Untitled
Jeneil Miller
2026-02-22
options(repos = c(CRAN = "https://cloud.r-project.org"))Reminder: All homework solutions must be written up independently, even though you are allowed to discuss with other students. You need to save your homework assignment in a pdf/html format and upload it with the R code (.R or .rmd) into the Canvas before 11:59pm CT on the due day. No late homework assignment will be graded in any circumstance.
Problem 1 (20 points): This exercise involves the Boston housing data set.
- To begin, load in the Boston data set. Since the Boston data set is part of the MASS library, you need to install the MASS package into R/Rstudio and then access the package as follows:
#install packages
install.packages("MASS") ## Installing package into 'C:/Users/jenei/AppData/Local/R/win-library/4.5'
## (as 'lib' is unspecified)## package 'MASS' successfully unpacked and MD5 sums checked## Warning: cannot remove prior installation of package 'MASS'## Warning in file.copy(savedcopy, lib, recursive = TRUE): problem copying
## C:\Users\jenei\AppData\Local\R\win-library\4.5\00LOCK\MASS\libs\x64\MASS.dll to
## C:\Users\jenei\AppData\Local\R\win-library\4.5\MASS\libs\x64\MASS.dll:
## Permission denied## Warning: restored 'MASS'##
## The downloaded binary packages are in
## C:\Users\jenei\AppData\Local\Temp\RtmpCoFuKb\downloaded_packagesinstall.packages("mlbench")## Installing package into 'C:/Users/jenei/AppData/Local/R/win-library/4.5'
## (as 'lib' is unspecified)## package 'mlbench' successfully unpacked and MD5 sums checked
##
## The downloaded binary packages are in
## C:\Users\jenei\AppData\Local\Temp\RtmpCoFuKb\downloaded_packagesinstall.packages("caret")## Installing package into 'C:/Users/jenei/AppData/Local/R/win-library/4.5'
## (as 'lib' is unspecified)## package 'caret' successfully unpacked and MD5 sums checked## Warning: cannot remove prior installation of package 'caret'## Warning in file.copy(savedcopy, lib, recursive = TRUE): problem copying
## C:\Users\jenei\AppData\Local\R\win-library\4.5\00LOCK\caret\libs\x64\caret.dll
## to C:\Users\jenei\AppData\Local\R\win-library\4.5\caret\libs\x64\caret.dll:
## Permission denied## Warning: restored 'caret'##
## The downloaded binary packages are in
## C:\Users\jenei\AppData\Local\Temp\RtmpCoFuKb\downloaded_packageslibrary(MASS)## Warning: package 'MASS' was built under R version 4.5.2library(mlbench)## Warning: package 'mlbench' was built under R version 4.5.2library(caret)## Warning: package 'caret' was built under R version 4.5.2## Loading required package: ggplot2## Loading required package: latticeLoad data and basic structure
# Load dataset
data(Boston)
# View dataset
Boston## crim zn indus chas nox rm age dis rad tax ptratio black
## 1 0.00632 18.0 2.31 0 0.5380 6.575 65.2 4.0900 1 296 15.3 396.90
## 2 0.02731 0.0 7.07 0 0.4690 6.421 78.9 4.9671 2 242 17.8 396.90
## 3 0.02729 0.0 7.07 0 0.4690 7.185 61.1 4.9671 2 242 17.8 392.83
## 4 0.03237 0.0 2.18 0 0.4580 6.998 45.8 6.0622 3 222 18.7 394.63
## 5 0.06905 0.0 2.18 0 0.4580 7.147 54.2 6.0622 3 222 18.7 396.90
## 6 0.02985 0.0 2.18 0 0.4580 6.430 58.7 6.0622 3 222 18.7 394.12
## 7 0.08829 12.5 7.87 0 0.5240 6.012 66.6 5.5605 5 311 15.2 395.60
## 8 0.14455 12.5 7.87 0 0.5240 6.172 96.1 5.9505 5 311 15.2 396.90
## 9 0.21124 12.5 7.87 0 0.5240 5.631 100.0 6.0821 5 311 15.2 386.63
## 10 0.17004 12.5 7.87 0 0.5240 6.004 85.9 6.5921 5 311 15.2 386.71
## 11 0.22489 12.5 7.87 0 0.5240 6.377 94.3 6.3467 5 311 15.2 392.52
## 12 0.11747 12.5 7.87 0 0.5240 6.009 82.9 6.2267 5 311 15.2 396.90
## 13 0.09378 12.5 7.87 0 0.5240 5.889 39.0 5.4509 5 311 15.2 390.50
## 14 0.62976 0.0 8.14 0 0.5380 5.949 61.8 4.7075 4 307 21.0 396.90
## 15 0.63796 0.0 8.14 0 0.5380 6.096 84.5 4.4619 4 307 21.0 380.02
## 16 0.62739 0.0 8.14 0 0.5380 5.834 56.5 4.4986 4 307 21.0 395.62
## 17 1.05393 0.0 8.14 0 0.5380 5.935 29.3 4.4986 4 307 21.0 386.85
## 18 0.78420 0.0 8.14 0 0.5380 5.990 81.7 4.2579 4 307 21.0 386.75
## 19 0.80271 0.0 8.14 0 0.5380 5.456 36.6 3.7965 4 307 21.0 288.99
## 20 0.72580 0.0 8.14 0 0.5380 5.727 69.5 3.7965 4 307 21.0 390.95
## 21 1.25179 0.0 8.14 0 0.5380 5.570 98.1 3.7979 4 307 21.0 376.57
## 22 0.85204 0.0 8.14 0 0.5380 5.965 89.2 4.0123 4 307 21.0 392.53
## 23 1.23247 0.0 8.14 0 0.5380 6.142 91.7 3.9769 4 307 21.0 396.90
## 24 0.98843 0.0 8.14 0 0.5380 5.813 100.0 4.0952 4 307 21.0 394.54
## 25 0.75026 0.0 8.14 0 0.5380 5.924 94.1 4.3996 4 307 21.0 394.33
## 26 0.84054 0.0 8.14 0 0.5380 5.599 85.7 4.4546 4 307 21.0 303.42
## 27 0.67191 0.0 8.14 0 0.5380 5.813 90.3 4.6820 4 307 21.0 376.88
## 28 0.95577 0.0 8.14 0 0.5380 6.047 88.8 4.4534 4 307 21.0 306.38
## 29 0.77299 0.0 8.14 0 0.5380 6.495 94.4 4.4547 4 307 21.0 387.94
## 30 1.00245 0.0 8.14 0 0.5380 6.674 87.3 4.2390 4 307 21.0 380.23
## 31 1.13081 0.0 8.14 0 0.5380 5.713 94.1 4.2330 4 307 21.0 360.17
## 32 1.35472 0.0 8.14 0 0.5380 6.072 100.0 4.1750 4 307 21.0 376.73
## 33 1.38799 0.0 8.14 0 0.5380 5.950 82.0 3.9900 4 307 21.0 232.60
## 34 1.15172 0.0 8.14 0 0.5380 5.701 95.0 3.7872 4 307 21.0 358.77
## 35 1.61282 0.0 8.14 0 0.5380 6.096 96.9 3.7598 4 307 21.0 248.31
## 36 0.06417 0.0 5.96 0 0.4990 5.933 68.2 3.3603 5 279 19.2 396.90
## 37 0.09744 0.0 5.96 0 0.4990 5.841 61.4 3.3779 5 279 19.2 377.56
## 38 0.08014 0.0 5.96 0 0.4990 5.850 41.5 3.9342 5 279 19.2 396.90
## 39 0.17505 0.0 5.96 0 0.4990 5.966 30.2 3.8473 5 279 19.2 393.43
## 40 0.02763 75.0 2.95 0 0.4280 6.595 21.8 5.4011 3 252 18.3 395.63
## 41 0.03359 75.0 2.95 0 0.4280 7.024 15.8 5.4011 3 252 18.3 395.62
## 42 0.12744 0.0 6.91 0 0.4480 6.770 2.9 5.7209 3 233 17.9 385.41
## 43 0.14150 0.0 6.91 0 0.4480 6.169 6.6 5.7209 3 233 17.9 383.37
## 44 0.15936 0.0 6.91 0 0.4480 6.211 6.5 5.7209 3 233 17.9 394.46
## 45 0.12269 0.0 6.91 0 0.4480 6.069 40.0 5.7209 3 233 17.9 389.39
## 46 0.17142 0.0 6.91 0 0.4480 5.682 33.8 5.1004 3 233 17.9 396.90
## 47 0.18836 0.0 6.91 0 0.4480 5.786 33.3 5.1004 3 233 17.9 396.90
## 48 0.22927 0.0 6.91 0 0.4480 6.030 85.5 5.6894 3 233 17.9 392.74
## 49 0.25387 0.0 6.91 0 0.4480 5.399 95.3 5.8700 3 233 17.9 396.90
## 50 0.21977 0.0 6.91 0 0.4480 5.602 62.0 6.0877 3 233 17.9 396.90
## 51 0.08873 21.0 5.64 0 0.4390 5.963 45.7 6.8147 4 243 16.8 395.56
## 52 0.04337 21.0 5.64 0 0.4390 6.115 63.0 6.8147 4 243 16.8 393.97
## 53 0.05360 21.0 5.64 0 0.4390 6.511 21.1 6.8147 4 243 16.8 396.90
## 54 0.04981 21.0 5.64 0 0.4390 5.998 21.4 6.8147 4 243 16.8 396.90
## 55 0.01360 75.0 4.00 0 0.4100 5.888 47.6 7.3197 3 469 21.1 396.90
## 56 0.01311 90.0 1.22 0 0.4030 7.249 21.9 8.6966 5 226 17.9 395.93
## 57 0.02055 85.0 0.74 0 0.4100 6.383 35.7 9.1876 2 313 17.3 396.90
## 58 0.01432 100.0 1.32 0 0.4110 6.816 40.5 8.3248 5 256 15.1 392.90
## 59 0.15445 25.0 5.13 0 0.4530 6.145 29.2 7.8148 8 284 19.7 390.68
## 60 0.10328 25.0 5.13 0 0.4530 5.927 47.2 6.9320 8 284 19.7 396.90
## 61 0.14932 25.0 5.13 0 0.4530 5.741 66.2 7.2254 8 284 19.7 395.11
## 62 0.17171 25.0 5.13 0 0.4530 5.966 93.4 6.8185 8 284 19.7 378.08
## 63 0.11027 25.0 5.13 0 0.4530 6.456 67.8 7.2255 8 284 19.7 396.90
## 64 0.12650 25.0 5.13 0 0.4530 6.762 43.4 7.9809 8 284 19.7 395.58
## 65 0.01951 17.5 1.38 0 0.4161 7.104 59.5 9.2229 3 216 18.6 393.24
## 66 0.03584 80.0 3.37 0 0.3980 6.290 17.8 6.6115 4 337 16.1 396.90
## 67 0.04379 80.0 3.37 0 0.3980 5.787 31.1 6.6115 4 337 16.1 396.90
## 68 0.05789 12.5 6.07 0 0.4090 5.878 21.4 6.4980 4 345 18.9 396.21
## 69 0.13554 12.5 6.07 0 0.4090 5.594 36.8 6.4980 4 345 18.9 396.90
## 70 0.12816 12.5 6.07 0 0.4090 5.885 33.0 6.4980 4 345 18.9 396.90
## 71 0.08826 0.0 10.81 0 0.4130 6.417 6.6 5.2873 4 305 19.2 383.73
## 72 0.15876 0.0 10.81 0 0.4130 5.961 17.5 5.2873 4 305 19.2 376.94
## 73 0.09164 0.0 10.81 0 0.4130 6.065 7.8 5.2873 4 305 19.2 390.91
## 74 0.19539 0.0 10.81 0 0.4130 6.245 6.2 5.2873 4 305 19.2 377.17
## 75 0.07896 0.0 12.83 0 0.4370 6.273 6.0 4.2515 5 398 18.7 394.92
## 76 0.09512 0.0 12.83 0 0.4370 6.286 45.0 4.5026 5 398 18.7 383.23
## 77 0.10153 0.0 12.83 0 0.4370 6.279 74.5 4.0522 5 398 18.7 373.66
## 78 0.08707 0.0 12.83 0 0.4370 6.140 45.8 4.0905 5 398 18.7 386.96
## 79 0.05646 0.0 12.83 0 0.4370 6.232 53.7 5.0141 5 398 18.7 386.40
## 80 0.08387 0.0 12.83 0 0.4370 5.874 36.6 4.5026 5 398 18.7 396.06
## 81 0.04113 25.0 4.86 0 0.4260 6.727 33.5 5.4007 4 281 19.0 396.90
## 82 0.04462 25.0 4.86 0 0.4260 6.619 70.4 5.4007 4 281 19.0 395.63
## 83 0.03659 25.0 4.86 0 0.4260 6.302 32.2 5.4007 4 281 19.0 396.90
## 84 0.03551 25.0 4.86 0 0.4260 6.167 46.7 5.4007 4 281 19.0 390.64
## 85 0.05059 0.0 4.49 0 0.4490 6.389 48.0 4.7794 3 247 18.5 396.90
## 86 0.05735 0.0 4.49 0 0.4490 6.630 56.1 4.4377 3 247 18.5 392.30
## 87 0.05188 0.0 4.49 0 0.4490 6.015 45.1 4.4272 3 247 18.5 395.99
## 88 0.07151 0.0 4.49 0 0.4490 6.121 56.8 3.7476 3 247 18.5 395.15
## 89 0.05660 0.0 3.41 0 0.4890 7.007 86.3 3.4217 2 270 17.8 396.90
## 90 0.05302 0.0 3.41 0 0.4890 7.079 63.1 3.4145 2 270 17.8 396.06
## 91 0.04684 0.0 3.41 0 0.4890 6.417 66.1 3.0923 2 270 17.8 392.18
## 92 0.03932 0.0 3.41 0 0.4890 6.405 73.9 3.0921 2 270 17.8 393.55
## 93 0.04203 28.0 15.04 0 0.4640 6.442 53.6 3.6659 4 270 18.2 395.01
## 94 0.02875 28.0 15.04 0 0.4640 6.211 28.9 3.6659 4 270 18.2 396.33
## 95 0.04294 28.0 15.04 0 0.4640 6.249 77.3 3.6150 4 270 18.2 396.90
## 96 0.12204 0.0 2.89 0 0.4450 6.625 57.8 3.4952 2 276 18.0 357.98
## 97 0.11504 0.0 2.89 0 0.4450 6.163 69.6 3.4952 2 276 18.0 391.83
## 98 0.12083 0.0 2.89 0 0.4450 8.069 76.0 3.4952 2 276 18.0 396.90
## 99 0.08187 0.0 2.89 0 0.4450 7.820 36.9 3.4952 2 276 18.0 393.53
## 100 0.06860 0.0 2.89 0 0.4450 7.416 62.5 3.4952 2 276 18.0 396.90
## 101 0.14866 0.0 8.56 0 0.5200 6.727 79.9 2.7778 5 384 20.9 394.76
## 102 0.11432 0.0 8.56 0 0.5200 6.781 71.3 2.8561 5 384 20.9 395.58
## 103 0.22876 0.0 8.56 0 0.5200 6.405 85.4 2.7147 5 384 20.9 70.80
## 104 0.21161 0.0 8.56 0 0.5200 6.137 87.4 2.7147 5 384 20.9 394.47
## 105 0.13960 0.0 8.56 0 0.5200 6.167 90.0 2.4210 5 384 20.9 392.69
## 106 0.13262 0.0 8.56 0 0.5200 5.851 96.7 2.1069 5 384 20.9 394.05
## 107 0.17120 0.0 8.56 0 0.5200 5.836 91.9 2.2110 5 384 20.9 395.67
## 108 0.13117 0.0 8.56 0 0.5200 6.127 85.2 2.1224 5 384 20.9 387.69
## 109 0.12802 0.0 8.56 0 0.5200 6.474 97.1 2.4329 5 384 20.9 395.24
## 110 0.26363 0.0 8.56 0 0.5200 6.229 91.2 2.5451 5 384 20.9 391.23
## 111 0.10793 0.0 8.56 0 0.5200 6.195 54.4 2.7778 5 384 20.9 393.49
## 112 0.10084 0.0 10.01 0 0.5470 6.715 81.6 2.6775 6 432 17.8 395.59
## 113 0.12329 0.0 10.01 0 0.5470 5.913 92.9 2.3534 6 432 17.8 394.95
## 114 0.22212 0.0 10.01 0 0.5470 6.092 95.4 2.5480 6 432 17.8 396.90
## 115 0.14231 0.0 10.01 0 0.5470 6.254 84.2 2.2565 6 432 17.8 388.74
## 116 0.17134 0.0 10.01 0 0.5470 5.928 88.2 2.4631 6 432 17.8 344.91
## 117 0.13158 0.0 10.01 0 0.5470 6.176 72.5 2.7301 6 432 17.8 393.30
## 118 0.15098 0.0 10.01 0 0.5470 6.021 82.6 2.7474 6 432 17.8 394.51
## 119 0.13058 0.0 10.01 0 0.5470 5.872 73.1 2.4775 6 432 17.8 338.63
## 120 0.14476 0.0 10.01 0 0.5470 5.731 65.2 2.7592 6 432 17.8 391.50
## 121 0.06899 0.0 25.65 0 0.5810 5.870 69.7 2.2577 2 188 19.1 389.15
## 122 0.07165 0.0 25.65 0 0.5810 6.004 84.1 2.1974 2 188 19.1 377.67
## 123 0.09299 0.0 25.65 0 0.5810 5.961 92.9 2.0869 2 188 19.1 378.09
## 124 0.15038 0.0 25.65 0 0.5810 5.856 97.0 1.9444 2 188 19.1 370.31
## 125 0.09849 0.0 25.65 0 0.5810 5.879 95.8 2.0063 2 188 19.1 379.38
## 126 0.16902 0.0 25.65 0 0.5810 5.986 88.4 1.9929 2 188 19.1 385.02
## 127 0.38735 0.0 25.65 0 0.5810 5.613 95.6 1.7572 2 188 19.1 359.29
## 128 0.25915 0.0 21.89 0 0.6240 5.693 96.0 1.7883 4 437 21.2 392.11
## 129 0.32543 0.0 21.89 0 0.6240 6.431 98.8 1.8125 4 437 21.2 396.90
## 130 0.88125 0.0 21.89 0 0.6240 5.637 94.7 1.9799 4 437 21.2 396.90
## 131 0.34006 0.0 21.89 0 0.6240 6.458 98.9 2.1185 4 437 21.2 395.04
## 132 1.19294 0.0 21.89 0 0.6240 6.326 97.7 2.2710 4 437 21.2 396.90
## 133 0.59005 0.0 21.89 0 0.6240 6.372 97.9 2.3274 4 437 21.2 385.76
## 134 0.32982 0.0 21.89 0 0.6240 5.822 95.4 2.4699 4 437 21.2 388.69
## 135 0.97617 0.0 21.89 0 0.6240 5.757 98.4 2.3460 4 437 21.2 262.76
## 136 0.55778 0.0 21.89 0 0.6240 6.335 98.2 2.1107 4 437 21.2 394.67
## 137 0.32264 0.0 21.89 0 0.6240 5.942 93.5 1.9669 4 437 21.2 378.25
## 138 0.35233 0.0 21.89 0 0.6240 6.454 98.4 1.8498 4 437 21.2 394.08
## 139 0.24980 0.0 21.89 0 0.6240 5.857 98.2 1.6686 4 437 21.2 392.04
## 140 0.54452 0.0 21.89 0 0.6240 6.151 97.9 1.6687 4 437 21.2 396.90
## 141 0.29090 0.0 21.89 0 0.6240 6.174 93.6 1.6119 4 437 21.2 388.08
## 142 1.62864 0.0 21.89 0 0.6240 5.019 100.0 1.4394 4 437 21.2 396.90
## 143 3.32105 0.0 19.58 1 0.8710 5.403 100.0 1.3216 5 403 14.7 396.90
## 144 4.09740 0.0 19.58 0 0.8710 5.468 100.0 1.4118 5 403 14.7 396.90
## 145 2.77974 0.0 19.58 0 0.8710 4.903 97.8 1.3459 5 403 14.7 396.90
## 146 2.37934 0.0 19.58 0 0.8710 6.130 100.0 1.4191 5 403 14.7 172.91
## 147 2.15505 0.0 19.58 0 0.8710 5.628 100.0 1.5166 5 403 14.7 169.27
## 148 2.36862 0.0 19.58 0 0.8710 4.926 95.7 1.4608 5 403 14.7 391.71
## 149 2.33099 0.0 19.58 0 0.8710 5.186 93.8 1.5296 5 403 14.7 356.99
## 150 2.73397 0.0 19.58 0 0.8710 5.597 94.9 1.5257 5 403 14.7 351.85
## 151 1.65660 0.0 19.58 0 0.8710 6.122 97.3 1.6180 5 403 14.7 372.80
## 152 1.49632 0.0 19.58 0 0.8710 5.404 100.0 1.5916 5 403 14.7 341.60
## 153 1.12658 0.0 19.58 1 0.8710 5.012 88.0 1.6102 5 403 14.7 343.28
## 154 2.14918 0.0 19.58 0 0.8710 5.709 98.5 1.6232 5 403 14.7 261.95
## 155 1.41385 0.0 19.58 1 0.8710 6.129 96.0 1.7494 5 403 14.7 321.02
## 156 3.53501 0.0 19.58 1 0.8710 6.152 82.6 1.7455 5 403 14.7 88.01
## 157 2.44668 0.0 19.58 0 0.8710 5.272 94.0 1.7364 5 403 14.7 88.63
## 158 1.22358 0.0 19.58 0 0.6050 6.943 97.4 1.8773 5 403 14.7 363.43
## 159 1.34284 0.0 19.58 0 0.6050 6.066 100.0 1.7573 5 403 14.7 353.89
## 160 1.42502 0.0 19.58 0 0.8710 6.510 100.0 1.7659 5 403 14.7 364.31
## 161 1.27346 0.0 19.58 1 0.6050 6.250 92.6 1.7984 5 403 14.7 338.92
## 162 1.46336 0.0 19.58 0 0.6050 7.489 90.8 1.9709 5 403 14.7 374.43
## 163 1.83377 0.0 19.58 1 0.6050 7.802 98.2 2.0407 5 403 14.7 389.61
## 164 1.51902 0.0 19.58 1 0.6050 8.375 93.9 2.1620 5 403 14.7 388.45
## 165 2.24236 0.0 19.58 0 0.6050 5.854 91.8 2.4220 5 403 14.7 395.11
## 166 2.92400 0.0 19.58 0 0.6050 6.101 93.0 2.2834 5 403 14.7 240.16
## 167 2.01019 0.0 19.58 0 0.6050 7.929 96.2 2.0459 5 403 14.7 369.30
## 168 1.80028 0.0 19.58 0 0.6050 5.877 79.2 2.4259 5 403 14.7 227.61
## 169 2.30040 0.0 19.58 0 0.6050 6.319 96.1 2.1000 5 403 14.7 297.09
## 170 2.44953 0.0 19.58 0 0.6050 6.402 95.2 2.2625 5 403 14.7 330.04
## 171 1.20742 0.0 19.58 0 0.6050 5.875 94.6 2.4259 5 403 14.7 292.29
## 172 2.31390 0.0 19.58 0 0.6050 5.880 97.3 2.3887 5 403 14.7 348.13
## 173 0.13914 0.0 4.05 0 0.5100 5.572 88.5 2.5961 5 296 16.6 396.90
## 174 0.09178 0.0 4.05 0 0.5100 6.416 84.1 2.6463 5 296 16.6 395.50
## 175 0.08447 0.0 4.05 0 0.5100 5.859 68.7 2.7019 5 296 16.6 393.23
## 176 0.06664 0.0 4.05 0 0.5100 6.546 33.1 3.1323 5 296 16.6 390.96
## 177 0.07022 0.0 4.05 0 0.5100 6.020 47.2 3.5549 5 296 16.6 393.23
## 178 0.05425 0.0 4.05 0 0.5100 6.315 73.4 3.3175 5 296 16.6 395.60
## 179 0.06642 0.0 4.05 0 0.5100 6.860 74.4 2.9153 5 296 16.6 391.27
## 180 0.05780 0.0 2.46 0 0.4880 6.980 58.4 2.8290 3 193 17.8 396.90
## 181 0.06588 0.0 2.46 0 0.4880 7.765 83.3 2.7410 3 193 17.8 395.56
## 182 0.06888 0.0 2.46 0 0.4880 6.144 62.2 2.5979 3 193 17.8 396.90
## 183 0.09103 0.0 2.46 0 0.4880 7.155 92.2 2.7006 3 193 17.8 394.12
## 184 0.10008 0.0 2.46 0 0.4880 6.563 95.6 2.8470 3 193 17.8 396.90
## 185 0.08308 0.0 2.46 0 0.4880 5.604 89.8 2.9879 3 193 17.8 391.00
## 186 0.06047 0.0 2.46 0 0.4880 6.153 68.8 3.2797 3 193 17.8 387.11
## 187 0.05602 0.0 2.46 0 0.4880 7.831 53.6 3.1992 3 193 17.8 392.63
## 188 0.07875 45.0 3.44 0 0.4370 6.782 41.1 3.7886 5 398 15.2 393.87
## 189 0.12579 45.0 3.44 0 0.4370 6.556 29.1 4.5667 5 398 15.2 382.84
## 190 0.08370 45.0 3.44 0 0.4370 7.185 38.9 4.5667 5 398 15.2 396.90
## 191 0.09068 45.0 3.44 0 0.4370 6.951 21.5 6.4798 5 398 15.2 377.68
## 192 0.06911 45.0 3.44 0 0.4370 6.739 30.8 6.4798 5 398 15.2 389.71
## 193 0.08664 45.0 3.44 0 0.4370 7.178 26.3 6.4798 5 398 15.2 390.49
## 194 0.02187 60.0 2.93 0 0.4010 6.800 9.9 6.2196 1 265 15.6 393.37
## 195 0.01439 60.0 2.93 0 0.4010 6.604 18.8 6.2196 1 265 15.6 376.70
## 196 0.01381 80.0 0.46 0 0.4220 7.875 32.0 5.6484 4 255 14.4 394.23
## 197 0.04011 80.0 1.52 0 0.4040 7.287 34.1 7.3090 2 329 12.6 396.90
## 198 0.04666 80.0 1.52 0 0.4040 7.107 36.6 7.3090 2 329 12.6 354.31
## 199 0.03768 80.0 1.52 0 0.4040 7.274 38.3 7.3090 2 329 12.6 392.20
## 200 0.03150 95.0 1.47 0 0.4030 6.975 15.3 7.6534 3 402 17.0 396.90
## 201 0.01778 95.0 1.47 0 0.4030 7.135 13.9 7.6534 3 402 17.0 384.30
## 202 0.03445 82.5 2.03 0 0.4150 6.162 38.4 6.2700 2 348 14.7 393.77
## 203 0.02177 82.5 2.03 0 0.4150 7.610 15.7 6.2700 2 348 14.7 395.38
## 204 0.03510 95.0 2.68 0 0.4161 7.853 33.2 5.1180 4 224 14.7 392.78
## 205 0.02009 95.0 2.68 0 0.4161 8.034 31.9 5.1180 4 224 14.7 390.55
## 206 0.13642 0.0 10.59 0 0.4890 5.891 22.3 3.9454 4 277 18.6 396.90
## 207 0.22969 0.0 10.59 0 0.4890 6.326 52.5 4.3549 4 277 18.6 394.87
## 208 0.25199 0.0 10.59 0 0.4890 5.783 72.7 4.3549 4 277 18.6 389.43
## 209 0.13587 0.0 10.59 1 0.4890 6.064 59.1 4.2392 4 277 18.6 381.32
## 210 0.43571 0.0 10.59 1 0.4890 5.344 100.0 3.8750 4 277 18.6 396.90
## 211 0.17446 0.0 10.59 1 0.4890 5.960 92.1 3.8771 4 277 18.6 393.25
## 212 0.37578 0.0 10.59 1 0.4890 5.404 88.6 3.6650 4 277 18.6 395.24
## 213 0.21719 0.0 10.59 1 0.4890 5.807 53.8 3.6526 4 277 18.6 390.94
## 214 0.14052 0.0 10.59 0 0.4890 6.375 32.3 3.9454 4 277 18.6 385.81
## 215 0.28955 0.0 10.59 0 0.4890 5.412 9.8 3.5875 4 277 18.6 348.93
## 216 0.19802 0.0 10.59 0 0.4890 6.182 42.4 3.9454 4 277 18.6 393.63
## 217 0.04560 0.0 13.89 1 0.5500 5.888 56.0 3.1121 5 276 16.4 392.80
## 218 0.07013 0.0 13.89 0 0.5500 6.642 85.1 3.4211 5 276 16.4 392.78
## 219 0.11069 0.0 13.89 1 0.5500 5.951 93.8 2.8893 5 276 16.4 396.90
## 220 0.11425 0.0 13.89 1 0.5500 6.373 92.4 3.3633 5 276 16.4 393.74
## 221 0.35809 0.0 6.20 1 0.5070 6.951 88.5 2.8617 8 307 17.4 391.70
## 222 0.40771 0.0 6.20 1 0.5070 6.164 91.3 3.0480 8 307 17.4 395.24
## 223 0.62356 0.0 6.20 1 0.5070 6.879 77.7 3.2721 8 307 17.4 390.39
## 224 0.61470 0.0 6.20 0 0.5070 6.618 80.8 3.2721 8 307 17.4 396.90
## 225 0.31533 0.0 6.20 0 0.5040 8.266 78.3 2.8944 8 307 17.4 385.05
## 226 0.52693 0.0 6.20 0 0.5040 8.725 83.0 2.8944 8 307 17.4 382.00
## 227 0.38214 0.0 6.20 0 0.5040 8.040 86.5 3.2157 8 307 17.4 387.38
## 228 0.41238 0.0 6.20 0 0.5040 7.163 79.9 3.2157 8 307 17.4 372.08
## 229 0.29819 0.0 6.20 0 0.5040 7.686 17.0 3.3751 8 307 17.4 377.51
## 230 0.44178 0.0 6.20 0 0.5040 6.552 21.4 3.3751 8 307 17.4 380.34
## 231 0.53700 0.0 6.20 0 0.5040 5.981 68.1 3.6715 8 307 17.4 378.35
## 232 0.46296 0.0 6.20 0 0.5040 7.412 76.9 3.6715 8 307 17.4 376.14
## 233 0.57529 0.0 6.20 0 0.5070 8.337 73.3 3.8384 8 307 17.4 385.91
## 234 0.33147 0.0 6.20 0 0.5070 8.247 70.4 3.6519 8 307 17.4 378.95
## 235 0.44791 0.0 6.20 1 0.5070 6.726 66.5 3.6519 8 307 17.4 360.20
## 236 0.33045 0.0 6.20 0 0.5070 6.086 61.5 3.6519 8 307 17.4 376.75
## 237 0.52058 0.0 6.20 1 0.5070 6.631 76.5 4.1480 8 307 17.4 388.45
## 238 0.51183 0.0 6.20 0 0.5070 7.358 71.6 4.1480 8 307 17.4 390.07
## 239 0.08244 30.0 4.93 0 0.4280 6.481 18.5 6.1899 6 300 16.6 379.41
## 240 0.09252 30.0 4.93 0 0.4280 6.606 42.2 6.1899 6 300 16.6 383.78
## 241 0.11329 30.0 4.93 0 0.4280 6.897 54.3 6.3361 6 300 16.6 391.25
## 242 0.10612 30.0 4.93 0 0.4280 6.095 65.1 6.3361 6 300 16.6 394.62
## 243 0.10290 30.0 4.93 0 0.4280 6.358 52.9 7.0355 6 300 16.6 372.75
## 244 0.12757 30.0 4.93 0 0.4280 6.393 7.8 7.0355 6 300 16.6 374.71
## 245 0.20608 22.0 5.86 0 0.4310 5.593 76.5 7.9549 7 330 19.1 372.49
## 246 0.19133 22.0 5.86 0 0.4310 5.605 70.2 7.9549 7 330 19.1 389.13
## 247 0.33983 22.0 5.86 0 0.4310 6.108 34.9 8.0555 7 330 19.1 390.18
## 248 0.19657 22.0 5.86 0 0.4310 6.226 79.2 8.0555 7 330 19.1 376.14
## 249 0.16439 22.0 5.86 0 0.4310 6.433 49.1 7.8265 7 330 19.1 374.71
## 250 0.19073 22.0 5.86 0 0.4310 6.718 17.5 7.8265 7 330 19.1 393.74
## 251 0.14030 22.0 5.86 0 0.4310 6.487 13.0 7.3967 7 330 19.1 396.28
## 252 0.21409 22.0 5.86 0 0.4310 6.438 8.9 7.3967 7 330 19.1 377.07
## 253 0.08221 22.0 5.86 0 0.4310 6.957 6.8 8.9067 7 330 19.1 386.09
## 254 0.36894 22.0 5.86 0 0.4310 8.259 8.4 8.9067 7 330 19.1 396.90
## 255 0.04819 80.0 3.64 0 0.3920 6.108 32.0 9.2203 1 315 16.4 392.89
## 256 0.03548 80.0 3.64 0 0.3920 5.876 19.1 9.2203 1 315 16.4 395.18
## 257 0.01538 90.0 3.75 0 0.3940 7.454 34.2 6.3361 3 244 15.9 386.34
## 258 0.61154 20.0 3.97 0 0.6470 8.704 86.9 1.8010 5 264 13.0 389.70
## 259 0.66351 20.0 3.97 0 0.6470 7.333 100.0 1.8946 5 264 13.0 383.29
## 260 0.65665 20.0 3.97 0 0.6470 6.842 100.0 2.0107 5 264 13.0 391.93
## 261 0.54011 20.0 3.97 0 0.6470 7.203 81.8 2.1121 5 264 13.0 392.80
## 262 0.53412 20.0 3.97 0 0.6470 7.520 89.4 2.1398 5 264 13.0 388.37
## 263 0.52014 20.0 3.97 0 0.6470 8.398 91.5 2.2885 5 264 13.0 386.86
## 264 0.82526 20.0 3.97 0 0.6470 7.327 94.5 2.0788 5 264 13.0 393.42
## 265 0.55007 20.0 3.97 0 0.6470 7.206 91.6 1.9301 5 264 13.0 387.89
## 266 0.76162 20.0 3.97 0 0.6470 5.560 62.8 1.9865 5 264 13.0 392.40
## 267 0.78570 20.0 3.97 0 0.6470 7.014 84.6 2.1329 5 264 13.0 384.07
## 268 0.57834 20.0 3.97 0 0.5750 8.297 67.0 2.4216 5 264 13.0 384.54
## 269 0.54050 20.0 3.97 0 0.5750 7.470 52.6 2.8720 5 264 13.0 390.30
## 270 0.09065 20.0 6.96 1 0.4640 5.920 61.5 3.9175 3 223 18.6 391.34
## 271 0.29916 20.0 6.96 0 0.4640 5.856 42.1 4.4290 3 223 18.6 388.65
## 272 0.16211 20.0 6.96 0 0.4640 6.240 16.3 4.4290 3 223 18.6 396.90
## 273 0.11460 20.0 6.96 0 0.4640 6.538 58.7 3.9175 3 223 18.6 394.96
## 274 0.22188 20.0 6.96 1 0.4640 7.691 51.8 4.3665 3 223 18.6 390.77
## 275 0.05644 40.0 6.41 1 0.4470 6.758 32.9 4.0776 4 254 17.6 396.90
## 276 0.09604 40.0 6.41 0 0.4470 6.854 42.8 4.2673 4 254 17.6 396.90
## 277 0.10469 40.0 6.41 1 0.4470 7.267 49.0 4.7872 4 254 17.6 389.25
## 278 0.06127 40.0 6.41 1 0.4470 6.826 27.6 4.8628 4 254 17.6 393.45
## 279 0.07978 40.0 6.41 0 0.4470 6.482 32.1 4.1403 4 254 17.6 396.90
## 280 0.21038 20.0 3.33 0 0.4429 6.812 32.2 4.1007 5 216 14.9 396.90
## 281 0.03578 20.0 3.33 0 0.4429 7.820 64.5 4.6947 5 216 14.9 387.31
## 282 0.03705 20.0 3.33 0 0.4429 6.968 37.2 5.2447 5 216 14.9 392.23
## 283 0.06129 20.0 3.33 1 0.4429 7.645 49.7 5.2119 5 216 14.9 377.07
## 284 0.01501 90.0 1.21 1 0.4010 7.923 24.8 5.8850 1 198 13.6 395.52
## 285 0.00906 90.0 2.97 0 0.4000 7.088 20.8 7.3073 1 285 15.3 394.72
## 286 0.01096 55.0 2.25 0 0.3890 6.453 31.9 7.3073 1 300 15.3 394.72
## 287 0.01965 80.0 1.76 0 0.3850 6.230 31.5 9.0892 1 241 18.2 341.60
## 288 0.03871 52.5 5.32 0 0.4050 6.209 31.3 7.3172 6 293 16.6 396.90
## 289 0.04590 52.5 5.32 0 0.4050 6.315 45.6 7.3172 6 293 16.6 396.90
## 290 0.04297 52.5 5.32 0 0.4050 6.565 22.9 7.3172 6 293 16.6 371.72
## 291 0.03502 80.0 4.95 0 0.4110 6.861 27.9 5.1167 4 245 19.2 396.90
## 292 0.07886 80.0 4.95 0 0.4110 7.148 27.7 5.1167 4 245 19.2 396.90
## 293 0.03615 80.0 4.95 0 0.4110 6.630 23.4 5.1167 4 245 19.2 396.90
## 294 0.08265 0.0 13.92 0 0.4370 6.127 18.4 5.5027 4 289 16.0 396.90
## 295 0.08199 0.0 13.92 0 0.4370 6.009 42.3 5.5027 4 289 16.0 396.90
## 296 0.12932 0.0 13.92 0 0.4370 6.678 31.1 5.9604 4 289 16.0 396.90
## 297 0.05372 0.0 13.92 0 0.4370 6.549 51.0 5.9604 4 289 16.0 392.85
## 298 0.14103 0.0 13.92 0 0.4370 5.790 58.0 6.3200 4 289 16.0 396.90
## 299 0.06466 70.0 2.24 0 0.4000 6.345 20.1 7.8278 5 358 14.8 368.24
## 300 0.05561 70.0 2.24 0 0.4000 7.041 10.0 7.8278 5 358 14.8 371.58
## 301 0.04417 70.0 2.24 0 0.4000 6.871 47.4 7.8278 5 358 14.8 390.86
## 302 0.03537 34.0 6.09 0 0.4330 6.590 40.4 5.4917 7 329 16.1 395.75
## 303 0.09266 34.0 6.09 0 0.4330 6.495 18.4 5.4917 7 329 16.1 383.61
## 304 0.10000 34.0 6.09 0 0.4330 6.982 17.7 5.4917 7 329 16.1 390.43
## 305 0.05515 33.0 2.18 0 0.4720 7.236 41.1 4.0220 7 222 18.4 393.68
## 306 0.05479 33.0 2.18 0 0.4720 6.616 58.1 3.3700 7 222 18.4 393.36
## 307 0.07503 33.0 2.18 0 0.4720 7.420 71.9 3.0992 7 222 18.4 396.90
## 308 0.04932 33.0 2.18 0 0.4720 6.849 70.3 3.1827 7 222 18.4 396.90
## 309 0.49298 0.0 9.90 0 0.5440 6.635 82.5 3.3175 4 304 18.4 396.90
## 310 0.34940 0.0 9.90 0 0.5440 5.972 76.7 3.1025 4 304 18.4 396.24
## 311 2.63548 0.0 9.90 0 0.5440 4.973 37.8 2.5194 4 304 18.4 350.45
## 312 0.79041 0.0 9.90 0 0.5440 6.122 52.8 2.6403 4 304 18.4 396.90
## 313 0.26169 0.0 9.90 0 0.5440 6.023 90.4 2.8340 4 304 18.4 396.30
## 314 0.26938 0.0 9.90 0 0.5440 6.266 82.8 3.2628 4 304 18.4 393.39
## 315 0.36920 0.0 9.90 0 0.5440 6.567 87.3 3.6023 4 304 18.4 395.69
## 316 0.25356 0.0 9.90 0 0.5440 5.705 77.7 3.9450 4 304 18.4 396.42
## 317 0.31827 0.0 9.90 0 0.5440 5.914 83.2 3.9986 4 304 18.4 390.70
## 318 0.24522 0.0 9.90 0 0.5440 5.782 71.7 4.0317 4 304 18.4 396.90
## 319 0.40202 0.0 9.90 0 0.5440 6.382 67.2 3.5325 4 304 18.4 395.21
## 320 0.47547 0.0 9.90 0 0.5440 6.113 58.8 4.0019 4 304 18.4 396.23
## 321 0.16760 0.0 7.38 0 0.4930 6.426 52.3 4.5404 5 287 19.6 396.90
## 322 0.18159 0.0 7.38 0 0.4930 6.376 54.3 4.5404 5 287 19.6 396.90
## 323 0.35114 0.0 7.38 0 0.4930 6.041 49.9 4.7211 5 287 19.6 396.90
## 324 0.28392 0.0 7.38 0 0.4930 5.708 74.3 4.7211 5 287 19.6 391.13
## 325 0.34109 0.0 7.38 0 0.4930 6.415 40.1 4.7211 5 287 19.6 396.90
## 326 0.19186 0.0 7.38 0 0.4930 6.431 14.7 5.4159 5 287 19.6 393.68
## 327 0.30347 0.0 7.38 0 0.4930 6.312 28.9 5.4159 5 287 19.6 396.90
## 328 0.24103 0.0 7.38 0 0.4930 6.083 43.7 5.4159 5 287 19.6 396.90
## 329 0.06617 0.0 3.24 0 0.4600 5.868 25.8 5.2146 4 430 16.9 382.44
## 330 0.06724 0.0 3.24 0 0.4600 6.333 17.2 5.2146 4 430 16.9 375.21
## 331 0.04544 0.0 3.24 0 0.4600 6.144 32.2 5.8736 4 430 16.9 368.57
## 332 0.05023 35.0 6.06 0 0.4379 5.706 28.4 6.6407 1 304 16.9 394.02
## 333 0.03466 35.0 6.06 0 0.4379 6.031 23.3 6.6407 1 304 16.9 362.25
## 334 0.05083 0.0 5.19 0 0.5150 6.316 38.1 6.4584 5 224 20.2 389.71
## 335 0.03738 0.0 5.19 0 0.5150 6.310 38.5 6.4584 5 224 20.2 389.40
## 336 0.03961 0.0 5.19 0 0.5150 6.037 34.5 5.9853 5 224 20.2 396.90
## 337 0.03427 0.0 5.19 0 0.5150 5.869 46.3 5.2311 5 224 20.2 396.90
## 338 0.03041 0.0 5.19 0 0.5150 5.895 59.6 5.6150 5 224 20.2 394.81
## 339 0.03306 0.0 5.19 0 0.5150 6.059 37.3 4.8122 5 224 20.2 396.14
## 340 0.05497 0.0 5.19 0 0.5150 5.985 45.4 4.8122 5 224 20.2 396.90
## 341 0.06151 0.0 5.19 0 0.5150 5.968 58.5 4.8122 5 224 20.2 396.90
## 342 0.01301 35.0 1.52 0 0.4420 7.241 49.3 7.0379 1 284 15.5 394.74
## 343 0.02498 0.0 1.89 0 0.5180 6.540 59.7 6.2669 1 422 15.9 389.96
## 344 0.02543 55.0 3.78 0 0.4840 6.696 56.4 5.7321 5 370 17.6 396.90
## 345 0.03049 55.0 3.78 0 0.4840 6.874 28.1 6.4654 5 370 17.6 387.97
## 346 0.03113 0.0 4.39 0 0.4420 6.014 48.5 8.0136 3 352 18.8 385.64
## 347 0.06162 0.0 4.39 0 0.4420 5.898 52.3 8.0136 3 352 18.8 364.61
## 348 0.01870 85.0 4.15 0 0.4290 6.516 27.7 8.5353 4 351 17.9 392.43
## 349 0.01501 80.0 2.01 0 0.4350 6.635 29.7 8.3440 4 280 17.0 390.94
## 350 0.02899 40.0 1.25 0 0.4290 6.939 34.5 8.7921 1 335 19.7 389.85
## 351 0.06211 40.0 1.25 0 0.4290 6.490 44.4 8.7921 1 335 19.7 396.90
## 352 0.07950 60.0 1.69 0 0.4110 6.579 35.9 10.7103 4 411 18.3 370.78
## 353 0.07244 60.0 1.69 0 0.4110 5.884 18.5 10.7103 4 411 18.3 392.33
## 354 0.01709 90.0 2.02 0 0.4100 6.728 36.1 12.1265 5 187 17.0 384.46
## 355 0.04301 80.0 1.91 0 0.4130 5.663 21.9 10.5857 4 334 22.0 382.80
## 356 0.10659 80.0 1.91 0 0.4130 5.936 19.5 10.5857 4 334 22.0 376.04
## 357 8.98296 0.0 18.10 1 0.7700 6.212 97.4 2.1222 24 666 20.2 377.73
## 358 3.84970 0.0 18.10 1 0.7700 6.395 91.0 2.5052 24 666 20.2 391.34
## 359 5.20177 0.0 18.10 1 0.7700 6.127 83.4 2.7227 24 666 20.2 395.43
## 360 4.26131 0.0 18.10 0 0.7700 6.112 81.3 2.5091 24 666 20.2 390.74
## 361 4.54192 0.0 18.10 0 0.7700 6.398 88.0 2.5182 24 666 20.2 374.56
## 362 3.83684 0.0 18.10 0 0.7700 6.251 91.1 2.2955 24 666 20.2 350.65
## 363 3.67822 0.0 18.10 0 0.7700 5.362 96.2 2.1036 24 666 20.2 380.79
## 364 4.22239 0.0 18.10 1 0.7700 5.803 89.0 1.9047 24 666 20.2 353.04
## 365 3.47428 0.0 18.10 1 0.7180 8.780 82.9 1.9047 24 666 20.2 354.55
## 366 4.55587 0.0 18.10 0 0.7180 3.561 87.9 1.6132 24 666 20.2 354.70
## 367 3.69695 0.0 18.10 0 0.7180 4.963 91.4 1.7523 24 666 20.2 316.03
## 368 13.52220 0.0 18.10 0 0.6310 3.863 100.0 1.5106 24 666 20.2 131.42
## 369 4.89822 0.0 18.10 0 0.6310 4.970 100.0 1.3325 24 666 20.2 375.52
## 370 5.66998 0.0 18.10 1 0.6310 6.683 96.8 1.3567 24 666 20.2 375.33
## 371 6.53876 0.0 18.10 1 0.6310 7.016 97.5 1.2024 24 666 20.2 392.05
## 372 9.23230 0.0 18.10 0 0.6310 6.216 100.0 1.1691 24 666 20.2 366.15
## 373 8.26725 0.0 18.10 1 0.6680 5.875 89.6 1.1296 24 666 20.2 347.88
## 374 11.10810 0.0 18.10 0 0.6680 4.906 100.0 1.1742 24 666 20.2 396.90
## 375 18.49820 0.0 18.10 0 0.6680 4.138 100.0 1.1370 24 666 20.2 396.90
## 376 19.60910 0.0 18.10 0 0.6710 7.313 97.9 1.3163 24 666 20.2 396.90
## 377 15.28800 0.0 18.10 0 0.6710 6.649 93.3 1.3449 24 666 20.2 363.02
## 378 9.82349 0.0 18.10 0 0.6710 6.794 98.8 1.3580 24 666 20.2 396.90
## 379 23.64820 0.0 18.10 0 0.6710 6.380 96.2 1.3861 24 666 20.2 396.90
## 380 17.86670 0.0 18.10 0 0.6710 6.223 100.0 1.3861 24 666 20.2 393.74
## 381 88.97620 0.0 18.10 0 0.6710 6.968 91.9 1.4165 24 666 20.2 396.90
## 382 15.87440 0.0 18.10 0 0.6710 6.545 99.1 1.5192 24 666 20.2 396.90
## 383 9.18702 0.0 18.10 0 0.7000 5.536 100.0 1.5804 24 666 20.2 396.90
## 384 7.99248 0.0 18.10 0 0.7000 5.520 100.0 1.5331 24 666 20.2 396.90
## 385 20.08490 0.0 18.10 0 0.7000 4.368 91.2 1.4395 24 666 20.2 285.83
## 386 16.81180 0.0 18.10 0 0.7000 5.277 98.1 1.4261 24 666 20.2 396.90
## 387 24.39380 0.0 18.10 0 0.7000 4.652 100.0 1.4672 24 666 20.2 396.90
## 388 22.59710 0.0 18.10 0 0.7000 5.000 89.5 1.5184 24 666 20.2 396.90
## 389 14.33370 0.0 18.10 0 0.7000 4.880 100.0 1.5895 24 666 20.2 372.92
## 390 8.15174 0.0 18.10 0 0.7000 5.390 98.9 1.7281 24 666 20.2 396.90
## 391 6.96215 0.0 18.10 0 0.7000 5.713 97.0 1.9265 24 666 20.2 394.43
## 392 5.29305 0.0 18.10 0 0.7000 6.051 82.5 2.1678 24 666 20.2 378.38
## 393 11.57790 0.0 18.10 0 0.7000 5.036 97.0 1.7700 24 666 20.2 396.90
## 394 8.64476 0.0 18.10 0 0.6930 6.193 92.6 1.7912 24 666 20.2 396.90
## 395 13.35980 0.0 18.10 0 0.6930 5.887 94.7 1.7821 24 666 20.2 396.90
## 396 8.71675 0.0 18.10 0 0.6930 6.471 98.8 1.7257 24 666 20.2 391.98
## 397 5.87205 0.0 18.10 0 0.6930 6.405 96.0 1.6768 24 666 20.2 396.90
## 398 7.67202 0.0 18.10 0 0.6930 5.747 98.9 1.6334 24 666 20.2 393.10
## 399 38.35180 0.0 18.10 0 0.6930 5.453 100.0 1.4896 24 666 20.2 396.90
## 400 9.91655 0.0 18.10 0 0.6930 5.852 77.8 1.5004 24 666 20.2 338.16
## 401 25.04610 0.0 18.10 0 0.6930 5.987 100.0 1.5888 24 666 20.2 396.90
## 402 14.23620 0.0 18.10 0 0.6930 6.343 100.0 1.5741 24 666 20.2 396.90
## 403 9.59571 0.0 18.10 0 0.6930 6.404 100.0 1.6390 24 666 20.2 376.11
## 404 24.80170 0.0 18.10 0 0.6930 5.349 96.0 1.7028 24 666 20.2 396.90
## 405 41.52920 0.0 18.10 0 0.6930 5.531 85.4 1.6074 24 666 20.2 329.46
## 406 67.92080 0.0 18.10 0 0.6930 5.683 100.0 1.4254 24 666 20.2 384.97
## 407 20.71620 0.0 18.10 0 0.6590 4.138 100.0 1.1781 24 666 20.2 370.22
## 408 11.95110 0.0 18.10 0 0.6590 5.608 100.0 1.2852 24 666 20.2 332.09
## 409 7.40389 0.0 18.10 0 0.5970 5.617 97.9 1.4547 24 666 20.2 314.64
## 410 14.43830 0.0 18.10 0 0.5970 6.852 100.0 1.4655 24 666 20.2 179.36
## 411 51.13580 0.0 18.10 0 0.5970 5.757 100.0 1.4130 24 666 20.2 2.60
## 412 14.05070 0.0 18.10 0 0.5970 6.657 100.0 1.5275 24 666 20.2 35.05
## 413 18.81100 0.0 18.10 0 0.5970 4.628 100.0 1.5539 24 666 20.2 28.79
## 414 28.65580 0.0 18.10 0 0.5970 5.155 100.0 1.5894 24 666 20.2 210.97
## 415 45.74610 0.0 18.10 0 0.6930 4.519 100.0 1.6582 24 666 20.2 88.27
## 416 18.08460 0.0 18.10 0 0.6790 6.434 100.0 1.8347 24 666 20.2 27.25
## 417 10.83420 0.0 18.10 0 0.6790 6.782 90.8 1.8195 24 666 20.2 21.57
## 418 25.94060 0.0 18.10 0 0.6790 5.304 89.1 1.6475 24 666 20.2 127.36
## 419 73.53410 0.0 18.10 0 0.6790 5.957 100.0 1.8026 24 666 20.2 16.45
## 420 11.81230 0.0 18.10 0 0.7180 6.824 76.5 1.7940 24 666 20.2 48.45
## 421 11.08740 0.0 18.10 0 0.7180 6.411 100.0 1.8589 24 666 20.2 318.75
## 422 7.02259 0.0 18.10 0 0.7180 6.006 95.3 1.8746 24 666 20.2 319.98
## 423 12.04820 0.0 18.10 0 0.6140 5.648 87.6 1.9512 24 666 20.2 291.55
## 424 7.05042 0.0 18.10 0 0.6140 6.103 85.1 2.0218 24 666 20.2 2.52
## 425 8.79212 0.0 18.10 0 0.5840 5.565 70.6 2.0635 24 666 20.2 3.65
## 426 15.86030 0.0 18.10 0 0.6790 5.896 95.4 1.9096 24 666 20.2 7.68
## 427 12.24720 0.0 18.10 0 0.5840 5.837 59.7 1.9976 24 666 20.2 24.65
## 428 37.66190 0.0 18.10 0 0.6790 6.202 78.7 1.8629 24 666 20.2 18.82
## 429 7.36711 0.0 18.10 0 0.6790 6.193 78.1 1.9356 24 666 20.2 96.73
## 430 9.33889 0.0 18.10 0 0.6790 6.380 95.6 1.9682 24 666 20.2 60.72
## 431 8.49213 0.0 18.10 0 0.5840 6.348 86.1 2.0527 24 666 20.2 83.45
## 432 10.06230 0.0 18.10 0 0.5840 6.833 94.3 2.0882 24 666 20.2 81.33
## 433 6.44405 0.0 18.10 0 0.5840 6.425 74.8 2.2004 24 666 20.2 97.95
## 434 5.58107 0.0 18.10 0 0.7130 6.436 87.9 2.3158 24 666 20.2 100.19
## 435 13.91340 0.0 18.10 0 0.7130 6.208 95.0 2.2222 24 666 20.2 100.63
## 436 11.16040 0.0 18.10 0 0.7400 6.629 94.6 2.1247 24 666 20.2 109.85
## 437 14.42080 0.0 18.10 0 0.7400 6.461 93.3 2.0026 24 666 20.2 27.49
## 438 15.17720 0.0 18.10 0 0.7400 6.152 100.0 1.9142 24 666 20.2 9.32
## 439 13.67810 0.0 18.10 0 0.7400 5.935 87.9 1.8206 24 666 20.2 68.95
## 440 9.39063 0.0 18.10 0 0.7400 5.627 93.9 1.8172 24 666 20.2 396.90
## 441 22.05110 0.0 18.10 0 0.7400 5.818 92.4 1.8662 24 666 20.2 391.45
## 442 9.72418 0.0 18.10 0 0.7400 6.406 97.2 2.0651 24 666 20.2 385.96
## 443 5.66637 0.0 18.10 0 0.7400 6.219 100.0 2.0048 24 666 20.2 395.69
## 444 9.96654 0.0 18.10 0 0.7400 6.485 100.0 1.9784 24 666 20.2 386.73
## 445 12.80230 0.0 18.10 0 0.7400 5.854 96.6 1.8956 24 666 20.2 240.52
## 446 10.67180 0.0 18.10 0 0.7400 6.459 94.8 1.9879 24 666 20.2 43.06
## 447 6.28807 0.0 18.10 0 0.7400 6.341 96.4 2.0720 24 666 20.2 318.01
## 448 9.92485 0.0 18.10 0 0.7400 6.251 96.6 2.1980 24 666 20.2 388.52
## 449 9.32909 0.0 18.10 0 0.7130 6.185 98.7 2.2616 24 666 20.2 396.90
## 450 7.52601 0.0 18.10 0 0.7130 6.417 98.3 2.1850 24 666 20.2 304.21
## 451 6.71772 0.0 18.10 0 0.7130 6.749 92.6 2.3236 24 666 20.2 0.32
## 452 5.44114 0.0 18.10 0 0.7130 6.655 98.2 2.3552 24 666 20.2 355.29
## 453 5.09017 0.0 18.10 0 0.7130 6.297 91.8 2.3682 24 666 20.2 385.09
## 454 8.24809 0.0 18.10 0 0.7130 7.393 99.3 2.4527 24 666 20.2 375.87
## 455 9.51363 0.0 18.10 0 0.7130 6.728 94.1 2.4961 24 666 20.2 6.68
## 456 4.75237 0.0 18.10 0 0.7130 6.525 86.5 2.4358 24 666 20.2 50.92
## 457 4.66883 0.0 18.10 0 0.7130 5.976 87.9 2.5806 24 666 20.2 10.48
## 458 8.20058 0.0 18.10 0 0.7130 5.936 80.3 2.7792 24 666 20.2 3.50
## 459 7.75223 0.0 18.10 0 0.7130 6.301 83.7 2.7831 24 666 20.2 272.21
## 460 6.80117 0.0 18.10 0 0.7130 6.081 84.4 2.7175 24 666 20.2 396.90
## 461 4.81213 0.0 18.10 0 0.7130 6.701 90.0 2.5975 24 666 20.2 255.23
## 462 3.69311 0.0 18.10 0 0.7130 6.376 88.4 2.5671 24 666 20.2 391.43
## 463 6.65492 0.0 18.10 0 0.7130 6.317 83.0 2.7344 24 666 20.2 396.90
## 464 5.82115 0.0 18.10 0 0.7130 6.513 89.9 2.8016 24 666 20.2 393.82
## 465 7.83932 0.0 18.10 0 0.6550 6.209 65.4 2.9634 24 666 20.2 396.90
## 466 3.16360 0.0 18.10 0 0.6550 5.759 48.2 3.0665 24 666 20.2 334.40
## 467 3.77498 0.0 18.10 0 0.6550 5.952 84.7 2.8715 24 666 20.2 22.01
## 468 4.42228 0.0 18.10 0 0.5840 6.003 94.5 2.5403 24 666 20.2 331.29
## 469 15.57570 0.0 18.10 0 0.5800 5.926 71.0 2.9084 24 666 20.2 368.74
## 470 13.07510 0.0 18.10 0 0.5800 5.713 56.7 2.8237 24 666 20.2 396.90
## 471 4.34879 0.0 18.10 0 0.5800 6.167 84.0 3.0334 24 666 20.2 396.90
## 472 4.03841 0.0 18.10 0 0.5320 6.229 90.7 3.0993 24 666 20.2 395.33
## 473 3.56868 0.0 18.10 0 0.5800 6.437 75.0 2.8965 24 666 20.2 393.37
## 474 4.64689 0.0 18.10 0 0.6140 6.980 67.6 2.5329 24 666 20.2 374.68
## 475 8.05579 0.0 18.10 0 0.5840 5.427 95.4 2.4298 24 666 20.2 352.58
## 476 6.39312 0.0 18.10 0 0.5840 6.162 97.4 2.2060 24 666 20.2 302.76
## 477 4.87141 0.0 18.10 0 0.6140 6.484 93.6 2.3053 24 666 20.2 396.21
## 478 15.02340 0.0 18.10 0 0.6140 5.304 97.3 2.1007 24 666 20.2 349.48
## 479 10.23300 0.0 18.10 0 0.6140 6.185 96.7 2.1705 24 666 20.2 379.70
## 480 14.33370 0.0 18.10 0 0.6140 6.229 88.0 1.9512 24 666 20.2 383.32
## 481 5.82401 0.0 18.10 0 0.5320 6.242 64.7 3.4242 24 666 20.2 396.90
## 482 5.70818 0.0 18.10 0 0.5320 6.750 74.9 3.3317 24 666 20.2 393.07
## 483 5.73116 0.0 18.10 0 0.5320 7.061 77.0 3.4106 24 666 20.2 395.28
## 484 2.81838 0.0 18.10 0 0.5320 5.762 40.3 4.0983 24 666 20.2 392.92
## 485 2.37857 0.0 18.10 0 0.5830 5.871 41.9 3.7240 24 666 20.2 370.73
## 486 3.67367 0.0 18.10 0 0.5830 6.312 51.9 3.9917 24 666 20.2 388.62
## 487 5.69175 0.0 18.10 0 0.5830 6.114 79.8 3.5459 24 666 20.2 392.68
## 488 4.83567 0.0 18.10 0 0.5830 5.905 53.2 3.1523 24 666 20.2 388.22
## 489 0.15086 0.0 27.74 0 0.6090 5.454 92.7 1.8209 4 711 20.1 395.09
## 490 0.18337 0.0 27.74 0 0.6090 5.414 98.3 1.7554 4 711 20.1 344.05
## 491 0.20746 0.0 27.74 0 0.6090 5.093 98.0 1.8226 4 711 20.1 318.43
## 492 0.10574 0.0 27.74 0 0.6090 5.983 98.8 1.8681 4 711 20.1 390.11
## 493 0.11132 0.0 27.74 0 0.6090 5.983 83.5 2.1099 4 711 20.1 396.90
## 494 0.17331 0.0 9.69 0 0.5850 5.707 54.0 2.3817 6 391 19.2 396.90
## 495 0.27957 0.0 9.69 0 0.5850 5.926 42.6 2.3817 6 391 19.2 396.90
## 496 0.17899 0.0 9.69 0 0.5850 5.670 28.8 2.7986 6 391 19.2 393.29
## 497 0.28960 0.0 9.69 0 0.5850 5.390 72.9 2.7986 6 391 19.2 396.90
## 498 0.26838 0.0 9.69 0 0.5850 5.794 70.6 2.8927 6 391 19.2 396.90
## 499 0.23912 0.0 9.69 0 0.5850 6.019 65.3 2.4091 6 391 19.2 396.90
## 500 0.17783 0.0 9.69 0 0.5850 5.569 73.5 2.3999 6 391 19.2 395.77
## 501 0.22438 0.0 9.69 0 0.5850 6.027 79.7 2.4982 6 391 19.2 396.90
## 502 0.06263 0.0 11.93 0 0.5730 6.593 69.1 2.4786 1 273 21.0 391.99
## 503 0.04527 0.0 11.93 0 0.5730 6.120 76.7 2.2875 1 273 21.0 396.90
## 504 0.06076 0.0 11.93 0 0.5730 6.976 91.0 2.1675 1 273 21.0 396.90
## 505 0.10959 0.0 11.93 0 0.5730 6.794 89.3 2.3889 1 273 21.0 393.45
## 506 0.04741 0.0 11.93 0 0.5730 6.030 80.8 2.5050 1 273 21.0 396.90
## lstat medv
## 1 4.98 24.0
## 2 9.14 21.6
## 3 4.03 34.7
## 4 2.94 33.4
## 5 5.33 36.2
## 6 5.21 28.7
## 7 12.43 22.9
## 8 19.15 27.1
## 9 29.93 16.5
## 10 17.10 18.9
## 11 20.45 15.0
## 12 13.27 18.9
## 13 15.71 21.7
## 14 8.26 20.4
## 15 10.26 18.2
## 16 8.47 19.9
## 17 6.58 23.1
## 18 14.67 17.5
## 19 11.69 20.2
## 20 11.28 18.2
## 21 21.02 13.6
## 22 13.83 19.6
## 23 18.72 15.2
## 24 19.88 14.5
## 25 16.30 15.6
## 26 16.51 13.9
## 27 14.81 16.6
## 28 17.28 14.8
## 29 12.80 18.4
## 30 11.98 21.0
## 31 22.60 12.7
## 32 13.04 14.5
## 33 27.71 13.2
## 34 18.35 13.1
## 35 20.34 13.5
## 36 9.68 18.9
## 37 11.41 20.0
## 38 8.77 21.0
## 39 10.13 24.7
## 40 4.32 30.8
## 41 1.98 34.9
## 42 4.84 26.6
## 43 5.81 25.3
## 44 7.44 24.7
## 45 9.55 21.2
## 46 10.21 19.3
## 47 14.15 20.0
## 48 18.80 16.6
## 49 30.81 14.4
## 50 16.20 19.4
## 51 13.45 19.7
## 52 9.43 20.5
## 53 5.28 25.0
## 54 8.43 23.4
## 55 14.80 18.9
## 56 4.81 35.4
## 57 5.77 24.7
## 58 3.95 31.6
## 59 6.86 23.3
## 60 9.22 19.6
## 61 13.15 18.7
## 62 14.44 16.0
## 63 6.73 22.2
## 64 9.50 25.0
## 65 8.05 33.0
## 66 4.67 23.5
## 67 10.24 19.4
## 68 8.10 22.0
## 69 13.09 17.4
## 70 8.79 20.9
## 71 6.72 24.2
## 72 9.88 21.7
## 73 5.52 22.8
## 74 7.54 23.4
## 75 6.78 24.1
## 76 8.94 21.4
## 77 11.97 20.0
## 78 10.27 20.8
## 79 12.34 21.2
## 80 9.10 20.3
## 81 5.29 28.0
## 82 7.22 23.9
## 83 6.72 24.8
## 84 7.51 22.9
## 85 9.62 23.9
## 86 6.53 26.6
## 87 12.86 22.5
## 88 8.44 22.2
## 89 5.50 23.6
## 90 5.70 28.7
## 91 8.81 22.6
## 92 8.20 22.0
## 93 8.16 22.9
## 94 6.21 25.0
## 95 10.59 20.6
## 96 6.65 28.4
## 97 11.34 21.4
## 98 4.21 38.7
## 99 3.57 43.8
## 100 6.19 33.2
## 101 9.42 27.5
## 102 7.67 26.5
## 103 10.63 18.6
## 104 13.44 19.3
## 105 12.33 20.1
## 106 16.47 19.5
## 107 18.66 19.5
## 108 14.09 20.4
## 109 12.27 19.8
## 110 15.55 19.4
## 111 13.00 21.7
## 112 10.16 22.8
## 113 16.21 18.8
## 114 17.09 18.7
## 115 10.45 18.5
## 116 15.76 18.3
## 117 12.04 21.2
## 118 10.30 19.2
## 119 15.37 20.4
## 120 13.61 19.3
## 121 14.37 22.0
## 122 14.27 20.3
## 123 17.93 20.5
## 124 25.41 17.3
## 125 17.58 18.8
## 126 14.81 21.4
## 127 27.26 15.7
## 128 17.19 16.2
## 129 15.39 18.0
## 130 18.34 14.3
## 131 12.60 19.2
## 132 12.26 19.6
## 133 11.12 23.0
## 134 15.03 18.4
## 135 17.31 15.6
## 136 16.96 18.1
## 137 16.90 17.4
## 138 14.59 17.1
## 139 21.32 13.3
## 140 18.46 17.8
## 141 24.16 14.0
## 142 34.41 14.4
## 143 26.82 13.4
## 144 26.42 15.6
## 145 29.29 11.8
## 146 27.80 13.8
## 147 16.65 15.6
## 148 29.53 14.6
## 149 28.32 17.8
## 150 21.45 15.4
## 151 14.10 21.5
## 152 13.28 19.6
## 153 12.12 15.3
## 154 15.79 19.4
## 155 15.12 17.0
## 156 15.02 15.6
## 157 16.14 13.1
## 158 4.59 41.3
## 159 6.43 24.3
## 160 7.39 23.3
## 161 5.50 27.0
## 162 1.73 50.0
## 163 1.92 50.0
## 164 3.32 50.0
## 165 11.64 22.7
## 166 9.81 25.0
## 167 3.70 50.0
## 168 12.14 23.8
## 169 11.10 23.8
## 170 11.32 22.3
## 171 14.43 17.4
## 172 12.03 19.1
## 173 14.69 23.1
## 174 9.04 23.6
## 175 9.64 22.6
## 176 5.33 29.4
## 177 10.11 23.2
## 178 6.29 24.6
## 179 6.92 29.9
## 180 5.04 37.2
## 181 7.56 39.8
## 182 9.45 36.2
## 183 4.82 37.9
## 184 5.68 32.5
## 185 13.98 26.4
## 186 13.15 29.6
## 187 4.45 50.0
## 188 6.68 32.0
## 189 4.56 29.8
## 190 5.39 34.9
## 191 5.10 37.0
## 192 4.69 30.5
## 193 2.87 36.4
## 194 5.03 31.1
## 195 4.38 29.1
## 196 2.97 50.0
## 197 4.08 33.3
## 198 8.61 30.3
## 199 6.62 34.6
## 200 4.56 34.9
## 201 4.45 32.9
## 202 7.43 24.1
## 203 3.11 42.3
## 204 3.81 48.5
## 205 2.88 50.0
## 206 10.87 22.6
## 207 10.97 24.4
## 208 18.06 22.5
## 209 14.66 24.4
## 210 23.09 20.0
## 211 17.27 21.7
## 212 23.98 19.3
## 213 16.03 22.4
## 214 9.38 28.1
## 215 29.55 23.7
## 216 9.47 25.0
## 217 13.51 23.3
## 218 9.69 28.7
## 219 17.92 21.5
## 220 10.50 23.0
## 221 9.71 26.7
## 222 21.46 21.7
## 223 9.93 27.5
## 224 7.60 30.1
## 225 4.14 44.8
## 226 4.63 50.0
## 227 3.13 37.6
## 228 6.36 31.6
## 229 3.92 46.7
## 230 3.76 31.5
## 231 11.65 24.3
## 232 5.25 31.7
## 233 2.47 41.7
## 234 3.95 48.3
## 235 8.05 29.0
## 236 10.88 24.0
## 237 9.54 25.1
## 238 4.73 31.5
## 239 6.36 23.7
## 240 7.37 23.3
## 241 11.38 22.0
## 242 12.40 20.1
## 243 11.22 22.2
## 244 5.19 23.7
## 245 12.50 17.6
## 246 18.46 18.5
## 247 9.16 24.3
## 248 10.15 20.5
## 249 9.52 24.5
## 250 6.56 26.2
## 251 5.90 24.4
## 252 3.59 24.8
## 253 3.53 29.6
## 254 3.54 42.8
## 255 6.57 21.9
## 256 9.25 20.9
## 257 3.11 44.0
## 258 5.12 50.0
## 259 7.79 36.0
## 260 6.90 30.1
## 261 9.59 33.8
## 262 7.26 43.1
## 263 5.91 48.8
## 264 11.25 31.0
## 265 8.10 36.5
## 266 10.45 22.8
## 267 14.79 30.7
## 268 7.44 50.0
## 269 3.16 43.5
## 270 13.65 20.7
## 271 13.00 21.1
## 272 6.59 25.2
## 273 7.73 24.4
## 274 6.58 35.2
## 275 3.53 32.4
## 276 2.98 32.0
## 277 6.05 33.2
## 278 4.16 33.1
## 279 7.19 29.1
## 280 4.85 35.1
## 281 3.76 45.4
## 282 4.59 35.4
## 283 3.01 46.0
## 284 3.16 50.0
## 285 7.85 32.2
## 286 8.23 22.0
## 287 12.93 20.1
## 288 7.14 23.2
## 289 7.60 22.3
## 290 9.51 24.8
## 291 3.33 28.5
## 292 3.56 37.3
## 293 4.70 27.9
## 294 8.58 23.9
## 295 10.40 21.7
## 296 6.27 28.6
## 297 7.39 27.1
## 298 15.84 20.3
## 299 4.97 22.5
## 300 4.74 29.0
## 301 6.07 24.8
## 302 9.50 22.0
## 303 8.67 26.4
## 304 4.86 33.1
## 305 6.93 36.1
## 306 8.93 28.4
## 307 6.47 33.4
## 308 7.53 28.2
## 309 4.54 22.8
## 310 9.97 20.3
## 311 12.64 16.1
## 312 5.98 22.1
## 313 11.72 19.4
## 314 7.90 21.6
## 315 9.28 23.8
## 316 11.50 16.2
## 317 18.33 17.8
## 318 15.94 19.8
## 319 10.36 23.1
## 320 12.73 21.0
## 321 7.20 23.8
## 322 6.87 23.1
## 323 7.70 20.4
## 324 11.74 18.5
## 325 6.12 25.0
## 326 5.08 24.6
## 327 6.15 23.0
## 328 12.79 22.2
## 329 9.97 19.3
## 330 7.34 22.6
## 331 9.09 19.8
## 332 12.43 17.1
## 333 7.83 19.4
## 334 5.68 22.2
## 335 6.75 20.7
## 336 8.01 21.1
## 337 9.80 19.5
## 338 10.56 18.5
## 339 8.51 20.6
## 340 9.74 19.0
## 341 9.29 18.7
## 342 5.49 32.7
## 343 8.65 16.5
## 344 7.18 23.9
## 345 4.61 31.2
## 346 10.53 17.5
## 347 12.67 17.2
## 348 6.36 23.1
## 349 5.99 24.5
## 350 5.89 26.6
## 351 5.98 22.9
## 352 5.49 24.1
## 353 7.79 18.6
## 354 4.50 30.1
## 355 8.05 18.2
## 356 5.57 20.6
## 357 17.60 17.8
## 358 13.27 21.7
## 359 11.48 22.7
## 360 12.67 22.6
## 361 7.79 25.0
## 362 14.19 19.9
## 363 10.19 20.8
## 364 14.64 16.8
## 365 5.29 21.9
## 366 7.12 27.5
## 367 14.00 21.9
## 368 13.33 23.1
## 369 3.26 50.0
## 370 3.73 50.0
## 371 2.96 50.0
## 372 9.53 50.0
## 373 8.88 50.0
## 374 34.77 13.8
## 375 37.97 13.8
## 376 13.44 15.0
## 377 23.24 13.9
## 378 21.24 13.3
## 379 23.69 13.1
## 380 21.78 10.2
## 381 17.21 10.4
## 382 21.08 10.9
## 383 23.60 11.3
## 384 24.56 12.3
## 385 30.63 8.8
## 386 30.81 7.2
## 387 28.28 10.5
## 388 31.99 7.4
## 389 30.62 10.2
## 390 20.85 11.5
## 391 17.11 15.1
## 392 18.76 23.2
## 393 25.68 9.7
## 394 15.17 13.8
## 395 16.35 12.7
## 396 17.12 13.1
## 397 19.37 12.5
## 398 19.92 8.5
## 399 30.59 5.0
## 400 29.97 6.3
## 401 26.77 5.6
## 402 20.32 7.2
## 403 20.31 12.1
## 404 19.77 8.3
## 405 27.38 8.5
## 406 22.98 5.0
## 407 23.34 11.9
## 408 12.13 27.9
## 409 26.40 17.2
## 410 19.78 27.5
## 411 10.11 15.0
## 412 21.22 17.2
## 413 34.37 17.9
## 414 20.08 16.3
## 415 36.98 7.0
## 416 29.05 7.2
## 417 25.79 7.5
## 418 26.64 10.4
## 419 20.62 8.8
## 420 22.74 8.4
## 421 15.02 16.7
## 422 15.70 14.2
## 423 14.10 20.8
## 424 23.29 13.4
## 425 17.16 11.7
## 426 24.39 8.3
## 427 15.69 10.2
## 428 14.52 10.9
## 429 21.52 11.0
## 430 24.08 9.5
## 431 17.64 14.5
## 432 19.69 14.1
## 433 12.03 16.1
## 434 16.22 14.3
## 435 15.17 11.7
## 436 23.27 13.4
## 437 18.05 9.6
## 438 26.45 8.7
## 439 34.02 8.4
## 440 22.88 12.8
## 441 22.11 10.5
## 442 19.52 17.1
## 443 16.59 18.4
## 444 18.85 15.4
## 445 23.79 10.8
## 446 23.98 11.8
## 447 17.79 14.9
## 448 16.44 12.6
## 449 18.13 14.1
## 450 19.31 13.0
## 451 17.44 13.4
## 452 17.73 15.2
## 453 17.27 16.1
## 454 16.74 17.8
## 455 18.71 14.9
## 456 18.13 14.1
## 457 19.01 12.7
## 458 16.94 13.5
## 459 16.23 14.9
## 460 14.70 20.0
## 461 16.42 16.4
## 462 14.65 17.7
## 463 13.99 19.5
## 464 10.29 20.2
## 465 13.22 21.4
## 466 14.13 19.9
## 467 17.15 19.0
## 468 21.32 19.1
## 469 18.13 19.1
## 470 14.76 20.1
## 471 16.29 19.9
## 472 12.87 19.6
## 473 14.36 23.2
## 474 11.66 29.8
## 475 18.14 13.8
## 476 24.10 13.3
## 477 18.68 16.7
## 478 24.91 12.0
## 479 18.03 14.6
## 480 13.11 21.4
## 481 10.74 23.0
## 482 7.74 23.7
## 483 7.01 25.0
## 484 10.42 21.8
## 485 13.34 20.6
## 486 10.58 21.2
## 487 14.98 19.1
## 488 11.45 20.6
## 489 18.06 15.2
## 490 23.97 7.0
## 491 29.68 8.1
## 492 18.07 13.6
## 493 13.35 20.1
## 494 12.01 21.8
## 495 13.59 24.5
## 496 17.60 23.1
## 497 21.14 19.7
## 498 14.10 18.3
## 499 12.92 21.2
## 500 15.10 17.5
## 501 14.33 16.8
## 502 9.67 22.4
## 503 9.08 20.6
## 504 5.64 23.9
## 505 6.48 22.0
## 506 7.88 11.9# Number of rows and columns
dim(Boston)## [1] 506 14# Structure and variable descriptions
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...?Boston## starting httpd help server ... doneHow many rows are in this Boston data set? How many columns? What do the rows and columns represent?
The Boston housing dataset contains 506 rows and 14 columns. Each row represents a census tract in Boston, while each column represents a variable describing housing characteristics, economic indicators, or demographic factors such as crime rate, property tax rate, average number of rooms, and median home value.
- Make some pairwise scatterplots of the predictors (columns) in this data set. Describe your findings.
pairs(Boston)
The pairwise scatterplots show several noticeable relationships among
predictors. For example, median home value tends to increase with the
number of rooms and decrease as the percentage of lower-status
population increases. There also appears to be correlation between tax
rate and accessibility to highways. Some variables show potential
multicollinearity, meaning that certain predictors may contain
overlapping information.
- Are any of the predictors associated with per capita crime rate? If so, explain the relationship.
cor(Boston)## crim zn indus chas nox
## crim 1.00000000 -0.20046922 0.40658341 -0.055891582 0.42097171
## zn -0.20046922 1.00000000 -0.53382819 -0.042696719 -0.51660371
## indus 0.40658341 -0.53382819 1.00000000 0.062938027 0.76365145
## chas -0.05589158 -0.04269672 0.06293803 1.000000000 0.09120281
## nox 0.42097171 -0.51660371 0.76365145 0.091202807 1.00000000
## rm -0.21924670 0.31199059 -0.39167585 0.091251225 -0.30218819
## age 0.35273425 -0.56953734 0.64477851 0.086517774 0.73147010
## dis -0.37967009 0.66440822 -0.70802699 -0.099175780 -0.76923011
## rad 0.62550515 -0.31194783 0.59512927 -0.007368241 0.61144056
## tax 0.58276431 -0.31456332 0.72076018 -0.035586518 0.66802320
## ptratio 0.28994558 -0.39167855 0.38324756 -0.121515174 0.18893268
## black -0.38506394 0.17552032 -0.35697654 0.048788485 -0.38005064
## lstat 0.45562148 -0.41299457 0.60379972 -0.053929298 0.59087892
## medv -0.38830461 0.36044534 -0.48372516 0.175260177 -0.42732077
## rm age dis rad tax ptratio
## crim -0.21924670 0.35273425 -0.37967009 0.625505145 0.58276431 0.2899456
## zn 0.31199059 -0.56953734 0.66440822 -0.311947826 -0.31456332 -0.3916785
## indus -0.39167585 0.64477851 -0.70802699 0.595129275 0.72076018 0.3832476
## chas 0.09125123 0.08651777 -0.09917578 -0.007368241 -0.03558652 -0.1215152
## nox -0.30218819 0.73147010 -0.76923011 0.611440563 0.66802320 0.1889327
## rm 1.00000000 -0.24026493 0.20524621 -0.209846668 -0.29204783 -0.3555015
## age -0.24026493 1.00000000 -0.74788054 0.456022452 0.50645559 0.2615150
## dis 0.20524621 -0.74788054 1.00000000 -0.494587930 -0.53443158 -0.2324705
## rad -0.20984667 0.45602245 -0.49458793 1.000000000 0.91022819 0.4647412
## tax -0.29204783 0.50645559 -0.53443158 0.910228189 1.00000000 0.4608530
## ptratio -0.35550149 0.26151501 -0.23247054 0.464741179 0.46085304 1.0000000
## black 0.12806864 -0.27353398 0.29151167 -0.444412816 -0.44180801 -0.1773833
## lstat -0.61380827 0.60233853 -0.49699583 0.488676335 0.54399341 0.3740443
## medv 0.69535995 -0.37695457 0.24992873 -0.381626231 -0.46853593 -0.5077867
## black lstat medv
## crim -0.38506394 0.4556215 -0.3883046
## zn 0.17552032 -0.4129946 0.3604453
## indus -0.35697654 0.6037997 -0.4837252
## chas 0.04878848 -0.0539293 0.1752602
## nox -0.38005064 0.5908789 -0.4273208
## rm 0.12806864 -0.6138083 0.6953599
## age -0.27353398 0.6023385 -0.3769546
## dis 0.29151167 -0.4969958 0.2499287
## rad -0.44441282 0.4886763 -0.3816262
## tax -0.44180801 0.5439934 -0.4685359
## ptratio -0.17738330 0.3740443 -0.5077867
## black 1.00000000 -0.3660869 0.3334608
## lstat -0.36608690 1.0000000 -0.7376627
## medv 0.33346082 -0.7376627 1.0000000cor(Boston$crim, Boston)## crim zn indus chas nox rm age
## [1,] 1 -0.2004692 0.4065834 -0.05589158 0.4209717 -0.2192467 0.3527343
## dis rad tax ptratio black lstat medv
## [1,] -0.3796701 0.6255051 0.5827643 0.2899456 -0.3850639 0.4556215 -0.3883046Several predictors appear to be associated with per capita crime rate. Crime rate shows a positive relationship with tax rate and percentage of lower-status population, meaning that higher values of these predictors are associated with higher crime rates. Crime rate shows a negative relationship with median home value and number of rooms, suggesting that areas with higher property values and larger homes tend to have lower crime rates.
- Do any of the census tracts of Boston appear to have particularly high crime rates? Tax rates? Comment on the range of each predictor.
range(Boston$crim)## [1] 0.00632 88.97620range(Boston$tax)## [1] 187 711summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00The dataset shows that crime rates vary widely across census tracts, with some areas experiencing significantly higher crime levels than others.The per capita crime rate ranges from 0.0063 to 88.9762, indicating substantial variability and the presence of extreme outliers. The tax rate ranges from 187 to 711, showing wide variation in local property taxation across towns. Overall, the predictors demonstrate substantial variability, which is useful for predictive modeling because it provides diverse information across observations.
e.How many of the census tracts in this data set bound the Charles river?
table(Boston$chas)##
## 0 1
## 471 35The Boston housing data set contains 506 census tracts in total. Based on the variable chas, which indicates whether a tract bounds the Charles River, 35 census tracts bound the Charles River, while 471 census tracts do not. This shows that only a small fraction of the census tracts are located along the river.
- What is the median pupil-teacher ratio among the towns in this data set?
median(Boston$ptratio)## [1] 19.05The median pupil-teacher ratio among towns in the Boston housing data set is 19.05 students per teacher. This value represents the typical pupil-teacher ratio across the census tracts included in the data.
- 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.
data(Soybean)
str(Soybean)## 'data.frame': 683 obs. of 36 variables:
## $ Class : Factor w/ 19 levels "2-4-d-injury",..: 11 11 11 11 11 11 11 11 11 11 ...
## $ date : Factor w/ 7 levels "0","1","2","3",..: 7 5 4 4 7 6 6 5 7 5 ...
## $ plant.stand : Ord.factor w/ 2 levels "0"<"1": 1 1 1 1 1 1 1 1 1 1 ...
## $ precip : Ord.factor w/ 3 levels "0"<"1"<"2": 3 3 3 3 3 3 3 3 3 3 ...
## $ temp : Ord.factor w/ 3 levels "0"<"1"<"2": 2 2 2 2 2 2 2 2 2 2 ...
## $ hail : Factor w/ 2 levels "0","1": 1 1 1 1 1 1 1 2 1 1 ...
## $ crop.hist : Factor w/ 4 levels "0","1","2","3": 2 3 2 2 3 4 3 2 4 3 ...
## $ area.dam : Factor w/ 4 levels "0","1","2","3": 2 1 1 1 1 1 1 1 1 1 ...
## $ sever : Factor w/ 3 levels "0","1","2": 2 3 3 3 2 2 2 2 2 3 ...
## $ seed.tmt : Factor w/ 3 levels "0","1","2": 1 2 2 1 1 1 2 1 2 1 ...
## $ germ : Ord.factor w/ 3 levels "0"<"1"<"2": 1 2 3 2 3 2 1 3 2 3 ...
## $ plant.growth : Factor w/ 2 levels "0","1": 2 2 2 2 2 2 2 2 2 2 ...
## $ leaves : Factor w/ 2 levels "0","1": 2 2 2 2 2 2 2 2 2 2 ...
## $ leaf.halo : Factor w/ 3 levels "0","1","2": 1 1 1 1 1 1 1 1 1 1 ...
## $ leaf.marg : Factor w/ 3 levels "0","1","2": 3 3 3 3 3 3 3 3 3 3 ...
## $ leaf.size : Ord.factor w/ 3 levels "0"<"1"<"2": 3 3 3 3 3 3 3 3 3 3 ...
## $ leaf.shread : Factor w/ 2 levels "0","1": 1 1 1 1 1 1 1 1 1 1 ...
## $ leaf.malf : Factor w/ 2 levels "0","1": 1 1 1 1 1 1 1 1 1 1 ...
## $ leaf.mild : Factor w/ 3 levels "0","1","2": 1 1 1 1 1 1 1 1 1 1 ...
## $ stem : Factor w/ 2 levels "0","1": 2 2 2 2 2 2 2 2 2 2 ...
## $ lodging : Factor w/ 2 levels "0","1": 2 1 1 1 1 1 2 1 1 1 ...
## $ stem.cankers : Factor w/ 4 levels "0","1","2","3": 4 4 4 4 4 4 4 4 4 4 ...
## $ canker.lesion : Factor w/ 4 levels "0","1","2","3": 2 2 1 1 2 1 2 2 2 2 ...
## $ fruiting.bodies: Factor w/ 2 levels "0","1": 2 2 2 2 2 2 2 2 2 2 ...
## $ ext.decay : Factor w/ 3 levels "0","1","2": 2 2 2 2 2 2 2 2 2 2 ...
## $ mycelium : Factor w/ 2 levels "0","1": 1 1 1 1 1 1 1 1 1 1 ...
## $ int.discolor : Factor w/ 3 levels "0","1","2": 1 1 1 1 1 1 1 1 1 1 ...
## $ sclerotia : Factor w/ 2 levels "0","1": 1 1 1 1 1 1 1 1 1 1 ...
## $ fruit.pods : Factor w/ 4 levels "0","1","2","3": 1 1 1 1 1 1 1 1 1 1 ...
## $ fruit.spots : Factor w/ 4 levels "0","1","2","4": 4 4 4 4 4 4 4 4 4 4 ...
## $ seed : Factor w/ 2 levels "0","1": 1 1 1 1 1 1 1 1 1 1 ...
## $ mold.growth : Factor w/ 2 levels "0","1": 1 1 1 1 1 1 1 1 1 1 ...
## $ seed.discolor : Factor w/ 2 levels "0","1": 1 1 1 1 1 1 1 1 1 1 ...
## $ seed.size : Factor w/ 2 levels "0","1": 1 1 1 1 1 1 1 1 1 1 ...
## $ shriveling : Factor w/ 2 levels "0","1": 1 1 1 1 1 1 1 1 1 1 ...
## $ roots : Factor w/ 3 levels "0","1","2": 1 1 1 1 1 1 1 1 1 1 ...summary(Soybean)## Class date plant.stand precip temp
## brown-spot : 92 5 :149 0 :354 0 : 74 0 : 80
## alternarialeaf-spot: 91 4 :131 1 :293 1 :112 1 :374
## frog-eye-leaf-spot : 91 3 :118 NA's: 36 2 :459 2 :199
## phytophthora-rot : 88 2 : 93 NA's: 38 NA's: 30
## anthracnose : 44 6 : 90
## brown-stem-rot : 44 (Other):101
## (Other) :233 NA's : 1
## hail crop.hist area.dam sever seed.tmt germ plant.growth
## 0 :435 0 : 65 0 :123 0 :195 0 :305 0 :165 0 :441
## 1 :127 1 :165 1 :227 1 :322 1 :222 1 :213 1 :226
## NA's:121 2 :219 2 :145 2 : 45 2 : 35 2 :193 NA's: 16
## 3 :218 3 :187 NA's:121 NA's:121 NA's:112
## NA's: 16 NA's: 1
##
##
## leaves leaf.halo leaf.marg leaf.size leaf.shread leaf.malf leaf.mild
## 0: 77 0 :221 0 :357 0 : 51 0 :487 0 :554 0 :535
## 1:606 1 : 36 1 : 21 1 :327 1 : 96 1 : 45 1 : 20
## 2 :342 2 :221 2 :221 NA's:100 NA's: 84 2 : 20
## NA's: 84 NA's: 84 NA's: 84 NA's:108
##
##
##
## stem lodging stem.cankers canker.lesion fruiting.bodies ext.decay
## 0 :296 0 :520 0 :379 0 :320 0 :473 0 :497
## 1 :371 1 : 42 1 : 39 1 : 83 1 :104 1 :135
## NA's: 16 NA's:121 2 : 36 2 :177 NA's:106 2 : 13
## 3 :191 3 : 65 NA's: 38
## NA's: 38 NA's: 38
##
##
## mycelium int.discolor sclerotia fruit.pods fruit.spots seed
## 0 :639 0 :581 0 :625 0 :407 0 :345 0 :476
## 1 : 6 1 : 44 1 : 20 1 :130 1 : 75 1 :115
## NA's: 38 2 : 20 NA's: 38 2 : 14 2 : 57 NA's: 92
## NA's: 38 3 : 48 4 :100
## NA's: 84 NA's:106
##
##
## mold.growth seed.discolor seed.size shriveling roots
## 0 :524 0 :513 0 :532 0 :539 0 :551
## 1 : 67 1 : 64 1 : 59 1 : 38 1 : 86
## NA's: 92 NA's:106 NA's: 92 NA's:106 2 : 15
## NA's: 31
##
##
## - Investigate the frequency distributions for the categorical predictors. Are any of the distributions degenerate in the ways discussed earlier in this chapter?
# Frequency for each predictor
lapply(Soybean, table)## $Class
##
## 2-4-d-injury alternarialeaf-spot
## 16 91
## anthracnose bacterial-blight
## 44 20
## bacterial-pustule brown-spot
## 20 92
## brown-stem-rot charcoal-rot
## 44 20
## cyst-nematode diaporthe-pod-&-stem-blight
## 14 15
## diaporthe-stem-canker downy-mildew
## 20 20
## frog-eye-leaf-spot herbicide-injury
## 91 8
## phyllosticta-leaf-spot phytophthora-rot
## 20 88
## powdery-mildew purple-seed-stain
## 20 20
## rhizoctonia-root-rot
## 20
##
## $date
##
## 0 1 2 3 4 5 6
## 26 75 93 118 131 149 90
##
## $plant.stand
##
## 0 1
## 354 293
##
## $precip
##
## 0 1 2
## 74 112 459
##
## $temp
##
## 0 1 2
## 80 374 199
##
## $hail
##
## 0 1
## 435 127
##
## $crop.hist
##
## 0 1 2 3
## 65 165 219 218
##
## $area.dam
##
## 0 1 2 3
## 123 227 145 187
##
## $sever
##
## 0 1 2
## 195 322 45
##
## $seed.tmt
##
## 0 1 2
## 305 222 35
##
## $germ
##
## 0 1 2
## 165 213 193
##
## $plant.growth
##
## 0 1
## 441 226
##
## $leaves
##
## 0 1
## 77 606
##
## $leaf.halo
##
## 0 1 2
## 221 36 342
##
## $leaf.marg
##
## 0 1 2
## 357 21 221
##
## $leaf.size
##
## 0 1 2
## 51 327 221
##
## $leaf.shread
##
## 0 1
## 487 96
##
## $leaf.malf
##
## 0 1
## 554 45
##
## $leaf.mild
##
## 0 1 2
## 535 20 20
##
## $stem
##
## 0 1
## 296 371
##
## $lodging
##
## 0 1
## 520 42
##
## $stem.cankers
##
## 0 1 2 3
## 379 39 36 191
##
## $canker.lesion
##
## 0 1 2 3
## 320 83 177 65
##
## $fruiting.bodies
##
## 0 1
## 473 104
##
## $ext.decay
##
## 0 1 2
## 497 135 13
##
## $mycelium
##
## 0 1
## 639 6
##
## $int.discolor
##
## 0 1 2
## 581 44 20
##
## $sclerotia
##
## 0 1
## 625 20
##
## $fruit.pods
##
## 0 1 2 3
## 407 130 14 48
##
## $fruit.spots
##
## 0 1 2 4
## 345 75 57 100
##
## $seed
##
## 0 1
## 476 115
##
## $mold.growth
##
## 0 1
## 524 67
##
## $seed.discolor
##
## 0 1
## 513 64
##
## $seed.size
##
## 0 1
## 532 59
##
## $shriveling
##
## 0 1
## 539 38
##
## $roots
##
## 0 1 2
## 551 86 15The soybean data set from the UC Irvine Machine Learning Repository contains 683 observations, 35 categorical predictors, and 19 disease classes. Examination of the frequency tables shows that several predictors have highly imbalanced distributions. For example, mycelium has 639 observations in category 0 and only 6 in category 1, sclerotia has 625 observations in category 0 and 20 in category 1, and shriveling has 539 observations in category 0 and only 38 in category 1. These predictors are considered degenerate because one level dominates the data, providing very limited information for distinguishing among disease classes.
- Roughly 18 % of the data are missing. Are there particular predictors that are more likely to be missing? Is the pattern of missing data related to the classes?
# Count missing values per variable
colSums(is.na(Soybean))## Class date plant.stand precip temp
## 0 1 36 38 30
## hail crop.hist area.dam sever seed.tmt
## 121 16 1 121 121
## germ plant.growth leaves leaf.halo leaf.marg
## 112 16 0 84 84
## leaf.size leaf.shread leaf.malf leaf.mild stem
## 84 100 84 108 16
## lodging stem.cankers canker.lesion fruiting.bodies ext.decay
## 121 38 38 106 38
## mycelium int.discolor sclerotia fruit.pods fruit.spots
## 38 38 38 84 106
## seed mold.growth seed.discolor seed.size shriveling
## 92 92 106 92 106
## roots
## 31# Missing by class
aggregate(is.na(Soybean), by=list(Class = Soybean$Class), sum)## Class Class date plant.stand precip temp hail
## 1 2-4-d-injury 0 1 16 16 16 16
## 2 alternarialeaf-spot 0 0 0 0 0 0
## 3 anthracnose 0 0 0 0 0 0
## 4 bacterial-blight 0 0 0 0 0 0
## 5 bacterial-pustule 0 0 0 0 0 0
## 6 brown-spot 0 0 0 0 0 0
## 7 brown-stem-rot 0 0 0 0 0 0
## 8 charcoal-rot 0 0 0 0 0 0
## 9 cyst-nematode 0 0 14 14 14 14
## 10 diaporthe-pod-&-stem-blight 0 0 6 0 0 15
## 11 diaporthe-stem-canker 0 0 0 0 0 0
## 12 downy-mildew 0 0 0 0 0 0
## 13 frog-eye-leaf-spot 0 0 0 0 0 0
## 14 herbicide-injury 0 0 0 8 0 8
## 15 phyllosticta-leaf-spot 0 0 0 0 0 0
## 16 phytophthora-rot 0 0 0 0 0 68
## 17 powdery-mildew 0 0 0 0 0 0
## 18 purple-seed-stain 0 0 0 0 0 0
## 19 rhizoctonia-root-rot 0 0 0 0 0 0
## crop.hist area.dam sever seed.tmt germ plant.growth leaves leaf.halo
## 1 16 1 16 16 16 16 0 0
## 2 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0
## 4 0 0 0 0 0 0 0 0
## 5 0 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0 0
## 7 0 0 0 0 0 0 0 0
## 8 0 0 0 0 0 0 0 0
## 9 0 0 14 14 14 0 0 14
## 10 0 0 15 15 6 0 0 15
## 11 0 0 0 0 0 0 0 0
## 12 0 0 0 0 0 0 0 0
## 13 0 0 0 0 0 0 0 0
## 14 0 0 8 8 8 0 0 0
## 15 0 0 0 0 0 0 0 0
## 16 0 0 68 68 68 0 0 55
## 17 0 0 0 0 0 0 0 0
## 18 0 0 0 0 0 0 0 0
## 19 0 0 0 0 0 0 0 0
## leaf.marg leaf.size leaf.shread leaf.malf leaf.mild stem lodging
## 1 0 0 16 0 16 16 16
## 2 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0
## 4 0 0 0 0 0 0 0
## 5 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0
## 7 0 0 0 0 0 0 0
## 8 0 0 0 0 0 0 0
## 9 14 14 14 14 14 0 14
## 10 15 15 15 15 15 0 15
## 11 0 0 0 0 0 0 0
## 12 0 0 0 0 0 0 0
## 13 0 0 0 0 0 0 0
## 14 0 0 0 0 8 0 8
## 15 0 0 0 0 0 0 0
## 16 55 55 55 55 55 0 68
## 17 0 0 0 0 0 0 0
## 18 0 0 0 0 0 0 0
## 19 0 0 0 0 0 0 0
## stem.cankers canker.lesion fruiting.bodies ext.decay mycelium int.discolor
## 1 16 16 16 16 16 16
## 2 0 0 0 0 0 0
## 3 0 0 0 0 0 0
## 4 0 0 0 0 0 0
## 5 0 0 0 0 0 0
## 6 0 0 0 0 0 0
## 7 0 0 0 0 0 0
## 8 0 0 0 0 0 0
## 9 14 14 14 14 14 14
## 10 0 0 0 0 0 0
## 11 0 0 0 0 0 0
## 12 0 0 0 0 0 0
## 13 0 0 0 0 0 0
## 14 8 8 8 8 8 8
## 15 0 0 0 0 0 0
## 16 0 0 68 0 0 0
## 17 0 0 0 0 0 0
## 18 0 0 0 0 0 0
## 19 0 0 0 0 0 0
## sclerotia fruit.pods fruit.spots seed mold.growth seed.discolor seed.size
## 1 16 16 16 16 16 16 16
## 2 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0
## 4 0 0 0 0 0 0 0
## 5 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0
## 7 0 0 0 0 0 0 0
## 8 0 0 0 0 0 0 0
## 9 14 0 14 0 0 14 0
## 10 0 0 0 0 0 0 0
## 11 0 0 0 0 0 0 0
## 12 0 0 0 0 0 0 0
## 13 0 0 0 0 0 0 0
## 14 8 0 8 8 8 8 8
## 15 0 0 0 0 0 0 0
## 16 0 68 68 68 68 68 68
## 17 0 0 0 0 0 0 0
## 18 0 0 0 0 0 0 0
## 19 0 0 0 0 0 0 0
## shriveling roots
## 1 16 16
## 2 0 0
## 3 0 0
## 4 0 0
## 5 0 0
## 6 0 0
## 7 0 0
## 8 0 0
## 9 14 0
## 10 0 15
## 11 0 0
## 12 0 0
## 13 0 0
## 14 8 0
## 15 0 0
## 16 68 0
## 17 0 0
## 18 0 0
## 19 0 0Approximately 18% of the data contains missing values. Some predictors contain significantly more missing observations than others, indicating uneven missingness across variables. The pattern of missing data appears to vary by disease class, For example, the 2-4-d-injury class alone contains 16 missing values for several early predictors, while other classes have none. This uneven distribution suggests that the missing data pattern is related to the disease class and is therefore not completely random, which may impact classification performance.
- Develop a strategy for handling missing data, either by eliminating predictors or imputation.
# Remove predictors with too many missing values
Soybean_clean <- Soybean[, colSums(is.na(Soybean)) < 0.5*nrow(Soybean)]
# Imputation example (most frequent category)
library(caret)
preProcess(Soybean, method = "medianImpute")## Warning in pre_process_options(method, column_types): The following
## pre-processing methods were eliminated: 'medianImpute'## Created from 562 samples and 36 variables
##
## Pre-processing:
## - ignored (36)A practical way to handle the missing data is to drop predictors with a large amount of missing values (for example, more than 50%, or over 341 observations) and impute the remaining missing entries using the most common category. This keeps most of the 683 observations in the data while limiting noise and helping the model remain stable.
- Chapter 5 introduces Quantitative Structure-Activity Relationship (QSAR) modeling where the characteristics of a chemical compound are used to predict other chemical properties. The caret package contains a QSAR data set from Mente and Lombardo (2005). Here, the ability of a chemical to permeate the blood-brain barrier was experimentally determined for 208 compounds. 134 descriptors were measured for each compound
a.Start R and use these commands to load the data: The numeric outcome is contained in the vector logBBB while the predictors are in the data frame bbbDescr.
data(BloodBrain)
str(logBBB)## num [1:208] 1.08 -0.4 0.22 0.14 0.69 0.44 -0.43 1.38 0.75 0.88 ...str(bbbDescr)## 'data.frame': 208 obs. of 134 variables:
## $ tpsa : num 12 49.3 50.5 37.4 37.4 ...
## $ nbasic : int 1 0 1 0 1 1 1 1 1 1 ...
## $ negative : int 0 0 0 0 0 0 0 0 0 0 ...
## $ vsa_hyd : num 167.1 92.6 295.2 319.1 299.7 ...
## $ a_aro : int 0 6 15 15 12 11 6 12 12 6 ...
## $ weight : num 156 151 366 383 326 ...
## $ peoe_vsa.0 : num 76.9 38.2 58.1 62.2 74.8 ...
## $ peoe_vsa.1 : num 43.4 25.5 124.7 124.7 118 ...
## $ peoe_vsa.2 : num 0 0 21.7 13.2 33 ...
## $ peoe_vsa.3 : num 0 8.62 8.62 21.79 0 ...
## $ peoe_vsa.4 : num 0 23.3 17.4 0 0 ...
## $ peoe_vsa.5 : num 0 0 0 0 0 0 0 0 0 0 ...
## $ peoe_vsa.6 : num 17.24 0 8.62 8.62 8.62 ...
## $ peoe_vsa.0.1 : num 18.7 49 83.8 83.8 83.8 ...
## $ peoe_vsa.1.1 : num 43.5 0 49 68.8 36.8 ...
## $ peoe_vsa.2.1 : num 0 0 0 0 0 ...
## $ peoe_vsa.3.1 : num 0 0 0 0 0 0 0 0 0 0 ...
## $ peoe_vsa.4.1 : num 0 0 5.68 5.68 5.68 ...
## $ peoe_vsa.5.1 : num 0 13.567 2.504 0 0.137 ...
## $ peoe_vsa.6.1 : num 0 7.9 2.64 2.64 2.5 ...
## $ a_acc : int 0 2 2 2 2 2 2 2 0 2 ...
## $ a_acid : int 0 0 0 0 0 0 0 0 0 0 ...
## $ a_base : int 1 0 1 1 1 1 1 1 1 1 ...
## $ vsa_acc : num 0 13.57 8.19 8.19 8.19 ...
## $ vsa_acid : num 0 0 0 0 0 0 0 0 0 0 ...
## $ vsa_base : num 5.68 0 0 0 0 ...
## $ vsa_don : num 5.68 5.68 5.68 5.68 5.68 ...
## $ vsa_other : num 0 28.1 43.6 28.3 19.6 ...
## $ vsa_pol : num 0 13.6 0 0 0 ...
## $ slogp_vsa0 : num 18 25.4 14.1 14.1 14.1 ...
## $ slogp_vsa1 : num 0 23.3 34.8 34.8 34.8 ...
## $ slogp_vsa2 : num 3.98 23.86 0 0 0 ...
## $ slogp_vsa3 : num 0 0 76.2 76.2 76.2 ...
## $ slogp_vsa4 : num 4.41 0 3.19 3.19 3.19 ...
## $ slogp_vsa5 : num 32.9 0 9.51 0 0 ...
## $ slogp_vsa6 : num 0 0 0 0 0 0 0 0 0 0 ...
## $ slogp_vsa7 : num 0 70.6 148.1 144 140.7 ...
## $ slogp_vsa8 : num 113.2 0 75.5 75.5 75.5 ...
## $ slogp_vsa9 : num 33.3 41.3 28.3 55.5 26 ...
## $ smr_vsa0 : num 0 23.86 12.63 3.12 3.12 ...
## $ smr_vsa1 : num 18 25.4 27.8 27.8 27.8 ...
## $ smr_vsa2 : num 4.41 0 0 0 0 ...
## $ smr_vsa3 : num 3.98 5.24 8.43 8.43 8.43 ...
## $ smr_vsa4 : num 0 20.8 29.6 21.4 20.3 ...
## $ smr_vsa5 : num 113.2 70.6 235.1 235.1 234.6 ...
## $ smr_vsa6 : num 0 5.26 76.25 76.25 76.25 ...
## $ smr_vsa7 : num 66.2 33.3 0 31.3 0 ...
## $ tpsa.1 : num 16.6 49.3 51.7 38.6 38.6 ...
## $ logp.o.w. : num 2.948 0.889 4.439 5.254 3.8 ...
## $ frac.anion7. : num 0 0.001 0 0 0 0 0.001 0 0 0 ...
## $ frac.cation7. : num 0.999 0 0.986 0.986 0.986 0.986 0.996 0.946 0.999 0.976 ...
## $ andrewbind : num 3.4 -3.3 12.8 12.8 10.3 10 10.4 15.9 12.9 9.5 ...
## $ rotatablebonds : int 3 2 8 8 8 8 8 7 4 5 ...
## $ mlogp : num 2.5 1.06 4.66 3.82 3.27 ...
## $ clogp : num 2.97 0.494 5.137 5.878 4.367 ...
## $ mw : num 155 151 365 382 325 ...
## $ nocount : int 1 3 5 4 4 4 4 3 2 4 ...
## $ hbdnr : int 1 2 1 1 1 1 2 1 1 0 ...
## $ rule.of.5violations : int 0 0 1 1 0 0 0 0 1 0 ...
## $ alert : int 0 0 0 0 0 0 0 0 0 0 ...
## $ prx : int 0 1 6 2 2 2 1 0 0 4 ...
## $ ub : num 0 3 5.3 5.3 4.2 3.6 3 4.7 4.2 3 ...
## $ pol : int 0 2 3 3 2 2 2 3 4 1 ...
## $ inthb : int 0 0 0 0 0 0 1 0 0 0 ...
## $ adistm : num 0 395 1365 703 746 ...
## $ adistd : num 0 10.9 25.7 10 10.6 ...
## $ polar_area : num 21.1 117.4 82.1 65.1 66.2 ...
## $ nonpolar_area : num 379 248 638 668 602 ...
## $ psa_npsa : num 0.0557 0.4743 0.1287 0.0974 0.11 ...
## $ tcsa : num 0.0097 0.0134 0.0111 0.0108 0.0118 0.0111 0.0123 0.0099 0.0106 0.0115 ...
## $ tcpa : num 0.1842 0.0417 0.0972 0.1218 0.1186 ...
## $ tcnp : num 0.0103 0.0198 0.0125 0.0119 0.013 0.0125 0.0162 0.011 0.0109 0.0122 ...
## $ ovality : num 1.1 1.12 1.3 1.3 1.27 ...
## $ surface_area : num 400 365 720 733 668 ...
## $ volume : num 656 555 1224 1257 1133 ...
## $ most_negative_charge: num -0.617 -0.84 -0.801 -0.761 -0.857 ...
## $ most_positive_charge: num 0.307 0.497 0.541 0.48 0.455 ...
## $ sum_absolute_charge : num 3.89 4.89 7.98 7.93 7.85 ...
## $ dipole_moment : num 1.19 4.21 3.52 3.15 3.27 ...
## $ homo : num -9.67 -8.96 -8.63 -8.56 -8.67 ...
## $ lumo : num 3.4038 0.1942 0.0589 -0.2651 0.3149 ...
## $ hardness : num 6.54 4.58 4.34 4.15 4.49 ...
## $ ppsa1 : num 349 223 518 508 509 ...
## $ ppsa2 : num 679 546 2066 2013 1999 ...
## $ ppsa3 : num 31 42.3 64 61.7 61.6 ...
## $ pnsa1 : num 51.1 141.8 202 225.4 158.8 ...
## $ pnsa2 : num -99.3 -346.9 -805.9 -894 -623.3 ...
## $ pnsa3 : num -10.5 -44 -43.8 -42 -39.8 ...
## $ fpsa1 : num 0.872 0.611 0.719 0.692 0.762 ...
## $ fpsa2 : num 1.7 1.5 2.87 2.75 2.99 ...
## $ fpsa3 : num 0.0774 0.1159 0.0888 0.0842 0.0922 ...
## $ fnsa1 : num 0.128 0.389 0.281 0.307 0.238 ...
## $ fnsa2 : num -0.248 -0.951 -1.12 -1.22 -0.933 ...
## $ fnsa3 : num -0.0262 -0.1207 -0.0608 -0.0573 -0.0596 ...
## $ wpsa1 : num 139.7 81.4 372.7 372.1 340.1 ...
## $ wpsa2 : num 272 199 1487 1476 1335 ...
## $ wpsa3 : num 12.4 15.4 46 45.2 41.1 ...
## $ wnsa1 : num 20.4 51.8 145.4 165.3 106 ...
## $ wnsa2 : num -39.8 -126.6 -580.1 -655.3 -416.3 ...
## [list output truncated]- Do any of the individual predictors have degenerate distributions?
nearZeroVar(bbbDescr)## [1] 3 16 17 22 25 50 60Several predictors exhibit near-zero variance, meaning they take almost the same value across most of the 208 compounds. Specifically, 7 predictors were identified as near-zero variance variables. These predictors contribute little useful information for modeling and can be safely removed to simplify the model and improve computational efficiency.
- Generally speaking, are there strong relationships between the predictor data? If so, how could correlations in the predictor set be reduced? Does this have a dramatic effect on the number of predictors available for modeling?
cor_matrix <- cor(bbbDescr)
highCorr <- findCorrelation(cor_matrix)
length(highCorr)## [1] 36The predictor variables show strong correlations among certain descriptors, which suggests redundancy in the dataset. These correlations can be reduced by removing predictors that exceed a specified correlation threshold. Applying correlation filtering substantially reduces the number of predictors while retaining most of the relevant information, helping to reduce multicollinearity without dramatically harming model performance.
- Brodnjak-Vonina et al. (2005) develop a methodology for food laboratories to determine the type of oil from a sample. In their procedure, they used a gas chromatograph (an instrument that separates chemicals in a sample) to measure seven different fatty acids in an oil. These measurements would then be used to predict the type of oil in food samples. To create their model, they used 96 samples2 of seven types of oils. These data can be found in the caret package using data(oil). The oil types are contained in a factor variable called oilType. The types are pumpkin (coded as A), sunflower (B), peanut (C), olive (D), soybean (E), rapeseed (F) and corn (G). In R,
data(oil)
str(oilType)## Factor w/ 7 levels "A","B","C","D",..: 1 1 1 1 1 1 1 1 1 1 ...table(oilType)## oilType
## A B C D E F G
## 37 26 3 7 11 10 2- Use the sample function in base R to create a completely random sample of 60 oils. How closely do the frequencies of the random sample match the original samples? Repeat this procedure several times of understand the variation in the sampling process.
set.seed(123)
randomSample <- sample(seq_along(oilType), 60)
table(oilType[randomSample])##
## A B C D E F G
## 24 17 3 3 6 5 2A completely random sample of 60 out of 96 oils does not preserve the original oil-type proportions. For example, pumpkin oil drops from 37 to 24 samples, while olive oil drops from 7 to 3. Repeating the sampling shows noticeable variation in class counts due to randomness, especially for rare oils like corn (2 total samples).
- Use the caret package function createDataPartition to create a stratified random sample. How does this compare to completely random samples?
stratSample <- createDataPartition(oilType, p = 60/length(oilType), list = FALSE)
table(oilType[stratSample])##
## A B C D E F G
## 24 17 2 5 7 7 2Using createDataPartition from the caret produces a stratified sample of 60 oils that closely matches the original distribution. For instance, pumpkin oil remains at 24 samples, and sunflower oil at 17, making the sample more representative and reducing sampling bias compared to random sampling.
- With such a small samples size, what are the options for determining performance of the model? Should a test set be used?
With a small dataset, using a separate test set may significantly reduce the amount of data available for model training. Alternative methods such as cross-validation, bootstrapping, or leave-one-out cross-validation (LOOCV) are often preferred. These techniques allow the model to be evaluated multiple times using different training and validation splits, which improves reliability while preserving limited data.
- One method for understanding the uncertainty of a test set is to use a confidence interval. To obtain a confidence interval for the overall accuracy, the based R function binom.test can be used. It requires the user to input the number of samples and the number correctly classified to calculate the interval. For example, suppose a test set sample of 20 oil samples was set aside and 76 were used for model training. For this test set size and a model that is about 80 % accurate (16 out of 20 correct), the confidence interval would be computed using
binom.test(16, 20)##
## Exact binomial test
##
## data: 16 and 20
## number of successes = 16, number of trials = 20, p-value = 0.01182
## alternative hypothesis: true probability of success is not equal to 0.5
## 95 percent confidence interval:
## 0.563386 0.942666
## sample estimates:
## probability of success
## 0.8In this case, the width of the 95% confidence interval is 37.9 %. Try different samples sizes and accuracy rates to understand the trade-off between the uncertainty in the results, the model performance, and the test set size.
binom.test(32, 40)##
## Exact binomial test
##
## data: 32 and 40
## number of successes = 32, number of trials = 40, p-value = 0.0001822
## alternative hypothesis: true probability of success is not equal to 0.5
## 95 percent confidence interval:
## 0.6435220 0.9094776
## sample estimates:
## probability of success
## 0.8binom.test(45, 50)##
## Exact binomial test
##
## data: 45 and 50
## number of successes = 45, number of trials = 50, p-value = 4.21e-09
## alternative hypothesis: true probability of success is not equal to 0.5
## 95 percent confidence interval:
## 0.7818646 0.9667249
## sample estimates:
## probability of success
## 0.9Confidence intervals show how certain we are about a model’s accuracy. When the test set is small, the confidence interval is wide, meaning the accuracy estimate is less reliable. For example, 80% accuracy on 20 samples (16/20) gives a wide 95% CI of 56.3%–94.3%. As the test set gets larger, the interval becomes narrower: 40 samples (32/40) gives 64.4%–90.9%, and 50 samples (45/50) gives 78.2%–96.7%. Larger test sets reduce uncertainty, but they leave less data for training, so there is a trade-off.
- Briefly discuss the bias–variance tradeoff in statistics and predictive modeling. According to your discussions, please draw a conceptual bias–variance tradeoff plot with x-axis being model complexity and y-axis being error.
complexity <- seq(1, 10, length = 100)
bias <- 1/(complexity)
variance <- complexity/10
total_error <- bias + variance
plot(complexity, bias, type="l", lty=2, ylim=c(0,2),
ylab="Error", xlab="Model Complexity")
lines(complexity, variance, lty=3)
lines(complexity, total_error, lty=1)
legend("topright",
legend=c("Bias","Variance","Total Error"),
lty=c(2,3,1))
The bias-variance tradeoff describes how model complexity impacts prediction error. Models with low complexity tend to have high bias because they oversimplify relationships in the data. Highly complex models have low bias but high variance because they may overfit the training data.In the visualization, the bias curve decreases as model complexity increases, while the variance curve increases. The total error curve is U-shaped and reaches its minimum showing the optimal model complexity. This point represents the best balance between bias and variance.