DATA 624 - HOMEWORK 8
Question 7.2
Friedman (1991) introduced several benchmark data sets create by simulation. One of these simulations used the following nonlinear equation to create data:
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:
library(mlbench)
set.seed(200)
trainingData <- mlbench.friedman1(200, sd = 1)
## We convert the 'x' data from a matrix to a data frame
## One reason is that this will give the columns names.
trainingData$x <- data.frame(trainingData$x)
## Look at the data using
featurePlot(trainingData$x, trainingData$y)
## or other methods.
## This creates 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. For example:
#library(caret)
set.seed(200)
knnModel <- train(x = trainingData$x,
y = trainingData$y,
method = "knn",
preProc = c("center", "scale"),
tuneLength = 10)
knnModel## 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 3.654912 0.4779838 2.958475
## 7 3.529432 0.5118581 2.861742
## 9 3.446330 0.5425096 2.780756
## 11 3.378049 0.5723793 2.719410
## 13 3.332339 0.5953773 2.692863
## 15 3.309235 0.6111389 2.663046
## 17 3.317408 0.6201421 2.678898
## 19 3.311667 0.6333800 2.682098
## 21 3.316340 0.6407537 2.688887
## 23 3.326040 0.6491480 2.705915
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 15.knnPred <- predict(knnModel, newdata = testData$x)
## The function 'postResample' can be used to get the test set
## perforamnce values
knn_metrics <- postResample(pred = knnPred, obs = testData$y)
knn_metrics## RMSE Rsquared MAE
## 3.1750657 0.6785946 2.5443169Which models appear to give the best performance? Does MARS select the informative predictors (those named X1–X5)?
Answer:
SVM-Linear
The final model trained by linear SVM is epsilon = 0.1 cost C = 1 with RMSE 2.7633860 & R2 0.6973384
set.seed(200)
SVMLinearMod <- train(x = trainingData$x,
y = trainingData$y,
method = "svmLinear",
preProc = c("center", "scale"),
tuneLength = 14,
trControl = trainControl(method = "cv"))
SVMLinearMod## Support Vector Machines with Linear 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:
##
## RMSE Rsquared MAE
## 2.429138 0.7585222 1.965661
##
## Tuning parameter 'C' was held constant at a value of 1## Support Vector Machine object of class "ksvm"
##
## SV type: eps-svr (regression)
## parameter : epsilon = 0.1 cost C = 1
##
## Linear (vanilla) kernel function.
##
## Number of Support Vectors : 172
##
## Objective Function Value : -54.9759
## Training error : 0.216921SVMLinearPred <- predict(SVMLinearMod, newdata = testData$x)
SVMLinear_metrics <- postResample(pred = SVMLinearPred, obs = testData$y)
SVMLinear_metrics## RMSE Rsquared MAE
## 2.7633860 0.6973384 2.0970616SVM-Radial
The final model trained by Radial SVM is epsilon = 0.1 cost C = 8, sigma = 0.0629932410345396 with RMSE 2.0541197 & R2 0.8290353
set.seed(200)
SVMRadialMod <- train(x = trainingData$x,
y = trainingData$y,
method = "svmRadial",
preProc = c("center", "scale"),
tuneLength = 14,
trControl = trainControl(method = "cv"))
SVMRadialMod## 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 2.525164 0.7810576 2.010680
## 0.50 2.270567 0.7944850 1.794902
## 1.00 2.099356 0.8155574 1.659376
## 2.00 2.005858 0.8302852 1.578799
## 4.00 1.934650 0.8435677 1.528373
## 8.00 1.915665 0.8475605 1.528648
## 16.00 1.923914 0.8463074 1.535991
## 32.00 1.923914 0.8463074 1.535991
## 64.00 1.923914 0.8463074 1.535991
## 128.00 1.923914 0.8463074 1.535991
## 256.00 1.923914 0.8463074 1.535991
## 512.00 1.923914 0.8463074 1.535991
## 1024.00 1.923914 0.8463074 1.535991
## 2048.00 1.923914 0.8463074 1.535991
##
## Tuning parameter 'sigma' was held constant at a value of 0.06299324
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.06299324 and C = 8.## Support Vector Machine object of class "ksvm"
##
## SV type: eps-svr (regression)
## parameter : epsilon = 0.1 cost C = 8
##
## Gaussian Radial Basis kernel function.
## Hyperparameter : sigma = 0.0629932410345396
##
## Number of Support Vectors : 152
##
## Objective Function Value : -72.63
## Training error : 0.009177SVMRadialPred <- predict(SVMRadialMod, newdata = testData$x)
SVMRadial_metrics <- postResample(pred = SVMRadialPred, obs = testData$y)
SVMRadial_metrics## RMSE Rsquared MAE
## 2.0541197 0.8290353 1.5586411SVM-Polynomial
The final model trained by Polynomial SVM is epsilon = 0.1 cost C = 0.5, degree = 3 scale = 0.1 offset = 1 with RMSE 2.0564650 & R2 0.8310884
set.seed(200)
SVMPolyMod <- train(x = trainingData$x,
y = trainingData$y,
method = "svmPoly",
preProc = c("center", "scale"),
tuneLength = 4,
trControl = trainControl(method = "cv"))
SVMPolyMod## Support Vector Machines with Polynomial 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:
##
## degree scale C RMSE Rsquared MAE
## 1 0.001 0.25 4.763881 0.7366269 3.912571
## 1 0.001 0.50 4.617330 0.7376064 3.787900
## 1 0.001 1.00 4.351479 0.7355789 3.561381
## 1 0.001 2.00 3.867351 0.7401050 3.158700
## 1 0.010 0.25 3.711577 0.7439878 3.037307
## 1 0.010 0.50 3.037734 0.7500121 2.494361
## 1 0.010 1.00 2.606886 0.7554606 2.126507
## 1 0.010 2.00 2.465015 0.7586805 2.018188
## 1 0.100 0.25 2.432377 0.7604321 1.991550
## 1 0.100 0.50 2.419285 0.7591638 1.971433
## 1 0.100 1.00 2.435724 0.7563615 1.973661
## 1 0.100 2.00 2.434895 0.7566897 1.969241
## 1 1.000 0.25 2.435493 0.7567619 1.969113
## 1 1.000 0.50 2.433348 0.7574620 1.967454
## 1 1.000 1.00 2.429940 0.7583999 1.966357
## 1 1.000 2.00 2.432347 0.7581066 1.967671
## 2 0.001 0.25 4.617339 0.7376255 3.787914
## 2 0.001 0.50 4.351486 0.7356099 3.561395
## 2 0.001 1.00 3.867297 0.7400445 3.158577
## 2 0.001 2.00 3.270622 0.7441841 2.694402
## 2 0.010 0.25 3.035230 0.7518007 2.492293
## 2 0.010 0.50 2.590422 0.7587008 2.115071
## 2 0.010 1.00 2.439896 0.7637583 1.997834
## 2 0.010 2.00 2.371906 0.7673980 1.927845
## 2 0.100 0.25 2.069776 0.8155998 1.650388
## 2 0.100 0.50 1.965305 0.8301488 1.550369
## 2 0.100 1.00 1.974104 0.8334526 1.579994
## 2 0.100 2.00 1.997227 0.8345576 1.609912
## 2 1.000 0.25 2.061148 0.8410088 1.688012
## 2 1.000 0.50 2.081844 0.8407738 1.693817
## 2 1.000 1.00 2.087780 0.8403437 1.698219
## 2 1.000 2.00 2.084166 0.8407541 1.694988
## 3 0.001 0.25 4.477852 0.7363407 3.668531
## 3 0.001 0.50 4.107151 0.7334257 3.357893
## 3 0.001 1.00 3.560177 0.7405641 2.923402
## 3 0.001 2.00 2.881611 0.7542154 2.369244
## 3 0.010 0.25 2.723753 0.7588724 2.225310
## 3 0.010 0.50 2.477358 0.7636596 2.023215
## 3 0.010 1.00 2.344012 0.7737494 1.903935
## 3 0.010 2.00 2.301737 0.7783092 1.859257
## 3 0.100 0.25 1.952669 0.8382025 1.543128
## 3 0.100 0.50 1.930412 0.8467547 1.526408
## 3 0.100 1.00 2.026823 0.8395150 1.616447
## 3 0.100 2.00 2.078726 0.8365432 1.687398
## 3 1.000 0.25 3.221462 0.6569500 2.603166
## 3 1.000 0.50 3.221462 0.6569500 2.603166
## 3 1.000 1.00 3.221462 0.6569500 2.603166
## 3 1.000 2.00 3.221462 0.6569500 2.603166
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were degree = 3, scale = 0.1 and C = 0.5.## Support Vector Machine object of class "ksvm"
##
## SV type: eps-svr (regression)
## parameter : epsilon = 0.1 cost C = 0.5
##
## Polynomial kernel function.
## Hyperparameters : degree = 3 scale = 0.1 offset = 1
##
## Number of Support Vectors : 146
##
## Objective Function Value : -8.0447
## Training error : 0.020293SVMPolyPred <- predict(SVMPolyMod, newdata = testData$x)
SVMPoly_metrics <- postResample(pred = SVMPolyPred, obs = testData$y)
SVMPoly_metrics## RMSE Rsquared MAE
## 2.0564650 0.8310884 1.5544685MARS
The final MARS model is nprune = 14 and degree = 2, with RMSE 1.1722635 & R2 0.9448890. MARS selected 14 of 18 terms, and 5 of 10 predictors, which are X1, X4, X2, X5 and X3 orded by variable importance.
set.seed(200)
MARSMod <- train(x = trainingData$x,
y = trainingData$y,
method ='earth',
tuneGrid = expand.grid(.degree = 1:2,
.nprune = 2:38),
trControl = trainControl(method = "cv"))## Loading required package: earth## Loading required package: Formula## Loading required package: plotmo## Loading required package: plotrix## Loading required package: TeachingDemos## 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 4.188280 0.3042527 3.460689
## 1 3 3.551182 0.4999832 2.837116
## 1 4 2.653143 0.7167280 2.128222
## 1 5 2.405769 0.7562160 1.948161
## 1 6 2.295006 0.7754603 1.853199
## 1 7 1.771950 0.8611767 1.391357
## 1 8 1.647182 0.8774867 1.299564
## 1 9 1.609816 0.8837307 1.299705
## 1 10 1.635035 0.8798236 1.309436
## 1 11 1.571915 0.8896147 1.260711
## 1 12 1.571561 0.8898750 1.253077
## 1 13 1.567577 0.8906927 1.250795
## 1 14 1.571673 0.8909652 1.245508
## 1 15 1.571673 0.8909652 1.245508
## 1 16 1.571673 0.8909652 1.245508
## 1 17 1.571673 0.8909652 1.245508
## 1 18 1.571673 0.8909652 1.245508
## 1 19 1.571673 0.8909652 1.245508
## 1 20 1.571673 0.8909652 1.245508
## 1 21 1.571673 0.8909652 1.245508
## 1 22 1.571673 0.8909652 1.245508
## 1 23 1.571673 0.8909652 1.245508
## 1 24 1.571673 0.8909652 1.245508
## 1 25 1.571673 0.8909652 1.245508
## 1 26 1.571673 0.8909652 1.245508
## 1 27 1.571673 0.8909652 1.245508
## 1 28 1.571673 0.8909652 1.245508
## 1 29 1.571673 0.8909652 1.245508
## 1 30 1.571673 0.8909652 1.245508
## 1 31 1.571673 0.8909652 1.245508
## 1 32 1.571673 0.8909652 1.245508
## 1 33 1.571673 0.8909652 1.245508
## 1 34 1.571673 0.8909652 1.245508
## 1 35 1.571673 0.8909652 1.245508
## 1 36 1.571673 0.8909652 1.245508
## 1 37 1.571673 0.8909652 1.245508
## 1 38 1.571673 0.8909652 1.245508
## 2 2 4.188280 0.3042527 3.460689
## 2 3 3.551182 0.4999832 2.837116
## 2 4 2.615256 0.7216809 2.128763
## 2 5 2.344223 0.7683855 1.890080
## 2 6 2.275048 0.7762472 1.807779
## 2 7 1.841464 0.8418935 1.457945
## 2 8 1.641647 0.8839822 1.288520
## 2 9 1.535119 0.9002991 1.214772
## 2 10 1.473254 0.9101555 1.158761
## 2 11 1.379476 0.9207735 1.080991
## 2 12 1.285380 0.9283193 1.033426
## 2 13 1.267261 0.9328905 1.014726
## 2 14 1.261797 0.9327541 1.009821
## 2 15 1.266663 0.9320714 1.005751
## 2 16 1.270858 0.9322465 1.009757
## 2 17 1.263778 0.9327687 1.007653
## 2 18 1.263778 0.9327687 1.007653
## 2 19 1.263778 0.9327687 1.007653
## 2 20 1.263778 0.9327687 1.007653
## 2 21 1.263778 0.9327687 1.007653
## 2 22 1.263778 0.9327687 1.007653
## 2 23 1.263778 0.9327687 1.007653
## 2 24 1.263778 0.9327687 1.007653
## 2 25 1.263778 0.9327687 1.007653
## 2 26 1.263778 0.9327687 1.007653
## 2 27 1.263778 0.9327687 1.007653
## 2 28 1.263778 0.9327687 1.007653
## 2 29 1.263778 0.9327687 1.007653
## 2 30 1.263778 0.9327687 1.007653
## 2 31 1.263778 0.9327687 1.007653
## 2 32 1.263778 0.9327687 1.007653
## 2 33 1.263778 0.9327687 1.007653
## 2 34 1.263778 0.9327687 1.007653
## 2 35 1.263778 0.9327687 1.007653
## 2 36 1.263778 0.9327687 1.007653
## 2 37 1.263778 0.9327687 1.007653
## 2 38 1.263778 0.9327687 1.007653
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 14 and degree = 2.## Selected 14 of 18 terms, and 5 of 10 predictors (nprune=14)
## Termination condition: Reached nk 21
## Importance: X1, X4, X2, X5, X3, X6-unused, X7-unused, X8-unused, X9-unused, ...
## Number of terms at each degree of interaction: 1 9 4
## GCV 1.62945 RSS 225.8601 GRSq 0.9338437 RSq 0.953688MARSModPred <- predict(MARSMod, newdata = testData$x)
MARS_metrics <- postResample(pred = MARSModPred, obs = testData$y)
MARS_metrics## RMSE Rsquared MAE
## 1.1722635 0.9448890 0.9324923Neural Networks
The final neural network model is size = 5, decay = 0.07, with RMSE 2.2480475 & R2 0.8031027.
#remove predictors with high correlation, however find that there are no significant correlation between each pair of predictors.
set.seed(200)
tooHigh <- findCorrelation(cor(trainingData$x), cutoff = .75)
tooHigh## integer(0)NeuralNetMod <- train(x = trainingData$x,
y = trainingData$y,
method ='avNNet',
preProc = c("center", "scale"),
tuneGrid = expand.grid(.decay = seq(0.01,0.1,0.01),
.size = c(1:10),
.bag = FALSE),
trControl = trainControl(method = "cv"),
trace = FALSE,
linout =TRUE#,
#MaxNWts = 10 * (ncol(trainingData$x) + 1) + 10 + 1,
#maxit = 500
)## Warning: executing %dopar% sequentially: no parallel backend registered## 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
## 0.01 1 2.404567 0.7609557 1.889455
## 0.01 2 2.436960 0.7534744 1.919892
## 0.01 3 2.187507 0.8006263 1.700457
## 0.01 4 2.027172 0.8239211 1.608293
## 0.01 5 2.174956 0.8040747 1.746873
## 0.01 6 2.247179 0.7962598 1.748556
## 0.01 7 2.495552 0.7516917 1.996630
## 0.01 8 2.479849 0.7525438 1.970609
## 0.01 9 2.455646 0.7551865 1.918742
## 0.01 10 2.380386 0.7666502 1.944705
## 0.02 1 2.388957 0.7626569 1.875134
## 0.02 2 2.457544 0.7543633 1.935851
## 0.02 3 2.150727 0.8047792 1.690877
## 0.02 4 2.037427 0.8227565 1.596348
## 0.02 5 2.082575 0.8180478 1.705675
## 0.02 6 2.414145 0.7730944 1.891141
## 0.02 7 2.409065 0.7745675 1.915445
## 0.02 8 2.435317 0.7650026 2.013716
## 0.02 9 2.584975 0.7503707 2.044301
## 0.02 10 2.405229 0.7562320 1.908182
## 0.03 1 2.383746 0.7633527 1.870546
## 0.03 2 2.454290 0.7514461 1.908840
## 0.03 3 2.217704 0.7918645 1.719262
## 0.03 4 2.050625 0.8289918 1.608316
## 0.03 5 2.102705 0.8161141 1.679154
## 0.03 6 2.211142 0.7992296 1.745093
## 0.03 7 2.391308 0.7723404 1.958513
## 0.03 8 2.639188 0.7333796 2.081299
## 0.03 9 2.193183 0.8034147 1.740475
## 0.03 10 2.386450 0.7629278 1.873561
## 0.04 1 2.384438 0.7629684 1.869899
## 0.04 2 2.450467 0.7539379 1.942115
## 0.04 3 2.199523 0.7993416 1.737917
## 0.04 4 2.016975 0.8198140 1.649942
## 0.04 5 2.069597 0.8195621 1.654101
## 0.04 6 2.315627 0.7842001 1.837216
## 0.04 7 2.331788 0.7731338 1.839560
## 0.04 8 2.411618 0.7677358 1.890454
## 0.04 9 2.360925 0.7745422 1.896727
## 0.04 10 2.238863 0.7864714 1.779224
## 0.05 1 2.386109 0.7627911 1.870035
## 0.05 2 2.360599 0.7701985 1.847652
## 0.05 3 2.176532 0.7942336 1.721067
## 0.05 4 1.977518 0.8319675 1.549494
## 0.05 5 2.083588 0.8206270 1.678024
## 0.05 6 2.201673 0.8021564 1.786855
## 0.05 7 2.339674 0.7797736 1.833052
## 0.05 8 2.372672 0.7664515 1.886467
## 0.05 9 2.419643 0.7611444 1.935209
## 0.05 10 2.252022 0.7894785 1.813213
## 0.06 1 2.388128 0.7623863 1.871239
## 0.06 2 2.377274 0.7637839 1.883250
## 0.06 3 2.145231 0.8023846 1.698228
## 0.06 4 2.030444 0.8220632 1.619815
## 0.06 5 2.019334 0.8290427 1.579891
## 0.06 6 2.104081 0.8172304 1.676594
## 0.06 7 2.141941 0.8121311 1.741943
## 0.06 8 2.264920 0.7973905 1.774783
## 0.06 9 2.504734 0.7454076 1.934063
## 0.06 10 2.407879 0.7646765 1.874436
## 0.07 1 2.388737 0.7621763 1.870702
## 0.07 2 2.473526 0.7461275 1.958137
## 0.07 3 2.111039 0.8094795 1.693624
## 0.07 4 2.074931 0.8166422 1.599258
## 0.07 5 1.969976 0.8312649 1.590712
## 0.07 6 2.150874 0.8053445 1.722743
## 0.07 7 2.407378 0.7672085 1.903367
## 0.07 8 2.592483 0.7373282 2.064595
## 0.07 9 2.426241 0.7517220 1.906753
## 0.07 10 2.345099 0.7739126 1.857714
## 0.08 1 2.388303 0.7621290 1.869889
## 0.08 2 2.501862 0.7459779 1.993047
## 0.08 3 2.082786 0.8151120 1.656598
## 0.08 4 1.997009 0.8299575 1.568438
## 0.08 5 2.069612 0.8170014 1.650303
## 0.08 6 2.192552 0.8067040 1.761656
## 0.08 7 2.209919 0.8034384 1.748902
## 0.08 8 2.498832 0.7637999 1.971576
## 0.08 9 2.353432 0.7861735 1.860627
## 0.08 10 2.438012 0.7506400 1.988893
## 0.09 1 2.391093 0.7617087 1.872667
## 0.09 2 2.448119 0.7535060 1.945496
## 0.09 3 2.138156 0.8069623 1.634501
## 0.09 4 2.023500 0.8279610 1.624832
## 0.09 5 2.059881 0.8263668 1.651843
## 0.09 6 2.142091 0.8139378 1.694527
## 0.09 7 2.224850 0.8037135 1.778493
## 0.09 8 2.314672 0.7774339 1.864141
## 0.09 9 2.332838 0.7753480 1.847349
## 0.09 10 2.385689 0.7711283 1.956335
## 0.10 1 2.392297 0.7614530 1.873813
## 0.10 2 2.424834 0.7584952 1.915808
## 0.10 3 2.174391 0.8008953 1.719494
## 0.10 4 2.036973 0.8242977 1.611291
## 0.10 5 1.973613 0.8371935 1.557636
## 0.10 6 2.257253 0.7915989 1.774801
## 0.10 7 2.213409 0.8004569 1.775937
## 0.10 8 2.359574 0.7756999 1.845723
## 0.10 9 2.425894 0.7553730 1.882439
## 0.10 10 2.343082 0.7739852 1.917299
##
## 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 = 5, decay = 0.07 and bag = FALSE.NeuralNetModPred <- predict(NeuralNetMod, newdata = testData$x)
NeuralNet_metrics <- postResample(pred = NeuralNetModPred, obs = testData$y)
NeuralNet_metrics## RMSE Rsquared MAE
## 2.2480475 0.8031027 1.6938278Model Comparison
The best model selected by both RMSE & R2 is MARS.
rbind(knn_metrics,
SVMLinear_metrics,
SVMRadial_metrics,
SVMPoly_metrics,
MARS_metrics,
NeuralNet_metrics) %>%
data.frame() %>%
arrange(RMSE)Question 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.
Load Data
library(AppliedPredictiveModeling)
data(ChemicalManufacturingProcess)
chem_predictors <- ChemicalManufacturingProcess %>% select(-Yield) %>% as.matrix()
chem_response <- ChemicalManufacturingProcess %>% select(Yield) %>% as.matrix()The matrix processPredictors contains the 57 predictors (12 describing the input biological material and 45 describing the process predictors) for the 176 manufacturing runs. yield contains the percent yield for each run.
Data Imputation
Imputd mssing values using missFroest package.
## missForest iteration 1 in progress...done!
## missForest iteration 2 in progress...done!
## missForest iteration 3 in progress...done!train_test_split
set.seed(200)
train_index <- createDataPartition(chem_response,
p = 0.75,
list = FALSE,
times = 1) %>%
as.vector()
data_train_X <- imp_chem_predictors[train_index,]
data_train_Y <- chem_response[train_index,]
data_test_X <- imp_chem_predictors[-train_index,]
data_test_Y <-chem_response[-train_index,](a)
Which nonlinear regression model gives the optimal resampling and test set performance?
Answer: The optimal model is radial SVM. See below:
KNN
set.seed(200)
# predictors with zero variance are removed
knnModel <- train(x = data_train_X[,-nearZeroVar(data_train_X)],
y = data_train_Y,
method = "knn",
preProc = c("center", "scale"),
tuneLength = 10)
knnModel## k-Nearest Neighbors
##
## 132 samples
## 56 predictor
##
## Pre-processing: centered (56), scaled (56)
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 132, 132, 132, 132, 132, 132, ...
## Resampling results across tuning parameters:
##
## k RMSE Rsquared MAE
## 5 1.551795 0.3241916 1.229662
## 7 1.509729 0.3451176 1.204782
## 9 1.491200 0.3583055 1.203081
## 11 1.466583 0.3820689 1.185985
## 13 1.463003 0.3894212 1.176979
## 15 1.472545 0.3814257 1.192809
## 17 1.479893 0.3784425 1.195397
## 19 1.480769 0.3841633 1.195457
## 21 1.479273 0.3919096 1.200877
## 23 1.483252 0.3992531 1.201789
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 13.knnPred <- predict(knnModel, newdata = data_test_X[,-nearZeroVar(data_train_X)])
knn_metrics <- postResample(pred = knnPred, obs = data_test_Y)
knn_metrics## RMSE Rsquared MAE
## 1.4678782 0.3897311 1.2356481SVM-Linear
set.seed(200)
SVMLinearMod <- train(x = data_train_X,
y = data_train_Y,
method = "svmLinear",
preProc = c("center", "scale"),
tuneLength = 14,
trControl = trainControl(method = "cv"))
SVMLinearMod## Support Vector Machines with Linear Kernel
##
## 132 samples
## 57 predictor
##
## Pre-processing: centered (57), scaled (57)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 117, 119, 119, 119, 120, 119, ...
## Resampling results:
##
## RMSE Rsquared MAE
## 4.256036 0.400964 1.961182
##
## Tuning parameter 'C' was held constant at a value of 1## Support Vector Machine object of class "ksvm"
##
## SV type: eps-svr (regression)
## parameter : epsilon = 0.1 cost C = 1
##
## Linear (vanilla) kernel function.
##
## Number of Support Vectors : 116
##
## Objective Function Value : -31.6826
## Training error : 0.237728SVMLinearPred <- predict(SVMLinearMod, newdata = data_test_X)
SVMLinear_metrics <- postResample(pred = SVMLinearPred, obs = data_test_Y)
SVMLinear_metrics## RMSE Rsquared MAE
## 2.5920541 0.2407341 1.3461138SVM-Radial
set.seed(200)
SVMRadialMod <- train(x = data_train_X,
y = data_train_Y,
method = "svmRadial",
preProc = c("center", "scale"),
tuneLength = 14,
trControl = trainControl(method = "cv"))
SVMRadialMod## 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: 117, 119, 119, 119, 120, 119, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 1.421498 0.4527061 1.1628997
## 0.50 1.314494 0.5092712 1.0781953
## 1.00 1.239404 0.5479787 1.0061873
## 2.00 1.188196 0.5684967 0.9653448
## 4.00 1.181656 0.5668905 0.9655505
## 8.00 1.173050 0.5699283 0.9611789
## 16.00 1.171625 0.5704403 0.9592864
## 32.00 1.171625 0.5704403 0.9592864
## 64.00 1.171625 0.5704403 0.9592864
## 128.00 1.171625 0.5704403 0.9592864
## 256.00 1.171625 0.5704403 0.9592864
## 512.00 1.171625 0.5704403 0.9592864
## 1024.00 1.171625 0.5704403 0.9592864
## 2048.00 1.171625 0.5704403 0.9592864
##
## Tuning parameter 'sigma' was held constant at a value of 0.01388288
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.01388288 and C = 16.## Support Vector Machine object of class "ksvm"
##
## SV type: eps-svr (regression)
## parameter : epsilon = 0.1 cost C = 16
##
## Gaussian Radial Basis kernel function.
## Hyperparameter : sigma = 0.0138828772878836
##
## Number of Support Vectors : 116
##
## Objective Function Value : -81.3542
## Training error : 0.009159SVMRadialPred <- predict(SVMRadialMod, newdata = data_test_X)
SVMRadial_metrics <- postResample(pred = SVMRadialPred, obs = data_test_Y)
SVMRadial_metrics## RMSE Rsquared MAE
## 1.1957521 0.6021279 0.9150740SVM-Polynomial
set.seed(200)
SVMPolyMod <- train(x = data_train_X,
y = data_train_Y,
method = "svmPoly",
preProc = c("center", "scale"),
tuneLength = 4,
trControl = trainControl(method = "cv"))
SVMPolyMod## Support Vector Machines with Polynomial Kernel
##
## 132 samples
## 57 predictor
##
## Pre-processing: centered (57), scaled (57)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 117, 119, 119, 119, 120, 119, ...
## Resampling results across tuning parameters:
##
## degree scale C RMSE Rsquared MAE
## 1 0.001 0.25 1.683264 0.3746725 1.377157
## 1 0.001 0.50 1.606614 0.3957547 1.318496
## 1 0.001 1.00 1.524060 0.4172733 1.243002
## 1 0.001 2.00 1.498827 0.4607211 1.179196
## 1 0.010 0.25 1.522099 0.4699540 1.169305
## 1 0.010 0.50 1.620679 0.4926258 1.179766
## 1 0.010 1.00 1.634714 0.5093664 1.138212
## 1 0.010 2.00 1.696997 0.4897924 1.152637
## 1 0.100 0.25 1.687421 0.4733905 1.154785
## 1 0.100 0.50 1.948745 0.4478448 1.242430
## 1 0.100 1.00 2.083795 0.4357293 1.280166
## 1 0.100 2.00 2.136063 0.4324949 1.284946
## 1 1.000 0.25 2.464643 0.4262428 1.387340
## 1 1.000 0.50 3.215053 0.4162814 1.630072
## 1 1.000 1.00 4.255734 0.4009670 1.961093
## 1 1.000 2.00 4.963163 0.3818179 2.189860
## 2 0.001 0.25 1.601207 0.4001036 1.314176
## 2 0.001 0.50 1.537789 0.4144653 1.248557
## 2 0.001 1.00 1.579477 0.4570835 1.203609
## 2 0.001 2.00 1.756289 0.4775600 1.226604
## 2 0.010 0.25 1.876932 0.4684323 1.243554
## 2 0.010 0.50 1.354650 0.5075909 1.052072
## 2 0.010 1.00 1.416567 0.4917575 1.060609
## 2 0.010 2.00 2.479736 0.4385817 1.356142
## 2 0.100 0.25 9.815126 0.4219278 3.438655
## 2 0.100 0.50 10.098146 0.4156893 3.512786
## 2 0.100 1.00 10.098146 0.4156893 3.512786
## 2 0.100 2.00 10.098146 0.4156893 3.512786
## 2 1.000 0.25 27.407013 0.2765334 8.469170
## 2 1.000 0.50 27.407013 0.2765334 8.469170
## 2 1.000 1.00 27.407013 0.2765334 8.469170
## 2 1.000 2.00 27.407013 0.2765334 8.469170
## 3 0.001 0.25 1.590442 0.3987141 1.285375
## 3 0.001 0.50 1.638400 0.4372220 1.251221
## 3 0.001 1.00 1.980782 0.4577857 1.304568
## 3 0.001 2.00 2.348464 0.4862026 1.379246
## 3 0.010 0.25 15.427151 0.4710504 4.968457
## 3 0.010 0.50 17.930107 0.4836357 5.635979
## 3 0.010 1.00 33.242438 0.4295094 9.890216
## 3 0.010 2.00 51.109826 0.4180338 14.852286
## 3 0.100 0.25 185.532776 0.4399172 52.180195
## 3 0.100 0.50 185.532776 0.4399172 52.180195
## 3 0.100 1.00 185.532776 0.4399172 52.180195
## 3 0.100 2.00 185.532776 0.4399172 52.180195
## 3 1.000 0.25 180.203946 0.3697730 50.768973
## 3 1.000 0.50 180.203946 0.3697730 50.768973
## 3 1.000 1.00 180.203946 0.3697730 50.768973
## 3 1.000 2.00 180.203946 0.3697730 50.768973
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were degree = 2, scale = 0.01 and C = 0.5.## Support Vector Machine object of class "ksvm"
##
## SV type: eps-svr (regression)
## parameter : epsilon = 0.1 cost C = 0.5
##
## Polynomial kernel function.
## Hyperparameters : degree = 2 scale = 0.01 offset = 1
##
## Number of Support Vectors : 120
##
## Objective Function Value : -23.7109
## Training error : 0.251737SVMPolyPred <- predict(SVMPolyMod, newdata = data_test_X)
SVMPoly_metrics <- postResample(pred = SVMPolyPred, obs = data_test_Y)
SVMPoly_metrics## RMSE Rsquared MAE
## 4.71511318 0.02416611 1.56883546MARS
set.seed(200)
MARSMod <- train(x = data_train_X,
y = data_train_Y,
method ='earth',
tuneGrid = expand.grid(.degree = 1:2,
.nprune = 2:38),
trControl = trainControl(method = "cv"))
MARSMod## Multivariate Adaptive Regression Spline
##
## 132 samples
## 57 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 117, 119, 119, 119, 120, 119, ...
## Resampling results across tuning parameters:
##
## degree nprune RMSE Rsquared MAE
## 1 2 1.375678 0.4502209 1.0968022
## 1 3 1.164807 0.5986966 0.9562711
## 1 4 1.163736 0.6264452 0.9468020
## 1 5 1.202425 0.5978173 0.9740852
## 1 6 1.189325 0.6083079 0.9544333
## 1 7 1.211895 0.5680295 0.9813838
## 1 8 1.181037 0.6154397 0.9635183
## 1 9 1.177400 0.6205997 0.9780710
## 1 10 1.207724 0.6024369 0.9701158
## 1 11 1.161506 0.6284988 0.9467581
## 1 12 1.158087 0.6209042 0.9402577
## 1 13 1.176768 0.6264094 0.9489019
## 1 14 1.190565 0.6103812 0.9586550
## 1 15 1.196750 0.6090927 0.9658895
## 1 16 1.196750 0.6090927 0.9658895
## 1 17 1.196750 0.6090927 0.9658895
## 1 18 1.196750 0.6090927 0.9658895
## 1 19 1.196750 0.6090927 0.9658895
## 1 20 1.196750 0.6090927 0.9658895
## 1 21 1.196750 0.6090927 0.9658895
## 1 22 1.196750 0.6090927 0.9658895
## 1 23 1.196750 0.6090927 0.9658895
## 1 24 1.196750 0.6090927 0.9658895
## 1 25 1.196750 0.6090927 0.9658895
## 1 26 1.196750 0.6090927 0.9658895
## 1 27 1.196750 0.6090927 0.9658895
## 1 28 1.196750 0.6090927 0.9658895
## 1 29 1.196750 0.6090927 0.9658895
## 1 30 1.196750 0.6090927 0.9658895
## 1 31 1.196750 0.6090927 0.9658895
## 1 32 1.196750 0.6090927 0.9658895
## 1 33 1.196750 0.6090927 0.9658895
## 1 34 1.196750 0.6090927 0.9658895
## 1 35 1.196750 0.6090927 0.9658895
## 1 36 1.196750 0.6090927 0.9658895
## 1 37 1.196750 0.6090927 0.9658895
## 1 38 1.196750 0.6090927 0.9658895
## 2 2 1.375678 0.4502209 1.0968022
## 2 3 1.282556 0.5430144 1.0304046
## 2 4 1.255658 0.5621046 1.0258003
## 2 5 1.276373 0.5279753 1.0307734
## 2 6 1.298958 0.5221774 1.0388639
## 2 7 1.283178 0.5500904 1.0179494
## 2 8 1.266318 0.5656429 0.9990201
## 2 9 1.246332 0.5835004 0.9978897
## 2 10 1.252962 0.5779382 1.0082517
## 2 11 1.308158 0.5420952 1.0459230
## 2 12 1.340706 0.5404366 1.0720183
## 2 13 3.517883 0.4704572 1.7114485
## 2 14 3.563824 0.4414633 1.7368568
## 2 15 3.637768 0.4160540 1.7820021
## 2 16 3.665263 0.4209541 1.7956450
## 2 17 3.668593 0.4168348 1.8078235
## 2 18 3.665984 0.4211707 1.8020251
## 2 19 3.670239 0.4189316 1.8017304
## 2 20 3.671178 0.4187977 1.8032385
## 2 21 3.671178 0.4187977 1.8032385
## 2 22 3.671178 0.4187977 1.8032385
## 2 23 3.671178 0.4187977 1.8032385
## 2 24 3.671178 0.4187977 1.8032385
## 2 25 3.671178 0.4187977 1.8032385
## 2 26 3.671178 0.4187977 1.8032385
## 2 27 3.671178 0.4187977 1.8032385
## 2 28 3.671178 0.4187977 1.8032385
## 2 29 3.671178 0.4187977 1.8032385
## 2 30 3.671178 0.4187977 1.8032385
## 2 31 3.671178 0.4187977 1.8032385
## 2 32 3.671178 0.4187977 1.8032385
## 2 33 3.671178 0.4187977 1.8032385
## 2 34 3.671178 0.4187977 1.8032385
## 2 35 3.671178 0.4187977 1.8032385
## 2 36 3.671178 0.4187977 1.8032385
## 2 37 3.671178 0.4187977 1.8032385
## 2 38 3.671178 0.4187977 1.8032385
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 12 and degree = 1.## Selected 11 of 22 terms, and 8 of 57 predictors (nprune=12)
## Termination condition: RSq changed by less than 0.001 at 22 terms
## Importance: ManufacturingProcess32, ManufacturingProcess09, ...
## Number of terms at each degree of interaction: 1 10 (additive model)
## GCV 0.9907559 RSS 92.47805 GRSq 0.7100006 RSq 0.7917905MARSModPred <- predict(MARSMod, newdata = data_test_X)
MARS_metrics <- postResample(pred = MARSModPred, obs = data_test_Y)
MARS_metrics## RMSE Rsquared MAE
## 1.4430421 0.4781265 1.0992813Neural Networks
#remove predictors with high correlation & zero variance
set.seed(200)
tooHigh <- findCorrelation(cor(data_train_X), cutoff = .75)
zeroVar <- nearZeroVar(data_train_X)
data_train_X_nnet <- data_train_X[,-c(tooHigh, zeroVar)]
data_test_X_nnet <- data_test_X[,-c(tooHigh, zeroVar)]
NeuralNetMod <- train(x = data_train_X_nnet,
y = data_train_Y,
method ='avNNet',
preProc = c("center", "scale"),
tuneGrid = expand.grid(.decay = seq(0.01,0.1,0.01),
.size = c(1:10),
.bag = FALSE),
trControl = trainControl(method = "cv"),
linout = TRUE,
trace = FALSE#,
#MaxNWts = 10 * (ncol(data_train_X_nnet) + 1) + 10 + 1,
#maxit = 500
)
NeuralNetMod## Model Averaged Neural Network
##
## 132 samples
## 34 predictor
##
## Pre-processing: centered (34), scaled (34)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 117, 119, 119, 119, 120, 119, ...
## Resampling results across tuning parameters:
##
## decay size RMSE Rsquared MAE
## 0.01 1 1.427451 0.3887460 1.130936
## 0.01 2 1.620553 0.3115423 1.273391
## 0.01 3 1.853994 0.3336499 1.392409
## 0.01 4 2.051651 0.2761899 1.630915
## 0.01 5 1.942580 0.3168311 1.601149
## 0.01 6 1.732200 0.3285281 1.411282
## 0.01 7 2.083488 0.2152749 1.640340
## 0.01 8 2.468403 0.2522522 1.839822
## 0.01 9 2.767764 0.2937365 2.088256
## 0.01 10 4.280530 0.1917860 2.858685
## 0.02 1 1.458798 0.3932435 1.155685
## 0.02 2 1.807918 0.3606960 1.359665
## 0.02 3 1.868123 0.3139632 1.450237
## 0.02 4 2.012758 0.3266285 1.583753
## 0.02 5 1.985378 0.3310082 1.528892
## 0.02 6 1.991248 0.2337125 1.560010
## 0.02 7 1.973326 0.3729903 1.524827
## 0.02 8 2.374413 0.2662972 1.840226
## 0.02 9 2.775930 0.2388662 2.006360
## 0.02 10 3.335813 0.2025044 2.431955
## 0.03 1 1.585242 0.3755848 1.227608
## 0.03 2 1.476757 0.4260359 1.174995
## 0.03 3 1.754745 0.2935537 1.349305
## 0.03 4 1.741521 0.3285667 1.313340
## 0.03 5 1.957692 0.3043973 1.494413
## 0.03 6 1.826346 0.3608158 1.490702
## 0.03 7 1.929869 0.3252210 1.505998
## 0.03 8 2.637620 0.3372921 1.978753
## 0.03 9 3.060518 0.1620072 2.362720
## 0.03 10 3.444545 0.2101038 2.405551
## 0.04 1 1.353552 0.4628707 1.076860
## 0.04 2 1.380306 0.4613384 1.087277
## 0.04 3 1.783514 0.3094794 1.458493
## 0.04 4 2.235161 0.2559330 1.661532
## 0.04 5 1.736656 0.3246556 1.494199
## 0.04 6 1.716645 0.3638087 1.441975
## 0.04 7 1.841973 0.3574257 1.492535
## 0.04 8 2.249882 0.2919924 1.753178
## 0.04 9 3.427715 0.2450564 2.382084
## 0.04 10 3.924570 0.2698836 2.694337
## 0.05 1 1.420437 0.4155736 1.118680
## 0.05 2 1.430306 0.4258002 1.142858
## 0.05 3 1.637061 0.3830650 1.330358
## 0.05 4 1.970178 0.3346820 1.585407
## 0.05 5 1.980557 0.3214543 1.591256
## 0.05 6 1.789672 0.3592004 1.464016
## 0.05 7 1.721217 0.3919702 1.431492
## 0.05 8 2.368348 0.2823875 1.906301
## 0.05 9 3.026458 0.2189595 2.285271
## 0.05 10 3.731834 0.2576779 2.408524
## 0.06 1 1.411942 0.4275929 1.141417
## 0.06 2 1.539923 0.3857977 1.220320
## 0.06 3 1.607675 0.3650410 1.236290
## 0.06 4 1.978821 0.2765915 1.565986
## 0.06 5 1.595173 0.4233449 1.285335
## 0.06 6 1.908092 0.2671835 1.533573
## 0.06 7 1.997003 0.2810350 1.614707
## 0.06 8 2.239182 0.2804457 1.782907
## 0.06 9 3.323601 0.1700343 2.358865
## 0.06 10 3.581164 0.2102110 2.659024
## 0.07 1 1.368715 0.4580374 1.102102
## 0.07 2 1.411459 0.4672564 1.136610
## 0.07 3 1.643834 0.3728565 1.327541
## 0.07 4 2.084341 0.2980537 1.565960
## 0.07 5 1.881413 0.3267361 1.520333
## 0.07 6 1.607724 0.3758301 1.274806
## 0.07 7 1.989687 0.3775952 1.547057
## 0.07 8 2.268318 0.2719314 1.768904
## 0.07 9 3.123147 0.2661343 2.110991
## 0.07 10 3.519479 0.2090979 2.466776
## 0.08 1 1.430320 0.4252740 1.177936
## 0.08 2 1.469452 0.4249909 1.201104
## 0.08 3 1.548482 0.3915762 1.178563
## 0.08 4 1.837766 0.2933293 1.433302
## 0.08 5 1.958125 0.2778207 1.557876
## 0.08 6 1.708796 0.3720530 1.344469
## 0.08 7 1.982597 0.3160837 1.541975
## 0.08 8 2.584283 0.2153725 2.136519
## 0.08 9 3.254148 0.1979960 2.223982
## 0.08 10 3.690224 0.2249961 2.357296
## 0.09 1 1.443055 0.4039372 1.169081
## 0.09 2 1.626104 0.3488965 1.262790
## 0.09 3 1.777321 0.3268011 1.339624
## 0.09 4 2.068568 0.2584575 1.690297
## 0.09 5 1.866494 0.3021558 1.456848
## 0.09 6 1.841506 0.2957539 1.521895
## 0.09 7 1.833741 0.3178940 1.460764
## 0.09 8 2.105591 0.3485138 1.628470
## 0.09 9 2.792763 0.2567766 2.040789
## 0.09 10 3.720309 0.2148692 2.518638
## 0.10 1 1.388405 0.4456244 1.137111
## 0.10 2 1.497681 0.4079854 1.210880
## 0.10 3 1.775107 0.3515419 1.395480
## 0.10 4 1.838054 0.3112919 1.452439
## 0.10 5 1.791878 0.2910324 1.473367
## 0.10 6 1.767500 0.3594731 1.397193
## 0.10 7 1.849339 0.3735993 1.456400
## 0.10 8 2.223509 0.3214700 1.662424
## 0.10 9 2.868613 0.3067745 2.076558
## 0.10 10 3.553018 0.2153736 2.439118
##
## 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 = 0.04 and bag = FALSE.NeuralNetModPred <- predict(NeuralNetMod, newdata = data_test_X_nnet)
NeuralNet_metrics <- postResample(pred = NeuralNetModPred, obs = data_test_Y)
NeuralNet_metrics## RMSE Rsquared MAE
## 1.2317267 0.5703354 0.9433186Model Comparison
The best model selected by both RMSE & R2 is Radial SVM.
rbind(knn_metrics,
SVMLinear_metrics,
SVMRadial_metrics,
SVMPoly_metrics,
MARS_metrics,
NeuralNet_metrics) %>%
data.frame() %>%
arrange(RMSE)(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??
Top Impoartant Predictors
Anwer: 1) The tap 10 most important of the optimal nonlinear model (here the radial SVM model).
Manufacturing Process variables domiate the list.
The top 10 important predoctors selected by the radial SVM and Elastic Net model are the same.
SVMRadial_vapImp <- varImp(SVMRadialMod)
SVMRadial_top10_Imp <- SVMRadial_vapImp$importance %>%
arrange(desc(Overall)) %>%
top_n(10, Overall)
SVMRadial_top10_ImpOptimal linear model in #6.3
set.seed(200)
enetGrid <- data.frame(.lambda = seq(0, 3, length = 50),
.fraction = seq(0.01, 1, length = 50))
enetTune <- train(data_train_X, data_train_Y,
method = 'enet',
tuneGrid = enetGrid,
trControl = trainControl(method = "cv"),
preProc = c('center', 'scale'))
enet_vapImp <- varImp(enetTune)
enet_Top10_Imp <- enet_vapImp$importance %>%
arrange(desc(Overall)) %>%
top_n(10, Overall)
enet_Top10_Imp(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?
Answer: As observed from the correlation plot, all biological material (BM) predictors have postive correlationship with the response variable Yield, while the manufacturing process (MP) predictors are overallly have overall smaller positive correlation with Yield than those of BMs, or have negative correslation with Yield. In future runs of manufacturing process, those individual MP predictors with small absolute value of correlation values can be further analysed and improvement actions can be taken to such MP steps in order to increase the yield so as to boost revenue.
top_pred <- SVMRadial_top10_Imp %>%
rownames_to_column() %>%
spread(key = rowname, value = Overall)
cor_df <- data.frame(imp_chem_predictors) %>%
select(names(top_pred)) %>%
cbind(chem_response)
names(cor_df) <- names(cor_df) %>%
str_replace('BiologicalMaterial', 'BM') %>%
str_replace('ManufacturingProcess', 'MP')
cor_df <- cor(cor_df)
corrplot(cor_df)