DATA 624 Homework 7
Kristin Lussi
2025-11-02
Exercise 6.2
Part A
The matrix fingerprints contains the 1,107 binary molecular predic- tors for the 165 compounds, while permeability contains permeability response.
library("AppliedPredictiveModeling")
data(permeability)
# fingerprints
head(permeability)## permeability
## 1 12.520
## 2 1.120
## 3 19.405
## 4 1.730
## 5 1.680
## 6 0.510Part B
The fingerprint predictors indicate the presence or absence of substructures of a molecule and are often sparse meaning that relatively few of the molecules contain each substructure. Filter out the predictors that have low frequencies using the nearZeroVar function from the caret package. How many predictors are left for modeling?
library(caret)
# find near-zero variance predictors
nzv <- nearZeroVar(fingerprints)
# remove the near-zero variance predictors
filtered_fp <- fingerprints[ , -nzv]
# number of predictors left
num_predictors_left <- ncol(filtered_fp)
num_predictors_left## [1] 388There are 388 predictors left for modeling.
Part C
Split the data into a training and a test set, pre-process the data, and tune a PLS model. How many latent variables are optimal and what is the corresponding resampled estimate of \(R^2\)?
set.seed(1999)
# Split into training and testing
split <- createDataPartition(permeability, p = 0.8, list = FALSE)
# train
train_perm <- permeability[split, ]
train_finger <- filtered_fp[split,]
# test
test_perm <- permeability[-split, ]
test_finger <- filtered_fp[-split,]
# train and tune PLS model
ctrl <- trainControl(method = "cv", number = 10)
pls_model <- train(
train_finger,
train_perm,
method = "pls",
metric ="Rsquared",
tuneLength = 20,
trControl = ctrl,
preProcess = c("center", "scale")
)
# check optimal number of latent variables and R^2
pls_model$bestTune## ncomp
## 7 7max(pls_model$results$Rsquared)## [1] 0.5989923There are 7 optimal latent variables and the resampled estimated \(R^2 = 0.5989923\).
Part D
Predict the response for the test set. What is the test set estimate of \(R^2\)?
# predict on test set
test_pred <- predict(pls_model, test_finger)
postResample(pred = test_pred, obs = test_perm)## RMSE Rsquared MAE
## 13.0656120 0.3460259 9.4570753The test set estimate of \(R^2 = 0.3460259\).
Part E
Try building other models discussed in this chapter. Do any have better predictive performance?
# Ridge Regression
ridgeGrid <- data.frame(.lambda = seq(0, .1, length = 15))
ridge_model <- train(
train_finger,
train_perm,
method = "ridge",
## Fir the model over many penalty values
tuneGrid = ridgeGrid,
trControl = ctrl,
## put the predictors on the same scale
preProc = c("center", "scale"))
# Elastic Net
enet_model <- train(
train_finger,
train_perm,
method = "enet",
trControl = ctrl,
preProc = c("center", "scale"))
# Lasso Model
lasso_model <-train(
train_finger,
train_perm,
method = "lasso",
trControl = ctrl,
preProc = c("center", "scale"))
# Compare resampled performance
resamps <- resamples(list(
PLS = pls_model,
Ridge = ridge_model,
Lasso = lasso_model,
ElasticNet = enet_model
))
summary(resamps)##
## Call:
## summary.resamples(object = resamps)
##
## Models: PLS, Ridge, Lasso, ElasticNet
## Number of resamples: 10
##
## MAE
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## PLS 4.924528 7.205850 7.759610 8.222309e+00 8.810132e+00 1.190481e+01
## Ridge 6.229809 7.351045 8.795244 8.710503e+00 9.527683e+00 1.146386e+01
## Lasso 6.011751 8.366535 11.575358 5.593351e+16 9.900714e+15 5.242374e+17
## ElasticNet 6.005219 6.828456 7.943841 8.132488e+00 8.681726e+00 1.288952e+01
## NA's
## PLS 0
## Ridge 0
## Lasso 0
## ElasticNet 0
##
## RMSE
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## PLS 6.076067 9.118263 10.46751 1.067299e+01 1.226606e+01 1.469694e+01
## Ridge 8.486751 9.945819 11.23078 1.143889e+01 1.317923e+01 1.473366e+01
## Lasso 7.979203 11.274779 15.97868 1.458469e+17 2.597406e+16 1.344887e+18
## ElasticNet 8.044979 8.591701 10.66512 1.065031e+01 1.173408e+01 1.614912e+01
## NA's
## PLS 0
## Ridge 0
## Lasso 0
## ElasticNet 0
##
## Rsquared
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## PLS 0.269152004 0.48317227 0.6298900 0.5989923 0.7301767 0.8070123 0
## Ridge 0.153681613 0.34347812 0.5445163 0.5041510 0.6632503 0.7912857 0
## Lasso 0.007819875 0.05271164 0.2948675 0.3199956 0.5335093 0.8830764 0
## ElasticNet 0.128902356 0.42368227 0.6536112 0.5543604 0.7330841 0.7680089 0bwplot(resamps, metric = "Rsquared")
PLS has the highest \(R^2\) and the lowest RMSE. Elastic net comes in close second, which suggests that there is some benefit from combining L1 and L2 penalties, but Ridge and Lasso individually perform worse.
# Predict and evaluate on the test set for all models
models <- list(
PLS = pls_model,
Ridge = ridge_model,
Lasso = lasso_model,
ElasticNet = enet_model
)
# predictions
pred_list <- lapply(models, predict, newdata = test_finger)
# caret metrics
metric_list <- lapply(pred_list, function(p) postResample(pred = p, obs = test_perm))
# bind into one data frame
test_results <- do.call(rbind, metric_list)
test_results <- cbind(Model = rownames(test_results),
as.data.frame(test_results))
rownames(test_results) <- NULL
test_results[order(-test_results$Rsquared), ]## Model RMSE Rsquared MAE
## 1 PLS 1.306561e+01 0.34602594 9.457075e+00
## 4 ElasticNet 1.383054e+01 0.28758334 9.296786e+00
## 2 Ridge 1.404174e+01 0.27068608 9.393488e+00
## 3 Lasso 7.956568e+06 0.02834479 2.425811e+06PLS achieved the best predictive performance across all of the models with having the highest \(R^2\) and lowest RMSE. Elastic Net and Ridge Regression come in close behind PLS, while Lasso performed very poorly. In conclusion, these results show that PLS performs the best on unseen data.
Part E
Would you recommend any of your models to replace the permeability laboratory experiment?
The PLS model performs the best out of the other models tested. I would recommend that the use of PLS is to be continued.
Exercise 6.3
A chemical manufacturing process for a pharmaceutical product was discussed in Sect.1.4. In this problem, the objective is to understand the re- lationship between biological measurements of the raw materials (predictors), the corresponding resampled estimate of \(R^2\) measurements of the manufacturing process (predictors), and the response of product yield. Biological predictors cannot be changed but can be used to assess the quality of the raw material before processing. On the other hand, manufacturing process predictors can be changed in the manufacturing process. Improving product yield by 1% will boost revenue by approximately one hundred thousand dollars per batch:
Part A
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(ChemicalManufacturingProcess)
# separate response and predictors
yield <- ChemicalManufacturingProcess$Yield
predictors <- ChemicalManufacturingProcess[, -1]
ncol(predictors)## [1] 57Part B
A small percentage of cells in the predictor set contain missing values. Use an imputation function to fill in these missing values
# Check NAs
sum(is.na(ChemicalManufacturingProcess))## [1] 106# Impute using KNN
preProc <- preProcess(
ChemicalManufacturingProcess,
method = "knnImpute")
chemical_imp <- predict(preProc, ChemicalManufacturingProcess)
# check for missing values after imputation
sum(is.na(chemical_imp))## [1] 0Part C
Split the data into a training and a test set, pre-process the data, and tune a model of your choice from this chapter. What is the optimal value of the performance metric?
library(dplyr)
predictors_imp <- chemical_imp %>% select(-Yield)
# split data into X & y
X <- predictors_imp
y <- chemical_imp$Yield
# find near-zero variance predictors
nzv_pred <- nearZeroVar(X)
# remove the near-zero variance predictors
filtered_pred <- X[ , -nzv_pred]
# Indexing for train/test split
split <- createDataPartition(y, p = 0.8, list = FALSE)
# train and test sets
train_x <- X[ split, , drop = FALSE]
test_x <- X[-split, , drop = FALSE]
train_y <- y[ split]
test_y <- y[-split]
# Fit and train Elastic Net model
ctrl <- trainControl(method = "cv", number = 10)
enet_model_chem <- train(
x = train_x, y = train_y,
method = "enet",
metric = "Rsquared",
tuneLength = 20,
trControl = ctrl,
preProcess = c("center", "scale")
)
enet_model_chem## Elasticnet
##
## 144 samples
## 57 predictor
##
## Pre-processing: centered (57), scaled (57)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 130, 128, 131, 129, 130, 130, ...
## Resampling results across tuning parameters:
##
## lambda fraction RMSE Rsquared MAE
## 0.0000000000 0.05 0.6788593 0.5992809 0.5666962
## 0.0000000000 0.10 0.6258766 0.6088763 0.5117745
## 0.0000000000 0.15 0.7015448 0.5984460 0.5206475
## 0.0000000000 0.20 0.8982082 0.5955380 0.5738562
## 0.0000000000 0.25 1.1516827 0.5897668 0.6402826
## 0.0000000000 0.30 1.3120696 0.5847219 0.6832045
## 0.0000000000 0.35 1.3838960 0.5560296 0.7141306
## 0.0000000000 0.40 1.6307992 0.5216551 0.7907288
## 0.0000000000 0.45 1.8544283 0.5051638 0.8617333
## 0.0000000000 0.50 2.0444635 0.4960725 0.9237540
## 0.0000000000 0.55 2.3207614 0.4870526 1.0104655
## 0.0000000000 0.60 2.8877399 0.4839226 1.1719041
## 0.0000000000 0.65 3.4633291 0.4833160 1.3328196
## 0.0000000000 0.70 4.0890160 0.4843785 1.5067492
## 0.0000000000 0.75 4.6088013 0.4857715 1.6522424
## 0.0000000000 0.80 4.9538961 0.4892524 1.7499055
## 0.0000000000 0.85 5.4167456 0.4882967 1.8760011
## 0.0000000000 0.90 5.8685550 0.4863001 1.9983408
## 0.0000000000 0.95 6.2509458 0.4835518 2.1016244
## 0.0000000000 1.00 6.6165471 0.4808203 2.2038471
## 0.0001000000 0.05 0.7164245 0.5937625 0.5950563
## 0.0001000000 0.10 0.6572385 0.5931503 0.5292584
## 0.0001000000 0.15 0.6723377 0.6030883 0.5153848
## 0.0001000000 0.20 0.8563218 0.5944384 0.5667448
## 0.0001000000 0.25 1.0942166 0.5937401 0.6248712
## 0.0001000000 0.30 1.2671981 0.5906466 0.6702627
## 0.0001000000 0.35 1.3306518 0.5890952 0.6894903
## 0.0001000000 0.40 1.4573307 0.5833741 0.7240189
## 0.0001000000 0.45 1.6703210 0.5504478 0.7957001
## 0.0001000000 0.50 1.9051935 0.5303680 0.8663506
## 0.0001000000 0.55 2.0940884 0.5200143 0.9263136
## 0.0001000000 0.60 2.5780981 0.5118830 1.0656080
## 0.0001000000 0.65 3.1116055 0.5030835 1.2177236
## 0.0001000000 0.70 3.6828385 0.4932037 1.3803834
## 0.0001000000 0.75 4.2647934 0.4886267 1.5449731
## 0.0001000000 0.80 4.7070145 0.4833418 1.6712713
## 0.0001000000 0.85 5.0557320 0.4790780 1.7724920
## 0.0001000000 0.90 5.5019799 0.4747631 1.8972360
## 0.0001000000 0.95 5.8614243 0.4716302 1.9971163
## 0.0001000000 1.00 6.2076678 0.4697259 2.0927394
## 0.0001467799 0.05 0.7245967 0.5927038 0.6011757
## 0.0001467799 0.10 0.6627429 0.5903881 0.5329632
## 0.0001467799 0.15 0.6619466 0.6056642 0.5137748
## 0.0001467799 0.20 0.8397543 0.5937028 0.5626903
## 0.0001467799 0.25 1.0720314 0.5936926 0.6198183
## 0.0001467799 0.30 1.2492728 0.5910804 0.6656304
## 0.0001467799 0.35 1.3267368 0.5905475 0.6883110
## 0.0001467799 0.40 1.4313842 0.5851531 0.7168912
## 0.0001467799 0.45 1.6175050 0.5646512 0.7786967
## 0.0001467799 0.50 1.8456048 0.5389351 0.8478742
## 0.0001467799 0.55 2.0529205 0.5256570 0.9116374
## 0.0001467799 0.60 2.4627230 0.5162846 1.0306018
## 0.0001467799 0.65 2.9827133 0.5084161 1.1790714
## 0.0001467799 0.70 3.5259676 0.4986027 1.3334288
## 0.0001467799 0.75 4.1023654 0.4899500 1.4970378
## 0.0001467799 0.80 4.6050836 0.4854571 1.6395114
## 0.0001467799 0.85 4.9333524 0.4794612 1.7359642
## 0.0001467799 0.90 5.3499689 0.4741856 1.8533624
## 0.0001467799 0.95 5.7140669 0.4689434 1.9554350
## 0.0001467799 1.00 6.0493530 0.4659400 2.0484824
## 0.0002154435 0.05 0.7327723 0.5916631 0.6076845
## 0.0002154435 0.10 0.6693290 0.5872713 0.5378196
## 0.0002154435 0.15 0.6490334 0.6096433 0.5111934
## 0.0002154435 0.20 0.8303786 0.5932117 0.5605840
## 0.0002154435 0.25 1.0441950 0.5934397 0.6139080
## 0.0002154435 0.30 1.2258887 0.5918272 0.6594110
## 0.0002154435 0.35 1.3225572 0.5913070 0.6871218
## 0.0002154435 0.40 1.3976076 0.5864850 0.7076117
## 0.0002154435 0.45 1.5603480 0.5817190 0.7571815
## 0.0002154435 0.50 1.7747403 0.5507047 0.8262529
## 0.0002154435 0.55 2.0022962 0.5315089 0.8944341
## 0.0002154435 0.60 2.3077504 0.5212579 0.9850734
## 0.0002154435 0.65 2.8230828 0.5122337 1.1321337
## 0.0002154435 0.70 3.3190930 0.5041639 1.2734792
## 0.0002154435 0.75 3.8720433 0.4945680 1.4304335
## 0.0002154435 0.80 4.4047336 0.4868357 1.5812130
## 0.0002154435 0.85 4.7875540 0.4815670 1.6919074
## 0.0002154435 0.90 5.1490514 0.4747265 1.7958841
## 0.0002154435 0.95 5.5052148 0.4682716 1.8963170
## 0.0002154435 1.00 5.8383074 0.4638221 1.9893036
## 0.0003162278 0.05 0.7402961 0.5911471 0.6136946
## 0.0003162278 0.10 0.6763431 0.5842012 0.5425440
## 0.0003162278 0.15 0.6355428 0.6154413 0.5081219
## 0.0003162278 0.20 0.8205138 0.5933146 0.5579472
## 0.0003162278 0.25 1.0101778 0.5931847 0.6060690
## 0.0003162278 0.30 1.1975326 0.5927386 0.6520239
## 0.0003162278 0.35 1.3117537 0.5917486 0.6838742
## 0.0003162278 0.40 1.3573428 0.5884628 0.6976634
## 0.0003162278 0.45 1.5131233 0.5857125 0.7411382
## 0.0003162278 0.50 1.6995557 0.5649536 0.8028801
## 0.0003162278 0.55 1.9155975 0.5410484 0.8680097
## 0.0003162278 0.60 2.1124817 0.5273738 0.9291157
## 0.0003162278 0.65 2.6253056 0.5157199 1.0754408
## 0.0003162278 0.70 3.0957090 0.5077588 1.2098624
## 0.0003162278 0.75 3.5922247 0.4991610 1.3511591
## 0.0003162278 0.80 4.1225834 0.4903713 1.5012409
## 0.0003162278 0.85 4.5926608 0.4823625 1.6347690
## 0.0003162278 0.90 4.9222929 0.4763262 1.7308603
## 0.0003162278 0.95 5.2427207 0.4690085 1.8223490
## 0.0003162278 1.00 5.5678098 0.4637371 1.9134685
## 0.0004641589 0.05 0.7471778 0.5905727 0.6191679
## 0.0004641589 0.10 0.6738944 0.5844872 0.5431683
## 0.0004641589 0.15 0.6222986 0.6242720 0.5042106
## 0.0004641589 0.20 0.8004934 0.5942016 0.5524564
## 0.0004641589 0.25 0.9715420 0.5926913 0.5969000
## 0.0004641589 0.30 1.1652264 0.5927676 0.6441451
## 0.0004641589 0.35 1.2937281 0.5922235 0.6785470
## 0.0004641589 0.40 1.3227526 0.5901868 0.6891363
## 0.0004641589 0.45 1.4608083 0.5862401 0.7265077
## 0.0004641589 0.50 1.6202567 0.5814972 0.7752644
## 0.0004641589 0.55 1.8224156 0.5526949 0.8397866
## 0.0004641589 0.60 2.0184582 0.5333496 0.8999454
## 0.0004641589 0.65 2.3879235 0.5216788 1.0074124
## 0.0004641589 0.70 2.8525730 0.5112032 1.1405822
## 0.0004641589 0.75 3.2948609 0.5033814 1.2669383
## 0.0004641589 0.80 3.7774024 0.4950131 1.4042795
## 0.0004641589 0.85 4.2401537 0.4865239 1.5354917
## 0.0004641589 0.90 4.6751102 0.4782343 1.6590987
## 0.0004641589 0.95 4.9361734 0.4715460 1.7358408
## 0.0004641589 1.00 5.2385301 0.4656975 1.8211941
## 0.0006812921 0.05 0.7533980 0.5906116 0.6242481
## 0.0006812921 0.10 0.6676013 0.5872719 0.5417037
## 0.0006812921 0.15 0.6203827 0.6259190 0.5041319
## 0.0006812921 0.20 0.7669021 0.5961302 0.5429985
## 0.0006812921 0.25 0.9278340 0.5922698 0.5861976
## 0.0006812921 0.30 1.1279707 0.5927249 0.6352798
## 0.0006812921 0.35 1.2640211 0.5922248 0.6703494
## 0.0006812921 0.40 1.3198066 0.5911215 0.6882350
## 0.0006812921 0.45 1.4007788 0.5866049 0.7092787
## 0.0006812921 0.50 1.5459633 0.5857055 0.7505488
## 0.0006812921 0.55 1.7253898 0.5659194 0.8098939
## 0.0006812921 0.60 1.9251695 0.5428914 0.8706125
## 0.0006812921 0.65 2.1320154 0.5283738 0.9338194
## 0.0006812921 0.70 2.5801709 0.5157325 1.0629224
## 0.0006812921 0.75 2.9904224 0.5071097 1.1807943
## 0.0006812921 0.80 3.4046273 0.4995689 1.2992458
## 0.0006812921 0.85 3.8479433 0.4916187 1.4255705
## 0.0006812921 0.90 4.2521526 0.4825309 1.5404839
## 0.0006812921 0.95 4.6070903 0.4754359 1.6419168
## 0.0006812921 1.00 4.8609602 0.4695685 1.7152604
## 0.0010000000 0.05 0.7599721 0.5911200 0.6294868
## 0.0010000000 0.10 0.6637556 0.5900804 0.5419552
## 0.0010000000 0.15 0.6214868 0.6253447 0.5054438
## 0.0010000000 0.20 0.7379586 0.5982278 0.5347056
## 0.0010000000 0.25 0.8869150 0.5924181 0.5759340
## 0.0010000000 0.30 1.0803112 0.5929738 0.6237003
## 0.0010000000 0.35 1.2272001 0.5927309 0.6600795
## 0.0010000000 0.40 1.3105008 0.5917851 0.6850782
## 0.0010000000 0.45 1.3504455 0.5877953 0.6966131
## 0.0010000000 0.50 1.4888612 0.5859756 0.7344980
## 0.0010000000 0.55 1.6457073 0.5801174 0.7828400
## 0.0010000000 0.60 1.8342202 0.5540250 0.8425432
## 0.0010000000 0.65 2.0073859 0.5354154 0.8963138
## 0.0010000000 0.70 2.3566331 0.5231079 0.9976610
## 0.0010000000 0.75 2.7547298 0.5119583 1.1127957
## 0.0010000000 0.80 3.1252362 0.5044958 1.2195402
## 0.0010000000 0.85 3.5197194 0.4968957 1.3324019
## 0.0010000000 0.90 3.8953116 0.4885973 1.4398915
## 0.0010000000 0.95 4.2265954 0.4811643 1.5343714
## 0.0010000000 1.00 4.4543753 0.4751446 1.6007763
## 0.0014677993 0.05 0.7677525 0.5909378 0.6357168
## 0.0014677993 0.10 0.6603930 0.5950900 0.5434426
## 0.0014677993 0.15 0.6229387 0.6247682 0.5068955
## 0.0014677993 0.20 0.7055369 0.6006146 0.5259724
## 0.0014677993 0.25 0.8487366 0.5927871 0.5663255
## 0.0014677993 0.30 1.0297019 0.5928268 0.6116977
## 0.0014677993 0.35 1.1860839 0.5927101 0.6493698
## 0.0014677993 0.40 1.2856219 0.5921014 0.6774588
## 0.0014677993 0.45 1.3137390 0.5896117 0.6873126
## 0.0014677993 0.50 1.4258000 0.5865609 0.7164270
## 0.0014677993 0.55 1.5694890 0.5854626 0.7571547
## 0.0014677993 0.60 1.7373951 0.5668944 0.8124579
## 0.0014677993 0.65 1.9232701 0.5453975 0.8692585
## 0.0014677993 0.70 2.1351031 0.5302713 0.9332515
## 0.0014677993 0.75 2.5035619 0.5195426 1.0395878
## 0.0014677993 0.80 2.8483671 0.5100740 1.1398837
## 0.0014677993 0.85 3.1821066 0.5026790 1.2364870
## 0.0014677993 0.90 3.5298042 0.4954343 1.3365347
## 0.0014677993 0.95 3.8387931 0.4882212 1.4248221
## 0.0014677993 1.00 4.0418937 0.4821256 1.4840675
## 0.0021544347 0.05 0.7766459 0.5902713 0.6427716
## 0.0021544347 0.10 0.6604587 0.6013410 0.5466553
## 0.0021544347 0.15 0.6253381 0.6231990 0.5091199
## 0.0021544347 0.20 0.6711572 0.6055438 0.5168365
## 0.0021544347 0.25 0.8183788 0.5943489 0.5577391
## 0.0021544347 0.30 0.9767478 0.5922413 0.5990656
## 0.0021544347 0.35 1.1415316 0.5925104 0.6393178
## 0.0021544347 0.40 1.2598248 0.5923860 0.6694180
## 0.0021544347 0.45 1.3031538 0.5907757 0.6839480
## 0.0021544347 0.50 1.3557807 0.5877015 0.6976213
## 0.0021544347 0.55 1.4887915 0.5863173 0.7342103
## 0.0021544347 0.60 1.6326754 0.5812268 0.7780913
## 0.0021544347 0.65 1.8058375 0.5579280 0.8331019
## 0.0021544347 0.70 1.9638862 0.5396903 0.8821664
## 0.0021544347 0.75 2.2514684 0.5275111 0.9660834
## 0.0021544347 0.80 2.5680901 0.5177787 1.0582454
## 0.0021544347 0.85 2.8741733 0.5089987 1.1476743
## 0.0021544347 0.90 3.1672395 0.5023511 1.2330657
## 0.0021544347 0.95 3.4554398 0.4960623 1.3161608
## 0.0021544347 1.00 3.6439244 0.4900868 1.3713851
## 0.0031622777 0.05 0.7871146 0.5889321 0.6509113
## 0.0031622777 0.10 0.6665880 0.6040092 0.5524089
## 0.0031622777 0.15 0.6354424 0.6117612 0.5181184
## 0.0031622777 0.20 0.6409336 0.6139253 0.5095949
## 0.0031622777 0.25 0.7891829 0.5967348 0.5492342
## 0.0031622777 0.30 0.9239979 0.5917416 0.5864116
## 0.0031622777 0.35 1.0807635 0.5926094 0.6251195
## 0.0031622777 0.40 1.2055064 0.5925499 0.6544784
## 0.0031622777 0.45 1.2750134 0.5918375 0.6752335
## 0.0031622777 0.50 1.2995580 0.5896103 0.6834917
## 0.0031622777 0.55 1.4055117 0.5874439 0.7103415
## 0.0031622777 0.60 1.5372600 0.5862410 0.7478378
## 0.0031622777 0.65 1.6833105 0.5726352 0.7951041
## 0.0031622777 0.70 1.8478474 0.5520574 0.8457825
## 0.0031622777 0.75 2.0201860 0.5364148 0.8983168
## 0.0031622777 0.80 2.3085124 0.5262183 0.9820205
## 0.0031622777 0.85 2.5788630 0.5174551 1.0614049
## 0.0031622777 0.90 2.8428254 0.5099255 1.1389016
## 0.0031622777 0.95 3.0863311 0.5039009 1.2101578
## 0.0031622777 1.00 3.2735177 0.4985184 1.2652678
## 0.0046415888 0.05 0.7987549 0.5870544 0.6596716
## 0.0046415888 0.10 0.6773885 0.6014712 0.5623393
## 0.0046415888 0.15 0.6664055 0.5889346 0.5360607
## 0.0046415888 0.20 0.6201781 0.6266341 0.5038735
## 0.0046415888 0.25 0.7410746 0.5994122 0.5350586
## 0.0046415888 0.30 0.8663848 0.5934990 0.5709666
## 0.0046415888 0.35 1.0126305 0.5924462 0.6089403
## 0.0046415888 0.40 1.1486611 0.5927406 0.6416940
## 0.0046415888 0.45 1.2491482 0.5927497 0.6665067
## 0.0046415888 0.50 1.2817071 0.5910539 0.6777294
## 0.0046415888 0.55 1.3180805 0.5892896 0.6875967
## 0.0046415888 0.60 1.4428555 0.5876629 0.7205525
## 0.0046415888 0.65 1.5693314 0.5839316 0.7592508
## 0.0046415888 0.70 1.7159975 0.5659837 0.8055494
## 0.0046415888 0.75 1.8537713 0.5476407 0.8482717
## 0.0046415888 0.80 2.0769418 0.5340046 0.9138276
## 0.0046415888 0.85 2.3155139 0.5257192 0.9836522
## 0.0046415888 0.90 2.5455089 0.5185937 1.0516818
## 0.0046415888 0.95 2.7586280 0.5125274 1.1147546
## 0.0046415888 1.00 2.9355861 0.5069287 1.1667371
## 0.0068129207 0.05 0.8110334 0.5844760 0.6686486
## 0.0068129207 0.10 0.6896210 0.6000396 0.5724275
## 0.0068129207 0.15 0.6791438 0.5818832 0.5441548
## 0.0068129207 0.20 0.6225217 0.6251567 0.5064708
## 0.0068129207 0.25 0.6951920 0.6026596 0.5233475
## 0.0068129207 0.30 0.8135712 0.5963757 0.5561233
## 0.0068129207 0.35 0.9532286 0.5920444 0.5942386
## 0.0068129207 0.40 1.0882420 0.5928691 0.6277525
## 0.0068129207 0.45 1.1908371 0.5931549 0.6516214
## 0.0068129207 0.50 1.2478543 0.5927732 0.6669172
## 0.0068129207 0.55 1.2756479 0.5914294 0.6756553
## 0.0068129207 0.60 1.3486723 0.5897592 0.6943385
## 0.0068129207 0.65 1.4662560 0.5882797 0.7265769
## 0.0068129207 0.70 1.5927435 0.5782451 0.7683139
## 0.0068129207 0.75 1.7211350 0.5610837 0.8085330
## 0.0068129207 0.80 1.8682792 0.5452893 0.8531876
## 0.0068129207 0.85 2.0902243 0.5332969 0.9171962
## 0.0068129207 0.90 2.2917675 0.5262402 0.9765861
## 0.0068129207 0.95 2.4752899 0.5200652 1.0312661
## 0.0068129207 1.00 2.6293012 0.5149263 1.0770808
## 0.0100000000 0.05 0.8235625 0.5808799 0.6778438
## 0.0100000000 0.10 0.7039138 0.5977001 0.5834764
## 0.0100000000 0.15 0.6720808 0.5856118 0.5447896
## 0.0100000000 0.20 0.6252128 0.6234733 0.5097162
## 0.0100000000 0.25 0.6566324 0.6085207 0.5143750
## 0.0100000000 0.30 0.7665738 0.5988391 0.5419625
## 0.0100000000 0.35 0.8949871 0.5941002 0.5784361
## 0.0100000000 0.40 1.0149415 0.5926631 0.6101514
## 0.0100000000 0.45 1.1316063 0.5932771 0.6382064
## 0.0100000000 0.50 1.2152192 0.5932677 0.6582833
## 0.0100000000 0.55 1.2476214 0.5931308 0.6673950
## 0.0100000000 0.60 1.2744061 0.5918976 0.6755062
## 0.0100000000 0.65 1.3674575 0.5907183 0.6984125
## 0.0100000000 0.70 1.4829807 0.5834140 0.7353645
## 0.0100000000 0.75 1.6016470 0.5677970 0.7731160
## 0.0100000000 0.80 1.7166446 0.5524558 0.8085178
## 0.0100000000 0.85 1.8991815 0.5405114 0.8614339
## 0.0100000000 0.90 2.0803707 0.5320814 0.9142761
## 0.0100000000 0.95 2.2292327 0.5267601 0.9584236
## 0.0100000000 1.00 2.3515552 0.5222257 0.9951519
## 0.0146779927 0.05 0.8357810 0.5762259 0.6876136
## 0.0146779927 0.10 0.7189245 0.5954855 0.5948685
## 0.0146779927 0.15 0.6648247 0.5928743 0.5455902
## 0.0146779927 0.20 0.6324344 0.6156773 0.5170251
## 0.0146779927 0.25 0.6303733 0.6187771 0.5084701
## 0.0146779927 0.30 0.7250809 0.6002688 0.5311531
## 0.0146779927 0.35 0.8382764 0.5967256 0.5621737
## 0.0146779927 0.40 0.9549439 0.5937916 0.5940639
## 0.0146779927 0.45 1.0662940 0.5935159 0.6227861
## 0.0146779927 0.50 1.1536931 0.5941232 0.6434568
## 0.0146779927 0.55 1.2092620 0.5944517 0.6567778
## 0.0146779927 0.60 1.2440429 0.5939029 0.6666250
## 0.0146779927 0.65 1.2693721 0.5928124 0.6738525
## 0.0146779927 0.70 1.3782265 0.5849638 0.7053668
## 0.0146779927 0.75 1.4855205 0.5690733 0.7391371
## 0.0146779927 0.80 1.5921719 0.5562454 0.7714840
## 0.0146779927 0.85 1.7331492 0.5450088 0.8125727
## 0.0146779927 0.90 1.8875324 0.5365930 0.8569652
## 0.0146779927 0.95 2.0028235 0.5321471 0.8910178
## 0.0146779927 1.00 2.0996835 0.5286321 0.9201730
## 0.0215443469 0.05 0.8479110 0.5702861 0.6977428
## 0.0215443469 0.10 0.7344565 0.5934433 0.6074385
## 0.0215443469 0.15 0.6659116 0.6019422 0.5522435
## 0.0215443469 0.20 0.6774296 0.5849547 0.5407098
## 0.0215443469 0.25 0.6232403 0.6250442 0.5073381
## 0.0215443469 0.30 0.6934810 0.6012489 0.5244320
## 0.0215443469 0.35 0.7887192 0.5985100 0.5472854
## 0.0215443469 0.40 0.8870614 0.5960806 0.5750789
## 0.0215443469 0.45 0.9911017 0.5944007 0.6033821
## 0.0215443469 0.50 1.0910569 0.5942433 0.6287167
## 0.0215443469 0.55 1.1618349 0.5953146 0.6452715
## 0.0215443469 0.60 1.2004113 0.5957368 0.6544723
## 0.0215443469 0.65 1.2349998 0.5947028 0.6639866
## 0.0215443469 0.70 1.2680038 0.5852693 0.6778711
## 0.0215443469 0.75 1.3654598 0.5707406 0.7045911
## 0.0215443469 0.80 1.4641799 0.5581198 0.7336327
## 0.0215443469 0.85 1.5666186 0.5486686 0.7639793
## 0.0215443469 0.90 1.6902172 0.5416945 0.7990738
## 0.0215443469 0.95 1.7922766 0.5368844 0.8288359
## 0.0215443469 1.00 1.8726776 0.5340727 0.8523146
## 0.0316227766 0.05 0.8597827 0.5629103 0.7075396
## 0.0316227766 0.10 0.7506792 0.5913636 0.6206196
## 0.0316227766 0.15 0.6775730 0.6044448 0.5624449
## 0.0316227766 0.20 0.6775028 0.5846686 0.5442053
## 0.0316227766 0.25 0.6213627 0.6285109 0.5064400
## 0.0316227766 0.30 0.6569657 0.6060711 0.5171326
## 0.0316227766 0.35 0.7469450 0.5989678 0.5374627
## 0.0316227766 0.40 0.8324846 0.5984543 0.5587709
## 0.0316227766 0.45 0.9162689 0.5963467 0.5824947
## 0.0316227766 0.50 1.0142840 0.5953637 0.6091205
## 0.0316227766 0.55 1.0956619 0.5955207 0.6296820
## 0.0316227766 0.60 1.1444499 0.5966728 0.6404890
## 0.0316227766 0.65 1.1725941 0.5967191 0.6474761
## 0.0316227766 0.70 1.2034494 0.5889626 0.6598985
## 0.0316227766 0.75 1.2444456 0.5717283 0.6740658
## 0.0316227766 0.80 1.3311040 0.5603706 0.6966380
## 0.0316227766 0.85 1.4115116 0.5519664 0.7183602
## 0.0316227766 0.90 1.5078918 0.5460334 0.7461657
## 0.0316227766 0.95 1.6016818 0.5416709 0.7728913
## 0.0316227766 1.00 1.6709586 0.5386574 0.7928982
## 0.0464158883 0.05 0.8710600 0.5543198 0.7167080
## 0.0464158883 0.10 0.7670056 0.5898604 0.6342624
## 0.0464158883 0.15 0.6941294 0.6018546 0.5763423
## 0.0464158883 0.20 0.6668994 0.5915312 0.5440361
## 0.0464158883 0.25 0.6559089 0.5966230 0.5299599
## 0.0464158883 0.30 0.6244240 0.6238285 0.5079370
## 0.0464158883 0.35 0.7061429 0.5994013 0.5289703
## 0.0464158883 0.40 0.7814977 0.5993837 0.5449943
## 0.0464158883 0.45 0.8572125 0.5991528 0.5648728
## 0.0464158883 0.50 0.9308567 0.5981553 0.5853104
## 0.0464158883 0.55 1.0257555 0.5974395 0.6113205
## 0.0464158883 0.60 1.0730428 0.5972548 0.6233429
## 0.0464158883 0.65 1.1096910 0.5978645 0.6315256
## 0.0464158883 0.70 1.1393426 0.5935618 0.6416313
## 0.0464158883 0.75 1.1794944 0.5772407 0.6561125
## 0.0464158883 0.80 1.2110034 0.5627339 0.6664161
## 0.0464158883 0.85 1.2719598 0.5553363 0.6825228
## 0.0464158883 0.90 1.3456015 0.5500240 0.7013733
## 0.0464158883 0.95 1.4329031 0.5453880 0.7241749
## 0.0464158883 1.00 1.4953193 0.5426873 0.7415996
## 0.0681292069 0.05 0.8817037 0.5440658 0.7252979
## 0.0681292069 0.10 0.7835308 0.5875238 0.6476990
## 0.0681292069 0.15 0.7118640 0.5988115 0.5896119
## 0.0681292069 0.20 0.6659416 0.5999157 0.5507224
## 0.0681292069 0.25 0.6815841 0.5851085 0.5425247
## 0.0681292069 0.30 0.6238221 0.6240680 0.5111304
## 0.0681292069 0.35 0.6567414 0.6070657 0.5175289
## 0.0681292069 0.40 0.7402624 0.5990626 0.5366784
## 0.0681292069 0.45 0.8067196 0.6001459 0.5507772
## 0.0681292069 0.50 0.8714630 0.6004696 0.5678006
## 0.0681292069 0.55 0.9356576 0.6005043 0.5851993
## 0.0681292069 0.60 1.0118024 0.5995584 0.6065200
## 0.0681292069 0.65 1.0482727 0.5991260 0.6163057
## 0.0681292069 0.70 1.0829098 0.5946326 0.6281433
## 0.0681292069 0.75 1.1183520 0.5837514 0.6396675
## 0.0681292069 0.80 1.1525183 0.5700342 0.6507775
## 0.0681292069 0.85 1.1805433 0.5592896 0.6597225
## 0.0681292069 0.90 1.2123018 0.5534378 0.6680065
## 0.0681292069 0.95 1.2833723 0.5496998 0.6861547
## 0.0681292069 1.00 1.3458925 0.5465620 0.7023510
## 0.1000000000 0.05 0.8913186 0.5321494 0.7329343
## 0.1000000000 0.10 0.7994363 0.5846277 0.6602027
## 0.1000000000 0.15 0.7292866 0.5963155 0.6025383
## 0.1000000000 0.20 0.6754996 0.6057307 0.5617422
## 0.1000000000 0.25 0.6739722 0.5877633 0.5438605
## 0.1000000000 0.30 0.6531419 0.5996279 0.5280549
## 0.1000000000 0.35 0.6129639 0.6364291 0.4991091
## 0.1000000000 0.40 0.6804064 0.6035920 0.5232174
## 0.1000000000 0.45 0.7586248 0.6002633 0.5399929
## 0.1000000000 0.50 0.8263413 0.6011419 0.5553249
## 0.1000000000 0.55 0.8807686 0.6023221 0.5693816
## 0.1000000000 0.60 0.9428068 0.6016159 0.5870308
## 0.1000000000 0.65 0.9933360 0.6005464 0.6014912
## 0.1000000000 0.70 1.0303581 0.5979245 0.6143594
## 0.1000000000 0.75 1.0662874 0.5890715 0.6273687
## 0.1000000000 0.80 1.0960848 0.5771348 0.6374875
## 0.1000000000 0.85 1.1253582 0.5661179 0.6463700
## 0.1000000000 0.90 1.1471871 0.5586036 0.6528692
## 0.1000000000 0.95 1.1780382 0.5535541 0.6613094
## 0.1000000000 1.00 1.2217660 0.5506177 0.6723645
##
## Rsquared was used to select the optimal model using the largest value.
## The final values used for the model were fraction = 0.35 and lambda = 0.1.max(enet_model_chem$results$Rsquared)## [1] 0.6364291The Elastic Net model achieved optimal tuning parameters of fraction = 0.15 and lambda = 0.001. This resulted in a cross-validated training \(R^2\) of 0.6625444.
Part D
Predict the response for the test set. What is the value of the performance metric and how does this compare with the resampled performance metric on the training set?
# Predict on test set
test_pred <- predict(enet_model_chem, newdata = test_x)
# Evaluate test performance
postResample(pred = test_pred, obs = test_y)## RMSE Rsquared MAE
## 0.6749481 0.4915518 0.5484792The Elastic Net model achieved a test-set \(R^2\) of 0.4785406, RMSE of 0.7282641, and MAE of 0.5623523. Comparing to the resampled training \(R^2\) of 0.6625444, the test performance is slightly lower but still strong, indicating that the model generalizes well to unseen data.
Part E
Which predictors are most important in the model you have trained? Do either the biological or process predictors dominate the list?
enet_imp <- varImp(enet_model_chem, scale = TRUE)
plot(enet_imp, top = 20)
enet_imp## loess r-squared variable importance
##
## only 20 most important variables shown (out of 57)
##
## Overall
## ManufacturingProcess32 100.00
## BiologicalMaterial06 97.08
## ManufacturingProcess13 91.16
## BiologicalMaterial03 82.20
## ManufacturingProcess36 82.07
## ManufacturingProcess17 75.42
## ManufacturingProcess09 74.87
## BiologicalMaterial02 74.26
## ManufacturingProcess06 68.55
## ManufacturingProcess31 68.18
## BiologicalMaterial12 67.41
## BiologicalMaterial04 56.75
## ManufacturingProcess11 52.87
## BiologicalMaterial11 52.81
## ManufacturingProcess33 52.63
## BiologicalMaterial08 51.55
## ManufacturingProcess29 46.89
## BiologicalMaterial01 45.98
## ManufacturingProcess30 42.39
## ManufacturingProcess18 34.47The top predictors in the Elastic Net model include both Manufacturing and Biological predictors, with the top predictor being ManufacturingProcess32, followed closely by BiologicalMaterial06 and ManufacturingProcess13. Among the top 10 predictors, 6 are manufacturing and 4 are biological. This suggests that both types of predictors are important to the predictive performance, but manufacturing processses slightly drive the performance.
Part F
Explore the relationships between each of the top predictors and the response. How could this information be helpful in improving yield in future runs of the manufacturing process?
library(tidyr)
library(ggplot2)
top_vars <- c("ManufacturingProcess32", "BiologicalMaterial06", "ManufacturingProcess13",
"ManufacturingProcess17", "BiologicalMaterial02")
chem_long <- chemical_imp %>%
select(Yield, all_of(top_vars)) %>%
pivot_longer(
cols = -Yield,
names_to = "Predictor",
values_to = "Value"
)
# Facet plot
ggplot(chem_long, aes(x = Value, y = Yield)) +
geom_point(alpha = 0.7) +
geom_smooth(method = "lm", se = FALSE, color = "blue") +
facet_wrap(~ Predictor, scales = "free_x") +
theme_minimal() +
labs(title = "Relationships Between Top Predictors and Yield",
x = "Predictor",
y = "Yield") +
theme(plot.title = element_text(hjust = 0.5))
sapply(top_vars, function(x) cor(chemical_imp[[x]], chemical_imp$Yield, use = "complete.obs"))## ManufacturingProcess32 BiologicalMaterial06 ManufacturingProcess13
## 0.6083321 0.4781634 -0.5036797
## ManufacturingProcess17 BiologicalMaterial02
## -0.4258069 0.4815158The scatterplots show that BiologicalMaterial02, BiologicalMaterial06, and ManufacturingProcess32 have positive correlations, or yield increases with higher values of these predictors. ManufacturingProcess17 and ManufacturingProcess13 have negative correlations. These relationships indicate that both manufacturing processes and biological material are key drivers of yield, with process control offering the most opportunity for improvement in future runs since they are controllable parameters.