1. Friedman (1991) introduced several benchmark data sets create by simulation. One of these simulations used the following nonlinear equation to create data: y = 10 sin(πx1x2) + 20(x3 − 0.5)2 + 10x4 + 5x5 + N(0, σ2) 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:

KNN

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  3.466085  0.5121775  2.816838
##    7  3.349428  0.5452823  2.727410
##    9  3.264276  0.5785990  2.660026
##   11  3.214216  0.6024244  2.603767
##   13  3.196510  0.6176570  2.591935
##   15  3.184173  0.6305506  2.577482
##   17  3.183130  0.6425367  2.567787
##   19  3.198752  0.6483184  2.592683
##   21  3.188993  0.6611428  2.588787
##   23  3.200458  0.6638353  2.604529
## 
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 17.
knnPred <- predict(knnModel, newdata = testData$x)
## The function 'postResample' can be used to get the test set performance values
postResample(pred = knnPred, obs = testData$y)
##      RMSE  Rsquared       MAE 
## 3.2040595 0.6819919 2.5683461

SVM

svmModel <- train(x=trainingData$x, y=trainingData$y, 
                  method="svmRadial", 
                  preProcess=c("center", "scale"), 
                  tuneLength=20)
svmModel
## Support Vector Machines with Radial Basis Function Kernel 
## 
## 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:
## 
##   C          RMSE      Rsquared   MAE     
##        0.25  2.545335  0.7804647  2.015121
##        0.50  2.319786  0.7965148  1.830009
##        1.00  2.188349  0.8119636  1.726027
##        2.00  2.103655  0.8241314  1.655842
##        4.00  2.066879  0.8294322  1.631051
##        8.00  2.052681  0.8313929  1.623550
##       16.00  2.049867  0.8318312  1.621820
##       32.00  2.049867  0.8318312  1.621820
##       64.00  2.049867  0.8318312  1.621820
##      128.00  2.049867  0.8318312  1.621820
##      256.00  2.049867  0.8318312  1.621820
##      512.00  2.049867  0.8318312  1.621820
##     1024.00  2.049867  0.8318312  1.621820
##     2048.00  2.049867  0.8318312  1.621820
##     4096.00  2.049867  0.8318312  1.621820
##     8192.00  2.049867  0.8318312  1.621820
##    16384.00  2.049867  0.8318312  1.621820
##    32768.00  2.049867  0.8318312  1.621820
##    65536.00  2.049867  0.8318312  1.621820
##   131072.00  2.049867  0.8318312  1.621820
## 
## Tuning parameter 'sigma' was held constant at a value of 0.06802164
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.06802164 and C = 16.
svmPred <- predict(svmModel, newdata=testData$x)
svmpr <- postResample(pred=svmPred, obs=testData$y)
svmpr
##      RMSE  Rsquared       MAE 
## 2.0864652 0.8236735 1.5854649

Neural Network

library(nnet)
nnetModel <- nnet(trainingData$x, trainingData$y,
                size = 5,
                decay = 0.01,
                linout = TRUE,
                trace = FALSE,
                maxit = 500,
                MaxNWts=5 * (ncol(trainingData$x) + 1) + 5 + 1)

nnetPred=predict(nnetModel,testData$x)
postResample(pred = nnetPred, obs = testData$y)
##      RMSE  Rsquared       MAE 
## 2.5842889 0.7416461 1.9705956

MARS: Multivariate Adaptive Regression Splines

marsGrid <- expand.grid(.degree=1:2,
                        .nprune=2:20)

marsModel <- train(x = trainingData$x,
                   y = trainingData$y,
                   method = "earth",
                   tuneGrid = marsGrid,
                   preProc = c("center", "scale"))
## Loading required package: earth
## Warning: package 'earth' was built under R version 4.1.3
## Loading required package: Formula
## Loading required package: plotmo
## Warning: package 'plotmo' was built under R version 4.1.3
## Loading required package: plotrix
## Loading required package: TeachingDemos
## Warning: package 'TeachingDemos' was built under R version 4.1.3
marsModel
## Multivariate Adaptive Regression Spline 
## 
## 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:
## 
##   degree  nprune  RMSE      Rsquared   MAE     
##   1        2      4.441521  0.2189670  3.657091
##   1        3      3.735661  0.4433125  3.009521
##   1        4      2.866002  0.6689285  2.293544
##   1        5      2.515705  0.7424904  2.010746
##   1        6      2.383991  0.7709143  1.894179
##   1        7      1.974017  0.8423134  1.553429
##   1        8      1.857331  0.8604192  1.463521
##   1        9      1.824625  0.8657419  1.419261
##   1       10      1.808885  0.8686792  1.401253
##   1       11      1.820357  0.8673487  1.397892
##   1       12      1.858969  0.8617098  1.430222
##   1       13      1.855403  0.8621244  1.432273
##   1       14      1.869544  0.8596946  1.446095
##   1       15      1.885361  0.8575057  1.462116
##   1       16      1.881835  0.8578251  1.458653
##   1       17      1.884816  0.8575150  1.462251
##   1       18      1.887079  0.8571628  1.464440
##   1       19      1.887079  0.8571628  1.464440
##   1       20      1.887079  0.8571628  1.464440
##   2        2      4.444235  0.2179564  3.660101
##   2        3      3.736840  0.4423882  3.010383
##   2        4      2.865763  0.6682012  2.287784
##   2        5      2.480514  0.7500061  1.977090
##   2        6      2.340100  0.7804955  1.840455
##   2        7      1.987852  0.8405024  1.558922
##   2        8      1.870327  0.8596864  1.466339
##   2        9      1.758141  0.8753405  1.376451
##   2       10      1.627200  0.8928529  1.266051
##   2       11      1.580031  0.8989829  1.238208
##   2       12      1.514494  0.9071887  1.181274
##   2       13      1.510248  0.9078432  1.181635
##   2       14      1.492638  0.9093650  1.168696
##   2       15      1.513457  0.9072019  1.186319
##   2       16      1.515728  0.9069050  1.186683
##   2       17      1.524082  0.9059094  1.195224
##   2       18      1.534602  0.9048036  1.203880
##   2       19      1.536397  0.9046605  1.205418
##   2       20      1.537450  0.9045338  1.206899
## 
## 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.
marsPred <- predict(marsModel, newdata = testData$x)
postResample(pred = marsPred, obs = testData$y)
##      RMSE  Rsquared       MAE 
## 1.2779993 0.9338365 1.0147070
rbind(
  "mars" = postResample(pred = marsPred, obs = testData$y),
  "svm" = postResample(pred = svmPred, obs = testData$y),
  "net" = postResample(pred = nnetPred, obs = testData$y),
  "knn" = postResample(pred = knnPred, obs = testData$y)
)
##          RMSE  Rsquared      MAE
## mars 1.277999 0.9338365 1.014707
## svm  2.086465 0.8236735 1.585465
## net  2.584289 0.7416461 1.970596
## knn  3.204059 0.6819919 2.568346

It seems that the MARS model is the most favorable, with the lowest test set RMSE.

Below the variable importance in the MARS model are calculated:

varImp(marsModel)
## earth variable importance
## 
##    Overall
## X1  100.00
## X4   75.40
## X2   49.00
## X5   15.72
## X3    0.00

The MARS model does select the informative predictors.

  1. Exercise 6.3 describes data for a chemical manufacturing process. Use the same data imputation, data splitting, and pre-processing steps as before and train several nonlinear regression models.

Data Pre-Processing

library(AppliedPredictiveModeling)
## Warning: package 'AppliedPredictiveModeling' was built under R version 4.1.3
data("ChemicalManufacturingProcess")
(cmpImpute <- preProcess(ChemicalManufacturingProcess[,-c(1)], method=c('bagImpute')))
## Created from 152 samples and 57 variables
## 
## Pre-processing:
##   - bagged tree imputation (57)
##   - ignored (0)
df <-  predict(cmpImpute, ChemicalManufacturingProcess[,-c(1)])

Nonlinear Regression Models

set.seed(999)
trainRow <- createDataPartition(ChemicalManufacturingProcess$Yield, p=0.8, list=FALSE)


trainx <- df[trainRow, ]
trainy <- ChemicalManufacturingProcess$Yield[trainRow]

testx <- df[-trainRow, ]
testy <- ChemicalManufacturingProcess$Yield[-trainRow]
## KNN 
knnModel <- train(x = trainx,
                  y = trainy,
                  method = "knn",
                  preProcess = c("center", "scale"),
                  tuneLength = 10)
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
knnPred <- predict(knnModel, newdata = testx)
## NN


library(nnet)
nnetModel <- nnet(x = trainx,
                   y = trainy,
                size = 5,
                decay = 0.01,
                linout = TRUE,
                trace = FALSE,
                maxit = 500,
                MaxNWts=5 * (ncol(trainx) + 1) + 5 + 1)

nnetPred=predict(nnetModel,testx)
postResample(pred = nnetPred, obs = testy)
##      RMSE  Rsquared       MAE 
## 2.0553609 0.4133609 1.6090328
## MARS 


marsGrid <- expand.grid(.degree=1:2,
                        .nprune=2:10)

marsModel <- train(trainx, trainy,
                   method = "earth",
                   tuneGrid = marsGrid,
                   preProc = c("center", "scale"))
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07

## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
marsPred <- predict(marsModel, newdata = testx)
## SVM 

svmModel <- train(trainx, trainy,
                        method = "svmRadial",
                        tuneLength=10,
                        preProc = c("center", "scale"))
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
svmPred <- predict(svmModel, newdata = testx)
  1. Which nonlinear regression model gives the optimal resampling and test set performance?
rbind(
  "mars" = postResample(pred = predict(marsModel), obs = trainy),
  "svm" = postResample(pred = predict(svmModel), obs = trainy),
  "net" = postResample(pred = predict(nnetModel), obs = trainy),
  "knn" = postResample(pred = predict(knnModel), obs = trainy)
)
##           RMSE  Rsquared       MAE
## mars 1.1756076 0.5863735 0.9399376
## svm  0.1732936 0.9932868 0.1686219
## net  0.7171163 0.8463978 0.4765185
## knn  1.3100524 0.5393043 1.0472135
rbind(
  "mars" = postResample(pred = marsPred, obs = testy),
  "svm" = postResample(pred = svmPred, obs = testy),
  "net" = postResample(pred = nnetPred, obs = testy),
  "knn" = postResample(pred = knnPred, obs = testy)
)
##           RMSE  Rsquared       MAE
## mars 1.0222880 0.7229700 0.8109983
## svm  0.9539216 0.7567363 0.7860088
## net  2.0553609 0.4133609 1.6090328
## knn  1.3240022 0.6163930 1.1042604

Looking at the lowest RMSE values, presented above for training and test set performances, SVM appears to be most optimal.

  1. 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?
varImp(svmModel)
## loess r-squared variable importance
## 
##   only 20 most important variables shown (out of 57)
## 
##                        Overall
## ManufacturingProcess32  100.00
## BiologicalMaterial06     88.14
## ManufacturingProcess36   80.40
## ManufacturingProcess13   76.90
## BiologicalMaterial03     75.15
## BiologicalMaterial02     64.46
## BiologicalMaterial12     63.47
## ManufacturingProcess17   57.78
## ManufacturingProcess09   55.72
## ManufacturingProcess31   53.54
## ManufacturingProcess33   52.96
## BiologicalMaterial04     49.38
## ManufacturingProcess29   47.25
## BiologicalMaterial11     44.88
## BiologicalMaterial01     43.98
## BiologicalMaterial08     41.02
## ManufacturingProcess06   38.80
## ManufacturingProcess02   31.54
## ManufacturingProcess11   30.83
## ManufacturingProcess18   23.57

There are slightly more process variables dominating the list rather than biological ones according to SVM.

varImp(marsModel)
## earth variable importance
## 
##                        Overall
## ManufacturingProcess32     100
## ManufacturingProcess09       0
varImp(nnetModel)
##        Overall
## X1  1.21254012
## X2  1.14134501
## X3  0.58379587
## X4  0.50716621
## X5  1.80570109
## X6  1.69893291
## X7  0.68814032
## X8  1.26935423
## X9  1.33005086
## X10 0.66427342
## X11 1.02469605
## X12 0.88458482
## X13 0.25238399
## X14 0.31177291
## X15 2.68219709
## X16 1.43565067
## X17 1.55226875
## X18 1.52591138
## X19 1.14069528
## X20 1.42743433
## X21 0.28296990
## X22 1.01866726
## X23 0.17436537
## X24 1.62462250
## X25 0.99443216
## X26 7.19032309
## X27 8.79450087
## X28 6.60120731
## X29 1.07978100
## X30 7.26186498
## X31 9.18558331
## X32 6.92734822
## X33 0.88763455
## X34 0.15567304
## X35 0.97000843
## X36 0.31073899
## X37 2.06673324
## X38 2.78043903
## X39 1.55108235
## X40 1.00162006
## X41 0.76588130
## X42 1.87976394
## X43 0.46038099
## X44 1.64218825
## X45 0.26381346
## X46 0.44678987
## X47 1.02879464
## X48 0.08198565
## X49 0.85026082
## X50 1.16539795
## X51 0.21705652
## X52 0.50210232
## X53 1.00108410
## X54 0.39914825
## X55 1.61829428
## X56 2.34392542
## X57 1.33464124
varImp(knnModel)
## loess r-squared variable importance
## 
##   only 20 most important variables shown (out of 57)
## 
##                        Overall
## ManufacturingProcess32  100.00
## BiologicalMaterial06     88.14
## ManufacturingProcess36   80.40
## ManufacturingProcess13   76.90
## BiologicalMaterial03     75.15
## BiologicalMaterial02     64.46
## BiologicalMaterial12     63.47
## ManufacturingProcess17   57.78
## ManufacturingProcess09   55.72
## ManufacturingProcess31   53.54
## ManufacturingProcess33   52.96
## BiologicalMaterial04     49.38
## ManufacturingProcess29   47.25
## BiologicalMaterial11     44.88
## BiologicalMaterial01     43.98
## BiologicalMaterial08     41.02
## ManufacturingProcess06   38.80
## ManufacturingProcess02   31.54
## ManufacturingProcess11   30.83
## ManufacturingProcess18   23.57

It appears that the ManufacturingProcess predictors are more important. The above examples show that process variables dominate the list as being the most important. However, different models selected different process variables in the top ten list of importance.

  1. 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?
yield= ChemicalManufacturingProcess$Yield
ggplot(df, aes(ManufacturingProcess32, yield)) +
  geom_point()

ggplot(df, aes(ManufacturingProcess13, yield)) +
  geom_point()

ggplot(df, aes(BiologicalMaterial06, yield)) +
  geom_point()

ggplot(df, aes(BiologicalMaterial03, yield)) +
  geom_point()

The plots above show the relationship between the top 10 predictors and the response. These plots suggest that for the SVM , the top predictors have much of a linear relationship with the response.