# 3.5 + 1.0 + 0.7 + 0.4 + 0.25 + 0.15 = 6
#we would want to keep at least 4 of the eigenvalues
#Which would give us a total of .9333 and which is 93.3% the
#is the least amount possible of added eigenvalues being higher
#than 90%Activity 3.3 - PCA implementation
SUBMISSION INSTRUCTIONS
- Render to html
- Publish your html to RPubs
- Submit a link to your published solutions
Problem 1
Consider the following 6 eigenvalues from a \(6\times 6\) correlation matrix:
\[\lambda_1 = 3.5, \lambda_2 = 1.0, \lambda_3 = 0.7, \lambda_4 = 0.4, \lambda_5 = 0.25, \lambda_6 = 0.15\]
If you want to retain enough principal components to explain at least 90% of the variability inherent in the data set, how many should you keep?
Problem 2
The iris data set is a classic data set often used to demonstrate PCA. Each iris in the data set contained a measurement of its sepal length, sepal width, petal length, and petal width. Consider the five irises below, following mean-centering and scaling:
library(tidyverse)
five_irises <- data.frame(
row.names = 1:5,
Sepal.Length = c(0.189, 0.551, -0.415, 0.310, -0.898),
Sepal.Width = c(-1.97, 0.786, 2.62, -0.590, 1.70),
Petal.Length = c(0.137, 1.04, -1.34, 0.534, -1.05),
Petal.Width = c(-0.262, 1.58, -1.31, 0.000875, -1.05)
) %>% as.matrix
five_irises Sepal.Length Sepal.Width Petal.Length Petal.Width
1 0.189 -1.970 0.137 -0.262000
2 0.551 0.786 1.040 1.580000
3 -0.415 2.620 -1.340 -1.310000
4 0.310 -0.590 0.534 0.000875
5 -0.898 1.700 -1.050 -1.050000Consider also the loadings for the first two principal components:
# Create the data frame
library(factoextra)Warning: package 'factoextra' was built under R version 4.5.2Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBafive_irises_pca <- prcomp(five_irises, scale. = TRUE)
five_irises_pcaStandard deviations (1, .., p=4):
[1] 1.7834925 0.8249853 0.3505249 0.1252439
Rotation (n x k) = (4 x 4):
PC1 PC2 PC3 PC4
Sepal.Length -0.5352000 0.06795288 0.8343787 0.1129407
Sepal.Width 0.3901261 0.86525013 0.2113644 -0.2333858
Petal.Length -0.5537354 0.10055074 -0.2570335 -0.7856211
Petal.Width -0.5047175 0.48643002 -0.4393985 0.5617606fviz_pca(five_irises_pca, axes=1:2)Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.
ℹ The deprecated feature was likely used in the ggpubr package.
Please report the issue at <https://github.com/kassambara/ggpubr/issues>.
#both five_irises_pca and pc_loadings yield the same thing
pc_loadings <- data.frame(
PC1 = c(0.5210659, -0.2693474, 0.5804131, 0.5648565),
PC2 = c(-0.37741762, -0.92329566, -0.02449161, -0.06694199),
row.names = c("Sepal.Length", "Sepal.Width", "Petal.Length", "Petal.Width")
) %>% as.matrix
pc_loadings PC1 PC2
Sepal.Length 0.5210659 -0.37741762
Sepal.Width -0.2693474 -0.92329566
Petal.Length 0.5804131 -0.02449161
Petal.Width 0.5648565 -0.06694199A plot of the first two PC scores for these five irises is shown in the plot below.
Match the ID of each iris (1-5) to the correct letter of its score coordinates on the plot.
#Iris 1
#PC1 = 0.189(0.52)-1.97(-0.269)+0.137(0.58)-0.262(0.565) = 0.559
#PC2 = 0.189(-0.377)-1.979(-0.924)+0.137(-0.0245) = 1.763
# Iris 1 = (0.559,1.76)
#Iris 2
#PC1 = 0.551(0.52) + 0.786(-0.269) + 1.04(0.58) + 1.58(0.56) = 1.563
#PC2 = 0.551(-0.377) + 0.786(-0.924) + 1.04(-0.0245) + 1.58(-0.067)
#= -1.065
#Iris 2 = (1.563, -1.065)
#Iris 3
#
#PC1 = -0.415(0.52) + 2.62(-.269) - 1.34(0.58) - 1.31(0.565)
# = -2.44
#PC2 = -0.415(-0.378) + 2.62(-0.924) + 1.04(-0.0245) + 1.58(-0.067)
#= 2.45
#Iris 3 = (-2.44,-2.39)
#Iris 4
#
#PC1 = 0.31(0.52)-0.59(-.269) + 0.534(0.58) + 0.00088(0.56)= 0.630
#
#PC2 = 0.31(-0.378) -0.59(-0.924)+0.534(-0.245) + 0.00088(-0.066) = #.2965
#
#Iris 4 = (0.63, 0.2965)
#Iris 5
#
#PC1 = -0.898(0.52) + 1.7(-0.269) - 1.05 (0.58) - 1.05(.565) = -2.1265
#
#PC2 = -0.898(-0.3) + 1.7(-0.924) - 1.05 (-0.0245) - 1.05(-0.067)
# = -1.135
#
#Iris 5 = (-2.13, -1.14)
#1-5 member that
# point a = iris 3
# point b = iris 1
# point c = iris 4
# point d = iris 2
# point e = iris 5Problem 3
These data are taken from the Places Rated Almanac, by Richard Boyer and David Savageau, copyrighted and published by Rand McNally. The nine rating criteria used by Places Rated Almanac are:
- Climate & Terrain
- Housing
- Health Care & Environment
- Crime
- Transportation
- Education
- The Arts
- Recreation
- Economics
For all but two of the above criteria, the higher the score, the better. For Housing and Crime, the lower the score the better. The scores are computed using the following component statistics for each criterion (see the Places Rated Almanac for details):
- Climate & Terrain: very hot and very cold months, seasonal temperature variation, heating- and cooling-degree days, freezing days, zero-degree days, ninety-degree days.
- Housing: utility bills, property taxes, mortgage payments.
- Health Care & Environment: per capita physicians, teaching hospitals, medical schools, cardiac rehabilitation centers, comprehensive cancer treatment centers, hospices, insurance/hospitalization costs index, flouridation of drinking water, air pollution.
- Crime: violent crime rate, property crime rate.
- Transportation: daily commute, public transportation, Interstate highways, air service, passenger rail service.
- Education: pupil/teacher ratio in the public K-12 system, effort index in K-12, accademic options in higher education.
- The Arts: museums, fine arts and public radio stations, public television stations, universities offering a degree or degrees in the arts, symphony orchestras, theatres, opera companies, dance companies, public libraries.
- Recreation: good restaurants, public golf courses, certified lanes for tenpin bowling, movie theatres, zoos, aquariums, family theme parks, sanctioned automobile race tracks, pari-mutuel betting attractions, major- and minor- league professional sports teams, NCAA Division I football and basketball teams, miles of ocean or Great Lakes coastline, inland water, national forests, national parks, or national wildlife refuges, Consolidated Metropolitan Statistical Area access.
- Economics: average household income adjusted for taxes and living costs, income growth, job growth.
In addition to these, latitude and longitude, population and state are also given, but should not be included in the PCA.
Use PCA to identify the major components of variation in the ratings among cities.
places <- read.csv('C:/Users/lr7273ow/OneDrive - Minnesota State/Documents/GitHub/DSCI_415/Activities/Data/Places.csv')
head(places) City Climate Housing HlthCare Crime Transp Educ Arts
1 AbileneTX 521 6200 237 923 4031 2757 996
2 AkronOH 575 8138 1656 886 4883 2438 5564
3 AlbanyGA 468 7339 618 970 2531 2560 237
4 Albany-Schenectady-TroyNY 476 7908 1431 610 6883 3399 4655
5 AlbuquerqueNM 659 8393 1853 1483 6558 3026 4496
6 AlexandriaLA 520 5819 640 727 2444 2972 334
Recreat Econ Long Lat Pop
1 1405 7633 -99.6890 32.5590 110932
2 2632 4350 -81.5180 41.0850 660328
3 859 5250 -84.1580 31.5750 112402
4 1617 5864 -73.7983 42.7327 835880
5 2612 5727 -106.6500 35.0830 419700
6 1018 5254 -92.4530 31.3020 135282places_num <- places %>% select('Climate','Housing','HlthCare',
'Crime','Transp','Educ','Arts','Recreat','Econ')A.
If you want to explore this data set in lower dimensional space using the first \(k\) principal components, how many would you use, and what percent of the total variability would these retained PCs explain? Use a scree plot to help you answer this question.
places_pca <- prcomp(places_num, scale. = TRUE)
places_pcaStandard deviations (1, .., p=9):
[1] 1.8461560 1.1018059 1.0684003 0.9596446 0.8679199 0.7940793 0.7021736
[8] 0.5639490 0.3469900
Rotation (n x k) = (9 x 9):
PC1 PC2 PC3 PC4 PC5 PC6
Climate 0.2064140 0.2178353 -0.689955982 0.13732125 -0.3691499 0.37460469
Housing 0.3565216 0.2506240 -0.208172230 0.51182871 0.2334878 -0.14163983
HlthCare 0.4602146 -0.2994653 -0.007324926 0.01470183 -0.1032405 -0.37384804
Crime 0.2812984 0.3553423 0.185104981 -0.53905047 -0.5239397 0.08092329
Transp 0.3511508 -0.1796045 0.146376283 -0.30290371 0.4043485 0.46759180
Educ 0.2752926 -0.4833821 0.229702548 0.33541103 -0.2088191 0.50216981
Arts 0.4630545 -0.1947899 -0.026484298 -0.10108039 -0.1050976 -0.46188072
Recreat 0.3278879 0.3844746 -0.050852640 -0.18980082 0.5295406 0.08991578
Econ 0.1354123 0.4712833 0.607314475 0.42176994 -0.1596201 0.03260813
PC7 PC8 PC9
Climate -0.08470577 -0.36230833 0.0013913515
Housing -0.23063862 0.61385513 0.0136003402
HlthCare 0.01386761 -0.18567612 -0.7163548935
Crime 0.01860646 0.43002477 -0.0586084614
Transp -0.58339097 -0.09359866 0.0036294527
Educ 0.42618186 0.18866756 0.1108401911
Arts -0.02152515 -0.20398969 0.6857582127
Recreat 0.62787789 -0.15059597 -0.0255062915
Econ -0.14974066 -0.40480926 0.0004377942fviz_eig(places_pca)Warning in geom_bar(stat = "identity", fill = barfill, color = barcolor, :
Ignoring empty aesthetic: `width`.
#ScreePlot should appear:
#We would use k=3 judging by the Scree Plot
#PC1 = 36%, PC2 = 13%, PC3 = 12%, PC4 = 10.5%, PC5 =7.5%,
#PC6 = 6.5% , PC7 = 5.5%, PC8 = 3.5% , PC9 = 1.5%
#In total this would explain ~61% of the variability for
#these PC values.
#
#B.
Interpret the retained principal components by examining the loadings (plot(s) of the loadings may be helpful). Which variables will be used to separate cities along the first and second principal axes, and how? Make sure to discuss the signs of the loadings, not just their contributions!
loadings function (x, ...)
x$loadings
<bytecode: 0x000001d187b9e0e8>
<environment: namespace:stats>#reference line is placed at 100*1/p, 8.333% for us
#first dimension contributions
fviz_contrib(places_pca, choice = 'var', axes = 1) +
theme_classic(base_size = 8) +
labs(x = 'Variable',
title = 'Contribution to first dimension')
#interpretion: For the first principal axes the variables that
#contribute the most to PC are Arts, Healthcare, housing, #Transportation, and Housing.
#As all loadings are >0, generally cities with more robust arts, #HC housing & transportation will have large PC1 values and cities with #less of those aspects will have smaller PC1 values
#
#second dimension contributions
fviz_contrib(places_pca, choice = 'var', axes = 2) +
theme_classic(base_size = 8) +
labs(x = 'Variable',
title = 'Contribution to second dimension')
#interpretion: For the second principal axes the variables that #contribute the most to principal components are Education, Economy Recreation, and Crime.
#As all loadings are >0, generally cities with those higher #metrics will have a larger PC2s and cities with metrics that
#are lower than that would have smaller PC2s. C.
Add the first two PC scores to the places data set. Create a biplot of the first 2 PCs, using repelled labeling to identify the cities. Which are the outlying cities and what characteristics make them unique?
places_pca$x PC1 PC2 PC3 PC4 PC5
[1,] -1.040179917 0.8937689684 1.4366540694 0.509834129 -5.651365e-01
[2,] 0.439813596 0.0750661769 -1.1547165398 -1.112207185 9.968838e-01
[3,] -1.875539260 0.0697916922 0.0733467565 -0.046231622 -6.795244e-01
[4,] 0.910741419 -1.8175821509 1.0961471968 0.573580789 7.690800e-01
[5,] 2.149247536 0.3288580786 0.0197383504 -1.031501541 -2.385498e-01
[6,] -1.787961101 -0.7812016710 0.0608349850 0.464818648 -8.239929e-01
[7,] -1.055448363 -1.0759557980 -0.4259539611 1.320925920 -3.345121e-01
[8,] -0.648321022 -0.7594404445 0.4080465874 0.231162972 -2.439728e-02
[9,] -1.804240224 -1.2134764019 -0.9165071462 0.062613595 4.964370e-01
[10,] -0.499566965 0.1022860267 0.3720736793 -0.095466883 -8.783725e-01
[11,] 2.973047786 2.0152753943 -2.3577097808 2.016339571 1.877389e-01
[12,] 0.447705316 2.2268258039 2.7026712755 0.003548290 2.066703e+00
[13,] -2.325173499 -0.8097023812 -1.2695267539 -1.083594618 -4.792700e-01
[14,] -1.815471687 -0.8181536195 -0.8550138336 -0.397388971 -1.344734e+00
[15,] 1.869943160 -1.0859874353 -0.5100809574 0.589741006 -2.856285e-01
[16,] -2.006028933 0.6188154454 -0.7375681985 -1.335312324 -7.002503e-01
[17,] -1.053951634 -0.3685144988 0.4031678212 0.446434184 1.182311e+00
[18,] -0.258106267 -0.8754437800 -0.8913924642 0.406310285 -3.665069e-02
[19,] -1.486790794 0.0652703329 -0.7139992448 0.122698736 -1.308464e+00
[20,] 3.025009160 -0.9309766612 0.3140421736 -0.620456023 -7.202042e-01
[21,] 1.369138378 3.0489754270 1.2270895184 0.391703994 -1.266965e+00
[22,] -1.026998436 -0.8345920434 0.0751086783 0.382249568 -1.204280e+00
[23,] -0.495600609 0.6627356544 -0.5833553117 -0.174722298 9.384964e-01
[24,] 0.002451547 0.1314013049 1.8302323410 1.099735952 -8.798192e-01
[25,] -0.455759052 1.4519459096 0.3164655016 -0.799710538 -1.082017e+00
[26,] 4.297358823 -0.9680198888 0.8235468853 -1.309209666 -3.571231e-01
[27,] -0.335278286 -0.7723509525 0.4531611272 -0.218725953 1.249516e+00
[28,] -0.478998124 1.0922358463 1.2219587439 -0.729185411 -1.035486e+00
[29,] -1.122352286 -0.2677294227 -0.7341054547 -1.475207783 2.625089e-01
[30,] -0.982173269 0.6330713237 1.1193550156 -0.658556383 -7.215045e-01
[31,] -1.830739805 -1.0811695729 -2.2034771761 0.087515367 7.874943e-01
[32,] -0.076348081 0.8415054047 -2.2550829013 -0.270128168 6.030321e-01
[33,] -1.511882599 -0.0162004424 -1.2284915896 -1.272745700 -5.696155e-01
[34,] 2.819651817 -1.0367095954 0.0379058203 2.607211774 -3.529655e-01
[35,] -0.035454028 -0.2835524455 0.9905653200 0.036511400 1.136500e+00
[36,] -1.373428901 0.8480378737 -0.3449391296 0.083767901 -1.660390e-01
[37,] -0.419610952 -0.8311277762 -0.0699921763 1.093860974 6.516072e-01
[38,] 0.267406582 -0.8149842215 -0.4290206999 -0.786474240 -7.369178e-01
[39,] -1.577584052 -0.7562501398 2.1611457443 0.513335884 1.975477e+00
[40,] -1.282610879 -0.2423605423 -0.7519446360 0.489382350 -1.972359e-01
[41,] -0.222897558 -0.4010209535 0.5332265895 0.351468133 9.216684e-01
[42,] -0.377734026 0.1285719676 -0.8793145247 -1.081142630 9.888063e-01
[43,] 6.301057205 -1.6087416195 0.3626613303 0.276039697 -1.732851e-01
[44,] 2.030150765 2.1142645523 1.0982432454 -0.260283954 2.557876e+00
[45,] -0.919912986 1.1220269904 1.6519986824 0.556145376 -3.994833e-01
[46,] -1.446961762 0.3236541676 0.6167768984 0.744083772 2.020609e-02
[47,] -1.132861573 1.4759648078 -2.1152166694 1.248330978 -6.148680e-01
[48,] 2.253046457 0.6479565276 -0.1336535036 2.218149977 -6.921827e-01
[49,] -0.873193216 0.0695816941 0.3060300564 1.405842520 2.800143e-01
[50,] -0.147814843 1.0354392546 0.7105650661 0.154568381 -1.285398e+00
[51,] -1.613062603 1.0226046210 0.0722909843 -1.400508720 9.683344e-01
[52,] -0.738116993 0.8289868408 2.3180053961 1.327332957 -7.591204e-01
[53,] 1.876472768 -1.1020733947 -0.1624560449 -0.388952788 1.931283e-01
[54,] -2.001245392 0.1334736650 -0.7702964268 0.592827036 -1.273164e+00
[55,] 1.619313300 -0.6211817386 1.7723036867 0.882772256 2.559644e+00
[56,] -1.291003494 -0.5776695187 -0.6352114598 0.149163466 7.867106e-02
[57,] -0.991952481 0.9697085136 0.9156212026 0.150074754 5.844215e-01
[58,] -0.959352041 -0.5209828377 0.2346267168 -0.371049330 9.859082e-01
[59,] 0.740169004 -0.7295567338 0.6455885962 -0.344109837 6.503480e-01
[60,] 0.985103642 0.0632051452 0.1813138141 -0.913800204 -6.978233e-01
[61,] -0.462647669 -0.8413176541 -0.4767477325 0.298646101 -2.282923e-01
[62,] 0.692940740 -0.4029771952 0.1232062532 -0.600253766 -9.372382e-01
[63,] 1.083025898 -0.9923045178 0.3192343247 1.255769966 9.732006e-02
[64,] -1.537912402 0.0013030207 -0.9351684111 -0.902695485 -4.346655e-01
[65,] 6.464911513 -3.0871344634 0.3916691774 -0.390163164 5.136286e-01
[66,] -0.365756207 -0.2184209571 -0.2747246486 0.506189161 -4.299635e-01
[67,] 1.823558457 -0.6502657615 -0.2283638592 -0.600839125 3.664458e-01
[68,] -1.707934635 0.2732411411 -0.6200200641 -0.193731283 1.214058e-01
[69,] 3.590349692 -0.8685391854 -0.3047822252 -1.304009984 1.094234e+00
[70,] -0.016946617 1.7057485941 1.1922295947 0.684315404 -2.112229e-01
[71,] -0.395038071 0.1121517557 -0.4318838950 -0.693372994 -3.803449e-01
[72,] 0.641097089 -0.2159941249 0.5565729601 -1.281441314 -6.793186e-01
[73,] -1.585037998 -0.5203912940 -0.3078887501 -0.363753173 -4.589915e-01
[74,] 0.905758517 -0.4660740592 -0.1609962083 -0.391643497 -2.856211e-01
[75,] -0.565657681 0.8018217307 1.9380493700 -0.498658175 -2.714447e-01
[76,] -1.294851099 -2.0140779298 -0.4492421184 1.126514515 -2.737946e-01
[77,] 3.416023434 0.1194977433 1.8473482926 0.019965744 -8.427859e-01
[78,] 1.496120491 0.1541749823 -0.0706576166 3.082428247 -6.980783e-02
[79,] -2.238668749 -1.6920972451 -0.5941302408 0.644334815 -3.980786e-01
[80,] -0.589746288 -0.5075009816 0.2642739147 -0.001210338 5.038755e-01
[81,] 0.601432820 -1.1352336229 -0.3833743810 -0.879145053 -7.029129e-01
[82,] 0.522570367 2.3305852750 0.4985493196 -1.034544654 6.938928e-01
[83,] -1.391476934 -0.6139962025 -0.3856772249 -0.525176932 5.704099e-02
[84,] 3.436432307 -0.0439973814 1.5780852910 -0.153143149 3.951040e-01
[85,] -0.137247862 -0.0453513063 0.5086807610 -0.816169134 6.529417e-01
[86,] 3.007985470 -1.4005755538 -0.4478319928 -1.668401663 -6.999493e-01
[87,] -2.788850085 0.1865804914 0.8060559754 -0.653247143 -3.652821e-01
[88,] -1.626253811 -0.9298225173 -0.4701747133 -0.128221884 1.008644e+00
[89,] -1.013280577 -1.2205780414 0.9180871806 -1.489816708 3.012189e+00
[90,] -0.354497511 0.3863967011 0.7055971735 -0.959059033 2.820046e-01
[91,] -1.723810234 -1.3785299292 1.3872074370 0.627799006 9.367789e-01
[92,] -0.779384424 0.5404239047 -0.5219209696 -0.968813691 -1.083794e+00
[93,] -1.935696209 -0.8972802456 -0.8359898962 -0.287245442 4.685350e-01
[94,] -1.350998399 -0.9611960904 0.0002077480 0.414236753 3.391358e-01
[95,] -1.423390619 1.0712923234 1.8857501205 1.378285210 -9.395224e-01
[96,] -0.785028346 -0.7310352720 -1.1048643286 -0.415507507 6.991830e-01
[97,] 0.924567974 -0.3285519572 -2.4425106519 -0.656114377 1.176291e+00
[98,] -0.927742412 -0.2010330320 -0.2140741782 -0.104187902 -2.658170e-01
[99,] -0.414404058 0.6598649535 -1.2888712243 -0.300119882 5.301198e-01
[100,] -1.200782011 -1.0281175705 2.0753697112 -0.408603186 2.396448e+00
[101,] -0.743120580 -0.8372680340 0.2756704404 -0.290187090 -1.269241e+00
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[303,] 1.201631848 1.5080046043 0.1054618668 -0.833862320 6.273364e-02
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[306,] -0.785314361 -0.3251808495 1.6739480187 1.424524751 -1.435590e+00
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[314,] 6.186471306 -2.2648738359 0.3755029458 0.656883134 -6.926983e-01
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[318,] 2.202577803 2.4942207082 2.0672389594 -0.088105627 9.562868e-02
[319,] -1.823472525 -1.2765601645 -0.6313246792 1.055725829 3.644746e-03
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[329,] -1.725960474 0.7303397379 -1.0761048262 -0.551422192 -1.017898e+00
PC6 PC7 PC8 PC9
[1,] 0.49785424 -0.4237513406 -1.001775696 0.3474043931
[2,] -0.62854651 -0.0118948797 -0.418745776 -0.1216854762
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[11,] 0.18993444 0.0764010470 0.234855870 -0.2206125700
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[18,] 1.16932824 -0.2741924354 -1.059346761 -0.1435005894
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[20,] 1.08242816 -1.9334151885 -0.960408756 -0.7294112470
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[32,] 1.14464537 1.1809030754 -0.380505425 0.3701559647
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[217,] -1.15725248 -0.4777955838 1.627914509 -0.1681201834
[218,] 0.76831900 -1.3726642023 0.138304377 -0.5277051948
[219,] 0.21576616 1.3111818792 0.152262032 -0.0229367486
[220,] 0.13015108 -0.0746222186 0.049399436 0.0083598396
[221,] 0.15341863 -0.2980677293 -1.005336913 -0.0867832197
[222,] 0.69670877 -0.5579544222 -0.371459236 0.2268811720
[223,] 0.13815316 -0.0418408114 -0.343354673 -0.8148326202
[224,] -1.14035236 0.0836348997 0.038160033 0.3013850917
[225,] 1.08442224 0.1888930905 0.248745744 0.1407296220
[226,] -0.89338394 -0.3491716065 -0.745643362 -0.1096266459
[227,] -0.85390102 -0.6588038790 -0.082281815 -0.1773419432
[228,] -0.10964290 0.4794590590 -0.097340402 0.1554362045
[229,] 0.21916124 -0.8951842978 -0.947463819 -0.0965853300
[230,] -1.85507162 -0.1049073123 -1.010859335 -0.3648449270
[231,] 0.82495869 -0.2376114588 0.323672666 -0.2072439815
[232,] 0.35830790 1.0316879970 0.205733105 0.1947397740
[233,] -1.10349804 0.1267281297 0.581897796 -0.2431708223
[234,] -0.50194981 0.6248778034 -1.128794257 -0.4326963310
[235,] 0.19963892 0.2852475490 -0.018107243 -0.0956813890
[236,] -0.92280204 0.3596850715 0.700851638 -0.1181486554
[237,] 0.10089656 0.6429615676 -0.914613335 -0.1310744373
[238,] 0.76603503 0.5346742758 0.250230166 0.1361993561
[239,] 0.77191503 0.9628338332 -0.462106741 0.1630092292
[240,] 1.82334872 0.3857172072 0.646260975 0.0952865103
[241,] -0.14914909 0.1658468047 -0.448673879 0.1520755102
[242,] 0.34935425 -0.3172289977 0.064423109 0.3410570145
[243,] 0.92948138 0.7374355648 0.188752899 -0.2253918212
[244,] 0.09805683 0.9186485244 0.390657914 0.5025300740
[245,] -0.06067322 0.2196684557 0.484987741 0.1430187799
[246,] -0.20902609 0.4203399964 0.905010462 -0.1591260255
[247,] 0.99070339 -0.4397014033 -0.925669802 -1.0224937104
[248,] 0.34366472 0.4862695256 -0.619254887 -0.1300602740
[249,] 0.91066245 -0.5864986218 -0.137402258 0.1909704585
[250,] 0.52728220 -1.3984620896 0.910203238 0.1571887950
[251,] -0.68509850 -2.2264025982 -0.763899005 -0.0768239030
[252,] 0.45877624 -0.8193490534 -0.579284863 -0.4972768144
[253,] -1.02277884 0.1193051054 0.620359773 -0.5599105213
[254,] 0.36765388 0.1966668866 -0.644922137 -0.1852289035
[255,] -2.33694560 -1.2379793560 -0.508898189 -1.7588965443
[256,] 0.91264792 1.1258451678 -0.369836457 0.2572322873
[257,] -0.57880284 -0.1264246166 0.908236779 -0.2174853161
[258,] 0.33989856 -0.1576345066 0.656172504 0.2332684374
[259,] -0.21467671 0.8923462161 0.720945118 -0.1444405819
[260,] -1.20238876 -0.4519462656 0.022868017 0.2196719345
[261,] 0.18753195 0.5535933787 -0.132384934 0.3876970368
[262,] 1.03407762 0.1385414588 -0.274429826 -0.1588819722
[263,] 1.81264364 0.5406512886 0.593791785 0.1018653939
[264,] -0.31421382 -0.0337847178 -0.420899586 0.0391403271
[265,] 0.72255949 -0.1995553328 0.410041593 0.1171581847
[266,] -0.34223135 0.1205076640 -0.470600949 -0.1858786223
[267,] -0.07470815 -0.2561653528 -0.694886997 0.3641145993
[268,] 0.10783829 -0.5097127118 -0.188655907 0.0690412875
[269,] 0.32042077 -0.2193836079 -0.189532358 -0.0764980625
[270,] 1.23906900 0.0669024853 0.338758630 -0.1160748156
[271,] 0.08300095 -1.6573476776 0.055542492 -0.1849954464
[272,] 1.34640007 -0.5433164013 0.592852975 0.3566928964
[273,] 0.50416539 -1.4712518397 0.551814998 0.1688386075
[274,] -0.04825834 -0.0540114970 0.413122383 0.1299717859
[275,] -0.46351400 0.1610118509 -0.124125433 -0.1484456591
[276,] 0.41746863 -0.7099263938 0.039750760 -0.3059740818
[277,] 0.60596766 0.6012509769 -0.596033241 0.0254770084
[278,] 0.52078378 0.7660413424 -1.557720772 -0.2869550113
[279,] 0.04386567 0.8952448942 -0.035826538 0.0638347823
[280,] -0.62246393 -0.2789671579 0.132589730 0.0043822104
[281,] -0.34397297 0.1733680637 -0.295654179 0.0497671949
[282,] 0.11149413 -0.3067336422 -0.421250922 -0.2523380202
[283,] -0.38152042 -0.2118324006 0.217888315 -0.0297526812
[284,] -0.22394214 -1.1988041169 0.339441379 -0.2671376149
[285,] 0.89436129 -0.2202340371 -0.335950603 0.1816913161
[286,] 0.74752767 -0.6323140068 -0.195514992 0.4745065107
[287,] 0.90554015 -0.5765639894 0.304528337 -0.6068117413
[288,] 1.65077907 0.0294455191 0.996376439 0.6030226922
[289,] 0.20870801 0.0571565219 -0.604329343 0.1767243064
[290,] -1.61106274 -1.0156687810 2.968071574 -0.1365328117
[291,] 0.39771717 0.6243184560 0.328294655 0.3406465046
[292,] -0.63840615 0.9777383766 -0.035197616 0.0769228420
[293,] -0.15358695 -1.2065472084 0.105256515 0.0803347313
[294,] 1.34825912 0.2954142880 -0.365378723 0.5183646171
[295,] 1.00889596 1.0591567970 -0.999530368 0.4381810780
[296,] 0.45818926 -0.4333327755 0.455146961 0.3952242090
[297,] 0.65569141 0.5397752855 -0.184456664 0.2318170108
[298,] 0.62478954 0.5846643131 -0.279716382 0.3574752234
[299,] -0.03751692 0.2736321737 0.320659656 0.2144822300
[300,] 0.04815721 -0.1044056678 0.101203735 -0.2092842205
[301,] 0.38542812 -0.8687649905 -0.016430687 -0.1929220110
[302,] 0.90774061 -0.0441999422 0.639993953 0.3016422400
[303,] 0.06173965 0.6262764607 -0.220377323 0.0493965885
[304,] -0.57682966 -0.0529164172 -0.877677661 -0.2153749933
[305,] -0.33037268 -1.0379666040 0.047040470 -0.0305480148
[306,] 0.81562183 -0.1357050853 -0.067520331 0.1546680098
[307,] 0.86731898 0.1381674770 -0.671642517 0.3694016590
[308,] 0.26889600 -1.0684829133 -0.498763645 -0.0370602803
[309,] 0.07792106 -0.8867133829 -0.404632462 0.1267726601
[310,] -0.63086736 0.2085258640 0.115525492 0.1469036256
[311,] 0.39675779 -0.2220016244 0.048408771 0.0690490126
[312,] -0.61549286 0.3320391606 -0.180546906 0.0054449867
[313,] -0.11911021 0.2989890979 0.203596960 0.1520547265
[314,] -0.41407743 -1.4208787266 0.201397434 0.7478453094
[315,] -0.16147414 0.0570180035 0.591750402 -0.3171120941
[316,] -0.70224081 0.8031892835 0.702164834 0.2232106391
[317,] -0.80781957 0.4284071971 0.559279716 -0.0308750605
[318,] 1.54203926 1.1457529897 0.694663256 0.5013517307
[319,] 0.02411639 0.7598343213 -0.434866031 0.0540074461
[320,] 0.28995160 0.1606643195 0.113117817 0.2734342366
[321,] -0.20510588 0.0034887195 -0.051368865 0.0262538576
[322,] 0.01767954 0.0795690009 0.429743883 -0.0649673806
[323,] 0.60203587 0.2404754512 -0.309126556 0.0950808419
[324,] 0.27232587 1.4225185576 0.221996369 -0.3365644435
[325,] 0.34811386 0.1590543834 0.309007802 -0.4005043131
[326,] 0.02781104 0.2578027139 -0.062501862 0.1448350292
[327,] -0.34488256 0.1708766916 0.136739228 0.1658618810
[328,] 0.02706346 0.4031505640 0.390746647 0.0576405432
[329,] -0.30628652 -0.5861626977 0.571340630 0.0878835920#adding PC1 and PC2 to the places data set
places <- places %>%
mutate(
PC1 = places_pca$x[, 1],
PC2 = places_pca$x[, 2]
)
#graphing the biplot of the first 2 PCs
library(ggplot2)
library(ggrepel)
#Putting the place values into the scores
scores <- as.data.frame(places_pca$x[, 1:2])
scores$City <- places$City
loadings <- as.data.frame(places_pca$rotation[, 1:2])
loadings$Variable <- rownames(loadings)
ggplot(scores, aes(PC1, PC2, label = City)) +
geom_point(color = "blue") +
geom_text_repel(size = 3) +
geom_segment(data = loadings,
inherit.aes = FALSE,
aes(x = 0, y = 0,
xend = PC1 * 3,
yend = PC2 * 3),
arrow = arrow(length = unit(0.2, "cm")),
color = "red") +
geom_text_repel(data = loadings,
inherit.aes = FALSE,
aes(x = PC1 * 3, y = PC2 * 3, label = Variable),
color = "red") +
theme_minimal() +
labs(title = "Biplot of First Two Principal Components")Warning: ggrepel: 305 unlabeled data points (too many overlaps). Consider
increasing max.overlaps
#some pretty obvious outliers are NYC (New York City), Chicago IL,
#San Francisco CA, Washington DC, Boston MA, Los Angeles CA, Philadelphia PA generally the largest metropolitans would be
#the outliers I spotted. The specific metrics that make them the
#largest outliers are the arts, health care and education for
#cities like NYC, Boston & Chicago. While metrics like Housing &
#Recreat make cities like Los Angeles CA and San Francisco CA #biggest outliers. Problem 4
The data we will look at here come from a study of malignant and benign breast cancer cells using fine needle aspiration conducted at the University of Wisconsin-Madison. The goal was determine if malignancy of a tumor could be established by using shape characteristics of cells obtained via fine needle aspiration (FNA) and digitized scanning of the cells.
The variables in the data file you will be using are:
- ID - patient identification number (not used in PCA)
- Diagnosis determined by biopsy - B = benign or M = malignant
- Radius: mean of distances from center to points on the perimeter
- Texture: standard deviation of gray-scale values
- Smoothness: local variation in radius lengths
- Compactness: perimeter^2 / area - 1.0
- Concavity: severity of concave portions of the contour
- Concavepts: number of concave portions of the contour
- Symmetry: measure of symmetry of the cell nucleus
- FracDim: fractal dimension; “coastline approximation” - 1
#reading in the dataset
bc_cells <- read.csv("C:/Users/lr7273ow/OneDrive - Minnesota State/Documents/GitHub/DSCI_415/Activities/Data/BreastDiag.csv")
head(bc_cells) Diagnosis Radius Texture Smoothness Compactness Concavity ConcavePts Symmetry
1 M 17.99 10.38 0.11840 0.27760 0.3001 0.14710 0.2419
2 M 20.57 17.77 0.08474 0.07864 0.0869 0.07017 0.1812
3 M 19.69 21.25 0.10960 0.15990 0.1974 0.12790 0.2069
4 M 11.42 20.38 0.14250 0.28390 0.2414 0.10520 0.2597
5 M 20.29 14.34 0.10030 0.13280 0.1980 0.10430 0.1809
6 M 12.45 15.70 0.12780 0.17000 0.1578 0.08089 0.2087
FracDim
1 0.07871
2 0.05667
3 0.05999
4 0.09744
5 0.05883
6 0.07613bc_cells_cols <- bc_cells %>%
select('Radius','Texture','Smoothness','Compactness','Concavity','ConcavePts','Symmetry','FracDim')
bc_cells_cols Radius Texture Smoothness Compactness Concavity ConcavePts Symmetry FracDim
1 17.990 10.38 0.11840 0.27760 0.3001000 0.147100 0.2419 0.07871
2 20.570 17.77 0.08474 0.07864 0.0869000 0.070170 0.1812 0.05667
3 19.690 21.25 0.10960 0.15990 0.1974000 0.127900 0.2069 0.05999
4 11.420 20.38 0.14250 0.28390 0.2414000 0.105200 0.2597 0.09744
5 20.290 14.34 0.10030 0.13280 0.1980000 0.104300 0.1809 0.05883
6 12.450 15.70 0.12780 0.17000 0.1578000 0.080890 0.2087 0.07613
7 18.250 19.98 0.09463 0.10900 0.1127000 0.074000 0.1794 0.05742
8 13.710 20.83 0.11890 0.16450 0.0936600 0.059850 0.2196 0.07451
9 13.000 21.82 0.12730 0.19320 0.1859000 0.093530 0.2350 0.07389
10 12.460 24.04 0.11860 0.23960 0.2273000 0.085430 0.2030 0.08243
11 16.020 23.24 0.08206 0.06669 0.0329900 0.033230 0.1528 0.05697
12 15.780 17.89 0.09710 0.12920 0.0995400 0.066060 0.1842 0.06082
13 19.170 24.80 0.09740 0.24580 0.2065000 0.111800 0.2397 0.07800
14 15.850 23.95 0.08401 0.10020 0.0993800 0.053640 0.1847 0.05338
15 13.730 22.61 0.11310 0.22930 0.2128000 0.080250 0.2069 0.07682
16 14.540 27.54 0.11390 0.15950 0.1639000 0.073640 0.2303 0.07077
17 14.680 20.13 0.09867 0.07200 0.0739500 0.052590 0.1586 0.05922
18 16.130 20.68 0.11700 0.20220 0.1722000 0.102800 0.2164 0.07356
19 19.810 22.15 0.09831 0.10270 0.1479000 0.094980 0.1582 0.05395
20 13.540 14.36 0.09779 0.08129 0.0666400 0.047810 0.1885 0.05766
21 13.080 15.71 0.10750 0.12700 0.0456800 0.031100 0.1967 0.06811
22 9.504 12.44 0.10240 0.06492 0.0295600 0.020760 0.1815 0.06905
23 15.340 14.26 0.10730 0.21350 0.2077000 0.097560 0.2521 0.07032
24 21.160 23.04 0.09428 0.10220 0.1097000 0.086320 0.1769 0.05278
25 16.650 21.38 0.11210 0.14570 0.1525000 0.091700 0.1995 0.06330
26 17.140 16.40 0.11860 0.22760 0.2229000 0.140100 0.3040 0.07413
27 14.580 21.53 0.10540 0.18680 0.1425000 0.087830 0.2252 0.06924
28 18.610 20.25 0.09440 0.10660 0.1490000 0.077310 0.1697 0.05699
29 15.300 25.27 0.10820 0.16970 0.1683000 0.087510 0.1926 0.06540
30 17.570 15.05 0.09847 0.11570 0.0987500 0.079530 0.1739 0.06149
31 18.630 25.11 0.10640 0.18870 0.2319000 0.124400 0.2183 0.06197
32 11.840 18.70 0.11090 0.15160 0.1218000 0.051820 0.2301 0.07799
33 17.020 23.98 0.11970 0.14960 0.2417000 0.120300 0.2248 0.06382
34 19.270 26.47 0.09401 0.17190 0.1657000 0.075930 0.1853 0.06261
35 16.130 17.88 0.10400 0.15590 0.1354000 0.077520 0.1998 0.06515
36 16.740 21.59 0.09610 0.13360 0.1348000 0.060180 0.1896 0.05656
37 14.250 21.72 0.09823 0.10980 0.1319000 0.055980 0.1885 0.06125
38 13.030 18.42 0.08983 0.03766 0.0256200 0.029230 0.1467 0.05863
39 14.990 25.20 0.09387 0.05131 0.0239800 0.028990 0.1565 0.05504
40 13.480 20.82 0.10160 0.12550 0.1063000 0.054390 0.1720 0.06419
41 13.440 21.58 0.08162 0.06031 0.0311000 0.020310 0.1784 0.05587
42 10.950 21.35 0.12270 0.12180 0.1044000 0.056690 0.1895 0.06870
43 19.070 24.81 0.09081 0.21900 0.2107000 0.099610 0.2310 0.06343
44 13.280 20.28 0.10410 0.14360 0.0984700 0.061580 0.1974 0.06782
45 13.170 21.81 0.09714 0.10470 0.0825900 0.052520 0.1746 0.06177
46 18.650 17.60 0.10990 0.16860 0.1974000 0.100900 0.1907 0.06049
47 8.196 16.84 0.08600 0.05943 0.0158800 0.005917 0.1769 0.06503
48 13.170 18.66 0.11580 0.12310 0.1226000 0.073400 0.2128 0.06777
49 12.050 14.63 0.10310 0.09092 0.0659200 0.027490 0.1675 0.06043
50 13.490 22.30 0.08752 0.07698 0.0475100 0.033840 0.1809 0.05718
51 11.760 21.60 0.08637 0.04966 0.0165700 0.011150 0.1495 0.05888
52 13.640 16.34 0.07685 0.06059 0.0185700 0.017230 0.1353 0.05953
53 11.940 18.24 0.08261 0.04751 0.0197200 0.013490 0.1868 0.06110
54 18.220 18.70 0.11480 0.14850 0.1772000 0.106000 0.2092 0.06310
55 15.100 22.02 0.09056 0.07081 0.0525300 0.033340 0.1616 0.05684
56 11.520 18.75 0.09524 0.05473 0.0303600 0.022780 0.1920 0.05907
57 19.210 18.57 0.10530 0.12670 0.1323000 0.089940 0.1917 0.05961
58 14.710 21.59 0.11370 0.13650 0.1293000 0.081230 0.2027 0.06758
59 13.050 19.31 0.08060 0.03789 0.0006920 0.004167 0.1819 0.05501
60 8.618 11.79 0.09752 0.05272 0.0206100 0.007799 0.1683 0.07187
61 10.170 14.88 0.11340 0.08061 0.0108400 0.012900 0.2743 0.06960
62 8.598 20.98 0.12430 0.08963 0.0300000 0.009259 0.1828 0.06757
63 14.250 22.15 0.10490 0.20080 0.2135000 0.086530 0.1949 0.07292
64 9.173 13.86 0.07721 0.08751 0.0598800 0.021800 0.2341 0.06963
65 12.680 23.84 0.11220 0.12620 0.1128000 0.068730 0.1905 0.06590
66 14.780 23.94 0.11720 0.14790 0.1267000 0.090290 0.1953 0.06654
67 9.465 21.01 0.10440 0.07773 0.0217200 0.015040 0.1717 0.06899
68 11.310 19.04 0.08139 0.04701 0.0370900 0.022300 0.1516 0.05667
69 9.029 17.33 0.10660 0.14130 0.3130000 0.043750 0.2111 0.08046
70 12.780 16.49 0.09831 0.05234 0.0365300 0.028640 0.1590 0.05653
71 18.940 21.31 0.09009 0.10290 0.1080000 0.079510 0.1582 0.05461
72 8.888 14.64 0.09783 0.15310 0.0860600 0.028720 0.1902 0.08980
73 17.200 24.52 0.10710 0.18300 0.1692000 0.079440 0.1927 0.06487
74 13.800 15.79 0.10070 0.12800 0.0778900 0.050690 0.1662 0.06566
75 12.310 16.52 0.09172 0.06829 0.0337200 0.022720 0.1720 0.05914
76 16.070 19.65 0.09168 0.08424 0.0976900 0.066380 0.1798 0.05391
77 13.530 10.94 0.12910 0.10470 0.0687700 0.065560 0.2403 0.06641
78 18.050 16.15 0.10650 0.21460 0.1684000 0.108000 0.2152 0.06673
79 20.180 23.97 0.12860 0.34540 0.3754000 0.160400 0.2906 0.08142
80 12.860 18.00 0.09934 0.09546 0.0388900 0.023150 0.1718 0.05997
81 11.450 20.97 0.11020 0.09362 0.0459100 0.022330 0.1842 0.07005
82 13.340 15.86 0.10780 0.15350 0.1169000 0.069870 0.1942 0.06902
83 25.220 24.91 0.10630 0.26650 0.3339000 0.184500 0.1829 0.06782
84 19.100 26.29 0.12150 0.17910 0.1937000 0.146900 0.1634 0.07224
85 12.000 15.65 0.09723 0.07165 0.0415100 0.018630 0.2079 0.05968
86 18.460 18.52 0.09874 0.10530 0.1335000 0.087950 0.2132 0.06022
87 14.480 21.46 0.09444 0.09947 0.1204000 0.049380 0.2075 0.05636
88 19.020 24.59 0.09029 0.12060 0.1468000 0.082710 0.1953 0.05629
89 12.360 21.80 0.08772 0.09445 0.0601500 0.037450 0.1930 0.06404
90 14.640 15.24 0.11320 0.13390 0.0996600 0.070640 0.2116 0.06346
91 14.620 24.02 0.08974 0.08606 0.0310200 0.029570 0.1685 0.05866
92 15.370 22.76 0.09200 0.10360 0.1122000 0.074830 0.1717 0.06097
93 13.270 14.76 0.07355 0.05055 0.0326100 0.026480 0.1386 0.05318
94 13.450 18.30 0.10220 0.08165 0.0397400 0.027800 0.1638 0.05710
95 15.060 19.83 0.10390 0.15530 0.1700000 0.088150 0.1855 0.06284
96 20.260 23.03 0.09078 0.13130 0.1465000 0.086830 0.2095 0.05649
97 12.180 17.84 0.10450 0.07057 0.0249000 0.029410 0.1900 0.06635
98 9.787 19.94 0.10240 0.05301 0.0068290 0.007937 0.1350 0.06890
99 11.600 12.84 0.08983 0.07525 0.0419600 0.033500 0.1620 0.06582
100 14.420 19.77 0.09752 0.11410 0.0938800 0.058390 0.1879 0.06390
101 13.610 24.98 0.09488 0.08511 0.0862500 0.044890 0.1609 0.05871
102 6.981 13.43 0.11700 0.07568 0.0000000 0.000000 0.1930 0.07818
103 12.180 20.52 0.08013 0.04038 0.0238300 0.017700 0.1739 0.05677
104 9.876 19.40 0.10050 0.09697 0.0615400 0.030290 0.1945 0.06322
105 10.490 19.29 0.09989 0.08578 0.0299500 0.012010 0.2217 0.06481
106 13.110 15.56 0.13980 0.17650 0.2071000 0.096010 0.1925 0.07692
107 11.640 18.33 0.11420 0.10170 0.0707000 0.034850 0.1801 0.06520
108 12.360 18.54 0.08477 0.06815 0.0264300 0.019210 0.1602 0.06066
109 22.270 19.67 0.13260 0.27680 0.4264000 0.182300 0.2556 0.07039
110 11.340 21.26 0.08759 0.06575 0.0513300 0.018990 0.1487 0.06529
111 9.777 16.99 0.10370 0.08404 0.0433400 0.017780 0.1584 0.07065
112 12.630 20.76 0.09933 0.12090 0.1065000 0.060210 0.1735 0.07070
113 14.260 19.65 0.07837 0.22330 0.3003000 0.077980 0.1704 0.07769
114 10.510 20.19 0.11220 0.13030 0.0647600 0.030680 0.1922 0.07782
115 8.726 15.83 0.11500 0.08201 0.0413200 0.019240 0.1649 0.07633
116 11.930 21.53 0.09768 0.07849 0.0332800 0.020080 0.1688 0.06194
117 8.950 15.76 0.09462 0.12430 0.0926300 0.023080 0.1305 0.07163
118 14.870 16.67 0.11620 0.16490 0.1690000 0.089230 0.2157 0.06768
119 15.780 22.91 0.11550 0.17520 0.2133000 0.094790 0.2096 0.07331
120 17.950 20.01 0.08402 0.06722 0.0729300 0.055960 0.2129 0.05025
121 11.410 10.82 0.09373 0.06685 0.0351200 0.026230 0.1667 0.06113
122 18.660 17.12 0.10540 0.11000 0.1457000 0.086650 0.1966 0.06213
123 24.250 20.20 0.14470 0.28670 0.4268000 0.201200 0.2655 0.06877
124 14.500 10.89 0.11010 0.10990 0.0884200 0.057780 0.1856 0.06402
125 13.370 16.39 0.07115 0.07325 0.0809200 0.028000 0.1422 0.05823
126 13.850 17.21 0.08785 0.06136 0.0142000 0.011410 0.1614 0.05890
127 13.610 24.69 0.09258 0.07862 0.0528500 0.030850 0.1761 0.06130
128 19.000 18.91 0.08217 0.08028 0.0927100 0.056270 0.1946 0.05044
129 15.100 16.39 0.11500 0.18070 0.1138000 0.085340 0.2001 0.06467
130 19.790 25.12 0.10150 0.15890 0.2545000 0.114900 0.2202 0.06113
131 12.190 13.29 0.10660 0.09509 0.0285500 0.028820 0.1880 0.06471
132 15.460 19.48 0.10920 0.12230 0.1466000 0.080870 0.1931 0.05796
133 16.160 21.54 0.10080 0.12840 0.1043000 0.056130 0.2160 0.05891
134 15.710 13.93 0.09462 0.09462 0.0713500 0.059330 0.1816 0.05723
135 18.450 21.91 0.09430 0.09709 0.1153000 0.068470 0.1692 0.05727
136 12.770 22.47 0.09055 0.05761 0.0471100 0.027040 0.1585 0.06065
137 11.710 16.67 0.10510 0.06095 0.0359200 0.026000 0.1339 0.05945
138 11.430 15.39 0.09639 0.06889 0.0350300 0.028750 0.1734 0.05865
139 14.950 17.57 0.11670 0.13050 0.1539000 0.086240 0.1957 0.06216
140 11.280 13.39 0.11640 0.11360 0.0463500 0.047960 0.1771 0.06072
141 9.738 11.97 0.09250 0.04102 0.0000000 0.000000 0.1903 0.06422
142 16.110 18.05 0.09721 0.11370 0.0944700 0.059430 0.1861 0.06248
143 11.430 17.31 0.10920 0.09486 0.0203100 0.018610 0.1645 0.06562
144 12.900 15.92 0.08677 0.09509 0.0489400 0.030880 0.1778 0.06235
145 10.750 14.97 0.07793 0.05139 0.0225100 0.007875 0.1399 0.05688
146 11.900 14.65 0.11520 0.12960 0.0371000 0.030030 0.1995 0.07839
147 11.800 16.58 0.10910 0.17000 0.1659000 0.074150 0.2678 0.07371
148 14.950 18.77 0.08138 0.11670 0.0905000 0.035620 0.1744 0.06493
149 14.440 15.18 0.09970 0.10210 0.0848700 0.055320 0.1724 0.06081
150 13.740 17.91 0.07944 0.06376 0.0288100 0.013290 0.1473 0.05580
151 13.000 20.78 0.11350 0.07589 0.0313600 0.026450 0.2540 0.06087
152 8.219 20.70 0.09405 0.13050 0.1321000 0.021680 0.2222 0.08261
153 9.731 15.34 0.10720 0.15990 0.4108000 0.078570 0.2548 0.09296
154 11.150 13.08 0.09754 0.05113 0.0198200 0.017860 0.1830 0.06105
155 13.150 15.34 0.09384 0.08498 0.0929300 0.034830 0.1822 0.06207
156 12.250 17.94 0.08654 0.06679 0.0388500 0.023310 0.1970 0.06228
157 17.680 20.74 0.11150 0.16650 0.1855000 0.105400 0.1971 0.06166
158 16.840 19.46 0.07445 0.07223 0.0515000 0.027710 0.1844 0.05268
159 12.060 12.74 0.09311 0.05241 0.0197200 0.019630 0.1590 0.05907
160 10.900 12.96 0.07515 0.03718 0.0030900 0.006588 0.1442 0.05743
161 11.750 20.18 0.10890 0.11410 0.0684300 0.037380 0.1993 0.06453
162 19.190 15.94 0.08694 0.11850 0.1193000 0.096670 0.1741 0.05176
163 19.590 18.15 0.11200 0.16660 0.2508000 0.128600 0.2027 0.06082
164 12.340 22.22 0.10120 0.10150 0.0537000 0.028220 0.1551 0.06761
165 23.270 22.04 0.08439 0.11450 0.1324000 0.097020 0.1801 0.05553
166 14.970 19.76 0.08421 0.05352 0.0194700 0.019390 0.1515 0.05266
167 10.800 9.71 0.09594 0.05736 0.0253100 0.016980 0.1381 0.06400
168 16.780 18.80 0.08865 0.09182 0.0842200 0.065760 0.1893 0.05534
169 17.470 24.68 0.10490 0.16030 0.2159000 0.104300 0.1538 0.06365
170 14.970 16.95 0.09855 0.07885 0.0260200 0.037810 0.1780 0.05650
171 12.320 12.39 0.10280 0.06981 0.0398700 0.037000 0.1959 0.05955
172 13.430 19.63 0.09048 0.06288 0.0585800 0.034380 0.1598 0.05671
173 15.460 11.89 0.12570 0.15550 0.2032000 0.109700 0.1966 0.07069
174 11.080 14.71 0.10060 0.05743 0.0236300 0.025830 0.1566 0.06669
175 10.660 15.15 0.08792 0.04302 0.0000000 0.000000 0.1928 0.05975
176 8.671 14.45 0.09138 0.04276 0.0000000 0.000000 0.1722 0.06724
177 9.904 18.06 0.09699 0.12940 0.1307000 0.037160 0.1669 0.08116
178 16.460 20.11 0.09831 0.15560 0.1793000 0.088660 0.1794 0.06323
179 13.010 22.22 0.06251 0.01938 0.0015950 0.001852 0.1395 0.05234
180 12.810 13.06 0.08739 0.03774 0.0091930 0.013300 0.1466 0.06133
181 27.220 21.87 0.10940 0.19140 0.2871000 0.187800 0.1800 0.05770
182 21.090 26.57 0.11410 0.28320 0.2487000 0.149600 0.2395 0.07398
183 15.700 20.31 0.09597 0.08799 0.0659300 0.051890 0.1618 0.05549
184 11.410 14.92 0.09059 0.08155 0.0618100 0.023610 0.1167 0.06217
185 15.280 22.41 0.09057 0.10520 0.0537500 0.032630 0.1727 0.06317
186 10.080 15.11 0.09267 0.04695 0.0015970 0.002404 0.1703 0.06048
187 18.310 18.58 0.08588 0.08468 0.0816900 0.058140 0.1621 0.05425
188 11.710 17.19 0.09774 0.06141 0.0380900 0.032390 0.1516 0.06095
189 11.810 17.39 0.10070 0.05562 0.0235300 0.015530 0.1718 0.05780
190 12.300 15.90 0.08080 0.07253 0.0384400 0.016540 0.1667 0.05474
191 14.220 23.12 0.10750 0.24130 0.1981000 0.066180 0.2384 0.07542
192 12.770 21.41 0.08749 0.06601 0.0311200 0.028640 0.1694 0.06287
193 9.720 18.22 0.06950 0.02344 0.0000000 0.000000 0.1653 0.06447
194 12.340 26.86 0.10340 0.13530 0.1085000 0.045620 0.1943 0.06937
195 14.860 23.21 0.10440 0.19800 0.1697000 0.088780 0.1737 0.06672
196 12.910 16.33 0.07941 0.05366 0.0387300 0.023770 0.1829 0.05667
197 13.770 22.29 0.12000 0.12670 0.1385000 0.065260 0.1834 0.06877
198 18.080 21.84 0.07371 0.08642 0.1103000 0.057780 0.1770 0.05340
199 19.180 22.49 0.08523 0.14280 0.1114000 0.067720 0.1767 0.05529
200 14.450 20.22 0.09872 0.12060 0.1180000 0.059800 0.1950 0.06466
201 12.230 19.56 0.09586 0.08087 0.0418700 0.041070 0.1979 0.06013
202 17.540 19.32 0.08968 0.11980 0.1036000 0.074880 0.1506 0.05491
203 23.290 26.67 0.11410 0.20840 0.3523000 0.162000 0.2200 0.06229
204 13.810 23.75 0.13230 0.17680 0.1558000 0.091760 0.2251 0.07421
205 12.470 18.60 0.09965 0.10580 0.0800500 0.038210 0.1925 0.06373
206 15.120 16.68 0.08876 0.09588 0.0755000 0.040790 0.1594 0.05986
207 9.876 17.27 0.10890 0.07232 0.0175600 0.019520 0.1934 0.06285
208 17.010 20.26 0.08772 0.07304 0.0695000 0.053900 0.2026 0.05223
209 13.110 22.54 0.10020 0.14830 0.0870500 0.051020 0.1850 0.07310
210 15.270 12.91 0.08182 0.06230 0.0589200 0.031570 0.1359 0.05526
211 20.580 22.14 0.09090 0.13480 0.1640000 0.095610 0.1765 0.05024
212 11.840 18.94 0.08871 0.06900 0.0266900 0.013930 0.1533 0.06057
213 28.110 18.47 0.11420 0.15160 0.3201000 0.159500 0.1648 0.05525
214 17.420 25.56 0.10060 0.11460 0.1682000 0.065970 0.1308 0.05866
215 14.190 23.81 0.09463 0.13060 0.1115000 0.064620 0.2235 0.06433
216 13.860 16.93 0.10260 0.15170 0.0990100 0.056020 0.2106 0.06916
217 11.890 18.35 0.09363 0.11540 0.0663600 0.031420 0.1967 0.06314
218 10.200 17.48 0.08054 0.05907 0.0577400 0.010710 0.1964 0.06315
219 19.800 21.56 0.09383 0.13060 0.1272000 0.086910 0.2094 0.05581
220 19.530 32.47 0.08420 0.11300 0.1145000 0.066370 0.1428 0.05313
221 13.650 13.16 0.09646 0.08711 0.0388800 0.025630 0.1360 0.06344
222 13.560 13.90 0.10510 0.11920 0.0786000 0.044510 0.1962 0.06303
223 10.180 17.53 0.10610 0.08502 0.0176800 0.019150 0.1910 0.06908
224 15.750 20.25 0.10250 0.12040 0.1147000 0.064620 0.1935 0.06303
225 13.270 17.02 0.08445 0.04994 0.0355400 0.024560 0.1496 0.05674
226 14.340 13.47 0.09906 0.07624 0.0572400 0.046030 0.2075 0.05448
227 10.440 15.46 0.10530 0.07722 0.0066430 0.012160 0.1788 0.06450
228 15.000 15.51 0.08371 0.10960 0.0650500 0.037800 0.1881 0.05907
229 12.620 23.97 0.07903 0.07529 0.0543800 0.020360 0.1514 0.06019
230 12.830 22.33 0.10880 0.17990 0.1695000 0.068610 0.2123 0.07254
231 17.050 19.08 0.11410 0.15720 0.1910000 0.109000 0.2131 0.06325
232 11.320 27.08 0.06883 0.03813 0.0163300 0.003125 0.1869 0.05628
233 11.220 33.81 0.07780 0.03574 0.0049670 0.006434 0.1845 0.05828
234 20.510 27.81 0.09159 0.10740 0.1554000 0.083400 0.1448 0.05592
235 9.567 15.91 0.08464 0.04087 0.0165200 0.016670 0.1551 0.06403
236 14.030 21.25 0.09070 0.06945 0.0146200 0.018960 0.1517 0.05835
237 23.210 26.97 0.09509 0.16820 0.1950000 0.123700 0.1909 0.06309
238 20.480 21.46 0.08355 0.08348 0.0904200 0.060220 0.1467 0.05177
239 14.220 27.85 0.08223 0.10390 0.1103000 0.044080 0.1342 0.06129
240 17.460 39.28 0.09812 0.12980 0.1417000 0.088110 0.1809 0.05966
241 13.640 15.60 0.09423 0.06630 0.0470500 0.037310 0.1717 0.05660
242 12.420 15.04 0.07926 0.03393 0.0105300 0.011080 0.1546 0.05754
243 11.300 18.19 0.09592 0.13250 0.1548000 0.028540 0.2054 0.07669
244 13.750 23.77 0.08043 0.06807 0.0469700 0.023440 0.1773 0.05429
245 19.400 23.50 0.10270 0.15580 0.2049000 0.088860 0.1978 0.06000
246 10.480 19.86 0.10700 0.05971 0.0483100 0.030700 0.1737 0.06440
247 13.200 17.43 0.07215 0.04524 0.0433600 0.011050 0.1487 0.05635
248 12.890 14.11 0.08760 0.13460 0.1374000 0.039800 0.1596 0.06409
249 10.650 25.22 0.09657 0.07234 0.0237900 0.016150 0.1897 0.06329
250 11.520 14.93 0.10130 0.07808 0.0432800 0.029290 0.1883 0.06168
251 20.940 23.56 0.10070 0.16060 0.2712000 0.131000 0.2205 0.05898
252 11.500 18.45 0.09345 0.05991 0.0263800 0.020690 0.1834 0.05934
253 19.730 19.82 0.10620 0.18490 0.2417000 0.097400 0.1733 0.06697
254 17.300 17.08 0.10080 0.10410 0.1266000 0.083530 0.1813 0.05613
255 19.450 19.33 0.10350 0.11880 0.1379000 0.085910 0.1776 0.05647
256 13.960 17.05 0.10960 0.12790 0.0978900 0.052460 0.1908 0.06130
257 19.550 28.77 0.09260 0.20630 0.1784000 0.114400 0.1893 0.06232
258 15.320 17.27 0.13350 0.22840 0.2448000 0.124200 0.2398 0.07596
259 15.660 23.20 0.11090 0.31140 0.3176000 0.137700 0.2495 0.08104
260 15.530 33.56 0.10630 0.16390 0.1751000 0.083990 0.2091 0.06650
261 20.310 27.06 0.10000 0.10880 0.1519000 0.093330 0.1814 0.05572
262 17.350 23.06 0.08662 0.06290 0.0289100 0.028370 0.1564 0.05307
263 17.290 22.13 0.08999 0.12730 0.0969700 0.075070 0.2108 0.05464
264 15.610 19.38 0.07840 0.05616 0.0420900 0.028470 0.1547 0.05443
265 17.190 22.07 0.09726 0.08995 0.0906100 0.065270 0.1867 0.05580
266 20.730 31.12 0.09469 0.11430 0.1367000 0.086460 0.1769 0.05674
267 10.600 18.95 0.09688 0.11470 0.0638700 0.026420 0.1922 0.06491
268 13.590 21.84 0.07956 0.08259 0.0407200 0.021420 0.1635 0.05859
269 12.870 16.21 0.09425 0.06219 0.0390000 0.016150 0.2010 0.05769
270 10.710 20.39 0.10820 0.12890 0.0844800 0.028670 0.1668 0.06862
271 14.290 16.82 0.06429 0.02675 0.0072500 0.006250 0.1508 0.05376
272 11.290 13.04 0.09834 0.07608 0.0326500 0.027550 0.1769 0.06270
273 21.750 20.99 0.09401 0.19610 0.2195000 0.108800 0.1721 0.06194
274 9.742 15.67 0.09037 0.04689 0.0110300 0.014070 0.2081 0.06312
275 17.930 24.48 0.08855 0.07027 0.0569900 0.047440 0.1538 0.05510
276 11.890 17.36 0.12250 0.07210 0.0592900 0.074040 0.2015 0.05875
277 11.330 14.16 0.09379 0.03872 0.0014870 0.003333 0.1954 0.05821
278 18.810 19.98 0.08923 0.05884 0.0802000 0.058430 0.1550 0.04996
279 13.590 17.84 0.07948 0.04052 0.0199700 0.012380 0.1573 0.05520
280 13.850 15.18 0.09516 0.07688 0.0447900 0.037110 0.2110 0.05853
281 19.160 26.60 0.10200 0.14530 0.1921000 0.096640 0.1902 0.06220
282 11.740 14.02 0.07813 0.04340 0.0224500 0.027630 0.2101 0.06113
283 19.400 18.18 0.10370 0.14420 0.1626000 0.094640 0.1893 0.05892
284 16.240 18.77 0.10660 0.18020 0.1948000 0.090520 0.1876 0.06684
285 12.890 15.70 0.07818 0.09580 0.1115000 0.033900 0.1432 0.05935
286 12.580 18.40 0.08393 0.04216 0.0018600 0.002924 0.1697 0.05855
287 11.940 20.76 0.08605 0.10110 0.0657400 0.037910 0.1588 0.06766
288 12.890 13.12 0.06955 0.03729 0.0226000 0.011710 0.1337 0.05581
289 11.260 19.96 0.08020 0.11810 0.0927400 0.055880 0.2595 0.06233
290 11.370 18.89 0.08713 0.05008 0.0239900 0.021730 0.2013 0.05955
291 14.410 19.73 0.08757 0.16760 0.1362000 0.066020 0.1714 0.07192
292 14.960 19.10 0.08992 0.09823 0.0594000 0.048190 0.1879 0.05852
293 12.950 16.02 0.10050 0.07943 0.0615500 0.033700 0.1730 0.06470
294 11.850 17.46 0.08372 0.05642 0.0268800 0.022800 0.1875 0.05715
295 12.720 13.78 0.09667 0.08393 0.0128800 0.019240 0.1638 0.06100
296 13.770 13.27 0.09198 0.06221 0.0106300 0.019170 0.1592 0.05912
297 10.910 12.35 0.08518 0.04721 0.0123600 0.013690 0.1449 0.06031
298 11.760 18.14 0.09968 0.05914 0.0268500 0.035150 0.1619 0.06287
299 14.260 18.17 0.06576 0.05220 0.0247500 0.013740 0.1635 0.05586
300 10.510 23.09 0.10150 0.06797 0.0249500 0.018750 0.1695 0.06556
301 19.530 18.90 0.11500 0.16420 0.2197000 0.106200 0.1792 0.06552
302 12.460 19.89 0.08451 0.10140 0.0683000 0.030990 0.1781 0.06249
303 20.090 23.86 0.10800 0.18380 0.2283000 0.128000 0.2249 0.07469
304 10.490 18.61 0.10680 0.06678 0.0229700 0.017800 0.1482 0.06600
305 11.460 18.16 0.08853 0.07694 0.0334400 0.015020 0.1411 0.06243
306 11.600 24.49 0.07474 0.05688 0.0197400 0.013130 0.1935 0.05878
307 13.200 15.82 0.08511 0.05251 0.0014610 0.003261 0.1632 0.05894
308 9.000 14.40 0.07005 0.03116 0.0036810 0.003472 0.1788 0.06833
309 13.500 12.71 0.07376 0.03614 0.0027580 0.004419 0.1365 0.05335
310 13.050 13.84 0.08352 0.03735 0.0045590 0.008829 0.1453 0.05518
311 11.700 19.11 0.08814 0.05253 0.0158300 0.011480 0.1936 0.06128
312 14.610 15.69 0.07618 0.03515 0.0144700 0.018770 0.1632 0.05255
313 12.760 13.37 0.08794 0.07948 0.0405200 0.025480 0.1601 0.06140
314 11.540 10.72 0.08597 0.05969 0.0136700 0.008907 0.1833 0.06100
315 8.597 18.60 0.10740 0.05847 0.0000000 0.000000 0.2163 0.07359
316 12.490 16.85 0.08511 0.03834 0.0044730 0.006423 0.1215 0.05673
317 12.180 14.08 0.07734 0.03212 0.0112300 0.005051 0.1673 0.05649
318 18.220 18.87 0.09746 0.11170 0.1130000 0.079500 0.1807 0.05664
319 9.042 18.90 0.09968 0.19720 0.1975000 0.049080 0.2330 0.08743
320 12.430 17.00 0.07557 0.03454 0.0134200 0.016990 0.1472 0.05561
321 10.250 16.18 0.10610 0.11110 0.0672600 0.039650 0.1743 0.07279
322 20.160 19.66 0.08020 0.08564 0.1155000 0.077260 0.1928 0.05096
323 12.860 13.32 0.11340 0.08834 0.0380000 0.034000 0.1543 0.06476
324 20.340 21.51 0.11700 0.18750 0.2565000 0.150400 0.2569 0.06670
325 12.200 15.21 0.08673 0.06545 0.0199400 0.016920 0.1638 0.06129
326 12.670 17.30 0.10280 0.07664 0.0319300 0.021070 0.1707 0.05984
327 14.110 12.88 0.09309 0.05306 0.0176500 0.027330 0.1373 0.05700
328 12.030 17.93 0.07683 0.03892 0.0015460 0.005592 0.1382 0.06070
329 16.270 20.71 0.11690 0.13190 0.1478000 0.084880 0.1948 0.06277
330 16.260 21.88 0.11650 0.12830 0.1799000 0.079810 0.1869 0.06532
331 16.030 15.51 0.09491 0.13710 0.1204000 0.070410 0.1782 0.05976
332 12.980 19.35 0.09579 0.11250 0.0710700 0.029500 0.1761 0.06540
333 11.220 19.86 0.10540 0.06779 0.0050060 0.007583 0.1940 0.06028
334 11.250 14.78 0.08306 0.04458 0.0009737 0.002941 0.1773 0.06081
335 12.300 19.02 0.08313 0.04202 0.0077560 0.008535 0.1539 0.05945
336 17.060 21.00 0.11190 0.10560 0.1508000 0.099340 0.1727 0.06071
337 12.990 14.23 0.09462 0.09965 0.0373800 0.020980 0.1652 0.07238
338 18.770 21.43 0.09116 0.14020 0.1060000 0.060900 0.1953 0.06083
339 10.050 17.53 0.10070 0.07326 0.0251100 0.017750 0.1890 0.06331
340 23.510 24.27 0.10690 0.12830 0.2308000 0.141000 0.1797 0.05506
341 14.420 16.54 0.09751 0.11390 0.0800700 0.042230 0.1912 0.06412
342 9.606 16.84 0.08481 0.09228 0.0842200 0.022920 0.2036 0.07125
343 11.060 14.96 0.10330 0.09097 0.0539700 0.033410 0.1776 0.06907
344 19.680 21.68 0.09797 0.13390 0.1863000 0.110300 0.2082 0.05715
345 11.710 15.45 0.11500 0.07281 0.0400600 0.032500 0.2009 0.06506
346 10.260 14.71 0.09882 0.09159 0.0358100 0.020370 0.1633 0.07005
347 12.060 18.90 0.08386 0.05794 0.0075100 0.008488 0.1555 0.06048
348 14.760 14.74 0.08875 0.07780 0.0460800 0.035280 0.1521 0.05912
349 11.470 16.03 0.09076 0.05886 0.0258700 0.023220 0.1634 0.06372
350 11.950 14.96 0.11580 0.12060 0.0117100 0.017870 0.2459 0.06581
351 11.660 17.07 0.07561 0.03630 0.0083060 0.011620 0.1671 0.05731
352 15.750 19.22 0.12430 0.23640 0.2914000 0.124200 0.2375 0.07603
353 25.730 17.46 0.11490 0.23630 0.3368000 0.191300 0.1956 0.06121
354 15.080 25.74 0.10240 0.09769 0.1235000 0.065530 0.1647 0.06464
355 11.140 14.07 0.07274 0.06064 0.0450500 0.014710 0.1690 0.06083
356 12.560 19.07 0.08760 0.10380 0.1030000 0.043910 0.1533 0.06184
357 13.050 18.59 0.10820 0.13040 0.0960300 0.056030 0.2035 0.06501
358 13.870 16.21 0.08743 0.05492 0.0150200 0.020880 0.1424 0.05883
359 8.878 15.49 0.08293 0.07698 0.0472100 0.023810 0.1930 0.06621
360 9.436 18.32 0.10090 0.05956 0.0271000 0.014060 0.1506 0.06959
361 12.540 18.07 0.07436 0.02650 0.0011940 0.005449 0.1528 0.05185
362 13.300 21.57 0.08582 0.06373 0.0334400 0.024240 0.1815 0.05696
363 12.760 18.84 0.09676 0.07952 0.0268800 0.017810 0.1759 0.06183
364 16.500 18.29 0.09686 0.08468 0.0586200 0.048350 0.1495 0.05593
365 13.400 16.95 0.07937 0.05696 0.0218100 0.014730 0.1650 0.05701
366 20.440 21.78 0.09150 0.11310 0.0979900 0.077850 0.1618 0.05557
367 20.200 26.83 0.09905 0.16690 0.1641000 0.126500 0.1875 0.06020
368 12.210 18.02 0.09231 0.07175 0.0439200 0.020270 0.1695 0.05916
369 21.710 17.25 0.09384 0.08562 0.1168000 0.084650 0.1717 0.05054
370 22.010 21.90 0.10630 0.19540 0.2448000 0.150100 0.1824 0.06140
371 16.350 23.29 0.09742 0.14970 0.1811000 0.087730 0.2175 0.06218
372 15.190 13.21 0.07963 0.06934 0.0339300 0.026570 0.1721 0.05544
373 21.370 15.10 0.10010 0.15150 0.1932000 0.125500 0.1973 0.06183
374 20.640 17.35 0.09446 0.10760 0.1527000 0.089410 0.1571 0.05478
375 13.690 16.07 0.08302 0.06374 0.0255600 0.020310 0.1872 0.05669
376 16.170 16.07 0.09880 0.14380 0.0665100 0.053970 0.1990 0.06572
377 10.570 20.22 0.09073 0.16600 0.2280000 0.059410 0.2188 0.08450
378 13.460 28.21 0.07517 0.04726 0.0127100 0.011170 0.1421 0.05763
379 13.660 15.15 0.08268 0.07548 0.0424900 0.024710 0.1792 0.05897
380 11.080 18.83 0.12160 0.21540 0.1689000 0.063670 0.2196 0.07950
381 11.270 12.96 0.12370 0.11110 0.0790000 0.055500 0.2018 0.06914
382 11.040 14.93 0.07987 0.07079 0.0354600 0.020740 0.2003 0.06246
383 12.050 22.72 0.06935 0.10730 0.0794300 0.029780 0.1203 0.06659
384 12.390 17.48 0.10420 0.12970 0.0589200 0.028800 0.1779 0.06588
385 13.280 13.72 0.08363 0.08575 0.0507700 0.028640 0.1617 0.05594
386 14.600 23.29 0.08682 0.06636 0.0839000 0.052710 0.1627 0.05416
387 12.210 14.09 0.08108 0.07823 0.0683900 0.025340 0.1646 0.06154
388 13.880 16.16 0.07026 0.04831 0.0204500 0.008507 0.1607 0.05474
389 11.270 15.50 0.08365 0.11140 0.1007000 0.027570 0.1810 0.07252
390 19.550 23.21 0.10100 0.13180 0.1856000 0.102100 0.1989 0.05884
391 10.260 12.22 0.09996 0.07542 0.0192300 0.019680 0.1800 0.06569
392 8.734 16.84 0.10390 0.07428 0.0000000 0.000000 0.1985 0.07098
393 15.490 19.97 0.11600 0.15620 0.1891000 0.091130 0.1929 0.06744
394 21.610 22.28 0.11670 0.20870 0.2810000 0.156200 0.2162 0.06606
395 12.100 17.72 0.10290 0.09758 0.0478300 0.033260 0.1937 0.06161
396 14.060 17.18 0.08045 0.05361 0.0268100 0.032510 0.1641 0.05764
397 13.510 18.89 0.10590 0.11470 0.0858000 0.053810 0.1806 0.06079
398 12.800 17.46 0.08044 0.08895 0.0739000 0.040830 0.1574 0.05750
399 11.060 14.83 0.07741 0.04768 0.0271200 0.007246 0.1535 0.06214
400 11.800 17.26 0.09087 0.06232 0.0285300 0.016380 0.1847 0.06019
401 17.910 21.02 0.12300 0.25760 0.3189000 0.119800 0.2113 0.07115
402 11.930 10.91 0.08872 0.05242 0.0260600 0.017960 0.1601 0.05541
403 12.960 18.29 0.07351 0.07899 0.0405700 0.018830 0.1874 0.05899
404 12.940 16.17 0.09879 0.08836 0.0329600 0.023900 0.1735 0.06200
405 12.340 14.95 0.08682 0.04571 0.0210900 0.020540 0.1571 0.05708
406 10.940 18.59 0.10040 0.07460 0.0494400 0.029320 0.1486 0.06615
407 16.140 14.86 0.09495 0.08501 0.0550000 0.045280 0.1735 0.05875
408 12.850 21.37 0.07551 0.08316 0.0612600 0.018670 0.1580 0.06114
409 17.990 20.66 0.10360 0.13040 0.1201000 0.088240 0.1992 0.06069
410 12.270 17.92 0.08685 0.06526 0.0321100 0.026530 0.1966 0.05597
411 11.360 17.57 0.08858 0.05313 0.0278300 0.021000 0.1601 0.05913
412 11.040 16.83 0.10770 0.07804 0.0304600 0.024800 0.1714 0.06340
413 9.397 21.68 0.07969 0.06053 0.0373500 0.005128 0.1274 0.06724
414 14.990 22.11 0.08515 0.10250 0.0685900 0.038760 0.1944 0.05913
415 15.130 29.81 0.08320 0.04605 0.0468600 0.027390 0.1852 0.05294
416 11.890 21.17 0.09773 0.08120 0.0255500 0.021790 0.2019 0.06290
417 9.405 21.70 0.10440 0.06159 0.0204700 0.012570 0.2025 0.06601
418 15.500 21.08 0.11200 0.15710 0.1522000 0.084810 0.2085 0.06864
419 12.700 12.17 0.08785 0.05794 0.0236000 0.024020 0.1583 0.06275
420 11.160 21.41 0.10180 0.05978 0.0089550 0.010760 0.1615 0.06144
421 11.570 19.04 0.08546 0.07722 0.0548500 0.014280 0.2031 0.06267
422 14.690 13.98 0.10310 0.18360 0.1450000 0.063000 0.2086 0.07406
423 11.610 16.02 0.10880 0.11680 0.0709700 0.044970 0.1886 0.06320
424 13.660 19.13 0.09057 0.11470 0.0965700 0.048120 0.1848 0.06181
425 9.742 19.12 0.10750 0.08333 0.0089340 0.019670 0.2538 0.07029
426 10.030 21.28 0.08117 0.03912 0.0024700 0.005159 0.1630 0.06439
427 10.480 14.98 0.09816 0.10130 0.0633500 0.022180 0.1925 0.06915
428 10.800 21.98 0.08801 0.05743 0.0361400 0.014040 0.2016 0.05977
429 11.130 16.62 0.08151 0.03834 0.0136900 0.013700 0.1511 0.06148
430 12.720 17.67 0.07896 0.04522 0.0140200 0.018350 0.1459 0.05544
431 14.900 22.53 0.09947 0.22250 0.2733000 0.097110 0.2041 0.06898
432 12.400 17.68 0.10540 0.13160 0.0774100 0.027990 0.1811 0.07102
433 20.180 19.54 0.11330 0.14890 0.2133000 0.125900 0.1724 0.06053
434 18.820 21.97 0.10180 0.13890 0.1594000 0.087440 0.1943 0.06132
435 14.860 16.94 0.08924 0.07074 0.0334600 0.028770 0.1573 0.05703
436 13.980 19.62 0.10600 0.11330 0.1126000 0.064630 0.1669 0.06544
437 12.870 19.54 0.09136 0.07883 0.0179700 0.020900 0.1861 0.06347
438 14.040 15.98 0.08458 0.05895 0.0353400 0.029440 0.1714 0.05898
439 13.850 19.60 0.08684 0.06330 0.0134200 0.022930 0.1555 0.05673
440 14.020 15.66 0.07966 0.05581 0.0208700 0.026520 0.1589 0.05586
441 10.970 17.20 0.08915 0.11130 0.0945700 0.036130 0.1489 0.06640
442 17.270 25.42 0.08331 0.11090 0.1204000 0.057360 0.1467 0.05407
443 13.780 15.79 0.08817 0.06718 0.0105500 0.009937 0.1405 0.05848
444 10.570 18.32 0.08142 0.04462 0.0199300 0.011110 0.2372 0.05768
445 18.030 16.85 0.08947 0.12320 0.1090000 0.062540 0.1720 0.05780
446 11.990 24.89 0.10300 0.09218 0.0544100 0.042740 0.1820 0.06850
447 17.750 28.03 0.09997 0.13140 0.1698000 0.082930 0.1713 0.05916
448 14.800 17.66 0.09179 0.08890 0.0406900 0.022600 0.1893 0.05886
449 14.530 19.34 0.08388 0.07800 0.0881700 0.029250 0.1473 0.05746
450 21.100 20.52 0.09684 0.11750 0.1572000 0.115500 0.1554 0.05661
451 11.870 21.54 0.06613 0.10640 0.0877700 0.023860 0.1349 0.06612
452 19.590 25.00 0.10320 0.09871 0.1655000 0.090630 0.1663 0.05391
453 12.000 28.23 0.08437 0.06450 0.0405500 0.019450 0.1615 0.06104
454 14.530 13.98 0.10990 0.09242 0.0689500 0.064950 0.1650 0.06121
455 12.620 17.15 0.08583 0.05430 0.0296600 0.022720 0.1799 0.05826
456 13.380 30.72 0.09245 0.07426 0.0281900 0.032640 0.1375 0.06016
457 11.630 29.29 0.09357 0.08574 0.0716000 0.020170 0.1799 0.06166
458 13.210 25.25 0.08791 0.05205 0.0277200 0.020680 0.1619 0.05584
459 13.000 25.13 0.08369 0.05073 0.0120600 0.017620 0.1667 0.05449
460 9.755 28.20 0.07984 0.04626 0.0154100 0.010430 0.1621 0.05952
461 17.080 27.15 0.09898 0.11100 0.1007000 0.064310 0.1793 0.06281
462 27.420 26.27 0.10840 0.19880 0.3635000 0.168900 0.2061 0.05623
463 14.400 26.99 0.06995 0.05223 0.0347600 0.017370 0.1707 0.05433
464 11.600 18.36 0.08508 0.05855 0.0336700 0.017770 0.1516 0.05859
465 13.170 18.22 0.07466 0.05994 0.0485900 0.028700 0.1454 0.05549
466 13.240 20.13 0.08284 0.12230 0.1010000 0.028330 0.1601 0.06432
467 13.140 20.74 0.08675 0.10890 0.1085000 0.035100 0.1562 0.06020
468 9.668 18.10 0.08311 0.05428 0.0147900 0.005769 0.1680 0.06412
469 17.600 23.33 0.09289 0.20040 0.2136000 0.100200 0.1696 0.07369
470 11.620 18.18 0.11750 0.14830 0.1020000 0.055640 0.1957 0.07255
471 9.667 18.49 0.08946 0.06258 0.0294800 0.015140 0.2238 0.06413
472 12.040 28.14 0.08752 0.06000 0.0236700 0.023770 0.1854 0.05698
473 14.920 14.93 0.08098 0.08549 0.0553900 0.032210 0.1687 0.05669
474 12.270 29.97 0.07699 0.03398 0.0000000 0.000000 0.1701 0.05960
475 10.880 15.62 0.10070 0.10690 0.0511500 0.015710 0.1861 0.06837
476 12.830 15.73 0.09040 0.08269 0.0583500 0.030780 0.1705 0.05913
477 14.200 20.53 0.08931 0.11080 0.0506300 0.030580 0.1506 0.06009
478 13.900 16.62 0.06828 0.05319 0.0222400 0.013390 0.1813 0.05536
479 11.490 14.59 0.10460 0.08228 0.0530800 0.019690 0.1779 0.06574
480 16.250 19.51 0.10260 0.18930 0.2236000 0.091940 0.2151 0.06578
481 12.160 18.03 0.09087 0.07838 0.0291600 0.015270 0.1464 0.06284
482 13.900 19.24 0.07991 0.05326 0.0299500 0.020700 0.1579 0.05594
483 13.470 14.06 0.10710 0.11550 0.0578600 0.052660 0.1779 0.06639
484 13.700 17.64 0.09950 0.07957 0.0454800 0.031600 0.1732 0.06088
485 15.730 11.28 0.10430 0.12990 0.1191000 0.062110 0.1784 0.06259
486 12.450 16.41 0.09514 0.15110 0.1544000 0.048460 0.2082 0.07325
487 14.640 16.85 0.08641 0.06698 0.0519200 0.027910 0.1409 0.05355
488 19.440 18.82 0.10890 0.14480 0.2256000 0.119400 0.1823 0.06115
489 11.680 16.17 0.11280 0.09263 0.0427900 0.031320 0.1853 0.06401
490 16.690 20.20 0.07497 0.07112 0.0364900 0.023070 0.1846 0.05325
491 12.250 22.44 0.08192 0.05200 0.0171400 0.012610 0.1544 0.05976
492 17.850 13.23 0.07838 0.06217 0.0444500 0.041780 0.1220 0.05243
493 18.010 20.56 0.10010 0.12890 0.1170000 0.077620 0.2116 0.06077
494 12.460 12.83 0.07372 0.04043 0.0071730 0.011490 0.1613 0.06013
495 13.160 20.54 0.07335 0.05275 0.0180000 0.012560 0.1713 0.05888
496 14.870 20.21 0.09587 0.08345 0.0682400 0.049510 0.1487 0.05748
497 12.650 18.17 0.10760 0.13340 0.0801700 0.050740 0.1641 0.06854
498 12.470 17.31 0.08928 0.07630 0.0360900 0.023690 0.1526 0.06046
499 18.490 17.52 0.10120 0.13170 0.1491000 0.091830 0.1832 0.06697
500 20.590 21.24 0.10850 0.16440 0.2188000 0.112100 0.1848 0.06222
501 15.040 16.74 0.09883 0.13640 0.0772100 0.061420 0.1668 0.06869
502 13.820 24.49 0.11620 0.16810 0.1357000 0.067590 0.2275 0.07237
503 12.540 16.32 0.11580 0.10850 0.0592800 0.032790 0.1943 0.06612
504 23.090 19.83 0.09342 0.12750 0.1676000 0.100300 0.1505 0.05484
505 9.268 12.87 0.16340 0.22390 0.0973000 0.052520 0.2378 0.09502
506 9.676 13.14 0.12550 0.22040 0.1188000 0.070380 0.2057 0.09575
507 12.220 20.04 0.10960 0.11520 0.0817500 0.021660 0.2124 0.06894
508 11.060 17.12 0.11940 0.10710 0.0406300 0.042680 0.1954 0.07976
509 16.300 15.70 0.09427 0.06712 0.0552600 0.045630 0.1711 0.05657
510 15.460 23.95 0.11830 0.18700 0.2030000 0.085200 0.1807 0.07083
511 11.740 14.69 0.08099 0.09661 0.0672600 0.026390 0.1499 0.06758
512 14.810 14.70 0.08472 0.05016 0.0341600 0.025410 0.1659 0.05348
513 13.400 20.52 0.11060 0.14690 0.1445000 0.081720 0.2116 0.07325
514 14.580 13.66 0.09832 0.08918 0.0822200 0.043490 0.1739 0.05640
515 15.050 19.07 0.09215 0.08597 0.0748600 0.043350 0.1561 0.05915
516 11.340 18.61 0.10490 0.08499 0.0430200 0.025940 0.1927 0.06211
517 18.310 20.58 0.10680 0.12480 0.1569000 0.094510 0.1860 0.05941
518 19.890 20.26 0.10370 0.13100 0.1411000 0.094310 0.1802 0.06188
519 12.880 18.22 0.12180 0.16610 0.0482500 0.053030 0.1709 0.07253
520 12.750 16.70 0.11250 0.11170 0.0388000 0.029950 0.2120 0.06623
521 9.295 13.90 0.13710 0.12250 0.0333200 0.024210 0.2197 0.07696
522 24.630 21.60 0.10300 0.21060 0.2310000 0.147100 0.1991 0.06739
523 11.260 19.83 0.08511 0.04413 0.0050670 0.005664 0.1637 0.06343
524 13.710 18.68 0.09916 0.10700 0.0538500 0.037830 0.1714 0.06843
525 9.847 15.68 0.09492 0.08419 0.0233000 0.024160 0.1387 0.06891
526 8.571 13.10 0.10360 0.07632 0.0256500 0.015100 0.1678 0.07126
527 13.460 18.75 0.10750 0.11380 0.0420100 0.031520 0.1723 0.06317
528 12.340 12.27 0.09003 0.06307 0.0295800 0.026470 0.1689 0.05808
529 13.940 13.17 0.12480 0.09755 0.1010000 0.066150 0.1976 0.06457
530 12.070 13.44 0.11000 0.09009 0.0378100 0.027980 0.1657 0.06608
531 11.750 17.56 0.10730 0.09713 0.0528200 0.044400 0.1598 0.06677
532 11.670 20.02 0.10160 0.09453 0.0420000 0.021570 0.1859 0.06461
533 13.680 16.33 0.09277 0.07255 0.0175200 0.018800 0.1631 0.06155
534 20.470 20.67 0.09156 0.13130 0.1523000 0.101500 0.2166 0.05419
535 10.960 17.62 0.09687 0.09752 0.0526300 0.027880 0.1619 0.06408
536 20.550 20.86 0.10460 0.17390 0.2085000 0.132200 0.2127 0.06251
537 14.270 22.55 0.10380 0.11540 0.1463000 0.061390 0.1926 0.05982
538 11.690 24.44 0.12360 0.15520 0.0451500 0.045310 0.2131 0.07405
539 7.729 25.49 0.08098 0.04878 0.0000000 0.000000 0.1870 0.07285
540 7.691 25.44 0.08668 0.11990 0.0925200 0.013640 0.2037 0.07751
541 11.540 14.44 0.09984 0.11200 0.0673700 0.025940 0.1818 0.06782
542 14.470 24.99 0.08837 0.12300 0.1009000 0.038900 0.1872 0.06341
543 14.740 25.42 0.08275 0.07214 0.0410500 0.030270 0.1840 0.05680
544 13.210 28.06 0.08671 0.06877 0.0298700 0.032750 0.1628 0.05781
545 13.870 20.70 0.09578 0.10180 0.0368800 0.023690 0.1620 0.06688
546 13.620 23.23 0.09246 0.06747 0.0297400 0.024430 0.1664 0.05801
547 10.320 16.35 0.09434 0.04994 0.0101200 0.005495 0.1885 0.06201
548 10.260 16.58 0.08877 0.08066 0.0435800 0.024380 0.1669 0.06714
549 9.683 19.34 0.08491 0.05030 0.0233700 0.009615 0.1580 0.06235
550 10.820 24.21 0.08192 0.06602 0.0154800 0.008160 0.1976 0.06328
551 10.860 21.48 0.07431 0.04227 0.0000000 0.000000 0.1661 0.05948
552 11.130 22.44 0.09566 0.08194 0.0482400 0.022570 0.2030 0.06552
553 12.770 29.43 0.08276 0.04234 0.0199700 0.014990 0.1539 0.05637
554 9.333 21.94 0.09240 0.05605 0.0399600 0.012820 0.1692 0.06576
555 12.880 28.92 0.08123 0.05824 0.0619500 0.023430 0.1566 0.05708
556 10.290 27.61 0.09030 0.07658 0.0599900 0.027380 0.1593 0.06127
557 10.160 19.59 0.10030 0.07504 0.0050250 0.011160 0.1791 0.06331
558 9.423 27.88 0.08123 0.04971 0.0000000 0.000000 0.1742 0.06059
559 14.590 22.68 0.08473 0.13300 0.1029000 0.037360 0.1454 0.06147
560 11.510 23.93 0.09261 0.10210 0.1112000 0.041050 0.1388 0.06570
561 14.050 27.15 0.09929 0.11260 0.0446200 0.043040 0.1537 0.06171
562 11.200 29.37 0.07449 0.03558 0.0000000 0.000000 0.1060 0.05502
563 15.220 30.62 0.10480 0.20870 0.2550000 0.094290 0.2128 0.07152
564 20.920 25.09 0.10990 0.22360 0.3174000 0.147400 0.2149 0.06879
565 21.560 22.39 0.11100 0.11590 0.2439000 0.138900 0.1726 0.05623
566 20.130 28.25 0.09780 0.10340 0.1440000 0.097910 0.1752 0.05533
567 16.600 28.08 0.08455 0.10230 0.0925100 0.053020 0.1590 0.05648
568 20.600 29.33 0.11780 0.27700 0.3514000 0.152000 0.2397 0.07016
569 7.760 24.54 0.05263 0.04362 0.0000000 0.000000 0.1587 0.05884A.
My analysis suggests 3 PCs should be retained. Support or refute this suggestion. What percent of variability is explained by the first 3 PCs?
#Getting the loadings
bc_cells_pca <- prcomp(bc_cells_cols, scale. = TRUE)
bc_cells_pca Standard deviations (1, .., p=8):
[1] 2.0705378 1.3503646 0.9086939 0.7061387 0.6101579 0.3035518 0.2622598
[8] 0.1783697
Rotation (n x k) = (8 x 8):
PC1 PC2 PC3 PC4 PC5
Radius -0.3003952 0.52850910 0.27751200 -0.0449523963 0.04245937
Texture -0.1432175 0.35378530 -0.89839046 -0.0002176232 0.21581443
Smoothness -0.3482386 -0.32661945 0.12684205 0.1097614573 0.84332416
Compactness -0.4584098 -0.07219238 -0.02956419 0.1825835334 -0.23762997
Concavity -0.4508935 0.12707085 0.04245883 0.1571126948 -0.30459047
ConcavePts -0.4459288 0.22823091 0.17458320 0.0608428515 0.01923459
Symmetry -0.3240333 -0.28112508 -0.08456832 -0.8897711849 -0.11359240
FracDim -0.2251375 -0.57996072 -0.24389523 0.3640273309 -0.27912206
PC6 PC7 PC8
Radius -0.518437923 0.36152546 -0.387460874
Texture -0.006127134 0.02418201 0.004590238
Smoothness 0.079444068 -0.04732075 -0.155456892
Compactness -0.388065805 -0.73686177 0.020239147
Concavity 0.700061530 0.02347868 -0.413095816
ConcavePts 0.125314641 0.21313047 0.808318445
Symmetry -0.018262848 0.05764443 -0.023810142
FracDim -0.261064577 0.52365191 -0.026129456fviz_eig(bc_cells_pca)Warning in geom_bar(stat = "identity", fill = barfill, color = barcolor, :
Ignoring empty aesthetic: `width`.
bc_cells_pca$sdev^2[1] 4.28712664 1.82348458 0.82572454 0.49863192 0.37229268 0.09214368 0.06878019
[8] 0.03181576#4.28712664 1.82348458 0.82572454 0.49863192 0.37229268 0.09214368
#0.06878019 0.03181576
#4.28712664 1.82348458 0.82572454 0.49863192 0.37229268 0.09214368
#0.06878019 0.03181576
#Total is 7.999
#First PCs are 4.287 + 1.824 + 0.826 = 6.937
#6.937 / 7.99 = 0.867 and 86.7%
#I would refute that suggestion that the first 3 PCs as they
#only explain 86.7% of the variability. Short of the 90%
#variability threshold which we want for, when we chose
#how many PCs we want to use.
#
#B.
Interpret the first 3 principal components by examining the eigenvectors/loadings. Discuss.
loadings <- bc_cells_pca$rotation
loadings PC1 PC2 PC3 PC4 PC5
Radius -0.3003952 0.52850910 0.27751200 -0.0449523963 0.04245937
Texture -0.1432175 0.35378530 -0.89839046 -0.0002176232 0.21581443
Smoothness -0.3482386 -0.32661945 0.12684205 0.1097614573 0.84332416
Compactness -0.4584098 -0.07219238 -0.02956419 0.1825835334 -0.23762997
Concavity -0.4508935 0.12707085 0.04245883 0.1571126948 -0.30459047
ConcavePts -0.4459288 0.22823091 0.17458320 0.0608428515 0.01923459
Symmetry -0.3240333 -0.28112508 -0.08456832 -0.8897711849 -0.11359240
FracDim -0.2251375 -0.57996072 -0.24389523 0.3640273309 -0.27912206
PC6 PC7 PC8
Radius -0.518437923 0.36152546 -0.387460874
Texture -0.006127134 0.02418201 0.004590238
Smoothness 0.079444068 -0.04732075 -0.155456892
Compactness -0.388065805 -0.73686177 0.020239147
Concavity 0.700061530 0.02347868 -0.413095816
ConcavePts 0.125314641 0.21313047 0.808318445
Symmetry -0.018262848 0.05764443 -0.023810142
FracDim -0.261064577 0.52365191 -0.026129456bc_cells_PC1 <- loadings[,1]
bc_cells_PC2 <- loadings[,2]
bc_cells_PC3 <- loadings[,3]
bc_cells_PC1 Radius Texture Smoothness Compactness Concavity ConcavePts
-0.3003952 -0.1432175 -0.3482386 -0.4584098 -0.4508935 -0.4459288
Symmetry FracDim
-0.3240333 -0.2251375 #bc_cells_PC1
# Radius Texture Smoothness Compactness Concavity
# -0.3003952 -0.1432175 -0.3482386 -0.4584098 -0.4508935
# ConcavePts Symmetry FracDim
# -0.4459288 -0.3240333 -0.2251375
#Our biggest contributers (we don't take into account the sign
#of the variable) to PC1 are compactness, concavity, concavepts, #symmetry. Tumors with rough texture & irregularity will
#have
bc_cells_PC2 Radius Texture Smoothness Compactness Concavity ConcavePts
0.52850910 0.35378530 -0.32661945 -0.07219238 0.12707085 0.22823091
Symmetry FracDim
-0.28112508 -0.57996072 # Radius Texture Smoothness Compactness Concavity
# 0.52850910 0.35378530 -0.32661945 -0.07219238 0.12707085
# ConcavePts Symmetry FracDim
# 0.22823091 -0.28112508 -0.57996072
#Our biggest contributers ( the sign of the variables don't matter again) to PC2 are Radius, FracDim, Texture, Smoothness,
#ConcavePts. We can infer that tumors with bigger radius and #fractal dimensions will contribute the most to PC2.
bc_cells_PC3 Radius Texture Smoothness Compactness Concavity ConcavePts
0.27751200 -0.89839046 0.12684205 -0.02956419 0.04245883 0.17458320
Symmetry FracDim
-0.08456832 -0.24389523 #Radius Texture Smoothness Compactness Concavity
# 0.27751200 -0.89839046 0.12684205 -0.02956419 0.04245883
#ConcavePts Symmetry FracDim
# 0.17458320 -0.08456832 -0.24389523
#Our biggest contributers to PC3 are Radius, Texture, FracDim, #ConcavePts. We can infer that this too means that for the PC3 #dimension tumors with less textures (because its negative) & larger radius, multiple #fractures (FracDim) contribute the most to the PC3 dimension.
#Texture plays a disproportiate role, meaning that less texture
#more easily sways the PC3 value dimension. C.
Examine a biplot of the first two PCs. Incorporate the third PC by sizing the points by this variable. (Hint: use fviz_pca to set up a biplot, but set col.ind='white'. Then use geom_point() to maintain full control over the point mapping.) Color-code by whether the cells are benign or malignant. Answer the following:
What characteristics distinguish malignant from benign cells?
Of the 3 PCs, which does the best job of differentiating malignant from benign cells?
scores <- as.data.frame(bc_cells_pca$x)
colnames(scores) <- paste0("PC", 1:ncol(scores))
scores$Diagnosis <- bc_cells$Diagnosis
# Build the base biplot using fviz_pca but hide its points
p <- fviz_pca_biplot(
bc_cells_pca,
col.ind = "white", # hide original points
geom = "point",
repel = TRUE,
col.var = "black"
)
# Add your own points scaled by PC3
p +
geom_point(
data = scores,
aes(x = PC1, y = PC2, color = Diagnosis, size = PC3),
alpha = 0.6
) +
scale_size_continuous(name = "PC3 (size)") +
scale_color_manual(values = c("B" = "steelblue", "M" = "firebrick")) +
theme_minimal() +
labs(title = "Biplot of First Two PCs (Size by PC3 & Diagnosis by Color)")
#Looking at the biplot and where the points are stationed by
#their color and size we can say that the characteristics that
#most underscore the differences between benign & malignant #tumors were Radius, Texture, ConcavePts, Concavity, Compactness
#
#We can say that for full certainity that PC1 is by far the strongest in terms of how much of a impact it has on #differentiating malignant & benign tumors. Numerically it
#explains 53.6% of the variability and visually we see a clearer
#split on the horizontal axis. On the other hand PC2 explains
#far less variability and it isn't as clear cut how different
#the two are on the vertical axis. Meanwhile PC3 comes out #slightly stronger than PC2 with the difference in size of the
#points being somewhat obvious (but hard to differentiate) and
#explaining slightly more in variability.