251 2.822168 3.927422 2.477121 2.891537 3.673390 3.237544 2.723456 3.080626
## 252 2.767156 3.921322 3.388811 3.031812 3.824776 3.468347 3.755646 3.288473
## 253 2.788875 3.989183 3.342620 3.168792 3.497068 3.414305 3.726483 3.282849
## 254 2.814248 3.873669 3.015360 2.894316 3.587935 3.435048 3.101403 3.308778
## 255 2.488551 3.963457 3.472171 2.640481 3.643354 3.329194 2.885926 3.176959
## 256 2.729165 3.934953 3.294246 2.951338 3.713070 3.554126 3.842360 3.424718
## 257 2.668386 3.879898 2.986324 3.062958 3.475235 3.428135 3.011147 2.969882
## 258 2.760422 3.993657 3.011570 3.134496 3.707315 3.446071 3.651569 3.362859
## 259 2.711807 3.867350 3.009451 3.028571 3.503246 3.442793 3.232488 3.313656
## 260 2.290035 3.859439 2.780317 2.535294 3.659441 3.398287 3.272074 3.196453
## 261 2.676694 3.747334 2.348305 2.986324 3.429591 3.466423 2.943989 3.102091
## 262 2.729974 3.881099 3.454845 3.115943 3.852419 3.547775 3.949195 3.350829
## 263 2.854913 3.923140 2.874482 3.006038 3.675045 3.515609 2.839478 3.272538
## 264 2.808886 4.065281 3.090611 2.690196 3.538951 3.436004 3.515344 3.349083
## 265 2.925828 4.141073 2.546543 3.044148 3.619093 3.387212 3.001734 3.502291
## 266 2.733197 3.976167 3.212454 2.986324 3.794349 3.369216 3.742568 3.591065
## 267 2.688420 3.800786 2.372912 3.013680 3.468052 3.432488 3.114277 3.055378
## 268 2.599883 3.838723 3.126131 3.078094 3.731347 3.468052 3.632963 3.178689
## 269 2.955688 4.160318 3.383097 3.040998 3.739493 3.446226 3.945370 3.524656
## 270 2.959041 4.234467 3.571243 3.209247 3.919026 3.527759 4.153083 3.662758
## 271 2.929419 4.205421 3.325721 3.027350 3.718003 3.432809 3.809290 3.293141
## 272 2.931966 4.191647 2.725912 3.011147 3.752970 3.434409 3.428783 3.518514
## 273 2.925828 4.155427 3.014940 2.984077 3.699838 3.416807 3.573800 3.231215
## 274 2.864511 4.111632 3.022016 2.959995 3.520221 3.434888 3.538699 3.353147
## 275 2.592177 3.980458 2.903633 2.972666 3.573104 3.419295 2.912222 3.403978
## 276 2.733999 3.838597 3.035029 3.207904 3.775100 3.390228 3.354493 3.349666
## 277 2.759668 3.825880 3.086004 2.570543 3.566202 3.509203 3.262925 3.141763
## 278 2.907411 4.007876 3.433770 3.068186 3.821775 3.432969 3.981229 3.681241
## 279 2.755875 3.825880 2.845098 2.584331 3.304706 3.480294 1.716003 3.041393
## 280 2.645422 3.909610 2.773055 2.653213 3.538825 3.407731 2.428135 3.119256
## 281 2.719331 3.757548 2.595496 3.014940 3.283753 3.423574 1.832509 2.971740
## 282 2.705864 3.815179 3.159868 3.078094 3.643551 3.456062 3.451172 3.142702
## 283 2.585461 3.814780 2.927370 2.880242 3.635081 3.426999 3.143951 3.133219
## 284 2.440909 3.902166 3.017451 2.745075 3.797337 3.423410 2.667453 3.121888
## 285 2.736397 3.773640 2.919078 3.016197 3.750817 3.458487 3.426836 3.259833
## 286 2.758912 3.840545 2.696356 2.939020 3.743039 3.443106 3.475381 3.180986
## 287 2.719331 3.896636 3.273464 3.088136 3.790426 3.488269 3.297323 3.186391
## 288 2.656098 3.905202 2.851258 3.083503 3.789510 3.547159 3.539829 3.180126
## 289 2.735599 3.802295 2.761176 2.950365 3.582972 3.432809 3.213252 3.239800
## 290 2.811575 4.373647 3.416641 2.921686 3.492760 3.481299 3.895699 3.237795
## 291 2.759668 3.924538 2.786751 2.732394 3.437751 3.500922 3.104146 3.079181
## 292 2.733999 3.818094 2.703291 2.621176 3.185259 3.426836 2.167317 3.164353
## 293 2.795880 3.928088 2.534026 3.144574 3.646110 3.333447 3.198382 3.212188
## 294 2.738781 3.884795 3.017033 2.838219 3.842047 3.497483 3.705864 3.454997
## 295 2.907411 3.890421 2.731589 3.065206 3.674861 3.405858 3.633165 3.602060
## 296 2.606381 3.904661 2.568202 3.064832 3.742725 3.445604 3.338656 3.286905
## 297 2.643453 3.871690 3.075182 3.174060 3.747256 3.483445 3.702431 3.468790
## 298 2.745855 3.742489 2.656098 2.799341 3.550228 3.478855 3.088490 3.146438
## 299 2.669317 3.757168 2.535294 2.914872 3.404320 3.462248 1.799341 2.825426
## 300 2.714330 3.890253 3.240050 2.999131 3.726156 3.455150 3.642366 3.290480
## 301 2.699838 3.851870 3.059942 2.999565 3.728191 3.446382 3.212720 3.057286
## 302 2.803457 4.025961 3.137354 3.072250 3.679973 3.519959 3.701482 3.216430
## 303 2.770115 3.931865 3.100026 3.146128 3.643156 3.428944 3.689220 3.495683
## 304 2.724276 3.874945 3.198932 3.033424 3.574957 3.419625 3.628185 3.306211
## 305 2.672098 3.810501 2.828660 3.006038 3.674218 3.378398 3.155943 3.037426
## 306 2.698970 3.863204 2.827369 2.980003 3.539076 3.516271 2.606381 2.800029
## 307 2.738781 3.828918 2.592177 2.602060 3.662002 3.472756 2.933487 3.243038
## 308 2.914343 4.021313 3.033021 2.984077 3.618362 3.397592 3.471585 3.192846
## 309 2.885361 3.954966 2.713491 2.876218 3.581722 3.367729 3.192289 3.165541
## 310 2.526339 3.853881 2.414973 3.038223 3.381476 3.430720 1.939519 3.149219
## 311 2.788875 3.863025 2.906874 3.054996 3.616265 3.438859 3.321598 3.168497
## 312 2.734800 3.890868 2.322219 3.053846 3.490520 3.327972 2.708421 3.447158
## 313 2.614897 3.785757 2.730782 3.066699 3.479719 3.457428 3.057286 3.096215
## 314 2.800029 4.137481 3.639586 3.119586 3.915716 3.560504 4.336480 3.198107
## 315 2.755112 4.001041 3.085647 2.897077 3.386321 3.476397 2.502427 2.975891
## 316 2.540329 3.896581 2.966142 2.845098 3.525174 3.460748 3.477121 3.278754
## 317 2.488551 3.883207 2.912753 2.645422 3.543571 3.439175 2.881385 3.218536
## 318 2.706718 4.021685 2.574031 3.251151 3.716087 3.508395 3.460597 3.576572
## 319 2.733999 3.817962 2.898176 2.488551 3.389166 3.477411 2.625312 3.104146
## 320 2.693727 3.848866 2.906335 3.065953 3.594724 3.474362 3.475235 3.178401
## 321 2.658965 3.806451 2.739572 3.071514 3.446071 3.438859 2.777427 3.051538
## 322 2.746634 3.862370 2.934498 2.666518 3.490941 3.463296 2.292256 2.860937
## 323 2.775974 3.899109 3.159868 3.047275 3.656290 3.493040 3.657534 3.283979
## 324 2.751279 3.836197 3.040998 3.153205 3.462997 3.458789 3.032216 3.426186
## 325 2.749736 3.940267 3.256477 2.832509 3.561459 3.518382 3.251395 2.959041
## 326 2.728354 3.808886 2.501059 3.043755 3.571825 3.396374 2.998259 3.330414
## 327 2.732394 3.922777 2.853090 2.643453 3.355452 3.462847 3.009451 2.925312
## 328 2.755875 3.846399 3.040207 2.972203 3.528145 3.465383 3.446692 3.122871
## 329 2.783904 3.896251 2.326336 3.071514 3.442166 3.377852 2.086360 2.962843
## V9
## 1 3.882695
## 2 3.638489
## 3 3.720159
## 4 3.768194
## 5 3.757927
## 6 3.720490
## 7 3.707315
## 8 3.763053
## 9 3.626340
## 10 3.795254
## 11 3.793790
## 12 3.884682
## 13 3.527630
## 14 3.629613
## 15 3.660771
## 16 3.674861
## 17 3.722222
## 18 3.715669
## 19 3.715418
## 20 3.800373
## 21 3.908860
## 22 3.734880
## 23 3.707315
## 24 3.871515
## 25 3.782186
## 26 3.740600
## 27 3.713659
## 28 3.799892
## 29 3.602928
## 30 3.779524
## 31 3.494015
## 32 3.628082
## 33 3.560504
## 34 3.779308
## 35 3.771808
## 36 3.763503
## 37 3.749272
## 38 3.685114
## 39 3.779524
## 40 3.683857
## 41 3.768120
## 42 3.665862
## 43 3.803662
## 44 3.852907
## 45 3.871398
## 46 3.779596
## 47 3.753660
## 48 3.857152
## 49 3.814714
## 50 3.838849
## 51 3.696182
## 52 3.905256
## 53 3.703033
## 54 3.754348
## 55 3.827757
## 56 3.685473
## 57 3.799547
## 58 3.699751
## 59 3.739177
## 60 3.712060
## 61 3.720738
## 62 3.753660
## 63 3.782114
## 64 3.651956
## 65 3.716421
## 66 3.724849
## 67 3.721811
## 68 3.688420
## 69 3.688331
## 70 3.886434
## 71 3.711554
## 72 3.722552
## 73 3.682326
## 74 3.713070
## 75 3.825426
## 76 3.675870
## 77 3.887054
## 78 3.848805
## 79 3.652343
## 80 3.698709
## 81 3.656290
## 82 3.819017
## 83 3.630835
## 84 3.855156
## 85 3.728029
## 86 3.591510
## 87 3.736635
## 88 3.596487
## 89 3.593508
## 90 3.786254
## 91 3.728191
## 92 3.705864
## 93 3.624695
## 94 3.682416
## 95 3.917400
## 96 3.633367
## 97 3.483587
## 98 3.716504
## 99 3.688776
## 100 3.747722
## 101 3.720573
## 102 3.665675
## 103 3.707400
## 104 3.571010
## 105 3.652440
## 106 3.685025
## 107 3.821579
## 108 3.853455
## 109 3.855277
## 110 3.901622
## 111 3.685652
## 112 3.802226
## 113 3.622732
## 114 3.836451
## 115 3.759366
## 116 3.625312
## 117 3.758836
## 118 3.780029
## 119 3.582745
## 120 3.713154
## 121 3.751741
## 122 3.685563
## 123 3.671821
## 124 3.735838
## 125 3.752048
## 126 3.716337
## 127 3.700011
## 128 3.650987
## 129 3.664454
## 130 3.726075
## 131 3.799823
## 132 3.674677
## 133 3.714916
## 134 3.823605
## 135 3.872972
## 136 3.641771
## 137 3.806994
## 138 3.706035
## 139 3.713070
## 140 3.582291
## 141 3.714833
## 142 3.788098
## 143 3.805229
## 144 3.640581
## 145 3.689220
## 146 3.690462
## 147 3.621280
## 148 3.665675
## 149 3.726238
## 150 3.680698
## 151 3.746556
## 152 3.790004
## 153 3.687083
## 154 3.713994
## 155 3.801472
## 156 3.743275
## 157 3.535167
## 158 3.751895
## 159 3.636789
## 160 3.986861
## 161 3.768786
## 162 3.760045
## 163 3.747722
## 164 3.709270
## 165 3.676419
## 166 3.721316
## 167 3.727623
## 168 3.780677
## 169 3.660201
## 170 3.814714
## 171 3.785686
## 172 3.647774
## 173 3.795532
## 174 3.634779
## 175 3.690905
## 176 3.722058
## 177 3.838723
## 178 3.518646
## 179 3.725585
## 180 3.732555
## 181 3.822887
## 182 3.772542
## 183 3.692494
## 184 3.752740
## 185 3.736237
## 186 3.851320
## 187 3.642069
## 188 3.758836
## 189 3.567262
## 190 3.850585
## 191 3.712650
## 192 3.766413
## 193 3.800992
## 194 3.799823
## 195 3.999131
## 196 3.697404
## 197 3.766636
## 198 3.659441
## 199 3.708166
## 200 3.789157
## 201 3.743275
## 202 3.689486
## 203 3.599883
## 204 3.569140
## 205 3.842734
## 206 3.773640
## 207 3.867526
## 208 3.693463
## 209 3.810904
## 210 3.738622
## 211 3.824256
## 212 3.809762
## 213 3.727379
## 214 3.755112
## 215 3.649627
## 216 3.768342
## 217 3.808346
## 218 3.736954
## 219 3.848805
## 220 3.878809
## 221 3.909503
## 222 3.691258
## 223 3.767379
## 224 3.765072
## 225 3.869994
## 226 3.782616
## 227 3.741467
## 228 3.784118
## 229 3.714916
## 230 3.652343
## 231 3.712144
## 232 3.780245
## 233 3.609914
## 234 3.751125
## 235 3.807197
## 236 3.628082
## 237 3.721068
## 238 3.739018
## 239 3.791410
## 240 3.656577
## 241 3.922570
## 242 3.829046
## 243 3.712144
## 244 3.538951
## 245 3.667079
## 246 3.644931
## 247 3.806519
## 248 3.747179
## 249 3.649237
## 250 3.754119
## 251 3.726401
## 252 3.768638
## 253 3.692230
## 254 3.723209
## 255 3.785259
## 256 3.799616
## 257 3.662002
## 258 3.725013
## 259 3.569257
## 260 3.739414
## 261 3.777499
## 262 3.763428
## 263 3.583765
## 264 3.799961
## 265 3.752509
## 266 3.693903
## 267 3.887617
## 268 3.837146
## 269 3.739493
## 270 3.782688
## 271 3.861534
## 272 3.764998
## 273 3.727134
## 274 3.756103
## 275 3.887336
## 276 3.747489
## 277 3.690816
## 278 3.770926
## 279 3.607991
## 280 3.678063
## 281 3.793301
## 282 3.818556
## 283 3.667266
## 284 3.716337
## 285 3.703807
## 286 3.674126
## 287 3.731105
## 288 3.723374
## 289 3.773201
## 290 3.789440
## 291 3.669967
## 292 3.573336
## 293 3.753736
## 294 3.650696
## 295 3.689042
## 296 3.779669
## 297 3.860697
## 298 3.638789
## 299 3.678700
## 300 3.656482
## 301 3.737511
## 302 3.761326
## 303 3.788663
## 304 3.852175
## 305 3.690196
## 306 3.864926
## 307 3.718169
## 308 3.764848
## 309 3.660011
## 310 3.880756
## 311 3.700963
## 312 3.745309
## 313 3.796505
## 314 3.783332
## 315 3.752509
## 316 3.644143
## 317 3.633468
## 318 3.902655
## 319 3.675778
## 320 3.780749
## 321 3.832828
## 322 3.516932
## 323 3.790567
## 324 3.731589
## 325 3.702431
## 326 3.697752
## 327 3.694254
## 328 3.590396
## 329 3.671543Principal Component Analysis
PCA <- princomp(placespc)
summary(PCA)## Importance of components:
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
## Standard deviation 0.6134452 0.22560372 0.1668374 0.15131845 0.12930691
## Proportion of Variance 0.7226740 0.09774248 0.0534537 0.04397184 0.03210956
## Cumulative Proportion 0.7226740 0.82041652 0.8738702 0.91784205 0.94995161
## Comp.6 Comp.7 Comp.8 Comp.9
## Standard deviation 0.10916208 0.09182048 0.062627954 0.042330504
## Proportion of Variance 0.02288413 0.01619086 0.007532295 0.003441103
## Cumulative Proportion 0.97283574 0.98902660 0.996558897 1.000000000cumulative proportion explains total variance.
Determine the eigenvalues to know how many principal components should be considered.
eigenvectors <- PCA$loadings
eigenvalues <- PCA$sdev*PCA$sdev
eigenvalues## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6
## 0.376315063 0.050897040 0.027834722 0.022897273 0.016720276 0.011916361
## Comp.7 Comp.8 Comp.9
## 0.008431000 0.003922261 0.001791872round(eigenvalues, 4)## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9
## 0.3763 0.0509 0.0278 0.0229 0.0167 0.0119 0.0084 0.0039 0.0018Eigenvalues and the proportion of variation explained by the principal components. Loadings are the eignevectors.
PCA$loadings##
## Loadings:
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9
## V1 0.141 0.153 0.398 0.831 0.315
## V2 0.129 -0.178 0.175 0.209 -0.696 -0.614
## V3 0.408 -0.859 0.276
## V4 0.100 0.220 0.593 0.724 -0.164 -0.182
## V5 0.150 0.221 -0.126 -0.870 0.372
## V6 0.204 -0.973
## V7 0.874 0.304 -0.363
## V8 0.159 0.334 0.584 -0.628 0.213 -0.142 0.235
## V9 0.121 -0.265 -0.664 0.662 0.168
##
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9
## SS loadings 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
## Proportion Var 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111
## Cumulative Var 0.111 0.222 0.333 0.444 0.556 0.667 0.778 0.889 1.000Produce the correlations with the variables in the data. For more than 1 principal component, the resulting score is,
round(cor(places[,1:9], PCA$scores), 3)## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9
## V1 0.236 0.027 0.204 0.130 0.439 0.782 -0.003 0.102 0.058
## V2 0.508 0.001 0.188 -0.232 0.254 0.225 -0.600 -0.372 -0.034
## V3 0.763 -0.425 0.178 0.003 0.046 -0.013 -0.018 -0.011 -0.063
## V4 0.373 0.268 0.592 0.595 -0.007 -0.123 0.042 -0.070 -0.027
## V5 0.594 0.054 0.244 -0.119 -0.677 0.245 -0.044 0.038 0.019
## V6 0.407 -0.281 0.031 -0.020 -0.119 0.059 -0.112 0.256 -0.806
## V7 0.724 -0.108 0.111 0.047 0.049 0.020 -0.025 -0.061 -0.068
## V8 0.469 0.359 0.558 -0.469 0.173 -0.062 0.098 0.015 0.008
## V9 0.112 0.165 0.243 0.086 0.049 -0.339 -0.723 0.478 0.077Another way for determining the number of principal components is to consider the Scree Plot.
screeplot(PCA, type = 'lines')
abline(1,0,col='red',lty=2)
Hence, from the results above, PCA 1 is the component that can be viewed as a measure of the quality of Arts, Health, Transportation, and Recreation, and the lack of quality in Housing (recall that high values for Housing are bad). Also, this component increases with increasing Arts, Health, Transportation, Housing and Recreation scores. This suggests that these five criteria vary together. If one increases, then the remaining ones tend to increase as well. Lastly, via screeplot, this component should be retained as significant.