HW 8 624
Iden Watanabe
April 3, 2019
require(mlbench)
require(caret)
require(neuralnet)
require(earth)
require(e1071)
require(kernlab)
require(elasticnet)
require(AppliedPredictiveModeling)7.2 Friedman (1991) introduced several benchmark data sets created by simulation. One of these simulations used the following nonlinear equation to create data:
\[\begin{equation} y = 10sin(\pi x_{1} x_{2}) + 20(x_{3} - 0.5)^2 + 10x_{4} + 5x_{5} + N(0, \sigma^2)\end{equation}\]
where the x values are random variables uniformly distributed between [0, 1] (there are also 5 other non-informative variables also created in the simulation). The package mlbench contains a function called mlbench.friedman1 that simulates these data:
set.seed(200)
trainingData <- mlbench.friedman1(200, sd = 1)
# Convert the 'x' data from a matrix to a data frame
trainingData$x <- data.frame(trainingData$x)
# Examine data using
featurePlot(trainingData$x, trainingData$y)
# Create a list with a vector 'y' and a matrix of predictors 'x'.
# Also simulate a large test set to estimate the true error rate with good precision:
testData <- mlbench.friedman1(5000, sd = 1)
testData$x <- data.frame(testData$x)Tune several models on these data. Which models appear to give the best performance? Does MARS select the informative predictors (those named X1 - X5)?
nnetGrid = expand.grid(.decay = c(0, 0.01, .1),
.size = c(1:10),
.bag = FALSE)
ctrl <- trainControl(method = "cv", number = 10)
nnetTune <- train(trainingData$x, trainingData$y,
method = "avNNet",
tuneGrid = nnetGrid,
trControl = ctrl,
preProc = c("center", "scale"),
linout = TRUE,
trace = FALSE,
MaxNWts = 10 * (ncol(trainingData$x) + 1) + 10 + 1,
maxit = 500)## Warning: executing %dopar% sequentially: no parallel backend registerednnetTune## Model Averaged Neural Network
##
## 200 samples
## 10 predictor
##
## Pre-processing: centered (10), scaled (10)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 180, 180, 180, 180, 180, 180, ...
## Resampling results across tuning parameters:
##
## decay size RMSE Rsquared MAE
## 0.00 1 2.427152 0.7727079 1.877410
## 0.00 2 2.434282 0.7673318 1.912576
## 0.00 3 2.055221 0.8290840 1.631535
## 0.00 4 1.962213 0.8453236 1.555980
## 0.00 5 2.450287 0.7765281 1.903426
## 0.00 6 3.053444 0.7112187 2.071329
## 0.00 7 3.585838 0.6846674 2.430196
## 0.00 8 5.332821 0.4212507 3.491635
## 0.00 9 5.978051 0.5547556 3.038404
## 0.00 10 3.215563 0.6188769 2.411575
## 0.01 1 2.398768 0.7703245 1.871203
## 0.01 2 2.388086 0.7723843 1.875899
## 0.01 3 2.068530 0.8262502 1.671387
## 0.01 4 1.996444 0.8370066 1.592990
## 0.01 5 1.989574 0.8437813 1.588518
## 0.01 6 2.270438 0.7957695 1.869313
## 0.01 7 2.139343 0.8244790 1.690482
## 0.01 8 2.231947 0.7999189 1.762487
## 0.01 9 2.415645 0.7749531 1.967882
## 0.01 10 2.457774 0.7645570 1.952877
## 0.10 1 2.403697 0.7687044 1.869519
## 0.10 2 2.429034 0.7668241 1.867627
## 0.10 3 2.089436 0.8255526 1.640594
## 0.10 4 2.015963 0.8373563 1.618945
## 0.10 5 2.091169 0.8212449 1.648915
## 0.10 6 2.161118 0.8140285 1.727497
## 0.10 7 2.346362 0.7889072 1.870386
## 0.10 8 2.274714 0.7939365 1.842615
## 0.10 9 2.366441 0.7818157 1.876866
## 0.10 10 2.318806 0.7794772 1.845449
##
## Tuning parameter 'bag' was held constant at a value of FALSE
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were size = 4, decay = 0 and bag
## = FALSE.nnetTune.pred <- predict(nnetTune, newdata = testData$x)
postResample(pred = nnetTune.pred, obs = testData$y)## RMSE Rsquared MAE
## 2.2427881 0.8038198 1.6630160marsGrid <- expand.grid(.degree = 1:2, .nprune = 2:38)
marsTuned <- train(trainingData$x, trainingData$y,
method = "earth",
tuneGrid = marsGrid,
trControl = trainControl(method = "cv"))
marsTuned## Multivariate Adaptive Regression Spline
##
## 200 samples
## 10 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 180, 180, 180, 180, 180, 180, ...
## Resampling results across tuning parameters:
##
## degree nprune RMSE Rsquared MAE
## 1 2 4.212446 0.3021419 3.4716858
## 1 3 3.472717 0.5129590 2.7594453
## 1 4 2.652065 0.7328246 2.0910260
## 1 5 2.490567 0.7690196 1.9306226
## 1 6 2.380927 0.7860229 1.8471215
## 1 7 1.880648 0.8646044 1.4552889
## 1 8 1.785272 0.8809243 1.4177585
## 1 9 1.718901 0.8866237 1.3399928
## 1 10 1.683972 0.8893395 1.2967812
## 1 11 1.664429 0.8920935 1.2693977
## 1 12 1.677539 0.8905058 1.2812632
## 1 13 1.658941 0.8941993 1.2597960
## 1 14 1.658941 0.8941993 1.2597960
## 1 15 1.658941 0.8941993 1.2597960
## 1 16 1.658941 0.8941993 1.2597960
## 1 17 1.658941 0.8941993 1.2597960
## 1 18 1.658941 0.8941993 1.2597960
## 1 19 1.658941 0.8941993 1.2597960
## 1 20 1.658941 0.8941993 1.2597960
## 1 21 1.658941 0.8941993 1.2597960
## 1 22 1.658941 0.8941993 1.2597960
## 1 23 1.658941 0.8941993 1.2597960
## 1 24 1.658941 0.8941993 1.2597960
## 1 25 1.658941 0.8941993 1.2597960
## 1 26 1.658941 0.8941993 1.2597960
## 1 27 1.658941 0.8941993 1.2597960
## 1 28 1.658941 0.8941993 1.2597960
## 1 29 1.658941 0.8941993 1.2597960
## 1 30 1.658941 0.8941993 1.2597960
## 1 31 1.658941 0.8941993 1.2597960
## 1 32 1.658941 0.8941993 1.2597960
## 1 33 1.658941 0.8941993 1.2597960
## 1 34 1.658941 0.8941993 1.2597960
## 1 35 1.658941 0.8941993 1.2597960
## 1 36 1.658941 0.8941993 1.2597960
## 1 37 1.658941 0.8941993 1.2597960
## 1 38 1.658941 0.8941993 1.2597960
## 2 2 4.212446 0.3021419 3.4716858
## 2 3 3.472717 0.5129590 2.7594453
## 2 4 2.693135 0.7245626 2.1305231
## 2 5 2.442820 0.7686431 1.9148821
## 2 6 2.344253 0.7846620 1.8172550
## 2 7 1.909792 0.8580800 1.4504219
## 2 8 1.733187 0.8809616 1.3349596
## 2 9 1.630642 0.8955227 1.2663309
## 2 10 1.511062 0.9113611 1.1657024
## 2 11 1.367457 0.9249030 1.0489712
## 2 12 1.303125 0.9319309 1.0039954
## 2 13 1.313746 0.9319897 1.0044841
## 2 14 1.304182 0.9341236 0.9987217
## 2 15 1.331727 0.9321630 1.0189439
## 2 16 1.335661 0.9327188 1.0209595
## 2 17 1.358706 0.9293399 1.0402368
## 2 18 1.358706 0.9293399 1.0402368
## 2 19 1.358706 0.9293399 1.0402368
## 2 20 1.358706 0.9293399 1.0402368
## 2 21 1.358706 0.9293399 1.0402368
## 2 22 1.358706 0.9293399 1.0402368
## 2 23 1.358706 0.9293399 1.0402368
## 2 24 1.358706 0.9293399 1.0402368
## 2 25 1.358706 0.9293399 1.0402368
## 2 26 1.358706 0.9293399 1.0402368
## 2 27 1.358706 0.9293399 1.0402368
## 2 28 1.358706 0.9293399 1.0402368
## 2 29 1.358706 0.9293399 1.0402368
## 2 30 1.358706 0.9293399 1.0402368
## 2 31 1.358706 0.9293399 1.0402368
## 2 32 1.358706 0.9293399 1.0402368
## 2 33 1.358706 0.9293399 1.0402368
## 2 34 1.358706 0.9293399 1.0402368
## 2 35 1.358706 0.9293399 1.0402368
## 2 36 1.358706 0.9293399 1.0402368
## 2 37 1.358706 0.9293399 1.0402368
## 2 38 1.358706 0.9293399 1.0402368
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 12 and degree = 2.marsTuned.pred <- predict(marsTuned, newdata = testData$x)
postResample(pred = marsTuned.pred, obs = testData$y)## RMSE Rsquared MAE
## 1.2803060 0.9335241 1.0168673svmRTuned <- train(trainingData$x, trainingData$y,
method = "svmRadial",
preProc = c("center", "scale"),
tuneLength = 14,
trControl = trainControl(method = "cv"))
svmRTuned## Support Vector Machines with Radial Basis Function Kernel
##
## 200 samples
## 10 predictor
##
## Pre-processing: centered (10), scaled (10)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 180, 180, 180, 180, 180, 180, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 2.510684 0.8103623 1.983875
## 0.50 2.244988 0.8234750 1.758509
## 1.00 2.055720 0.8428596 1.609290
## 2.00 1.952241 0.8516736 1.515053
## 4.00 1.896963 0.8579892 1.495612
## 8.00 1.896443 0.8580178 1.498113
## 16.00 1.893653 0.8585728 1.496280
## 32.00 1.893653 0.8585728 1.496280
## 64.00 1.893653 0.8585728 1.496280
## 128.00 1.893653 0.8585728 1.496280
## 256.00 1.893653 0.8585728 1.496280
## 512.00 1.893653 0.8585728 1.496280
## 1024.00 1.893653 0.8585728 1.496280
## 2048.00 1.893653 0.8585728 1.496280
##
## Tuning parameter 'sigma' was held constant at a value of 0.07105405
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.07105405 and C = 16.svmRTuned.pred <- predict(svmRTuned, newdata = testData$x)
postResample(pred = svmRTuned.pred, obs = testData$y)## RMSE Rsquared MAE
## 2.0954656 0.8222975 1.5925227knn.model <- train(x = trainingData$x,
y = trainingData$y,
method = "knn",
preProc = c("center", "scale"),
tuneLength = 10)
knn.model## k-Nearest Neighbors
##
## 200 samples
## 10 predictor
##
## Pre-processing: centered (10), scaled (10)
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 200, 200, 200, 200, 200, 200, ...
## Resampling results across tuning parameters:
##
## k RMSE Rsquared MAE
## 5 3.586485 0.4975146 2.896836
## 7 3.500326 0.5230692 2.823611
## 9 3.392740 0.5664701 2.737414
## 11 3.326507 0.5955090 2.682394
## 13 3.319794 0.6100535 2.668172
## 15 3.306687 0.6246524 2.668343
## 17 3.298392 0.6384923 2.659521
## 19 3.301742 0.6489522 2.676516
## 21 3.321578 0.6503496 2.707272
## 23 3.332881 0.6574033 2.719900
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 17.knn.pred <- predict(knn.model, newdata = testData$x)
postResample(pred = knn.pred, obs = testData$y)## RMSE Rsquared MAE
## 3.2040595 0.6819919 2.5683461Looking at the lowest RMSE score, the MARS model performed the best with a score of 1.28.
varImp(marsTuned)## earth variable importance
##
## Overall
## X1 100.00
## X4 85.05
## X2 69.03
## X5 48.88
## X3 39.40
## X7 0.00
## X9 0.00
## X10 0.00
## X8 0.00
## X6 0.00We can see that the model did indeed select the informative predictors X1 - X5.
7.5 Exercise 6.3 describes data for a chemical manufacturing process. Use the same data imputation, data splitting, and pre-processing steps as before and train several nonlinear regression models.
data("ChemicalManufacturingProcess")
chem <- ChemicalManufacturingProcess
# Impute NA's with mean
# We could simply use complete.cases(), but the question asks
# us to impute the data instead.
for(i in 1:ncol(chem)){
chem[is.na(chem[, i]), i] <- mean(chem[, i], na.rm = TRUE)
}
smpl <- floor(0.75 * nrow(chem))
set.seed(500)
indices <- sample(seq_len(nrow(chem)),
size = smpl)
chem.train <- chem[indices, ]
chem.test <- chem[-indices, ]c.nnetTune <- train(chem.train[, c(2:58)], chem.train$Yield,
method = "avNNet",
tuneGrid = nnetGrid,
trControl = ctrl,
preProc = c("center", "scale"),
linout = TRUE,
trace = FALSE,
MaxNWts = 10 * (ncol(chem.train[, c(2:58)]) + 1) + 10 + 1,
maxit = 500)
c.nnetTune## Model Averaged Neural Network
##
## 132 samples
## 57 predictor
##
## Pre-processing: centered (57), scaled (57)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 118, 118, 119, 120, 118, 120, ...
## Resampling results across tuning parameters:
##
## decay size RMSE Rsquared MAE
## 0.00 1 1.540484 0.3710816 1.282657
## 0.00 2 1.500332 0.3851505 1.234164
## 0.00 3 1.582423 0.3752830 1.290678
## 0.00 4 2.002771 0.2815988 1.686315
## 0.00 5 2.216152 0.2942267 1.778448
## 0.00 6 1.990009 0.3151599 1.562415
## 0.00 7 2.228044 0.2957739 1.793974
## 0.00 8 2.933695 0.2799302 2.233089
## 0.00 9 3.810005 0.1845336 2.823672
## 0.00 10 5.601943 0.1454208 3.893492
## 0.01 1 1.399486 0.4389382 1.122154
## 0.01 2 1.527744 0.4826590 1.257682
## 0.01 3 1.634446 0.4373145 1.294632
## 0.01 4 1.995117 0.3829491 1.486498
## 0.01 5 1.982764 0.3428744 1.581890
## 0.01 6 1.946596 0.3435212 1.461820
## 0.01 7 1.465854 0.4476063 1.176903
## 0.01 8 1.483262 0.4492530 1.242289
## 0.01 9 1.470047 0.4590768 1.206807
## 0.01 10 1.942021 0.2952918 1.582033
## 0.10 1 1.530769 0.4610690 1.170213
## 0.10 2 1.735851 0.3901363 1.377958
## 0.10 3 1.855225 0.4407307 1.449525
## 0.10 4 2.121765 0.4142139 1.670333
## 0.10 5 2.173475 0.4110713 1.577619
## 0.10 6 2.170186 0.3827452 1.617210
## 0.10 7 2.146464 0.3955249 1.598996
## 0.10 8 2.205528 0.3407110 1.578478
## 0.10 9 2.182696 0.3808684 1.532016
## 0.10 10 1.931678 0.3404045 1.524651
##
## Tuning parameter 'bag' was held constant at a value of FALSE
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were size = 1, decay = 0.01 and bag
## = FALSE.c.nnetTune.pred <- predict(c.nnetTune, newdata = chem.test[, c(2:58)])
postResample(pred = c.nnetTune.pred, obs = chem.test$Yield)## RMSE Rsquared MAE
## 1.4274537 0.5065262 1.1813058c.marsTuned <- train(chem.train[, c(2:58)], chem.train$Yield,
method = "earth",
tuneGrid = marsGrid,
trControl = trainControl(method = "cv"))
c.marsTuned## Multivariate Adaptive Regression Spline
##
## 132 samples
## 57 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 119, 119, 119, 118, 119, 119, ...
## Resampling results across tuning parameters:
##
## degree nprune RMSE Rsquared MAE
## 1 2 1.405640 0.4198687 1.1027672
## 1 3 1.204084 0.5527844 0.9411505
## 1 4 1.225335 0.5373171 0.9765592
## 1 5 1.283595 0.5129750 1.0365319
## 1 6 1.370793 0.4656111 1.1177620
## 1 7 1.376112 0.4761595 1.0992856
## 1 8 1.355913 0.4938236 1.0739463
## 1 9 1.407084 0.4719873 1.1167820
## 1 10 1.441237 0.4453103 1.1541519
## 1 11 1.438092 0.4521872 1.1519743
## 1 12 1.433341 0.4657832 1.1378658
## 1 13 1.442149 0.4606450 1.1454513
## 1 14 1.436164 0.4706398 1.1403077
## 1 15 1.436953 0.4662332 1.1444445
## 1 16 1.436953 0.4662332 1.1444445
## 1 17 1.434065 0.4696881 1.1445996
## 1 18 1.438190 0.4675597 1.1466872
## 1 19 1.458473 0.4507862 1.1704898
## 1 20 1.436769 0.4682910 1.1565554
## 1 21 1.449003 0.4596280 1.1619368
## 1 22 1.457259 0.4543248 1.1653079
## 1 23 1.493276 0.4394513 1.1962647
## 1 24 1.493276 0.4394513 1.1962647
## 1 25 1.493276 0.4394513 1.1962647
## 1 26 1.493276 0.4394513 1.1962647
## 1 27 1.493276 0.4394513 1.1962647
## 1 28 1.493276 0.4394513 1.1962647
## 1 29 1.493276 0.4394513 1.1962647
## 1 30 1.493276 0.4394513 1.1962647
## 1 31 1.493276 0.4394513 1.1962647
## 1 32 1.493276 0.4394513 1.1962647
## 1 33 1.493276 0.4394513 1.1962647
## 1 34 1.493276 0.4394513 1.1962647
## 1 35 1.493276 0.4394513 1.1962647
## 1 36 1.493276 0.4394513 1.1962647
## 1 37 1.493276 0.4394513 1.1962647
## 1 38 1.493276 0.4394513 1.1962647
## 2 2 1.405640 0.4198687 1.1027672
## 2 3 1.341527 0.4757920 1.0317014
## 2 4 1.244937 0.5363141 0.9895591
## 2 5 1.188321 0.5653586 0.9700304
## 2 6 1.255971 0.5242005 1.0006949
## 2 7 1.308966 0.5137854 1.0407030
## 2 8 1.248297 0.5533733 0.9902366
## 2 9 1.324066 0.5254465 1.0380904
## 2 10 1.322544 0.5414039 1.0367127
## 2 11 1.340566 0.5486054 1.0542886
## 2 12 1.338294 0.5414304 1.0573853
## 2 13 1.310544 0.5694702 1.0379755
## 2 14 1.320437 0.5780592 1.0501376
## 2 15 1.296171 0.5957070 1.0337207
## 2 16 1.298088 0.5845528 1.0362980
## 2 17 1.285853 0.5936393 1.0253206
## 2 18 1.273424 0.6069209 1.0156772
## 2 19 1.268778 0.5986489 0.9857518
## 2 20 1.476297 0.5586363 1.0852079
## 2 21 1.487704 0.5616790 1.1046292
## 2 22 1.477946 0.5731485 1.1089639
## 2 23 1.456536 0.5835862 1.0938680
## 2 24 1.473457 0.5803368 1.1098869
## 2 25 1.473457 0.5803368 1.1098869
## 2 26 1.473457 0.5803368 1.1098869
## 2 27 1.473457 0.5803368 1.1098869
## 2 28 1.473457 0.5803368 1.1098869
## 2 29 1.473457 0.5803368 1.1098869
## 2 30 1.473457 0.5803368 1.1098869
## 2 31 1.473457 0.5803368 1.1098869
## 2 32 1.473457 0.5803368 1.1098869
## 2 33 1.473457 0.5803368 1.1098869
## 2 34 1.473457 0.5803368 1.1098869
## 2 35 1.473457 0.5803368 1.1098869
## 2 36 1.473457 0.5803368 1.1098869
## 2 37 1.473457 0.5803368 1.1098869
## 2 38 1.473457 0.5803368 1.1098869
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 5 and degree = 2.c.marsTuned.pred <- predict(c.marsTuned, newdata = chem.test[, c(2:58)])
postResample(pred = c.marsTuned.pred, obs = chem.test$Yield)## RMSE Rsquared MAE
## 1.2474725 0.6218236 1.0000345c.svmRTuned <- train(chem.train[, c(2:58)], chem.train$Yield,
method = "svmRadial",
preProc = c("center", "scale"),
tuneLength = 14,
trControl = trainControl(method = "cv"))
c.svmRTuned## Support Vector Machines with Radial Basis Function Kernel
##
## 132 samples
## 57 predictor
##
## Pre-processing: centered (57), scaled (57)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 119, 117, 119, 120, 119, 119, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 1.388537 0.4851079 1.1372444
## 0.50 1.279626 0.5334872 1.0577804
## 1.00 1.198902 0.5755363 0.9878005
## 2.00 1.129038 0.6250191 0.9332085
## 4.00 1.097768 0.6544295 0.9005805
## 8.00 1.116161 0.6441875 0.9166731
## 16.00 1.106722 0.6493945 0.9088355
## 32.00 1.106722 0.6493945 0.9088355
## 64.00 1.106722 0.6493945 0.9088355
## 128.00 1.106722 0.6493945 0.9088355
## 256.00 1.106722 0.6493945 0.9088355
## 512.00 1.106722 0.6493945 0.9088355
## 1024.00 1.106722 0.6493945 0.9088355
## 2048.00 1.106722 0.6493945 0.9088355
##
## Tuning parameter 'sigma' was held constant at a value of 0.01184455
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.01184455 and C = 4.c.svmRTuned.pred <- predict(c.svmRTuned, newdata = chem.test[, c(2:58)])
postResample(pred = c.svmRTuned.pred, obs = chem.test$Yield)## RMSE Rsquared MAE
## 1.1248444 0.6890303 0.7871013c.knn <- train(x = chem.test[, c(2:58)],
y = chem.test$Yield,
method = "knn",
preProc = c("center", "scale"),
tuneLength = 10)
c.knn## k-Nearest Neighbors
##
## 44 samples
## 57 predictors
##
## Pre-processing: centered (57), scaled (57)
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 44, 44, 44, 44, 44, 44, ...
## Resampling results across tuning parameters:
##
## k RMSE Rsquared MAE
## 5 1.707041 0.4030052 1.384239
## 7 1.731494 0.3945085 1.408923
## 9 1.733158 0.4044277 1.407988
## 11 1.730961 0.4323474 1.403720
## 13 1.755071 0.4229475 1.418744
## 15 1.765401 0.4383027 1.428595
## 17 1.786617 0.4476249 1.451869
## 19 1.800604 0.4606678 1.462370
## 21 1.824636 0.4670124 1.478055
## 23 1.842358 0.4740013 1.490642
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 5.predict.cknn <- predict(c.knn, newdata = chem.test[, c(2:58)])
postResample(pred = predict.cknn, obs = chem.test$Yield)## RMSE Rsquared MAE
## 1.2070967 0.6928683 0.9762727a) Which nonlinear regression model gives the optimal resampling and test set performance?
This time, the SVM model gives us the best RSME score with 1.125.
b) Which predictors are most important in the optimal nonlinear regression model? Do either the biological or process variables dominate the list? How do the top ten important predictors compare to the top ten predictors from the optimal linear model?
c.svmRTuned$finalModel## Support Vector Machine object of class "ksvm"
##
## SV type: eps-svr (regression)
## parameter : epsilon = 0.1 cost C = 4
##
## Gaussian Radial Basis kernel function.
## Hyperparameter : sigma = 0.0118445493876638
##
## Number of Support Vectors : 120
##
## Objective Function Value : -95.674
## Training error : 0.056733It doesn’t seem like the SVM object allows us to examine which predictors were most important. We can see that the final model uses 118 data points from the training set, or 67%.
lmFit <- train(x = chem.train[, c(2:58)], y = chem.train$Yield,
method = 'lm', trControl = ctrl)
predict.lmFit <- predict(lmFit, newdata = chem.test[, c(2:58)])
postResample(pred = predict.lmFit, obs = chem.test$Yield)## RMSE Rsquared MAE
## 1.5716518 0.4338667 1.2734679plsTune <- train(chem.train[, c(2:58)], chem.train$Yield,
method = "pls",
tuneLength = 20,
trControl = ctrl,
preProc = c("center", "scale"))
predict.plsTune <- predict(plsTune, newdata = chem.test[, c(2:58)])
postResample(pred = predict.plsTune, obs = chem.test$Yield)## RMSE Rsquared MAE
## 1.1789877 0.6568608 0.9391095ridgeGrid <- data.frame(.lambda = seq(0, .1, length = 15))
ridgeRegFit <- train(chem.train[, c(2:58)], chem.train$Yield,
method = "ridge",
tuneGrid = ridgeGrid,
trControl = ctrl,
preProc = c("center", "scale"))
predict.ridgeRegFit <- predict(ridgeRegFit, newdata = chem.test[, c(2:58)])
postResample(pred = predict.ridgeRegFit, obs = chem.test$Yield)## RMSE Rsquared MAE
## 1.1220704 0.6925314 0.8673341However, we can look at the linear models, of which the penalized model seemed to perform best.
ridgeRegFit$finalModel##
## Call:
## elasticnet::enet(x = as.matrix(x), y = y, lambda = param$lambda)
## Sequence of moves:
## ManufacturingProcess32 ManufacturingProcess09 ManufacturingProcess13
## Var 44 21 25
## Step 1 2 3
## ManufacturingProcess17 ManufacturingProcess36 BiologicalMaterial06
## Var 29 48 6
## Step 4 5 6
## ManufacturingProcess06 ManufacturingProcess15 BiologicalMaterial03
## Var 18 27 3
## Step 7 8 9
## ManufacturingProcess44 ManufacturingProcess11 ManufacturingProcess37
## Var 56 23 49
## Step 10 11 12
## ManufacturingProcess39 ManufacturingProcess45 ManufacturingProcess34
## Var 51 57 46
## Step 13 14 15
## BiologicalMaterial07 ManufacturingProcess04 ManufacturingProcess43
## Var 7 16 55
## Step 16 17 18
## ManufacturingProcess23 ManufacturingProcess38 ManufacturingProcess07
## Var 35 50 19
## Step 19 20 21
## ManufacturingProcess24 ManufacturingProcess30 ManufacturingProcess12
## Var 36 42 24
## Step 22 23 24
## ManufacturingProcess28 BiologicalMaterial10 ManufacturingProcess14
## Var 40 10 26
## Step 25 26 27
## ManufacturingProcess18 BiologicalMaterial12 ManufacturingProcess19
## Var 30 12 31
## Step 28 29 30
## BiologicalMaterial09 ManufacturingProcess29 ManufacturingProcess05
## Var 9 41 17
## Step 31 32 33
## ManufacturingProcess02 BiologicalMaterial01 ManufacturingProcess22
## Var 14 1 34
## Step 34 35 36
## ManufacturingProcess35 ManufacturingProcess16 ManufacturingProcess20
## Var 47 28 32
## Step 37 38 39
## ManufacturingProcess30 ManufacturingProcess31 ManufacturingProcess41
## Var -42 43 53
## Step 40 41 42
## ManufacturingProcess03 ManufacturingProcess30 BiologicalMaterial02
## Var 15 42 2
## Step 43 44 45
## ManufacturingProcess40 ManufacturingProcess21 ManufacturingProcess27
## Var 52 33 39
## Step 46 47 48
## ManufacturingProcess25 ManufacturingProcess42 BiologicalMaterial08
## Var 37 54 8
## Step 49 50 51
## BiologicalMaterial05 ManufacturingProcess08 ManufacturingProcess01
## Var 5 20 13
## Step 52 53 54
## ManufacturingProcess10 BiologicalMaterial04 ManufacturingProcess26
## Var 22 4 38
## Step 55 56 57
## ManufacturingProcess33 BiologicalMaterial11
## Var 45 11 60
## Step 58 59 60The top ten of that model appears to be 9 ManufacturingProcesses and 1 BiologicalProcess.
c) Explore the relationships between the top predictors and the response for the predictors that are unique to the optimal nonlinear regression model. Do these plots reveal intuition about the biological or process predictors and their relationship with yield?
This seems like a strange question as only one of the four models in this chapter allow you to examine the predictors, at least when using the train function. This is unlike the vanilla linear regression where you can see how each predictor has a specific weight, or the MARS model which does seem to rank the predictors.