6.2

Developing a model to predict permeability (see Sect. 1.4) could save significant resources for a pharmaceutical company, while at the same time more rapidly identifying molecules that have a sufficient permeability to become a drug:

  1. Start R and use these commands to load the data:
library(AppliedPredictiveModeling)
## Warning: package 'AppliedPredictiveModeling' was built under R version 4.0.5
data(permeability)

The matrix fingerprints contains the 1,107 binary molecular predictors for the 165 compounds, while permeability contains permeability response.

  1. 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?
dim(fingerprints)
## [1]  165 1107
fingerprints <- fingerprints[, -nearZeroVar(fingerprints)]

dim(fingerprints)
## [1] 165 388

There were 1,107 predictors and now there are only 388 predictors left for modeling.

  1. 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(624)

# index for training
index <- createDataPartition(permeability, p = .8, list = FALSE)

# train 
train_perm <- permeability[index, ]
train_fp <- fingerprints[index, ]
# test
test_perm <- permeability[-index, ]
test_fp <- fingerprints [-index, ]

# 10-fold cross-validation to make reasonable estimates
ctrl <- trainControl(method = "cv", number = 10)

plsTune <- train(train_fp, train_perm, method = "pls", metric = "Rsquared",
             tuneLength = 20, trControl = ctrl, preProc = c("center", "scale"))

plot(plsTune) 

plsTune
## Partial Least Squares 
## 
## 133 samples
## 388 predictors
## 
## Pre-processing: centered (388), scaled (388) 
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 118, 119, 120, 120, 121, 120, ... 
## Resampling results across tuning parameters:
## 
##   ncomp  RMSE      Rsquared   MAE      
##    1     13.25656  0.2953172  10.202424
##    2     11.90191  0.4662745   8.710959
##    3     12.05579  0.4624570   9.271134
##    4     12.03297  0.4793759   9.314195
##    5     12.06645  0.4870447   8.986610
##    6     11.82964  0.5012972   8.788270
##    7     11.91363  0.5011672   9.153386
##    8     11.79990  0.4960881   9.119966
##    9     11.81946  0.4959475   9.301787
##   10     11.85288  0.4924486   9.176136
##   11     11.79654  0.5025443   9.128199
##   12     11.62869  0.5131115   8.965070
##   13     11.78348  0.5080595   8.920097
##   14     12.01377  0.4935108   9.101865
##   15     12.09297  0.4862359   9.131109
##   16     12.10087  0.4953868   9.053161
##   17     12.38093  0.4847366   9.218277
##   18     12.59348  0.4768971   9.402569
##   19     12.61895  0.4807222   9.338592
##   20     12.77045  0.4682401   9.549745
## 
## Rsquared was used to select the optimal model using the largest value.
## The final value used for the model was ncomp = 12.

The optimal tuning had 12 components with a corresponding \(R^2\) of 0.5297497.

  1. Predict the response for the test set. What is the test set estimate of \(R^2\)?
fp_predict <- predict(plsTune, test_fp)

postResample(fp_predict, test_perm)
##       RMSE   Rsquared        MAE 
## 11.4895371  0.4741832  9.3113125

The test set estimate of \(R^2\) is 0.4741832.

  1. Try building other models discussed in this chapter. Do any have better predictive performance?

enet

The \(R^2\) is 0.5307087 and the RMSE is lower using a penalized Elastic Net regression model.

set.seed(624)

# grid of penalties
enetGrid <- expand.grid(.lambda = c(0, 0.01, .1), .fraction = seq(.05, 1, length = 20))

# tuning penalized regression model
enetTune <- train(train_fp, train_perm, method = "enet",
                  tuneGrid = enetGrid, trControl = ctrl, preProc = c("center", "scale"))

plot(enetTune)

enetTune
## Elasticnet 
## 
## 133 samples
## 388 predictors
## 
## Pre-processing: centered (388), scaled (388) 
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 121, 118, 119, 121, 119, 120, ... 
## Resampling results across tuning parameters:
## 
##   lambda  fraction  RMSE      Rsquared   MAE      
##   0.00    0.05      12.53022  0.4195116   9.219840
##   0.00    0.10      11.83087  0.4482164   8.465318
##   0.00    0.15      11.94876  0.4475951   8.738893
##   0.00    0.20      11.98463  0.4484245   8.890257
##   0.00    0.25      11.79106  0.4640731   8.861128
##   0.00    0.30      11.66486  0.4742758   8.848355
##   0.00    0.35      11.72130  0.4749181   8.912993
##   0.00    0.40      11.90989  0.4695334   8.998669
##   0.00    0.45      12.30625  0.4529253   9.193462
##   0.00    0.50      12.68544  0.4362691   9.340897
##   0.00    0.55      13.00847  0.4214423   9.471613
##   0.00    0.60      13.29088  0.4089528   9.590155
##   0.00    0.65      13.44531  0.4004275   9.646439
##   0.00    0.70      13.61111  0.3907010   9.737890
##   0.00    0.75      13.82251  0.3811970   9.865902
##   0.00    0.80      13.94077  0.3755526   9.951804
##   0.00    0.85      14.00446  0.3726513  10.006198
##   0.00    0.90      14.15593  0.3660339  10.096242
##   0.00    0.95      14.35377  0.3580099  10.182112
##   0.00    1.00      14.48818  0.3547683  10.231509
##   0.01    0.05      12.91277  0.3962042   9.210662
##   0.01    0.10      14.49519  0.3998262  10.363051
##   0.01    0.15      16.02608  0.4115378  11.216868
##   0.01    0.20      17.63731  0.4189926  12.226957
##   0.01    0.25      19.54774  0.4096559  13.401832
##   0.01    0.30      21.65430  0.3895637  14.606572
##   0.01    0.35      23.60756  0.3748645  15.739320
##   0.01    0.40      25.60311  0.3611846  16.904536
##   0.01    0.45      27.57928  0.3504770  18.090540
##   0.01    0.50      29.55483  0.3408871  19.320808
##   0.01    0.55      31.52744  0.3320255  20.544216
##   0.01    0.60      33.47925  0.3280021  21.731405
##   0.01    0.65      35.46538  0.3231206  22.929380
##   0.01    0.70      37.43374  0.3183345  24.106466
##   0.01    0.75      39.37733  0.3156452  25.290183
##   0.01    0.80      41.31187  0.3139363  26.468389
##   0.01    0.85      43.29420  0.3110890  27.711822
##   0.01    0.90      45.37309  0.3072853  28.998914
##   0.01    0.95      47.41765  0.3041533  30.254470
##   0.01    1.00      49.32673  0.3034756  31.415359
##   0.10    0.05      12.55808  0.4101914   9.475823
##   0.10    0.10      12.00149  0.4328341   8.560342
##   0.10    0.15      12.08155  0.4311345   8.705317
##   0.10    0.20      12.22936  0.4267428   8.969534
##   0.10    0.25      12.11297  0.4349270   8.932117
##   0.10    0.30      12.04598  0.4378837   8.978154
##   0.10    0.35      12.04834  0.4369033   9.003314
##   0.10    0.40      12.08992  0.4341602   9.037805
##   0.10    0.45      12.16888  0.4291846   9.099988
##   0.10    0.50      12.25788  0.4233290   9.143842
##   0.10    0.55      12.32683  0.4187918   9.177390
##   0.10    0.60      12.41168  0.4130270   9.253065
##   0.10    0.65      12.47522  0.4078620   9.287724
##   0.10    0.70      12.53494  0.4039400   9.335168
##   0.10    0.75      12.58185  0.4016082   9.363731
##   0.10    0.80      12.61736  0.4001737   9.380885
##   0.10    0.85      12.65290  0.3987804   9.398010
##   0.10    0.90      12.69026  0.3974567   9.419637
##   0.10    0.95      12.73757  0.3953611   9.442814
##   0.10    1.00      12.77339  0.3939567   9.457362
## 
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were fraction = 0.3 and lambda = 0.
enet_predict <- predict(enetTune, test_fp)

postResample(enet_predict, test_perm)
##       RMSE   Rsquared        MAE 
## 11.9836978  0.4157479  9.7314675

lars

The Least Angle Regression is slightly worse than the PLS method as the \(R^2\) is lower and the RMSE is higher.

set.seed(624)

larsTune <- train(train_fp, train_perm, method = "lars", metric = "Rsquared",
                    tuneLength = 20, trControl = ctrl, preProc = c("center", "scale"))

plot(larsTune)

larsTune
## Least Angle Regression 
## 
## 133 samples
## 388 predictors
## 
## Pre-processing: centered (388), scaled (388) 
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 121, 118, 119, 121, 119, 120, ... 
## Resampling results across tuning parameters:
## 
##   fraction  RMSE          Rsquared   MAE         
##   0.05      1.323844e+15  0.3888424  3.538123e+14
##   0.10      2.647681e+15  0.3369696  7.076224e+14
##   0.15      3.971517e+15  0.3062702  1.061433e+15
##   0.20      5.295353e+15  0.2648578  1.415243e+15
##   0.25      6.619190e+15  0.2353928  1.769053e+15
##   0.30      7.943026e+15  0.2272336  2.122863e+15
##   0.35      9.266863e+15  0.2434762  2.476673e+15
##   0.40      1.059070e+16  0.2504526  2.830483e+15
##   0.45      1.191454e+16  0.2461329  3.184293e+15
##   0.50      1.323837e+16  0.2432880  3.538104e+15
##   0.55      1.456221e+16  0.2394726  3.891914e+15
##   0.60      1.588604e+16  0.2358817  4.245724e+15
##   0.65      1.719694e+16  0.2336453  4.596075e+15
##   0.70      1.847642e+16  0.2314378  4.938030e+15
##   0.75      1.975589e+16  0.2295532  5.279985e+15
##   0.80      2.103537e+16  0.2305318  5.621940e+15
##   0.85      2.231485e+16  0.2306874  5.963895e+15
##   0.90      2.359433e+16  0.2303752  6.305851e+15
##   0.95      2.487381e+16  0.2301607  6.647806e+15
##   1.00      2.615329e+16  0.2299920  6.989761e+15
## 
## Rsquared was used to select the optimal model using the largest value.
## The final value used for the model was fraction = 0.05.
lars_predict <- predict(larsTune, test_fp)

postResample(lars_predict, test_perm)
##       RMSE   Rsquared        MAE 
## 11.3841183  0.4548995  9.3269128

  1. Would you recommend any of your models to replace the permeability laboratory experiment?

I would recommend the Elastic Net regression model as it produced better statistics. It had a higher \(R^2\) and lower RMSE and MAE.

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 relationship between biological measurements of the raw materials (predictors), 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:

  1. Start R and use these commands to load the data:
library(AppliedPredictiveModeling)
data(ChemicalManufacturingProcess)

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.

  1. A small percentage of cells in the predictor set contain missing values. Use an imputation function to fill in these missing values (e.g., see Sect. 3.8).
sum(is.na(ChemicalManufacturingProcess))
## [1] 106
miss <- preProcess(ChemicalManufacturingProcess, method = "bagImpute")
Chemical <- predict(miss, ChemicalManufacturingProcess)

sum(is.na(Chemical))
## [1] 0

There were 106 missing values in ChemicalManufacturingProcess. Bagged trees were used to impute the data. Bagged trees are made using all the other variables.

  1. 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?
# filtering low frequencies
Chemical <- Chemical[, -nearZeroVar(Chemical)]
set.seed(624)

# index for training
index <- createDataPartition(Chemical$Yield, p = .8, list = FALSE)

# train 
train_chem <- Chemical[index, ]

# test
test_chem <- Chemical[-index, ]

pls

Optimal tuning has 3 components with \(R^2\) of 0.5623262.

set.seed(624)

plsTune <- train(Yield ~ ., Chemical , method = "pls", 
             tuneLength = 20, trControl = ctrl, preProc = c("center", "scale"))

plot(plsTune) 

plsTune
## Partial Least Squares 
## 
## 176 samples
##  56 predictor
## 
## Pre-processing: centered (56), scaled (56) 
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 160, 157, 158, 159, 158, 159, ... 
## Resampling results across tuning parameters:
## 
##   ncomp  RMSE      Rsquared   MAE     
##    1     1.436891  0.4568805  1.147590
##    2     1.872742  0.4711564  1.185897
##    3     1.292614  0.5633698  1.020010
##    4     1.480526  0.5381868  1.085319
##    5     1.707358  0.5156812  1.131007
##    6     1.821904  0.4903840  1.156300
##    7     2.006142  0.4802835  1.211850
##    8     2.092370  0.4622998  1.253598
##    9     2.220647  0.4485854  1.290999
##   10     2.322021  0.4410360  1.315081
##   11     2.446697  0.4264842  1.352393
##   12     2.475260  0.4206118  1.367989
##   13     2.464162  0.4197124  1.377783
##   14     2.418661  0.4227280  1.364288
##   15     2.375812  0.4242080  1.350175
##   16     2.368363  0.4267259  1.337946
##   17     2.386174  0.4254577  1.339398
##   18     2.376321  0.4271412  1.334877
##   19     2.400843  0.4255697  1.348122
##   20     2.422785  0.4231162  1.359970
## 
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was ncomp = 3.

enet

The optimal model has a fraction of 0.1 and \(\lambda\) of 0. The \(R^2\) is 0.6253182.

set.seed(624)

enetTune <- train(Yield ~ ., Chemical , method = "enet", 
                  tuneGrid = enetGrid, trControl = ctrl, preProc = c("center", "scale"))

plot(enetTune)

enetTune
## Elasticnet 
## 
## 176 samples
##  56 predictor
## 
## Pre-processing: centered (56), scaled (56) 
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 160, 157, 158, 159, 158, 159, ... 
## Resampling results across tuning parameters:
## 
##   lambda  fraction  RMSE      Rsquared   MAE      
##   0.00    0.05      1.263954  0.6225860  1.0294423
##   0.00    0.10      1.171467  0.6195893  0.9388695
##   0.00    0.15      1.285779  0.6043223  0.9537550
##   0.00    0.20      1.455526  0.5547255  1.0089814
##   0.00    0.25      1.743737  0.4948981  1.1067214
##   0.00    0.30      1.939587  0.4680604  1.1795594
##   0.00    0.35      1.908244  0.4619370  1.1924860
##   0.00    0.40      1.850254  0.4606806  1.1877837
##   0.00    0.45      1.794856  0.4594057  1.1771407
##   0.00    0.50      2.101187  0.4481459  1.2573220
##   0.00    0.55      2.407435  0.4725106  1.3304398
##   0.00    0.60      2.744013  0.5217990  1.4001566
##   0.00    0.65      3.225543  0.4863491  1.5339631
##   0.00    0.70      3.771484  0.4747361  1.6693508
##   0.00    0.75      4.094773  0.4681231  1.7557563
##   0.00    0.80      4.135408  0.4589587  1.7764653
##   0.00    0.85      4.225706  0.4419317  1.8076852
##   0.00    0.90      4.318549  0.4257147  1.8358006
##   0.00    0.95      4.415063  0.4120784  1.8636370
##   0.00    1.00      4.501021  0.4019904  1.8870890
##   0.01    0.05      1.536442  0.5973476  1.2453271
##   0.01    0.10      1.309336  0.6258493  1.0618555
##   0.01    0.15      1.201950  0.6195671  0.9778981
##   0.01    0.20      1.176854  0.6175939  0.9523896
##   0.01    0.25      1.161852  0.6236987  0.9319051
##   0.01    0.30      1.175811  0.6198597  0.9301295
##   0.01    0.35      1.246551  0.6032199  0.9540424
##   0.01    0.40      1.296291  0.5952774  0.9647967
##   0.01    0.45      1.362599  0.5626257  0.9909640
##   0.01    0.50      1.460207  0.5489330  1.0197566
##   0.01    0.55      1.600805  0.5172511  1.0731396
##   0.01    0.60      1.793769  0.4868596  1.1322410
##   0.01    0.65      1.885961  0.4712944  1.1672050
##   0.01    0.70      1.953361  0.4609687  1.1927301
##   0.01    0.75      2.016199  0.4534714  1.2156241
##   0.01    0.80      2.076369  0.4480579  1.2356802
##   0.01    0.85      2.143715  0.4433120  1.2568727
##   0.01    0.90      2.179751  0.4401770  1.2700016
##   0.01    0.95      2.129338  0.4398117  1.2610389
##   0.01    1.00      2.043851  0.4421279  1.2426724
##   0.10    0.05      1.649315  0.5363459  1.3354459
##   0.10    0.10      1.487591  0.6095911  1.2076167
##   0.10    0.15      1.354079  0.6236486  1.0996757
##   0.10    0.20      1.258077  0.6219261  1.0220867
##   0.10    0.25      1.200809  0.6201101  0.9778990
##   0.10    0.30      1.181463  0.6179041  0.9615112
##   0.10    0.35      1.171335  0.6193380  0.9468400
##   0.10    0.40      1.165095  0.6229576  0.9424421
##   0.10    0.45      1.167185  0.6235463  0.9390511
##   0.10    0.50      1.212130  0.6118562  0.9527442
##   0.10    0.55      1.310136  0.6005375  0.9787766
##   0.10    0.60      1.383384  0.5963431  0.9968869
##   0.10    0.65      1.442303  0.5805640  1.0192090
##   0.10    0.70      1.511715  0.5573960  1.0504530
##   0.10    0.75      1.598893  0.5315716  1.0836263
##   0.10    0.80      1.707558  0.5079105  1.1181461
##   0.10    0.85      1.772034  0.4960139  1.1387951
##   0.10    0.90      1.812766  0.4895472  1.1528253
##   0.10    0.95      1.846664  0.4841682  1.1650245
##   0.10    1.00      1.873437  0.4803403  1.1741747
## 
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were fraction = 0.25 and lambda = 0.01.

lars

The optimal model has a fraction of 0.05 and \(R^2\) of 0.6256310.

set.seed(624)

larsTune <- train(Yield ~ ., Chemical , method = "lars", metric = "Rsquared",
                    tuneLength = 20, trControl = ctrl, preProc = c("center", "scale"))

plot(larsTune)

larsTune
## Least Angle Regression 
## 
## 176 samples
##  56 predictor
## 
## Pre-processing: centered (56), scaled (56) 
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 160, 157, 158, 159, 158, 159, ... 
## Resampling results across tuning parameters:
## 
##   fraction  RMSE      Rsquared   MAE      
##   0.05      1.267315  0.6253620  1.0366493
##   0.10      1.157153  0.6231683  0.9356714
##   0.15      1.164004  0.6233060  0.9224129
##   0.20      1.425402  0.5565977  1.0057847
##   0.25      1.723565  0.4944026  1.1037771
##   0.30      1.938997  0.4673650  1.1768402
##   0.35      1.922474  0.4611011  1.1891826
##   0.40      1.874238  0.4586337  1.1868903
##   0.45      1.837601  0.4578589  1.1827928
##   0.50      1.724214  0.4650098  1.1591639
##   0.55      1.499832  0.4951473  1.1015248
##   0.60      1.282991  0.5698676  1.0257689
##   0.65      1.635598  0.4997657  1.1374580
##   0.70      2.069322  0.4784353  1.2473513
##   0.75      2.479127  0.4697471  1.3564650
##   0.80      2.856763  0.4576260  1.4618496
##   0.85      3.235797  0.4398416  1.5638245
##   0.90      3.625417  0.4236105  1.6648378
##   0.95      4.063903  0.4102988  1.7768984
##   1.00      4.501021  0.4019904  1.8870890
## 
## Rsquared was used to select the optimal model using the largest value.
## The final value used for the model was fraction = 0.05.

lm

The ordinary linear regression model has a Multiple \(R^2\) of 0.7813 and an Adjusted \(R^2\) of 0.6811.

lm_model <- lm(Yield ~ ., Chemical)

summary(lm_model)
## 
## Call:
## lm(formula = Yield ~ ., data = Chemical)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.17844 -0.53656 -0.02842  0.50526  2.00415 
## 
## Coefficients: (1 not defined because of singularities)
##                          Estimate Std. Error t value Pr(>|t|)    
## (Intercept)             4.234e+00  8.608e+01   0.049  0.96085    
## BiologicalMaterial01    2.483e-01  3.342e-01   0.743  0.45900    
## BiologicalMaterial02   -1.120e-01  1.281e-01  -0.874  0.38375    
## BiologicalMaterial03    1.636e-01  2.354e-01   0.695  0.48843    
## BiologicalMaterial04   -1.044e-01  5.235e-01  -0.199  0.84233    
## BiologicalMaterial05    1.513e-01  1.061e-01   1.426  0.15641    
## BiologicalMaterial06    3.336e-03  3.014e-01   0.011  0.99119    
## BiologicalMaterial08    3.808e-01  6.358e-01   0.599  0.55034    
## BiologicalMaterial09   -8.180e-01  1.370e+00  -0.597  0.55162    
## BiologicalMaterial10    7.954e-02  1.367e+00   0.058  0.95370    
## BiologicalMaterial11   -8.954e-02  8.230e-02  -1.088  0.27874    
## BiologicalMaterial12    3.493e-01  6.346e-01   0.551  0.58300    
## ManufacturingProcess01  6.695e-02  9.596e-02   0.698  0.48672    
## ManufacturingProcess02  1.343e-02  4.311e-02   0.311  0.75601    
## ManufacturingProcess03 -3.377e+00  5.103e+00  -0.662  0.50934    
## ManufacturingProcess04  6.282e-02  2.940e-02   2.137  0.03464 *  
## ManufacturingProcess05  7.326e-04  3.859e-03   0.190  0.84974    
## ManufacturingProcess06  3.261e-02  4.341e-02   0.751  0.45401    
## ManufacturingProcess07 -1.810e-01  2.126e-01  -0.851  0.39623    
## ManufacturingProcess08 -6.282e-02  2.522e-01  -0.249  0.80374    
## ManufacturingProcess09  2.614e-01  1.812e-01   1.443  0.15176    
## ManufacturingProcess10 -1.166e-01  5.742e-01  -0.203  0.83950    
## ManufacturingProcess11  1.942e-01  7.132e-01   0.272  0.78590    
## ManufacturingProcess12  3.761e-05  1.013e-04   0.371  0.71120    
## ManufacturingProcess13 -2.670e-01  3.843e-01  -0.695  0.48859    
## ManufacturingProcess14  3.058e-04  1.115e-02   0.027  0.97816    
## ManufacturingProcess15  1.972e-03  8.903e-03   0.222  0.82506    
## ManufacturingProcess16 -4.937e-05  3.190e-04  -0.155  0.87728    
## ManufacturingProcess17 -1.402e-01  3.011e-01  -0.466  0.64240    
## ManufacturingProcess18  4.245e-03  4.450e-03   0.954  0.34211    
## ManufacturingProcess19 -2.233e-03  7.301e-03  -0.306  0.76021    
## ManufacturingProcess20 -4.517e-03  4.721e-03  -0.957  0.34062    
## ManufacturingProcess21         NA         NA      NA       NA    
## ManufacturingProcess22 -1.666e-02  4.209e-02  -0.396  0.69299    
## ManufacturingProcess23 -4.181e-02  8.289e-02  -0.504  0.61495    
## ManufacturingProcess24 -1.931e-02  2.340e-02  -0.825  0.41100    
## ManufacturingProcess25 -6.493e-03  1.365e-02  -0.476  0.63506    
## ManufacturingProcess26  6.101e-03  1.041e-02   0.586  0.55909    
## ManufacturingProcess27 -7.061e-03  7.781e-03  -0.907  0.36601    
## ManufacturingProcess28 -7.882e-02  3.094e-02  -2.547  0.01212 *  
## ManufacturingProcess29  1.393e+00  8.961e-01   1.555  0.12261    
## ManufacturingProcess30 -3.693e-01  6.233e-01  -0.592  0.55463    
## ManufacturingProcess31  4.783e-02  1.203e-01   0.398  0.69168    
## ManufacturingProcess32  3.333e-01  6.833e-02   4.877 3.34e-06 ***
## ManufacturingProcess33 -4.068e-01  1.286e-01  -3.164  0.00197 ** 
## ManufacturingProcess34 -1.496e+00  2.753e+00  -0.543  0.58792    
## ManufacturingProcess35 -1.879e-02  1.765e-02  -1.064  0.28926    
## ManufacturingProcess36  2.833e+02  3.132e+02   0.904  0.36765    
## ManufacturingProcess37 -6.935e-01  2.889e-01  -2.401  0.01789 *  
## ManufacturingProcess38 -1.900e-01  2.417e-01  -0.786  0.43333    
## ManufacturingProcess39  7.077e-02  1.307e-01   0.542  0.58907    
## ManufacturingProcess40  4.605e-01  6.545e+00   0.070  0.94403    
## ManufacturingProcess41  2.549e-01  4.736e+00   0.054  0.95716    
## ManufacturingProcess42  4.372e-02  2.102e-01   0.208  0.83557    
## ManufacturingProcess43  2.268e-01  1.182e-01   1.919  0.05741 .  
## ManufacturingProcess44 -4.385e-01  1.186e+00  -0.370  0.71222    
## ManufacturingProcess45  9.547e-01  5.444e-01   1.754  0.08204 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.039 on 120 degrees of freedom
## Multiple R-squared:  0.7826, Adjusted R-squared:  0.683 
## F-statistic: 7.854 on 55 and 120 DF,  p-value: < 2.2e-16

ridge

The optimal model has \(\lambda\) of 0.1 and \(R^2\) of 0.4785667.

set.seed(624)

## Define the candidate set of values
ridgeGrid <- data.frame(.lambda = seq(0, .1, length = 15))

ridgeTune <- train(Yield ~ ., Chemical , method = "ridge",
                     tuneGrid = ridgeGrid, trControl = ctrl, preProc = c("center", "scale"))

plot(ridgeTune)

ridgeTune
## Ridge Regression 
## 
## 176 samples
##  56 predictor
## 
## Pre-processing: centered (56), scaled (56) 
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 160, 157, 158, 159, 158, 159, ... 
## Resampling results across tuning parameters:
## 
##   lambda       RMSE      Rsquared   MAE     
##   0.000000000  4.501021  0.4019904  1.887089
##   0.007142857  1.951471  0.4493837  1.224896
##   0.014285714  2.093121  0.4429601  1.247997
##   0.021428571  2.098017  0.4475006  1.242244
##   0.028571429  2.077066  0.4519785  1.232565
##   0.035714286  2.050861  0.4560167  1.222873
##   0.042857143  2.024673  0.4596597  1.213933
##   0.050000000  1.999988  0.4629748  1.206382
##   0.057142857  1.977156  0.4660169  1.200121
##   0.064285714  1.956157  0.4688278  1.194388
##   0.071428571  1.936855  0.4714397  1.189526
##   0.078571429  1.919085  0.4738781  1.185097
##   0.085714286  1.902686  0.4761635  1.180984
##   0.092857143  1.887513  0.4783129  1.177421
##   0.100000000  1.873437  0.4803403  1.174175
## 
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was lambda = 0.1.

  1. 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?

Ordinary linear model had the highest \(R^2\), but it comes with consequences. Therefore, the lars method was chosen as it had the highest \(R^2\).

The \(R^2\) is 0.7170636, which is higher than the training set.

lars_predict <- predict(larsTune, test_chem[ ,-1])

postResample(lars_predict, test_chem[ ,1])
##     RMSE Rsquared      MAE 
## 1.399505 0.718109 1.095894
  1. Which predictors are most important in the model you have trained? Do either the biological or process predictors dominate the list?
varImp(larsTune)
## loess r-squared variable importance
## 
##   only 20 most important variables shown (out of 56)
## 
##                        Overall
## ManufacturingProcess32  100.00
## ManufacturingProcess13   90.02
## BiologicalMaterial06     84.56
## ManufacturingProcess36   76.03
## ManufacturingProcess17   74.88
## BiologicalMaterial03     73.53
## ManufacturingProcess09   70.37
## BiologicalMaterial12     67.98
## BiologicalMaterial02     65.33
## ManufacturingProcess31   60.38
## ManufacturingProcess06   58.03
## ManufacturingProcess33   49.39
## BiologicalMaterial11     48.11
## BiologicalMaterial04     47.13
## ManufacturingProcess11   42.47
## BiologicalMaterial08     41.88
## BiologicalMaterial01     39.14
## ManufacturingProcess12   33.02
## ManufacturingProcess30   32.91
## BiologicalMaterial09     32.41

The 5 most important variables used in the modeling are ManufacturingProcess32, ManufacturingProcess13, BiologicalMaterial06, ManufacturingProcess36, and ManufacturingProcess17.

Process predictors dominate the list. The ratio of process to biological predictors is 11:9.

  1. 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?
top10 <- varImp(larsTune)$importance %>%
  arrange(-Overall) %>%
  head(10)


Chemical %>%
  select(c("Yield", row.names(top10))) %>%
  cor() %>%
  corrplot()

According to the correlation plot, ManufacturingProcess32 has the highest positive correlation with Yield. Three of the top ten variables are negatively correlated with Yield. This information can be helfup in the future runs of the manufacturing process, as these were the predictors that affected the yield. If they want to maximize or improve their yield, they way want to improve their measurements of the manufacturing process and biological measurements of the raw materials.