Homework 7
J John
6.2
(a)
## Loading required package: lattice## Loading required package: ggplot2## Loading required package: Rcpp## ##
## ## Amelia II: Multiple Imputation
## ## (Version 1.7.6, built: 2019-11-24)
## ## Copyright (C) 2005-2020 James Honaker, Gary King and Matthew Blackwell
## ## Refer to http://gking.harvard.edu/amelia/ for more information
## ##(b)
fingerprints dataset contains 1107 columns and 165 rows. 179 columns have near zero variance and 388 columns have non zero variance.
near0 <- nearZeroVar(fingerprints)
length(near0)## [1] 719ncol(fingerprints[, -near0])## [1] 388(c)
The optimal number of variable included in the PLS model is 10. This captures 72.48% of the variation in the predictors and 84.54% of the variation in the outcome variable. R^2 value for 10 components is 0.5534189
finger_data <- fingerprints[, -near0]
set.seed(1)
dp <- createDataPartition(permeability, p = 0.8, list = F, times = 1)
xtrain <- finger_data[dp, ]
xtest <- finger_data[-dp, ]
ytrain <- permeability[dp, ]
ytest <- permeability[-dp, ]
ctrl <- trainControl(method = "cv", number = 10)
pmod <- train(x = xtrain, y = ytrain,
method = "pls", tuneLength = 20,
trControl = ctrl,
preProc = c("center","scale"))
plot(pmod)
pmod$bestTunesummary(pmod$finalModel)## Data: X dimension: 133 388
## Y dimension: 133 1
## Fit method: oscorespls
## Number of components considered: 10
## TRAINING: % variance explained
## 1 comps 2 comps 3 comps 4 comps 5 comps 6 comps 7 comps
## X 21.09 34.70 40.33 45.02 51.49 58.92 62.07
## .outcome 34.57 51.73 59.70 66.80 72.46 76.41 79.79
## 8 comps 9 comps 10 comps
## X 66.69 69.49 72.48
## .outcome 81.40 83.47 84.54(d)
R^2 value is 0.3274115, which is worse than the train set R^2. Model might be overfitting to the train set.
predictions <- predict(pmod, xtest)
#RMSE
RMSE(predictions, ytest)## [1] 14.6758#R2
R2(predictions, ytest)## [1] 0.3274115(e)
Based on the R2 values below model, elastic net seems to be a better fit followed by ridge regression.
Elastic Net R2 : 0.5260718 Ridge Regression R2 : 0.5174488
set.seed(100)
ridgeGrid <- data.frame(.lambda = seq(0, .1, length = 15))
dp <- createDataPartition(permeability, p=0.8, list=FALSE)
xtrain <- finger_data[dp, ]
ytrain <- permeability[dp, ]
xtest <- finger_data[-dp, ]
ytest <- permeability[-dp, ]
enetmodel<- train(x=xtrain,y=ytrain,method='enet',
metric='Rsquared',
tuneLength=20,
trControl=trainControl(method='cv'),
preProcess=c('center', 'scale','knnImpute')
)## Warning: model fit failed for Fold08: lambda=0.0000000, fraction=1 Error in if (zmin < gamhat) { : missing value where TRUE/FALSE needed## Warning in nominalTrainWorkflow(x = x, y = y, wts = weights, info = trainInfo, :
## There were missing values in resampled performance measures.enetmodel## Elasticnet
##
## 133 samples
## 388 predictors
##
## Pre-processing: centered (388), scaled (388), nearest neighbor imputation (388)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 119, 121, 119, 120, 120, 120, ...
## Resampling results across tuning parameters:
##
## lambda fraction RMSE Rsquared MAE
## 0.0000000000 0.05 11.73026 0.5112930 8.880953e+00
## 0.0000000000 0.10 10.96060 0.5356455 8.156204e+00
## 0.0000000000 0.15 10.57113 0.5514554 7.781082e+00
## 0.0000000000 0.20 10.46411 0.5488368 7.722965e+00
## 0.0000000000 0.25 10.43059 0.5472689 7.662265e+00
## 0.0000000000 0.30 10.53928 0.5346824 7.712603e+00
## 0.0000000000 0.35 10.62082 0.5278685 7.739882e+00
## 0.0000000000 0.40 10.68500 0.5234071 7.873670e+00
## 0.0000000000 0.45 10.80291 0.5180283 8.019113e+00
## 0.0000000000 0.50 10.84283 0.5167099 8.058093e+00
## 0.0000000000 0.55 10.91289 0.5142547 8.107383e+00
## 0.0000000000 0.60 11.08055 0.5068495 8.252246e+00
## 0.0000000000 0.65 11.22801 0.5017611 8.395168e+00
## 0.0000000000 0.70 11.41195 0.4954862 8.578705e+00
## 0.0000000000 0.75 11.56735 0.4899989 8.715175e+00
## 0.0000000000 0.80 11.74535 0.4825328 8.842491e+00
## 0.0000000000 0.85 11.94446 0.4747595 8.975029e+00
## 0.0000000000 0.90 12.16190 0.4665919 9.122924e+00
## 0.0000000000 0.95 12.40922 0.4572871 9.300176e+00
## 0.0000000000 1.00 12.58892 0.4519220 9.346084e+00
## 0.0001000000 0.05 129.49363 0.2214589 6.781962e+01
## 0.0001000000 0.10 231.65992 0.2410610 1.209907e+02
## 0.0001000000 0.15 318.11068 0.2255764 1.720461e+02
## 0.0001000000 0.20 406.33944 0.2220196 2.255537e+02
## 0.0001000000 0.25 499.61126 0.2128017 2.792479e+02
## 0.0001000000 0.30 596.18842 0.1975489 3.325437e+02
## 0.0001000000 0.35 695.90779 0.1774370 3.863829e+02
## 0.0001000000 0.40 796.35336 0.1644050 4.400536e+02
## 0.0001000000 0.45 897.17326 0.1470067 4.933855e+02
## 0.0001000000 0.50 999.01415 0.1316774 5.462772e+02
## 0.0001000000 0.55 1100.79542 0.1236729 5.987292e+02
## 0.0001000000 0.60 1199.30351 0.1191616 6.491226e+02
## 0.0001000000 0.65 1294.23367 0.1184123 6.977653e+02
## 0.0001000000 0.70 1389.17023 0.1163315 7.462313e+02
## 0.0001000000 0.75 1484.12888 0.1137081 7.948504e+02
## 0.0001000000 0.80 1575.93324 0.1196733 8.416750e+02
## 0.0001000000 0.85 1666.04132 0.1239209 8.876706e+02
## 0.0001000000 0.90 1751.68372 0.1249225 9.315195e+02
## 0.0001000000 0.95 1832.48501 0.1260581 9.730975e+02
## 0.0001000000 1.00 1913.11248 0.1249479 1.014437e+03
## 0.0001467799 0.05 1000.48299 0.2530552 5.423035e+02
## 0.0001467799 0.10 1884.87769 0.2512423 1.027855e+03
## 0.0001467799 0.15 2707.69410 0.2422539 1.481979e+03
## 0.0001467799 0.20 3494.54974 0.2317828 1.914931e+03
## 0.0001467799 0.25 4279.83810 0.2210954 2.345487e+03
## 0.0001467799 0.30 4984.55002 0.2069287 2.728794e+03
## 0.0001467799 0.35 5683.97125 0.1936977 3.102438e+03
## 0.0001467799 0.40 6372.22299 0.1839923 3.458577e+03
## 0.0001467799 0.45 7039.66830 0.1693264 3.792883e+03
## 0.0001467799 0.50 7680.72308 0.1609887 4.086602e+03
## 0.0001467799 0.55 8322.05743 0.1529655 4.357807e+03
## 0.0001467799 0.60 8973.26649 0.1486384 4.740340e+03
## 0.0001467799 0.65 9602.54193 0.1422412 5.093666e+03
## 0.0001467799 0.70 10239.69853 0.1331670 5.433680e+03
## 0.0001467799 0.75 10865.53893 0.1255718 5.764683e+03
## 0.0001467799 0.80 11482.30187 0.1168392 6.080062e+03
## 0.0001467799 0.85 12097.53780 0.1121511 6.394341e+03
## 0.0001467799 0.90 12702.13036 0.1091484 6.700405e+03
## 0.0001467799 0.95 13291.80405 0.1055558 6.995199e+03
## 0.0001467799 1.00 13879.52277 0.1033603 7.288950e+03
## 0.0002154435 0.05 12699.28735 0.3166914 7.629310e+03
## 0.0002154435 0.10 22810.92097 0.3332481 1.381147e+04
## 0.0002154435 0.15 33379.05994 0.3231521 2.009753e+04
## 0.0002154435 0.20 43079.20289 0.3193254 2.598324e+04
## 0.0002154435 0.25 52769.40167 0.3112939 3.189438e+04
## 0.0002154435 0.30 62607.21143 0.2993253 3.785886e+04
## 0.0002154435 0.35 72528.98732 0.2863848 4.386765e+04
## 0.0002154435 0.40 82457.12288 0.2676579 4.987409e+04
## 0.0002154435 0.45 92063.17913 0.2425733 5.568503e+04
## 0.0002154435 0.50 100507.42922 0.2184252 6.095065e+04
## 0.0002154435 0.55 108270.27255 0.1913246 6.588700e+04
## 0.0002154435 0.60 116032.73320 0.1725215 7.080471e+04
## 0.0002154435 0.65 122977.82080 0.1583546 7.533387e+04
## 0.0002154435 0.70 129932.57414 0.1453684 7.977296e+04
## 0.0002154435 0.75 137238.13384 0.1340609 8.432719e+04
## 0.0002154435 0.80 144574.11743 0.1240208 8.882496e+04
## 0.0002154435 0.85 151551.05517 0.1130956 9.312778e+04
## 0.0002154435 0.90 158886.59065 0.1120048 9.762228e+04
## 0.0002154435 0.95 166981.91515 0.1130281 1.024669e+05
## 0.0002154435 1.00 175449.36812 0.1112229 1.074482e+05
## 0.0003162278 0.05 31.30934 0.4607036 1.912979e+01
## 0.0003162278 0.10 43.88456 0.4630086 2.847367e+01
## 0.0003162278 0.15 58.27657 0.4464020 3.903070e+01
## 0.0003162278 0.20 75.05823 0.4236235 5.174140e+01
## 0.0003162278 0.25 92.34641 0.3987150 6.480044e+01
## 0.0003162278 0.30 109.40882 0.3736252 7.710401e+01
## 0.0003162278 0.35 126.67606 0.3437560 8.933810e+01
## 0.0003162278 0.40 146.29682 0.3151006 1.028918e+02
## 0.0003162278 0.45 178.15888 0.2853031 1.229209e+02
## 0.0003162278 0.50 210.92601 0.2588831 1.421932e+02
## 0.0003162278 0.55 244.38817 0.2348774 1.610821e+02
## 0.0003162278 0.60 277.55044 0.2153709 1.793726e+02
## 0.0003162278 0.65 311.01230 0.2008296 1.985066e+02
## 0.0003162278 0.70 342.53784 0.1880802 2.163537e+02
## 0.0003162278 0.75 372.12129 0.1739390 2.327578e+02
## 0.0003162278 0.80 401.32922 0.1663224 2.490218e+02
## 0.0003162278 0.85 430.09411 0.1640584 2.650725e+02
## 0.0003162278 0.90 458.65386 0.1626614 2.808157e+02
## 0.0003162278 0.95 486.81258 0.1600761 2.961530e+02
## 0.0003162278 1.00 514.66187 0.1571215 3.112145e+02
## 0.0004641589 0.05 1449.98992 0.4175101 7.603304e+02
## 0.0004641589 0.10 2816.99025 0.4391738 1.480176e+03
## 0.0004641589 0.15 4142.28405 0.4256555 2.177221e+03
## 0.0004641589 0.20 5460.28428 0.4112536 2.868295e+03
## 0.0004641589 0.25 6779.37064 0.3984380 3.559323e+03
## 0.0004641589 0.30 8098.79019 0.3843961 4.249961e+03
## 0.0004641589 0.35 9394.84411 0.3641405 4.927150e+03
## 0.0004641589 0.40 10672.55710 0.3466935 5.593874e+03
## 0.0004641589 0.45 11989.60269 0.3268408 6.279629e+03
## 0.0004641589 0.50 13362.25602 0.3034455 6.989181e+03
## 0.0004641589 0.55 14686.62687 0.2818788 7.672847e+03
## 0.0004641589 0.60 16003.74729 0.2562062 8.353101e+03
## 0.0004641589 0.65 17321.42590 0.2401037 9.033376e+03
## 0.0004641589 0.70 18636.84667 0.2260205 9.711193e+03
## 0.0004641589 0.75 19917.12225 0.2111692 1.037077e+04
## 0.0004641589 0.80 21180.79170 0.2044938 1.101872e+04
## 0.0004641589 0.85 22438.70500 0.2019479 1.166299e+04
## 0.0004641589 0.90 23701.10393 0.1986715 1.230961e+04
## 0.0004641589 0.95 24966.15560 0.1952596 1.295612e+04
## 0.0004641589 1.00 26269.36403 0.1924903 1.362076e+04
## 0.0006812921 0.05 15202.39436 0.4568734 7.739529e+03
## 0.0006812921 0.10 27793.82374 0.4907443 1.434650e+04
## 0.0006812921 0.15 40492.69007 0.4840522 2.091724e+04
## 0.0006812921 0.20 53041.93118 0.4719306 2.748762e+04
## 0.0006812921 0.25 65406.46570 0.4618247 3.414555e+04
## 0.0006812921 0.30 77346.31015 0.4520317 4.064400e+04
## 0.0006812921 0.35 89037.49818 0.4330823 4.701525e+04
## 0.0006812921 0.40 100732.43530 0.4109373 5.338498e+04
## 0.0006812921 0.45 112425.25454 0.3898246 5.977154e+04
## 0.0006812921 0.50 123913.88433 0.3660525 6.604759e+04
## 0.0006812921 0.55 135399.08604 0.3363738 7.231555e+04
## 0.0006812921 0.60 146779.12135 0.3103539 7.853528e+04
## 0.0006812921 0.65 157219.60211 0.2863484 8.429266e+04
## 0.0006812921 0.70 167534.72350 0.2681377 8.994914e+04
## 0.0006812921 0.75 177708.94878 0.2519002 9.553771e+04
## 0.0006812921 0.80 187952.00754 0.2387073 1.011566e+05
## 0.0006812921 0.85 198291.83491 0.2257589 1.068013e+05
## 0.0006812921 0.90 208258.93692 0.2169298 1.122287e+05
## 0.0006812921 0.95 218021.33095 0.2126031 1.175453e+05
## 0.0006812921 1.00 227716.52056 0.2077198 1.228097e+05
## 0.0010000000 0.05 246.66872 0.4892400 1.181342e+02
## 0.0010000000 0.10 448.04571 0.5215124 2.155686e+02
## 0.0010000000 0.15 639.18278 0.5187544 3.081160e+02
## 0.0010000000 0.20 820.87656 0.5033338 3.960252e+02
## 0.0010000000 0.25 1002.59902 0.4858980 4.837712e+02
## 0.0010000000 0.30 1184.62117 0.4642999 5.717771e+02
## 0.0010000000 0.35 1367.06023 0.4436638 6.600518e+02
## 0.0010000000 0.40 1550.30411 0.4227287 7.487830e+02
## 0.0010000000 0.45 1732.29720 0.4059457 8.366983e+02
## 0.0010000000 0.50 1910.66306 0.3869240 9.219627e+02
## 0.0010000000 0.55 2089.07965 0.3649237 1.007230e+03
## 0.0010000000 0.60 2275.91882 0.3443069 1.097332e+03
## 0.0010000000 0.65 2459.91385 0.3216433 1.186626e+03
## 0.0010000000 0.70 2638.53644 0.2979140 1.271849e+03
## 0.0010000000 0.75 2812.47551 0.2804666 1.354641e+03
## 0.0010000000 0.80 2986.41973 0.2676600 1.437496e+03
## 0.0010000000 0.85 3159.74388 0.2515377 1.520850e+03
## 0.0010000000 0.90 3326.46211 0.2402413 1.601640e+03
## 0.0010000000 0.95 3492.39436 0.2315944 1.682008e+03
## 0.0010000000 1.00 3657.24133 0.2243417 1.759983e+03
## 0.0014677993 0.05 775.97009 0.3875490 5.212216e+02
## 0.0014677993 0.10 1662.65668 0.4141710 1.120952e+03
## 0.0014677993 0.15 2532.24969 0.4108568 1.700955e+03
## 0.0014677993 0.20 3365.47482 0.3981198 2.255316e+03
## 0.0014677993 0.25 4204.20528 0.3883312 2.814143e+03
## 0.0014677993 0.30 5051.88915 0.3811476 3.378064e+03
## 0.0014677993 0.35 5890.81313 0.3720068 3.935281e+03
## 0.0014677993 0.40 6726.29864 0.3639599 4.489978e+03
## 0.0014677993 0.45 7589.11924 0.3511131 5.061549e+03
## 0.0014677993 0.50 8454.65705 0.3386886 5.634972e+03
## 0.0014677993 0.55 9322.02428 0.3278018 6.209742e+03
## 0.0014677993 0.60 10189.93745 0.3120999 6.784621e+03
## 0.0014677993 0.65 11043.04686 0.2973018 7.349880e+03
## 0.0014677993 0.70 11899.28776 0.2824318 7.917763e+03
## 0.0014677993 0.75 12766.08153 0.2671000 8.494642e+03
## 0.0014677993 0.80 13644.61059 0.2555518 9.079919e+03
## 0.0014677993 0.85 14520.44665 0.2484996 9.663155e+03
## 0.0014677993 0.90 15348.19851 0.2410916 1.021362e+04
## 0.0014677993 0.95 16150.58069 0.2325383 1.074769e+04
## 0.0014677993 1.00 16947.29789 0.2249940 1.127704e+04
## 0.0021544347 0.05 205.24610 0.4456061 1.114135e+02
## 0.0021544347 0.10 392.19285 0.4850667 2.121032e+02
## 0.0021544347 0.15 577.27185 0.4897111 3.119985e+02
## 0.0021544347 0.20 758.69648 0.4789021 4.107415e+02
## 0.0021544347 0.25 940.45135 0.4667007 5.091743e+02
## 0.0021544347 0.30 1121.91770 0.4500824 6.073445e+02
## 0.0021544347 0.35 1302.60148 0.4373027 7.046563e+02
## 0.0021544347 0.40 1480.99046 0.4257033 8.021905e+02
## 0.0021544347 0.45 1658.48672 0.4073388 8.999217e+02
## 0.0021544347 0.50 1835.79788 0.3896755 9.974852e+02
## 0.0021544347 0.55 2012.91122 0.3770597 1.094838e+03
## 0.0021544347 0.60 2190.09589 0.3598401 1.192180e+03
## 0.0021544347 0.65 2367.31729 0.3425238 1.289501e+03
## 0.0021544347 0.70 2544.50407 0.3248600 1.386731e+03
## 0.0021544347 0.75 2721.64294 0.3093395 1.483941e+03
## 0.0021544347 0.80 2898.69976 0.2970171 1.581088e+03
## 0.0021544347 0.85 3075.58971 0.2884081 1.678141e+03
## 0.0021544347 0.90 3252.48584 0.2805309 1.775184e+03
## 0.0021544347 0.95 3429.26214 0.2719010 1.872118e+03
## 0.0021544347 1.00 3605.96702 0.2645014 1.968988e+03
## 0.0031622777 0.05 188.67302 0.4346400 1.196818e+02
## 0.0031622777 0.10 363.57651 0.4617011 2.311745e+02
## 0.0031622777 0.15 530.90825 0.4701175 3.378439e+02
## 0.0031622777 0.20 687.88264 0.4626100 4.346974e+02
## 0.0031622777 0.25 827.29123 0.4440860 5.208155e+02
## 0.0031622777 0.30 965.08596 0.4316220 6.061613e+02
## 0.0031622777 0.35 1102.67160 0.4213565 6.919498e+02
## 0.0031622777 0.40 1240.35469 0.4104015 7.799244e+02
## 0.0031622777 0.45 1378.82725 0.3932768 8.687357e+02
## 0.0031622777 0.50 1519.15270 0.3758102 9.581927e+02
## 0.0031622777 0.55 1659.91847 0.3572522 1.047278e+03
## 0.0031622777 0.60 1800.66386 0.3422289 1.136258e+03
## 0.0031622777 0.65 1941.43624 0.3295896 1.225193e+03
## 0.0031622777 0.70 2082.33812 0.3137405 1.314218e+03
## 0.0031622777 0.75 2223.24901 0.3008764 1.403217e+03
## 0.0031622777 0.80 2364.18677 0.2894144 1.492230e+03
## 0.0031622777 0.85 2505.15470 0.2794706 1.581231e+03
## 0.0031622777 0.90 2646.10641 0.2718022 1.670239e+03
## 0.0031622777 0.95 2787.05113 0.2664072 1.759234e+03
## 0.0031622777 1.00 2928.05132 0.2597736 1.848225e+03
## 0.0046415888 0.05 318.32099 0.4296604 2.344044e+02
## 0.0046415888 0.10 614.23175 0.4617003 4.784158e+02
## 0.0046415888 0.15 848.33140 0.4701866 6.750916e+02
## 0.0046415888 0.20 1094.86347 0.4662475 8.719617e+02
## 0.0046415888 0.25 1340.81841 0.4605304 1.064184e+03
## 0.0046415888 0.30 1584.53102 0.4478026 1.252077e+03
## 0.0046415888 0.35 1828.35093 0.4317517 1.441328e+03
## 0.0046415888 0.40 2072.80367 0.4161804 1.629844e+03
## 0.0046415888 0.45 2305.66048 0.4030888 1.811358e+03
## 0.0046415888 0.50 2539.32084 0.3858688 1.991711e+03
## 0.0046415888 0.55 2777.74596 0.3717939 2.169618e+03
## 0.0046415888 0.60 3014.40143 0.3594848 2.342953e+03
## 0.0046415888 0.65 3254.48506 0.3494057 2.520749e+03
## 0.0046415888 0.70 3497.74819 0.3399113 2.702955e+03
## 0.0046415888 0.75 3743.12265 0.3292750 2.886047e+03
## 0.0046415888 0.80 4004.19349 0.3181564 3.076278e+03
## 0.0046415888 0.85 4260.60871 0.3074471 3.262650e+03
## 0.0046415888 0.90 4495.76207 0.2985095 3.436852e+03
## 0.0046415888 0.95 4722.40695 0.2917455 3.606372e+03
## 0.0046415888 1.00 4953.74771 0.2846376 3.777918e+03
## 0.0068129207 0.05 603.13491 0.4827650 3.908896e+02
## 0.0068129207 0.10 1137.95832 0.5249078 7.377858e+02
## 0.0068129207 0.15 1674.12432 0.5322026 1.084942e+03
## 0.0068129207 0.20 2212.26423 0.5210346 1.431009e+03
## 0.0068129207 0.25 2750.45893 0.5136112 1.777070e+03
## 0.0068129207 0.30 3288.95376 0.4993016 2.123235e+03
## 0.0068129207 0.35 3827.63315 0.4821651 2.469499e+03
## 0.0068129207 0.40 4366.30629 0.4673629 2.815771e+03
## 0.0068129207 0.45 4904.97181 0.4556873 3.162020e+03
## 0.0068129207 0.50 5438.63882 0.4456865 3.505443e+03
## 0.0068129207 0.55 5963.50426 0.4319849 3.843796e+03
## 0.0068129207 0.60 6488.40950 0.4201895 4.182112e+03
## 0.0068129207 0.65 7013.30617 0.4102068 4.520453e+03
## 0.0068129207 0.70 7538.21752 0.4022256 4.858779e+03
## 0.0068129207 0.75 8063.11394 0.3949867 5.197046e+03
## 0.0068129207 0.80 8588.06942 0.3848236 5.535327e+03
## 0.0068129207 0.85 9113.03955 0.3761710 5.873632e+03
## 0.0068129207 0.90 9635.44732 0.3685180 6.210530e+03
## 0.0068129207 0.95 10140.10454 0.3614760 6.536466e+03
## 0.0068129207 1.00 10641.02425 0.3550964 6.859991e+03
## 0.0100000000 0.05 11.29834 0.4796733 8.076048e+00
## 0.0100000000 0.10 10.87626 0.5036502 7.780892e+00
## 0.0100000000 0.15 10.68853 0.5202723 7.903218e+00
## 0.0100000000 0.20 10.80087 0.5154918 8.089678e+00
## 0.0100000000 0.25 10.97174 0.5071605 8.203831e+00
## 0.0100000000 0.30 11.15696 0.4949747 8.434626e+00
## 0.0100000000 0.35 11.41231 0.4802925 8.640441e+00
## 0.0100000000 0.40 11.63302 0.4686331 8.850980e+00
## 0.0100000000 0.45 11.83618 0.4586729 9.032995e+00
## 0.0100000000 0.50 12.08237 0.4480545 9.235739e+00
## 0.0100000000 0.55 12.34545 0.4364731 9.427570e+00
## 0.0100000000 0.60 12.61046 0.4239076 9.640066e+00
## 0.0100000000 0.65 12.83792 0.4145138 9.864298e+00
## 0.0100000000 0.70 13.02992 0.4074144 1.006788e+01
## 0.0100000000 0.75 13.23825 0.3991008 1.023228e+01
## 0.0100000000 0.80 13.44113 0.3905248 1.040632e+01
## 0.0100000000 0.85 13.64563 0.3819838 1.057811e+01
## 0.0100000000 0.90 13.83363 0.3744453 1.073160e+01
## 0.0100000000 0.95 14.01363 0.3676065 1.087797e+01
## 0.0100000000 1.00 14.17464 0.3612613 1.100950e+01
## 0.0146779927 0.05 11.40594 0.4771448 8.271002e+00
## 0.0146779927 0.10 10.95729 0.4965565 7.791458e+00
## 0.0146779927 0.15 10.69917 0.5204905 7.856950e+00
## 0.0146779927 0.20 10.74037 0.5221544 8.029186e+00
## 0.0146779927 0.25 10.91493 0.5144603 8.158566e+00
## 0.0146779927 0.30 10.99702 0.5102279 8.225515e+00
## 0.0146779927 0.35 11.23636 0.4945261 8.452339e+00
## 0.0146779927 0.40 11.48136 0.4794289 8.661418e+00
## 0.0146779927 0.45 11.66890 0.4689460 8.823483e+00
## 0.0146779927 0.50 11.80194 0.4629783 8.911880e+00
## 0.0146779927 0.55 11.98273 0.4551318 9.073445e+00
## 0.0146779927 0.60 12.17358 0.4462614 9.271517e+00
## 0.0146779927 0.65 12.36920 0.4371468 9.463801e+00
## 0.0146779927 0.70 12.53216 0.4301381 9.628363e+00
## 0.0146779927 0.75 12.70427 0.4217997 9.800881e+00
## 0.0146779927 0.80 12.86940 0.4137668 9.953121e+00
## 0.0146779927 0.85 13.01690 0.4065135 1.006999e+01
## 0.0146779927 0.90 13.15913 0.3998447 1.020093e+01
## 0.0146779927 0.95 13.31056 0.3931108 1.033477e+01
## 0.0146779927 1.00 13.45090 0.3868010 1.046140e+01
## 0.0215443469 0.05 11.53570 0.4711752 8.474404e+00
## 0.0215443469 0.10 11.02967 0.4911782 7.768018e+00
## 0.0215443469 0.15 10.77500 0.5140922 7.818451e+00
## 0.0215443469 0.20 10.66846 0.5275783 7.926709e+00
## 0.0215443469 0.25 10.82474 0.5177926 8.107751e+00
## 0.0215443469 0.30 10.92077 0.5149211 8.198932e+00
## 0.0215443469 0.35 11.03158 0.5085305 8.312899e+00
## 0.0215443469 0.40 11.27002 0.4928566 8.490469e+00
## 0.0215443469 0.45 11.45692 0.4810452 8.647676e+00
## 0.0215443469 0.50 11.60504 0.4721592 8.778292e+00
## 0.0215443469 0.55 11.75377 0.4647739 8.903048e+00
## 0.0215443469 0.60 11.90192 0.4588195 9.049630e+00
## 0.0215443469 0.65 12.05233 0.4518967 9.204383e+00
## 0.0215443469 0.70 12.19845 0.4457200 9.345317e+00
## 0.0215443469 0.75 12.34000 0.4396185 9.485869e+00
## 0.0215443469 0.80 12.46937 0.4335937 9.600438e+00
## 0.0215443469 0.85 12.58611 0.4280684 9.704150e+00
## 0.0215443469 0.90 12.69986 0.4222588 9.798415e+00
## 0.0215443469 0.95 12.82048 0.4162308 9.901575e+00
## 0.0215443469 1.00 12.92701 0.4108419 9.998926e+00
## 0.0316227766 0.05 11.57860 0.4768285 8.568532e+00
## 0.0316227766 0.10 11.02619 0.4933007 7.710450e+00
## 0.0316227766 0.15 10.78639 0.5105980 7.756459e+00
## 0.0316227766 0.20 10.67788 0.5236193 7.857510e+00
## 0.0316227766 0.25 10.77487 0.5212057 8.032874e+00
## 0.0316227766 0.30 10.91919 0.5121925 8.146848e+00
## 0.0316227766 0.35 11.00861 0.5092312 8.255047e+00
## 0.0316227766 0.40 11.12213 0.5026624 8.364224e+00
## 0.0316227766 0.45 11.30673 0.4900217 8.507484e+00
## 0.0316227766 0.50 11.46235 0.4806370 8.624876e+00
## 0.0316227766 0.55 11.59829 0.4733022 8.741139e+00
## 0.0316227766 0.60 11.72230 0.4677667 8.844155e+00
## 0.0316227766 0.65 11.85604 0.4629895 8.992123e+00
## 0.0316227766 0.70 11.98829 0.4576117 9.141715e+00
## 0.0316227766 0.75 12.10576 0.4519980 9.262573e+00
## 0.0316227766 0.80 12.22263 0.4460353 9.377797e+00
## 0.0316227766 0.85 12.34614 0.4396814 9.495285e+00
## 0.0316227766 0.90 12.45809 0.4338052 9.590540e+00
## 0.0316227766 0.95 12.55731 0.4283673 9.670794e+00
## 0.0316227766 1.00 12.64413 0.4234744 9.745868e+00
## 0.0464158883 0.05 11.75045 0.4676929 8.774863e+00
## 0.0464158883 0.10 11.15701 0.4865227 7.784390e+00
## 0.0464158883 0.15 10.92231 0.5024384 7.755724e+00
## 0.0464158883 0.20 10.75540 0.5206385 7.841510e+00
## 0.0464158883 0.25 10.75649 0.5237036 7.967077e+00
## 0.0464158883 0.30 10.87256 0.5177596 8.112499e+00
## 0.0464158883 0.35 10.98034 0.5122208 8.216075e+00
## 0.0464158883 0.40 11.08009 0.5074762 8.303757e+00
## 0.0464158883 0.45 11.19232 0.5014066 8.390451e+00
## 0.0464158883 0.50 11.32431 0.4938889 8.485132e+00
## 0.0464158883 0.55 11.42710 0.4888996 8.567765e+00
## 0.0464158883 0.60 11.51728 0.4851493 8.651735e+00
## 0.0464158883 0.65 11.60452 0.4822867 8.745361e+00
## 0.0464158883 0.70 11.70284 0.4785988 8.855946e+00
## 0.0464158883 0.75 11.79221 0.4748525 8.962282e+00
## 0.0464158883 0.80 11.87249 0.4712862 9.053282e+00
## 0.0464158883 0.85 11.95418 0.4674740 9.135571e+00
## 0.0464158883 0.90 12.03289 0.4635807 9.214207e+00
## 0.0464158883 0.95 12.10071 0.4598085 9.275421e+00
## 0.0464158883 1.00 12.16115 0.4560467 9.326185e+00
## 0.0681292069 0.05 11.83764 0.4677199 8.902602e+00
## 0.0681292069 0.10 11.19073 0.4862580 7.835392e+00
## 0.0681292069 0.15 11.00591 0.4975823 7.720331e+00
## 0.0681292069 0.20 10.83825 0.5141903 7.808698e+00
## 0.0681292069 0.25 10.76865 0.5244606 7.906879e+00
## 0.0681292069 0.30 10.83362 0.5220736 8.038093e+00
## 0.0681292069 0.35 10.95142 0.5157257 8.162365e+00
## 0.0681292069 0.40 11.06430 0.5107080 8.266528e+00
## 0.0681292069 0.45 11.16608 0.5063722 8.353434e+00
## 0.0681292069 0.50 11.26208 0.5015397 8.426030e+00
## 0.0681292069 0.55 11.35680 0.4969997 8.494122e+00
## 0.0681292069 0.60 11.42652 0.4949126 8.566354e+00
## 0.0681292069 0.65 11.49245 0.4930455 8.630782e+00
## 0.0681292069 0.70 11.54532 0.4918683 8.713480e+00
## 0.0681292069 0.75 11.60353 0.4897735 8.787104e+00
## 0.0681292069 0.80 11.66567 0.4874048 8.853747e+00
## 0.0681292069 0.85 11.72960 0.4847556 8.919609e+00
## 0.0681292069 0.90 11.78494 0.4822123 8.976844e+00
## 0.0681292069 0.95 11.83245 0.4795878 9.025969e+00
## 0.0681292069 1.00 11.88503 0.4769072 9.069016e+00
## 0.1000000000 0.05 12.07598 0.4534109 9.207363e+00
## 0.1000000000 0.10 12.54107 0.4278885 8.990073e+00
## 0.1000000000 0.15 13.41247 0.4214700 9.575646e+00
## 0.1000000000 0.20 14.26380 0.4291633 1.042240e+01
## 0.1000000000 0.25 15.10993 0.4408117 1.124950e+01
## 0.1000000000 0.30 16.02829 0.4432588 1.209341e+01
## 0.1000000000 0.35 17.02230 0.4400193 1.292993e+01
## 0.1000000000 0.40 18.02820 0.4359747 1.376510e+01
## 0.1000000000 0.45 19.05102 0.4315621 1.458239e+01
## 0.1000000000 0.50 20.07065 0.4283229 1.540002e+01
## 0.1000000000 0.55 21.08059 0.4261541 1.621078e+01
## 0.1000000000 0.60 22.08132 0.4253826 1.701786e+01
## 0.1000000000 0.65 23.07332 0.4248939 1.781498e+01
## 0.1000000000 0.70 24.04011 0.4256316 1.859662e+01
## 0.1000000000 0.75 24.99168 0.4274110 1.937462e+01
## 0.1000000000 0.80 25.95704 0.4280401 2.014386e+01
## 0.1000000000 0.85 26.92573 0.4280844 2.092443e+01
## 0.1000000000 0.90 27.89232 0.4278613 2.170966e+01
## 0.1000000000 0.95 28.86043 0.4272247 2.248800e+01
## 0.1000000000 1.00 29.83380 0.4256524 2.326592e+01
##
## Rsquared was used to select the optimal model using the largest value.
## The final values used for the model were fraction = 0.15 and lambda = 0.plot(enetmodel)
predictions <- predict(enetmodel, xtest)
#RMSE
RMSE(predictions, ytest)## [1] 11.19975#R2
R2(predictions, ytest)## [1] 0.5260718ridgeGrid <- data.frame(.lambda = seq(0, .1, length = 15))
set.seed(100)
ridgeModel <- train(xtrain, ytrain,
method = "ridge", tuneGrid = ridgeGrid,
trControl = ctrl, preProc = c("center", "scale", "knnImpute"))
ridgeModel## Ridge Regression
##
## 133 samples
## 388 predictors
##
## Pre-processing: centered (388), scaled (388), nearest neighbor imputation (388)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 120, 120, 119, 121, 120, 119, ...
## Resampling results across tuning parameters:
##
## lambda RMSE Rsquared MAE
## 0.000000000 13.56846 0.3858837 10.092990
## 0.007142857 15.60215 0.2689935 11.569288
## 0.014285714 115.84292 0.3196042 82.969549
## 0.021428571 13.70226 0.3589300 10.306764
## 0.028571429 13.43111 0.3751570 10.118236
## 0.035714286 13.10860 0.3906398 9.825295
## 0.042857143 12.83212 0.4061188 9.623303
## 0.050000000 12.70364 0.4137372 9.503849
## 0.057142857 12.57114 0.4223368 9.407583
## 0.064285714 12.39352 0.4333250 9.282137
## 0.071428571 12.32787 0.4382032 9.225255
## 0.078571429 12.22797 0.4449271 9.148902
## 0.085714286 12.16429 0.4496660 9.092823
## 0.092857143 12.11078 0.4540054 9.042764
## 0.100000000 12.07131 0.4575539 9.008745
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was lambda = 0.1.plot(ridgeModel)
predictions <- predict(ridgeModel, xtest)
#RMSE
RMSE(predictions, ytest)## [1] 11.81689#R2
R2(predictions, ytest)## [1] 0.5174488(f)
Based on the R2 values of different regression methods, we see that these model only explains about 50% variation in the data. Better approaches would be to fit different models with better R2 values and see if they perform any better. With the limited knowledge about the problem, we dont know enough about it come to a conclusion.
6.3
(a)
data("ChemicalManufacturingProcess")(b)
Below figures display missing values before and after data imputation.
Centering and scaling the data KNN to replace missing values Remove highly correlated predictors Remove near zero variance predictors
missmap(ChemicalManufacturingProcess, col = c("red", "blue"))
pdata <- preProcess(ChemicalManufacturingProcess[,-1], method = c("center", "scale", "knnImpute", "corr", "nzv"))
chemdata <- predict(pdata, ChemicalManufacturingProcess[,-1])
missmap(chemdata, col = c("red", "blue"))
### (c) Optimal value:
The lowest point in the curve indicates the optimal lambda is : 0.007142857 and R2 : 0.4517271
set.seed(100)
dp <- createDataPartition(ChemicalManufacturingProcess$Yield, p=0.8, list=FALSE)
xtrain <- chemdata[dp, ]
ytrain <- ChemicalManufacturingProcess$Yield[dp]
xtest <- chemdata[-dp, ]
ytest <- ChemicalManufacturingProcess$Yield[-dp]
ridgeGrid <- data.frame(.lambda = seq(0, .1, length = 15))
ridge <- train(x=xtrain ,
y=ytrain,
method='ridge',
metric='RMSE',
tuneGrid=ridgeGrid,
trControl=trainControl(method='cv'),
preProcess=c('center','scale', 'knnImpute')
)
ridge## Ridge Regression
##
## 144 samples
## 56 predictor
##
## Pre-processing: centered (56), scaled (56), nearest neighbor imputation (56)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 130, 130, 130, 130, 130, 129, ...
## Resampling results across tuning parameters:
##
## lambda RMSE Rsquared MAE
## 0.000000000 2.465189 0.3529796 1.590211
## 0.007142857 1.844616 0.4517271 1.288468
## 0.014285714 1.965186 0.4633138 1.291750
## 0.021428571 2.026802 0.4738772 1.292381
## 0.028571429 2.062682 0.4808774 1.295481
## 0.035714286 2.084540 0.4855553 1.298670
## 0.042857143 2.097834 0.4887993 1.300154
## 0.050000000 2.105530 0.4911412 1.300773
## 0.057142857 2.109403 0.4928983 1.300648
## 0.064285714 2.110588 0.4942650 1.300083
## 0.071428571 2.109848 0.4953636 1.299394
## 0.078571429 2.107714 0.4962733 1.298385
## 0.085714286 2.104563 0.4970469 1.297062
## 0.092857143 2.100673 0.4977201 1.295515
## 0.100000000 2.096249 0.4983178 1.294485
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was lambda = 0.007142857.plot(ridge)
ridge$bestTune(d)
Test R2 is 0.16 and train R2 0.45. Model might be overfitting to the training set. RMSE = 3.202444 R2 = 0.1689975
predictions <- predict(ridge, xtest)
#RMSE
RMSE(predictions, ytest)## [1] 3.202444#R2
R2(predictions, ytest)## [1] 0.1689975(e)
Based on table below we see that ManufacturingProcess13 and ManufacturingProcess32 dominate the list
varImp(ridge)## loess r-squared variable importance
##
## only 20 most important variables shown (out of 56)
##
## Overall
## ManufacturingProcess32 100.00
## ManufacturingProcess13 97.84
## BiologicalMaterial06 82.22
## ManufacturingProcess17 77.27
## BiologicalMaterial03 76.21
## ManufacturingProcess36 70.77
## BiologicalMaterial02 68.79
## ManufacturingProcess09 67.86
## BiologicalMaterial12 63.36
## ManufacturingProcess06 55.15
## BiologicalMaterial04 54.31
## ManufacturingProcess33 49.26
## ManufacturingProcess31 47.73
## ManufacturingProcess11 45.72
## BiologicalMaterial11 42.44
## BiologicalMaterial08 41.89
## ManufacturingProcess29 41.28
## BiologicalMaterial01 41.19
## BiologicalMaterial09 39.70
## ManufacturingProcess02 36.69plot(varImp(ridge))