library(mlbench)
library(corrplot)
## corrplot 0.92 loaded
library(ggplot2)
library(mlbench)
library(dplyr)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library(tsibble)
## 
## Attaching package: 'tsibble'
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, union
library(tidyr)
library(caret)
## Loading required package: lattice
data(Glass)
str(Glass)
## 'data.frame':    214 obs. of  10 variables:
##  $ RI  : num  1.52 1.52 1.52 1.52 1.52 ...
##  $ Na  : num  13.6 13.9 13.5 13.2 13.3 ...
##  $ Mg  : num  4.49 3.6 3.55 3.69 3.62 3.61 3.6 3.61 3.58 3.6 ...
##  $ Al  : num  1.1 1.36 1.54 1.29 1.24 1.62 1.14 1.05 1.37 1.36 ...
##  $ Si  : num  71.8 72.7 73 72.6 73.1 ...
##  $ K   : num  0.06 0.48 0.39 0.57 0.55 0.64 0.58 0.57 0.56 0.57 ...
##  $ Ca  : num  8.75 7.83 7.78 8.22 8.07 8.07 8.17 8.24 8.3 8.4 ...
##  $ Ba  : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ Fe  : num  0 0 0 0 0 0.26 0 0 0 0.11 ...
##  $ Type: Factor w/ 6 levels "1","2","3","5",..: 1 1 1 1 1 1 1 1 1 1 ...

(a) Using visualizations, explore the predictor variables to understand their distributions as well as the relationships between predictors.

# Histograms
Glass %>%
  dplyr::select(-10)%>% 
  gather() %>%
  ggplot(aes(x=value))+geom_histogram()+facet_wrap(~key,scales = "free")
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

# Adjust the following for more variables
ggplot(Glass, aes(x = Mg)) + geom_histogram() 
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

# Boxplots
ggplot(Glass, aes(y = Na)) + geom_boxplot()  # Add more variables similarly

# Correlation Matrix

corrplot(cor(Glass%>%dplyr::select(-10)))

Some distributions follow what seems to be a normal distribution, while others are either left or right skewed; with the exception of K and Mg.

Seems to be some outliers present in the data, espercially as number 0, we need to remove the 0 to se ehow the istribution looks like. also as previously presented, some predictors are moderately skewed. ### (c) Are there any relevant transformations of one or more predictors that might improve the classification model?

transformations <- preProcess(Glass, method = c("BoxCox", "pca"))
transformations
## Created from 214 samples and 10 variables
## 
## Pre-processing:
##   - Box-Cox transformation (5)
##   - centered (9)
##   - ignored (1)
##   - principal component signal extraction (9)
##   - scaled (9)
## 
## Lambda estimates for Box-Cox transformation:
## -2, -0.1, 0.5, 2, -1.1
## PCA needed 7 components to capture 95 percent of the variance
transformed <- predict(transformations, Glass)
transformed
##     Type          PC1         PC2           PC3          PC4           PC5
## 1      1 -1.212644363 -0.39421392  0.1730755538  1.719385187 -0.1913387480
## 2      1  0.617907347 -0.70204758  0.5507034242  0.857534996 -0.1566312150
## 3      1  0.990702729 -0.88768855  0.6452945956  0.302771619 -0.1363025332
## 4      1  0.151021153 -0.90423365  0.1622361388  0.452156680 -0.4291846111
## 5      1  0.358284900 -1.01609650  0.5763959403  0.166783141 -0.3634192496
## 6      1  0.340801720 -1.35656371 -0.7451274975 -1.056833328  1.7762845375
## 7      1  0.234723869 -1.03070725  0.6391935695  0.190160928 -0.3911092786
## 8      1  0.078428804 -1.12675935  0.7481313379  0.010227397 -0.4319625918
## 9      1  0.022838065 -0.26358724 -0.0004617044  1.366893723 -0.3242774917
## 10     1 -0.004600609 -1.01077177 -0.0373025978 -0.403512405  0.4491990627
## 11     1  0.424855316 -1.39973505 -0.5286573751 -1.257759119  1.5638407937
## 12     1 -0.019955889 -1.03609638  0.2661694670 -0.219486261 -0.5938813346
## 13     1  0.352923374 -1.41756352 -0.3372690293 -1.134950362  1.5953243194
## 14     1 -0.168121939 -1.17401699 -0.0430557016 -0.805677214  0.9614960642
## 15     1  0.058794290 -1.11389606  0.3691541268 -0.584484651 -0.6216073170
## 16     1  0.088044990 -1.08260030  0.4707030797 -0.358877708 -0.5585567478
## 17     1 -0.229845421 -1.10538579  0.3396856541 -0.350596046 -0.6529245635
## 18     1 -1.582942183  0.22395103  0.1922621434  2.368705198 -0.2480093279
## 19     1 -0.634566341 -0.14620545  0.5753834287  1.277274663 -0.1694331272
## 20     1  0.300652679 -0.74962888 -0.2411185342 -0.242231318  0.1028281346
## 21     1 -0.204821667 -0.97624552 -0.5798562366 -0.668408623  1.0845331664
## 22     1 -1.743917799 -0.38581123  1.6450014249  2.446615014  0.0205505353
## 23     1 -0.078843148 -0.96660620  0.1205937521 -0.127259376 -0.6182591619
## 24     1  0.083171994 -0.96687380  0.2159476590 -0.293823757 -0.6138851308
## 25     1  0.100494767 -0.83413899  0.5933759349  0.326651523 -0.3868826920
## 26     1  0.014827614 -0.97326045  0.3285474436 -0.091198454 -0.5842291544
## 27     1  0.149359997 -0.68246810  0.0784793851  0.277246490 -0.5100467500
## 28     1  0.196113806 -0.97208050  0.3354177273 -0.249772433 -0.5407988179
## 29     1  0.091017248 -1.00284514  0.1798142427 -0.587522462 -0.6508261463
## 30     1  0.041855826 -0.83026747  0.2581914013  0.062149660 -0.5445950941
## 31     1 -0.327604750 -1.11575976 -0.2171047592 -0.849925646  0.5499151437
## 32     1 -0.034477698 -1.06910363  0.5979620369 -0.374255623 -0.5654669562
## 33     1 -0.409030071 -1.02331471 -0.4471147228 -0.793856139  1.3464112111
## 34     1  0.040602661 -1.10860536  0.1785962634 -0.943415599 -0.1365579522
## 35     1 -0.121640258 -0.90237516  0.1510164720 -0.343972720 -0.6591447117
## 36     1  0.337279495 -0.92821678  0.4842576650  0.204282090 -0.4214989286
## 37     1 -0.285519360 -0.02729177 -0.1796511192  1.395793899 -0.3687087143
## 38     1 -0.057482320 -0.88489340  0.0984205764 -0.321799906 -0.6833413123
## 39     1 -2.260098657 -0.05164764  0.8316472624  2.148931299 -0.3072986497
## 40     1 -2.260098657 -0.05164764  0.8316472624  2.148931299 -0.3072986497
## 41     1 -0.276675057 -0.98547582  0.3271062612 -0.253941489 -0.6762913455
## 42     1 -0.067966290 -1.02058846  0.4211591644 -0.478493191 -0.6425272538
## 43     1  0.069336753 -0.67330872  0.2262783600  0.150843388 -0.5379470856
## 44     1 -2.079912760 -0.02504404  0.3459191987  1.657462748 -0.4710267415
## 45     1 -0.781508406 -1.15407580 -0.7213586143 -1.150113860  1.9237364972
## 46     1 -0.447859817 -0.23199419 -0.2578229737  0.927149638 -0.5670011778
## 47     1 -0.646980404 -0.70469410 -0.2433308876 -0.246905364  0.7915060271
## 48     1 -3.113151934  0.58102509 -0.1302422980  1.840726564  0.3561588991
## 49     1 -2.205884051 -0.08999474  0.2648756066  1.047856773 -0.6136645318
## 50     1 -0.434785952 -0.26620938 -0.0923065966  0.921854740 -0.5622326251
## 51     1 -2.978273720 -0.05285174  0.0422996881  1.300116535  0.8766088403
## 52     1 -0.758381965 -0.41654893 -0.4531451284  0.076499761  0.2621050126
## 53     1 -0.182273950 -0.30711314  0.4942969263  0.077501451 -0.5980779950
## 54     1 -0.241202324 -0.31761225  0.3025643045 -0.164408770 -0.6869810434
## 55     1 -0.232306178 -0.41723722  0.2065869222 -0.501600235  0.1610509969
## 56     1 -0.516019602 -0.90838809 -0.1261013486 -2.050002773  1.2595255810
## 57     1  0.396337774 -1.79210837 -0.2254256705 -1.130790742  2.3392367304
## 58     1 -0.029268876 -0.89206035  0.1969069479 -0.121833203 -0.5976683749
## 59     1  0.331343942 -1.02698565  0.6034303400  0.447777442 -0.3188131932
## 60     1 -0.046717152 -0.98554835  0.0944491009  0.132431960  0.5693233261
## 61     1 -0.547649456 -0.39513534  0.7891821438  0.715091038 -0.2551023127
## 62     1 -0.240571094  0.68836982 -0.1317749922  1.654640386  0.0900703188
## 63     1 -1.989793074 -0.19535052 -0.2358361546  1.106390540  0.4428380326
## 64     1 -2.066767537  0.31195848  0.3314633318  2.231788619 -0.2980543466
## 65     1 -1.878697444 -0.11636451  0.0600334368  0.985399895  0.0963465166
## 66     1 -1.292567783  0.12327682  0.2902708851  1.228623884 -0.3877989238
## 67     1 -2.199889723 -0.33545202 -0.2565104209  0.193955161  0.8317216621
## 68     1 -2.173754819 -0.35139047 -0.1869162899  0.117475673  0.8344598541
## 69     1 -2.101130114 -0.27184770 -0.2702872544  0.251715987  0.7354092932
## 70     1 -2.363769142  0.12894139  0.1519447863  0.968592004 -0.3884972905
## 71     2  1.220946743 -0.41136909  0.1170384589  1.714860247  1.4087649468
## 72     2 -0.717848937 -0.84808128 -1.1187000387  0.368073987  2.4317782348
## 73     2  0.997263679 -1.19969911  0.3133934917 -0.104215258 -0.3799247243
## 74     2  0.932493244 -0.96121175  0.2612354102  0.216262716 -0.3227666795
## 75     2  0.976357249 -1.19768271  0.2339971300 -0.198741839 -0.4410065424
## 76     2  0.919528512 -1.19750870  0.2925699288 -0.205913629 -0.4320696678
## 77     2  0.729375045 -0.83822905 -0.0629619387  0.632946276 -0.3648849512
## 78     2  0.726723909 -1.05443139  0.1150800168 -0.006260591 -0.4312948304
## 79     2  0.472978386 -0.91126565  0.4966284781  0.299414556  1.1504063842
## 80     2  1.109161746 -1.02984107 -0.1961340723 -0.310320680 -0.5017903982
## 81     2  1.229894754 -0.89092652 -0.4337564964 -0.207977682 -0.5006449569
## 82     2  0.956732186 -1.10558904  0.5394342029 -0.053843465 -0.3291891530
## 83     2  0.546162222 -1.01813236  0.3816950287  0.380277802 -0.3870429969
## 84     2  0.692435556 -1.11883060 -0.1490100868 -0.281373109  0.3409402630
## 85     2  2.525594650 -0.81474773 -0.4071989693  1.043857005 -0.1946452981
## 86     2  0.618192624 -0.82058716  0.3740994800  0.304078029 -0.2948553011
## 87     2  0.873005590 -1.00642910  0.7987458052 -0.165299323 -0.2306286309
## 88     2  0.574917640 -0.91634798 -0.2076787182  0.081085855  0.4754656173
## 89     2  0.672721964 -1.03596978  0.2144554572 -0.066496289 -0.4422442481
## 90     2  0.774379273 -1.15122843 -0.2938043086 -1.043014835  0.2143259411
## 91     2 -1.026571202 -0.98607178 -0.8402039632 -0.176115004  1.2354597679
## 92     2  0.552655892 -1.00463359  0.4632932018 -0.303813214 -0.4104801405
## 93     2  0.268667910 -0.87219318  0.3773091710 -0.924553948  1.4744017158
## 94     2  0.720087172 -0.84530348  0.6769453717 -0.131719562 -0.2990290517
## 95     2  0.659104222 -1.11521608  0.2848953941 -0.652492372 -0.5864577805
## 96     2 -0.107913791 -0.39597872 -0.0992702789  0.639552412 -0.4997635804
## 97     2 -0.963219155 -0.84661848 -0.5008591086 -0.003095428  0.6117052768
## 98     2 -0.672664277 -1.36029028 -0.2981867357 -1.892363823  1.2562108490
## 99     2  0.632366231 -0.70981255  0.0051893289 -0.886824560 -0.7552740728
## 100    2  0.104599482 -0.34557105  0.1027953744 -0.294118926 -0.6254488341
## 101    2 -0.005654399 -0.76508881 -0.3050276937 -1.446022705  1.2883891805
## 102    2  0.028365736 -0.52366101 -0.3089055710 -0.991633457 -0.9660368776
## 103    2 -1.147608111 -0.90292132  0.6527118633 -1.692387217  1.0118340168
## 104    2 -3.834052098  1.28410757 -0.5881102395  2.390649943 -0.8844362021
## 105    2 -2.330239861  1.18587097 -0.4000933063  1.620174090 -0.7169642547
## 106    2 -2.911525034  1.52075755 -2.9163479591 -3.168508038  0.7333140346
## 107    2 -2.985759496  5.53675979 -5.9240005340 -0.835850419  1.3441563417
## 108    2 -6.017064426  3.21142695 -2.9509654715 -0.128372689 -0.0506138110
## 109    2 -1.695997212  2.12469723  0.8447359593 -0.242519213 -0.1915299159
## 110    2 -1.009871912  0.92319996  2.7741159994 -1.665679664 -0.7551993557
## 111    2 -4.008221929  1.55640483  0.1145896079 -2.597163233 -1.9702670287
## 112    2 -4.306996490  1.56812054 -0.1088550893 -2.638058782 -2.0721244058
## 113    2 -4.121451182  2.31425144 -0.0959632735 -0.534556474 -1.6802439982
## 114    2 -0.304296457 -0.86176212 -0.2716469696  0.362101571  0.8237456135
## 115    2 -0.272505734 -0.95681497 -0.0047653535  0.604590088 -0.5058878260
## 116    2 -0.022668531 -0.74032065  0.0689079402  0.816401954 -0.3717709046
## 117    2 -0.165178671 -0.85529341 -0.4251178154  0.403368118  0.4368048730
## 118    2  0.897064738 -0.57316310 -0.3568970523  1.047146374 -0.2859121580
## 119    2  0.130631912 -1.11331873 -0.9346149637 -0.378573163  2.1301408409
## 120    2  0.737535596 -0.82917937  0.0809243684  0.707931349 -0.3174502769
## 121    2 -0.095969162 -0.73965682 -0.0362706911  0.624061536 -0.4838334347
## 122    2  0.380151550 -1.18310672 -0.5789652368 -0.787858169  1.3522778601
## 123    2  0.594185375 -0.88772847  0.2658658873  0.166884778 -0.3845678134
## 124    2  0.823866077 -0.65155002 -0.0588188873  0.499894081 -0.3612291917
## 125    2 -0.813706092 -0.50529127  0.1308503423  0.502847048 -0.5988579641
## 126    2 -0.266731272 -0.74470852 -0.6326414230 -0.167358450  0.4409511354
## 127    2  0.083155983 -0.99409510  0.2486124698  0.019552742 -0.5261605368
## 128    2 -1.136117771  0.66123524 -0.7381479057  0.160453486  0.6998155820
## 129    2 -0.650219612  0.92110066 -0.9568420645 -0.172163467  0.7745134090
## 130    2 -1.056030359  1.34991524 -0.8289332478 -0.026442427  0.7083129118
## 131    2 -1.334500840  1.61723294  0.0680625492  0.035910241 -1.0134788454
## 132    2 -2.976821226  2.85965708 -1.0356018008  0.111230276 -0.5450496853
## 133    2  0.008702502 -0.96110813  0.2049432445  0.850134097 -0.3536584721
## 134    2 -0.052948992 -0.64496542 -0.8020096463  0.925092941  1.0023640183
## 135    2  0.129670487 -0.94291909  0.3726803376  0.510430376 -0.3478709366
## 136    2 -0.599702918 -1.06378221 -0.9109799738 -0.152425405  1.9843718429
## 137    2 -0.372312947 -1.21331117  0.1519071604 -0.285007611  0.5571697034
## 138    2  0.568711519 -1.01589207  0.0916284025 -0.165281661 -0.4913190039
## 139    2  0.817675768 -1.21063897  0.3155712617 -0.516866190 -0.4897464954
## 140    2  0.814169701 -1.08321151  0.1504860233 -0.347811584 -0.4849990616
## 141    2  0.691509140 -0.77949591 -0.0889942620  0.395519295 -0.4358904846
## 142    2 -0.503429911 -0.93525834 -0.1442333690 -0.119382925  1.0324261565
## 143    2  0.096335960 -1.21780507 -0.6423820226 -0.974890813  1.6526927221
## 144    2  0.720013150 -0.77437367 -0.1962834899 -0.057151076 -0.5290926805
## 145    2 -0.344565395 -0.98522480 -0.3181374953 -0.961781103  1.4914325047
## 146    2 -0.968006861 -1.12745733 -1.1585205492 -0.841003178  2.4000196207
## 147    3 -0.208873026 -0.57835784  1.0078709589  0.639684972 -0.1491258856
## 148    3  0.478783177 -0.89439783  0.3325976505  0.325273120 -0.3803691028
## 149    3  0.154993070 -0.91338587 -0.0558677842 -0.055278192  0.4348293783
## 150    3  0.048916902 -1.28166946 -0.0565447791 -0.683553060 -0.7159775326
## 151    3  0.324331374 -0.77784411 -0.7304717559 -0.331255052  0.9867611348
## 152    3 -1.693565616  0.19503677  0.5101416757  2.199917893 -0.1924863213
## 153    3 -0.850240998 -0.78736691  1.5778093741  0.638380799 -0.1672761830
## 154    3  0.424513508 -0.88605006  0.4331024305  0.388825629 -0.3855392833
## 155    3 -0.036724325 -0.90522307  0.0275785814  0.004488282 -0.6189595466
## 156    3  0.229212718 -0.92311245  0.5209586252 -0.168046730 -0.4876540107
## 157    3  0.212977201 -0.68580487  0.4019384980  0.334312352 -0.4252667185
## 158    3 -2.005393130 -0.07549828  0.7216025336  1.875289657 -0.3363915978
## 159    3  0.056649931 -0.30474973 -0.2401306460  0.784890289 -0.5080897051
## 160    3 -0.092455323 -0.26703312 -0.6712514185  0.529020865  0.2540475650
## 161    3 -0.137946465 -0.24266934 -0.2768885357  0.534240101 -0.5992693502
## 162    3 -1.379165479 -0.54198477  0.2599749606  0.225025838  1.8723037355
## 163    3 -2.269364897 -0.08431182 -1.1318067405  1.167612621  2.8823014074
## 164    5  4.997088693  1.61521815 -4.0712144336  3.183267578  0.6998845300
## 165    5 -0.298174633  0.51119427 -0.4022202688 -0.976978279 -1.0970363320
## 166    5 -1.725985308  0.38869115 -0.6665275552 -1.896281876 -1.5419804089
## 167    5 -1.710847703  0.10981758 -0.6058667809 -2.842263363 -1.7626891714
## 168    5 -0.623484101  1.35293810  0.6016175634 -2.423653537 -1.3485621355
## 169    5  0.743260642  0.76289099  0.3147645615 -2.431812252 -1.3618750904
## 170    5 -0.496598387  1.71509456  0.2020705370 -1.456937684 -1.2524290970
## 171    5 -1.722880485  2.36966847 -0.2709344489 -0.593123247 -1.3553672073
## 172    5  4.832006342 -1.30863566 -7.2087952724  0.009185106 -3.4839345325
## 173    5  4.881658593 -1.36480212 -7.0642799932 -0.148878756 -3.4767342937
## 174    5 -1.373107140  1.97316824  0.0306364126 -0.749922692 -1.2839871533
## 175    5 -1.006724672  0.72474872 -2.7787830326 -1.981325621  3.3592096703
## 176    5 -1.574016654  1.32258689 -0.1729684659 -2.463528206  1.3038822236
## 177    6 -0.349658255  0.76620620  0.6173167932  0.499644009 -0.3610089488
## 178    6 -0.766724621  0.51256012  1.0225739996  0.211945527 -0.4266931240
## 179    6  0.084493536  0.86820089  0.8074407666  0.738972127 -0.2095632522
## 180    6  0.036037147  0.77082237  0.8288628359  0.260893183 -0.3056450822
## 181    6  2.315480942 -0.32681903  2.6316258221 -0.877759023  0.2454341885
## 182    6  0.136081922  1.88730328  0.9934664209  0.430036232 -0.3698579815
## 183    6 -0.006918605  2.23394338  0.6500647556 -0.722325264 -0.7496298575
## 184    6 -1.362450713  1.55263450  2.3264840565 -0.360107133 -0.6665600415
## 185    6  2.957615109 -0.15631589  5.3192939467  0.560135587  0.9770542416
## 186    7  5.117324405 -1.47051113 -0.9993385196  1.303927439  0.5613860785
## 187    7  3.684754705  0.61026272 -2.3079929895  2.971995964  0.6485121688
## 188    7 -1.082659766  0.01829648 -0.2130442549  0.797107937 -0.7183333903
## 189    7 -0.618058790  1.78036820 -1.6701663984  2.401800512 -0.7891270303
## 190    7  0.050844987  3.53017221 -0.9737521616  3.730068777  0.5335978868
## 191    7  1.322613316  1.21524058  0.8814204669 -0.140873819  0.1483213533
## 192    7  1.967427493  2.31542432  0.8318699188 -0.673359847  0.8723049499
## 193    7  1.848308733  2.05570886  0.5498833088 -1.444511942  0.5471851728
## 194    7  2.031608084  3.17565609  0.4771248993 -0.046640603  1.2116642746
## 195    7  2.112912387  3.01082560  0.6643595620 -0.349323950  1.0954502009
## 196    7  2.071412600  2.16248777  0.6043009057 -1.273672923  0.2855624017
## 197    7  1.922209110  2.34065406  0.4639526931 -1.183610892 -0.1138498353
## 198    7  1.780036766  2.50776469  0.9981678489 -0.472564566 -0.0040785306
## 199    7  2.159553047  2.35388715  0.6964746510 -0.749808078 -0.0661558738
## 200    7  2.118770796  2.41773803  0.9349189513 -0.228436698  0.0232841720
## 201    7  2.521100365  2.17780635  1.4906946808 -0.306197552  0.2553391896
## 202    7  1.379554415 -1.01122949 -0.3779288259 -3.761189060 -2.2795553250
## 203    7  2.549107831  2.06023671  1.4712632560 -0.733891937  0.1524948907
## 204    7  2.493792289  3.22139239  0.8097743432  0.213192631  0.6486796950
## 205    7  2.097888608  2.40119302  1.1985916725 -0.308426602  0.1288555775
## 206    7  1.988198150  3.21630649  0.8945192604  0.375429815  0.5227211142
## 207    7  2.105652104  2.98684313  1.0426766248  0.159314072  0.4613249649
## 208    7  4.174381645  3.18863408 -1.2469740232  1.046823328  0.7370441359
## 209    7  1.743826410  2.52611226  0.4889148580 -0.636587111 -0.1908332321
## 210    7  2.151665360  2.94212078 -0.0495504477 -0.475093306 -0.0006797061
## 211    7  2.343113546  3.21082810  0.8417563356  0.274686836  0.5857915063
## 212    7  1.660576516  3.40825084  0.6154882045 -0.180684838  0.3657523368
## 213    7  2.321777345  2.89188315  1.0798380545 -0.492471709  0.4734031923
## 214    7  2.208037102  3.13422639  0.7106647354 -0.425174513  0.4363112725
##               PC6          PC7
## 1   -0.3686905326  0.479564119
## 2    0.0618974992  0.085255818
## 3   -0.1739301924  0.394120584
## 4   -0.2951884046  0.101784417
## 5   -0.3289072329 -0.139342912
## 6    0.1433597985  0.238997521
## 7   -0.2798260584 -0.314118846
## 8   -0.4096846624 -0.477686523
## 9    0.3005108512  0.247504438
## 10  -0.1512216707  0.026884918
## 11   0.0695939199  0.075396310
## 12  -0.5587011377  0.018933822
## 13   0.1640820705 -0.226971204
## 14  -0.1366169969 -0.203844586
## 15  -0.7348504094  0.008724878
## 16  -0.5953755650 -0.146101889
## 17  -0.6499617544 -0.163957979
## 18   0.3393843589  0.008639062
## 19   0.0432626848  0.421685799
## 20  -0.1704674870  0.628320065
## 21  -0.0152084865  0.265642160
## 22   0.5154285880 -1.515935442
## 23  -0.5022013514  0.146606343
## 24  -0.5091961753  0.113885514
## 25  -0.1476012103 -0.148461823
## 26  -0.3902292628 -0.152937554
## 27  -0.2135812672  0.243180710
## 28  -0.4893859649  0.083625875
## 29  -0.7306476104  0.233827804
## 30  -0.3242378853  0.001930407
## 31  -0.2721212755 -0.078351279
## 32  -0.5491603715 -0.267795431
## 33  -0.0322063032 -0.341534023
## 34  -0.5722565992  0.006266303
## 35  -0.5913963755  0.175364143
## 36  -0.0568561099  0.007621214
## 37   0.1725127444  0.296429737
## 38  -0.5305093930  0.107605949
## 39   0.1948893483 -0.927980428
## 40   0.1948893483 -0.927980428
## 41  -0.5053796637 -0.262098365
## 42  -0.5880551585 -0.161069626
## 43  -0.1805246204  0.099119107
## 44  -0.0527812973 -0.329013124
## 45   0.2403183365 -0.394352721
## 46   0.0904099462  0.405279404
## 47   0.2017413579 -0.253015141
## 48   0.1576244102 -0.465352347
## 49  -0.4060751166 -0.162417910
## 50   0.1604000923  0.110519808
## 51   0.3110749250 -0.906819317
## 52   0.1761899278  0.051451571
## 53   0.1037483431 -0.181591946
## 54  -0.0942679917 -0.022463543
## 55   0.1774565732 -0.102837956
## 56  -0.0267928895 -0.423948917
## 57   0.6413589594 -0.424852073
## 58  -0.5117061853 -0.010291205
## 59  -0.1988921614 -0.261682798
## 60   0.1014868483 -0.224232821
## 61  -0.1890851827  0.090262695
## 62  -0.8188277729  0.210511839
## 63   0.1011003925 -0.154998943
## 64   0.2131574988 -0.027344133
## 65  -0.0353146202 -0.122483262
## 66  -0.0510160621  0.351750836
## 67  -0.0434978186 -0.279104542
## 68  -0.0567002485 -0.309257476
## 69   0.0137643475 -0.259975384
## 70  -0.2398458356 -0.179172120
## 71   0.9559654043  0.880113861
## 72   0.8270171982 -0.032500090
## 73  -0.3579516596  0.205871467
## 74  -0.1788536380  0.355803376
## 75  -0.3665996059  0.208776115
## 76  -0.3722398448  0.200150250
## 77  -0.0144146292  0.439589891
## 78  -0.3833122111  0.390570385
## 79   0.4753606808 -0.132551592
## 80  -0.4451663987  0.811841482
## 81  -0.3807274169  1.137210214
## 82  -0.2501701324  0.060582839
## 83  -0.0721995749 -0.125635857
## 84  -0.0336502796  0.250869324
## 85   0.7748899758  0.584026164
## 86  -0.1572676663  0.454901535
## 87  -0.3296003688  0.299950043
## 88   0.2179141490  0.223449962
## 89  -0.3549033152  0.294627405
## 90  -0.4822426473  0.644761578
## 91   0.2094612151 -0.217712040
## 92  -0.4800234429  0.341104623
## 93   0.0153473336  0.541678964
## 94  -0.2471030358  0.334997558
## 95  -0.5364669843  0.158221749
## 96  -0.0856971479  0.421810930
## 97   0.1349794131 -0.264240083
## 98  -0.3038641621 -0.510196055
## 99  -0.3969444621  0.388238070
## 100 -0.4302738479  0.069151339
## 101  0.0579362715 -0.134883043
## 102 -0.4814354491  0.490425827
## 103 -0.1239729869 -1.036371462
## 104  0.2125699895 -0.159362185
## 105  0.3150862966  0.585701166
## 106  0.4715667259  0.345854258
## 107 -4.2663822352 -0.566693800
## 108  0.5185690429 -0.010441208
## 109  1.4983903137 -0.709162938
## 110  0.6577187035 -1.829398244
## 111 -1.1302250789 -0.750110967
## 112 -1.3083362611 -0.734189290
## 113  0.0641700561 -0.852339694
## 114  0.1478976883 -0.098717795
## 115 -0.4047485963  0.044607978
## 116 -0.2215158203  0.273222614
## 117 -0.0436440637  0.315768987
## 118  0.1486955156  0.854620899
## 119  0.5861057574  0.132266844
## 120  0.0386434694  0.302910606
## 121 -0.2505371797  0.217826769
## 122  0.0924484646  0.232802247
## 123 -0.2572582501  0.296515687
## 124 -0.0222384300  0.606442991
## 125 -0.4837296535 -0.224879326
## 126 -0.1397666826  0.447399307
## 127 -0.3784361353  0.122165324
## 128  0.9841210570  0.190251446
## 129  0.5113003115  0.231664315
## 130  1.4532381662  0.501620771
## 131  0.8024796824  0.160888066
## 132  1.2824822443  0.291472588
## 133 -0.2282142505 -0.037615848
## 134  0.4649398978  0.568556647
## 135 -0.3336988860  0.010088890
## 136  0.4168214189  0.021555394
## 137 -0.2288977455 -0.423741153
## 138 -0.5129666406  0.396823255
## 139 -0.6188082193  0.168054445
## 140 -0.5222628399  0.393452402
## 141 -0.0886401139  0.474691679
## 142 -0.0162082126 -0.540235329
## 143  0.1087384659 -0.103533054
## 144 -0.3233802071  0.702068750
## 145  0.3781532784 -0.280779437
## 146  0.4170793165 -0.244120532
## 147 -0.1552646251  0.094727010
## 148 -0.0918951186  0.190595105
## 149  0.0810403872  0.162357963
## 150 -0.9394133684  0.278995624
## 151  0.2746499244  0.645441448
## 152  0.2956169010  0.035035317
## 153 -0.1968069987 -0.758829268
## 154 -0.0001108286 -0.055479161
## 155 -0.3677603321  0.226952618
## 156 -0.3072768235  0.026207635
## 157  0.0096412703  0.134947934
## 158  0.1525201282 -0.624989032
## 159  0.1703481824  0.598447317
## 160  0.3986513877  0.694694624
## 161  0.0320429045  0.627570639
## 162  0.2712369538 -0.907252967
## 163  1.2171598356 -0.356561115
## 164 -1.8190026830  0.581349155
## 165  0.0015743947  0.754503857
## 166 -0.9695935579  0.573852119
## 167 -1.3650086394  0.377355107
## 168  0.2137182548  0.094345332
## 169  0.6267967430 -0.170962865
## 170  0.7707586980  0.304775281
## 171  0.8211732188  0.345435545
## 172  3.1326792854 -1.859394865
## 173  3.0791979765 -1.950158445
## 174  0.9460964080  0.198162814
## 175  1.1790285921  0.406960544
## 176  0.9299921576 -0.116796030
## 177  0.4633209025  0.785832988
## 178  0.2582880671  0.155240972
## 179  0.7886433251  0.780428095
## 180  0.5224171742  0.774574245
## 181  0.5264899174 -0.159674940
## 182  1.4900103923  0.648360137
## 183  1.2047655283  1.034941637
## 184  1.2848356738 -1.642836253
## 185  2.2852352445 -3.434470269
## 186 -1.2985522740 -1.429499415
## 187 -1.5445990434 -0.679148285
## 188 -0.1792301513 -0.115013268
## 189  1.5443966922  1.320534454
## 190 -0.3792040213 -0.820829473
## 191 -0.5028423528  0.272260940
## 192  0.9062776597  0.519729301
## 193  0.8357703084  1.125721901
## 194 -0.4352532558 -0.501020058
## 195 -0.5765137701 -0.557765857
## 196  0.4989522749  0.941603462
## 197  0.0569503812  0.772723878
## 198  0.5263315410  0.545444659
## 199  0.5176139936  1.046360609
## 200  0.9771196065  0.844658948
## 201  0.8122819480  0.335528457
## 202  0.3348788284 -2.736081348
## 203  0.6989077761  0.539643998
## 204 -0.7838392030 -0.565060274
## 205  0.6783728297  0.431786377
## 206 -0.4644774907 -0.656185215
## 207 -0.2207900401 -0.458241870
## 208 -2.3581526690 -2.810887099
## 209  0.6543578081  1.309756391
## 210 -0.0960649594  1.137154405
## 211 -0.5459217047 -0.475570128
## 212 -1.1813409632 -0.639844344
## 213 -0.9057082202 -0.618457245
## 214 -1.0912922763 -0.417915158

This function prepares the necessary components for data transformation. After the preProcess function is executed, the predict method applies the transformation to a dataset. PCA key advantage is its ability to generate uncorrelated components, ensuring its continued use for data reduction. Each subsequent principal component maximizes the captured variance while remaining uncorrelated with previously derived PCs.

3.2. The soybean data can also be found at the UC Irvine Machine Learning Repository. Data were collected to predict disease in 683 soybeans. The 35 predictors are mostly categorical and include information on the environmental conditions (e.g., temperature, precipitation) and plant conditions (e.g., left spots, mold growth). The outcome labels consist of 19 distinct classes. 6 http://archive.ics.uci.edu/ml/index.html. 3.8 Computing 59 The data can be loaded via:

data(Soybean)

(a) Investigate the frequency distributions for the categorical predictors. Are any of the distributions degenerate in the ways discussed earlier in this chapter?

soybeans <- Soybean %>% 
  dplyr::select(-1)

par(mfrow = c(2, 2))  # Set plot layout

lapply(2:ncol(soybeans), function(col) {
  hist(as.numeric(soybeans[,col]), 
       main = colnames(soybeans)[col], 
       xlab = colnames(soybeans)[col])
})

## [[1]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1] 354   0   0   0   0   0   0   0   0 293
## 
## $density
##  [1] 5.471406 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9] 0.000000 4.528594
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[2]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
## 
## $counts
##  [1]  74   0   0   0 112   0   0   0   0 459
## 
## $density
##  [1] 0.5736434 0.0000000 0.0000000 0.0000000 0.8682171 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 3.5581395
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[3]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
## 
## $counts
##  [1]  80   0   0   0 374   0   0   0   0 199
## 
## $density
##  [1] 0.6125574 0.0000000 0.0000000 0.0000000 2.8637060 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 1.5237366
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[4]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1] 435   0   0   0   0   0   0   0   0 127
## 
## $density
##  [1] 7.740214 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9] 0.000000 2.259786
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[5]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
## 
## $counts
##  [1]  65   0   0   0 165   0   0   0   0 219   0   0   0   0 218
## 
## $density
##  [1] 0.4872564 0.0000000 0.0000000 0.0000000 1.2368816 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 1.6416792 0.0000000 0.0000000 0.0000000 0.0000000
## [15] 1.6341829
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[6]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
## 
## $counts
##  [1] 123   0   0   0 227   0   0   0   0 145   0   0   0   0 187
## 
## $density
##  [1] 0.9017595 0.0000000 0.0000000 0.0000000 1.6642229 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 1.0630499 0.0000000 0.0000000 0.0000000 0.0000000
## [15] 1.3709677
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[7]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
## 
## $counts
##  [1] 195   0   0   0 322   0   0   0   0  45
## 
## $density
##  [1] 1.7348754 0.0000000 0.0000000 0.0000000 2.8647687 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 0.4003559
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[8]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
## 
## $counts
##  [1] 305   0   0   0 222   0   0   0   0  35
## 
## $density
##  [1] 2.7135231 0.0000000 0.0000000 0.0000000 1.9750890 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 0.3113879
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[9]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
## 
## $counts
##  [1] 165   0   0   0 213   0   0   0   0 193
## 
## $density
##  [1] 1.444834 0.000000 0.000000 0.000000 1.865149 0.000000 0.000000 0.000000
##  [9] 0.000000 1.690018
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[10]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1] 441   0   0   0   0   0   0   0   0 226
## 
## $density
##  [1] 6.611694 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9] 0.000000 3.388306
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[11]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1]  77   0   0   0   0   0   0   0   0 606
## 
## $density
##  [1] 1.127379 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9] 0.000000 8.872621
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[12]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
## 
## $counts
##  [1] 221   0   0   0  36   0   0   0   0 342
## 
## $density
##  [1] 1.8447412 0.0000000 0.0000000 0.0000000 0.3005008 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 2.8547579
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[13]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
## 
## $counts
##  [1] 357   0   0   0  21   0   0   0   0 221
## 
## $density
##  [1] 2.9799666 0.0000000 0.0000000 0.0000000 0.1752922 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 1.8447412
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[14]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
## 
## $counts
##  [1]  51   0   0   0 327   0   0   0   0 221
## 
## $density
##  [1] 0.4257095 0.0000000 0.0000000 0.0000000 2.7295492 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 1.8447412
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[15]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1] 487   0   0   0   0   0   0   0   0  96
## 
## $density
##  [1] 8.353345 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9] 0.000000 1.646655
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[16]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1] 554   0   0   0   0   0   0   0   0  45
## 
## $density
##  [1] 9.2487479 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 0.7512521
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[17]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
## 
## $counts
##  [1] 535   0   0   0  20   0   0   0   0  20
## 
## $density
##  [1] 4.652174 0.000000 0.000000 0.000000 0.173913 0.000000 0.000000 0.000000
##  [9] 0.000000 0.173913
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[18]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1] 296   0   0   0   0   0   0   0   0 371
## 
## $density
##  [1] 4.437781 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9] 0.000000 5.562219
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[19]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1] 520   0   0   0   0   0   0   0   0  42
## 
## $density
##  [1] 9.252669 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9] 0.000000 0.747331
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[20]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
## 
## $counts
##  [1] 379   0   0   0  39   0   0   0   0  36   0   0   0   0 191
## 
## $density
##  [1] 2.9379845 0.0000000 0.0000000 0.0000000 0.3023256 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 0.2790698 0.0000000 0.0000000 0.0000000 0.0000000
## [15] 1.4806202
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[21]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
## 
## $counts
##  [1] 320   0   0   0  83   0   0   0   0 177   0   0   0   0  65
## 
## $density
##  [1] 2.4806202 0.0000000 0.0000000 0.0000000 0.6434109 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 1.3720930 0.0000000 0.0000000 0.0000000 0.0000000
## [15] 0.5038760
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[22]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1] 473   0   0   0   0   0   0   0   0 104
## 
## $density
##  [1] 8.197574 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9] 0.000000 1.802426
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[23]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
## 
## $counts
##  [1] 497   0   0   0 135   0   0   0   0  13
## 
## $density
##  [1] 3.8527132 0.0000000 0.0000000 0.0000000 1.0465116 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 0.1007752
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[24]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1] 639   0   0   0   0   0   0   0   0   6
## 
## $density
##  [1] 9.90697674 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
##  [7] 0.00000000 0.00000000 0.00000000 0.09302326
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[25]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
## 
## $counts
##  [1] 581   0   0   0  44   0   0   0   0  20
## 
## $density
##  [1] 4.5038760 0.0000000 0.0000000 0.0000000 0.3410853 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 0.1550388
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[26]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1] 625   0   0   0   0   0   0   0   0  20
## 
## $density
##  [1] 9.6899225 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 0.3100775
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[27]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
## 
## $counts
##  [1] 407   0   0   0 130   0   0   0   0  14   0   0   0   0  48
## 
## $density
##  [1] 3.3973289 0.0000000 0.0000000 0.0000000 1.0851419 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 0.1168614 0.0000000 0.0000000 0.0000000 0.0000000
## [15] 0.4006678
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[28]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
## 
## $counts
##  [1] 345   0   0   0  75   0   0   0   0  57   0   0   0   0 100
## 
## $density
##  [1] 2.9896014 0.0000000 0.0000000 0.0000000 0.6499133 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 0.4939341 0.0000000 0.0000000 0.0000000 0.0000000
## [15] 0.8665511
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[29]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1] 476   0   0   0   0   0   0   0   0 115
## 
## $density
##  [1] 8.054146 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9] 0.000000 1.945854
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[30]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1] 524   0   0   0   0   0   0   0   0  67
## 
## $density
##  [1] 8.866328 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9] 0.000000 1.133672
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[31]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1] 513   0   0   0   0   0   0   0   0  64
## 
## $density
##  [1] 8.890815 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9] 0.000000 1.109185
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[32]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1] 532   0   0   0   0   0   0   0   0  59
## 
## $density
##  [1] 9.001692 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
##  [9] 0.000000 0.998308
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[33]]
## $breaks
##  [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
## 
## $counts
##  [1] 539   0   0   0   0   0   0   0   0  38
## 
## $density
##  [1] 9.3414211 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 0.6585789
## 
## $mids
##  [1] 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"
## 
## [[34]]
## $breaks
##  [1] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
## 
## $counts
##  [1] 551   0   0   0  86   0   0   0   0  15
## 
## $density
##  [1] 4.2254601 0.0000000 0.0000000 0.0000000 0.6595092 0.0000000 0.0000000
##  [8] 0.0000000 0.0000000 0.1150307
## 
## $mids
##  [1] 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9
## 
## $xname
## [1] "as.numeric(soybeans[, col])"
## 
## $equidist
## [1] TRUE
## 
## attr(,"class")
## [1] "histogram"

leaf.mild, mycelium and sclerotia appears to be degenerate, considering the low frequency count and minimal distinct values