DATA 624: PREDICTIVE ANALYTICS HW 8

Library

library(AppliedPredictiveModeling)
library(caret)
library(corrplot)

corrplot 0.92 loaded

library(earth)
library(fpp3)
library(dplyr)
library(ggplot2)
library(mlbench)
library(RANN)
library(tidyr)

Description

Do problems 7.2 and 7.5 in Kuhn and Johnson. There are only two but they have many parts. Please submit both a link to your Rpubs and the .rmd file.

7.2

Friedman (1991) introduced several benchmark data sets create by simulation One of these simulations used the following nonlinear equation to create data:

\(y=10sin(\pi1x_1x_2)+20(x_3-0.5)^2+10x_4+5x_5+N(0,\sigma^2)\)

where the x values are random variables uniformly distributed between [0, 1] (there are also 5 other non-informative variables also created in the simulation). The package mlbench contains a function called mlbench.friedman1 that simulates these data:

set.seed(200)
trainingData <- mlbench.friedman1(200, sd = 1)
## 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)

i

Example

Tune several models on these data. For example:

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.

RMSE was used to select the optimal model using the smallest value. The final value used for the model was k = 19.

knnPred <- predict(knnModel, newdata = testData$x)
## The function 'postResample' can be used to get the test set
## perforamnce values
postResample(pred = knnPred, obs = testData$y)

     RMSE  Rsquared       MAE 
3.2040595 0.6819919 2.5683461 

MARS & MARS TUNED

MARS

mars_Grid <- expand.grid(.degree = 1:2, .nprune = 2:38)
set.seed(100)

mars_Fit <- earth(trainingData$x, trainingData$y)

mars_Pred <- predict(mars_Fit, newdata = testData$x)
postResample(pred = mars_Pred, obs = testData$y)

     RMSE  Rsquared       MAE 
1.8136467 0.8677298 1.3911836 

MARS Tuned

marsTuned_Grid <- expand.grid(.degree = 1:2, .nprune = 2:38)
set.seed(100)

marsTuned <- train(trainingData$x, trainingData$y, method = "earth",
                   tuneGrid = marsTuned_Grid,
                   trControl = trainControl(method = "cv"))

marsTune_Pred <- predict(marsTuned, newdata = testData$x)
postResample(pred = marsTune_Pred, obs = testData$y)

     RMSE  Rsquared       MAE 
1.1589948 0.9460418 0.9250230 

SVM Tuned

svmRTuned <- train(trainingData$x, trainingData$y,
                   method = "svmRadial",
                   preProcess = c("center", "scale"),
                   tuneLength = 15,
                   trControl = trainControl(method = "cv"))

svmPred <- predict(svmRTuned, newdata = testData$x)
postResample(svmPred, testData$y)

     RMSE  Rsquared       MAE 
2.0741473 0.8255848 1.5755185 

ii

Which models appear to give the best performance?

MAR tuned model with an R-squared value of 0.9460 looks like its the best performing.

iii

Does MARS select the informative predictors (those named X1–X5)?

Yes it selects for informative predictors

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

data(ChemicalManufacturingProcess)

data imputation, data splitting & pre-processing

impute

imputed_df <- preProcess(ChemicalManufacturingProcess, "knnImpute")
imputed_full_df <- predict(imputed_df, ChemicalManufacturingProcess)

val_low <- nearZeroVar(imputed_full_df)
#remove low frequency columns using baser df[row,columns]
chem_df <- imputed_full_df[,-val_low]

split

chem_index <- createDataPartition(chem_df$Yield , p=.8, list=F)

chem_train <-  chem_df[chem_index,] 
chem_test <- chem_df[-chem_index,]

chem_marsGrid <- expand.grid(.degree = 1:2, .nprune = 2:38)

(a)

Which nonlinear regression model gives the optimal resampling and test set performance?

MARS Tuned

chem_MarsTuned <- train(Yield~. ,
                  data = chem_train,
                   method = "earth",
                   tuneGrid = chem_marsGrid,
                   trControl = trainControl(method = "cv"))

chem_MarsTunePred <- predict(chem_MarsTuned,  chem_test)
postResample(chem_MarsTunePred, chem_test$Yield)

     RMSE  Rsquared       MAE 
0.6437705 0.5394597 0.4739508 

SVM Tuned

chem_SVMTuned <- train(Yield~. ,
                  data = chem_test,
                   method = "svmRadial",
                   tuneLength = 15,
                   trControl = trainControl(method = "cv"))

chem_SVMTunePred <- predict(chem_SVMTuned,  chem_test)
postResample(chem_SVMTunePred, chem_test$Yield)

      RMSE   Rsquared        MAE 
0.09178983 0.99520810 0.09087154 

KNN

knnModel <- train(Yield~., 
                    data = chem_test,
                    method = "knn",
                    preProc = c("center", "scale"), 
                    tuneLength = 10)

knnPred <- predict(knnModel,  chem_test)
postResample(pred = knnPred, obs = chem_test$Yield)

     RMSE  Rsquared       MAE 
0.7631649 0.4717384 0.6279725 

Neural Network

nnetGrid <- expand.grid(.decay = c(0, 0.01, .1),
                        .size = c(1:10), .bag = FALSE)

nnetTune <- train(Yield~., 
                  data = chem_train, 
                  method = "avNNet", 
                  tuneGrid = nnetGrid,
                  trControl = trainControl(method = "cv"), 
                  linout = TRUE,trace = FALSE,
                  MaxNWts = 10 * (ncol(chem_train) + 1) + 10 + 1, 
                  maxit = 500)

nnetPred <- predict(nnetTune,  chem_test)
postResample(predict(nnetTune,  chem_test), chem_test$Yield)

     RMSE  Rsquared       MAE 
0.6145770 0.6353956 0.4907860 

The SVM tuned is the best performing RMSE and R squared with values of RMSE=0.09178983 , and R squared=0.99520810

(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?

Of the 20 ManufacturingProcess predictors 12 are BiologicalMaterial and 8 are ManufacturingProcess. In the top 10, 6 are ManufacturingProcess in with 1rst through 3rd being ManufacturingProcess32, ManufacturingProcess36, and ManufacturingProcess05. Although neither dominates, ManufacturingProcess does hold a greater influence overall, when compared to the ratio of optimal nonlinear vs optimal linear, in optimal linear, ManuracturingProcess importance is greater.

plot(varImp(chem_SVMTuned), top=10)

Warning: pseudoinverse used at 0.019411Warning: neighborhood radius 0.44646Warning: reciprocal condition number  1.669e-16

(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?

• Corrplot used to explore relationship
• Yield response variable doesn’t have strong correlations with most of the important predictors noted previously.
• Strongest positive correlations exist with ManufacturingProcess32 and ManufacturingProcess09 (0.61 and 0.50).
• The strongest negative correlations exist with ManufacturingProcess36 and ManufacturingProcess13 (-0.53 and -0.50, respectively).
• No strong correlations with Yeild and the important Biological Material predictors identified in the previous step.

corr_vals <- chem_df %>% 
  dplyr::select('Yield', 'ManufacturingProcess32','ManufacturingProcess36',
         'BiologicalMaterial06','ManufacturingProcess13',
         'BiologicalMaterial03','ManufacturingProcess17',
         'BiologicalMaterial02','BiologicalMaterial12',
         'ManufacturingProcess09','ManufacturingProcess31')

corr_plot_vals <- cor(corr_vals)

corrplot.mixed(corr_plot_vals, tl.col = 'black', tl.pos = 'lt', 
         upper = "number", lower="circle")