BrianSingh_Data624_HW8
2024-04-15
##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 = 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)## Warning: package 'mlbench' was built under R version 4.2.3library(caret)## Warning: package 'caret' was built under R version 4.2.3## Loading required package: ggplot2## Warning: package 'ggplot2' was built under R version 4.2.3## Loading required package: lattice## Warning: package 'lattice' was built under R version 4.2.3set.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)
## 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)
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$x)
## The function 'postResample' can be used to get the test set
## perforamnce values
postResample(pred = knnPred, obs = testData$y)## RMSE Rsquared MAE
## 3.2040595 0.6819919 2.5683461Which models appear to give the best performance? Does MARS select the informative predictors (those named X1–X5)?
mars_model<- train(x = trainingData$x, y = trainingData$y, method = "earth",preProcess = c("center", "scale"), tuneLength = 10)## Loading required package: earth## Warning: package 'earth' was built under R version 4.2.3## Loading required package: Formula## Loading required package: plotmo## Warning: package 'plotmo' was built under R version 4.2.3## Loading required package: plotrix## Warning: package 'plotrix' was built under R version 4.2.3mars_pred <- predict(mars_model, newdata = testData$x)
postResample(pred = mars_pred, obs = testData$y)## RMSE Rsquared MAE
## 1.776575 0.872700 1.358367varImp(mars_model)## earth variable importance
##
## Overall
## X1 100.00
## X4 82.78
## X2 64.18
## X5 40.21
## X3 28.14
## X6 0.00The MARS model has a lower RMSE and higher Rsquared than KNN and only selected X1-X5 as its predictors. It appears to be a better fit.
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.
Which nonlinear regression model gives the optimal resampling and test set performance?
library(AppliedPredictiveModeling)## Warning: package 'AppliedPredictiveModeling' was built under R version 4.2.3library(RANN)## Warning: package 'RANN' was built under R version 4.2.3library(dplyr)## Warning: package 'dplyr' was built under R version 4.2.3##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, uniondata(ChemicalManufacturingProcess)
set.seed(1234)
knn_pre <- preProcess(ChemicalManufacturingProcess, "knnImpute")
knn_pred <- predict(knn_pre, ChemicalManufacturingProcess)
knn_pred <- knn_pred %>% select_at(vars(-one_of(nearZeroVar(., names = TRUE))))
training_data <- createDataPartition(knn_pred$Yield, times = 1, p = 0.8, list = FALSE)
train_data <- knn_pred[training_data, ]
test_data <- knn_pred[-training_data, ]
knn_model <- train(Yield ~ ., data = train_data, method = "knn", center = TRUE, scale = TRUE, trControl = trainControl("cv", number = 10),tuneLength = 10)
knn_model## k-Nearest Neighbors
##
## 144 samples
## 56 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 130, 129, 128, 131, 130, 130, ...
## Resampling results across tuning parameters:
##
## k RMSE Rsquared MAE
## 5 0.7237610 0.5219385 0.6000154
## 7 0.7298434 0.5103148 0.6038800
## 9 0.7401387 0.4957407 0.6110017
## 11 0.7218398 0.5368142 0.5865431
## 13 0.7379855 0.5135531 0.6008255
## 15 0.7487626 0.5025050 0.6110746
## 17 0.7555478 0.5016355 0.6163655
## 19 0.7515744 0.5160700 0.6101203
## 21 0.7582183 0.5071037 0.6173736
## 23 0.7639953 0.5085869 0.6198149
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 11.knn_predict <- predict(knn_model, test_data)
knn_results <- data.frame(t(postResample(pred = knn_predict, obs = test_data$Yield)))
knn_results## RMSE Rsquared MAE
## 1 0.6994187 0.3547056 0.605811mars_grid <- expand.grid(degree = c(1:2), nprune = c(2:10))
mars_model <- train(Yield ~ ., data = train_data, method = "earth",tuneGrid = mars_grid,trControl = trainControl("cv", number = 10),tuneLength = 25)
mars_model## Multivariate Adaptive Regression Spline
##
## 144 samples
## 56 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 129, 129, 132, 130, 130, 128, ...
## Resampling results across tuning parameters:
##
## degree nprune RMSE Rsquared MAE
## 1 2 0.7789194 0.4401603 0.6179795
## 1 3 0.6804064 0.5664303 0.5508993
## 1 4 0.6761158 0.5806914 0.5507581
## 1 5 0.6402040 0.6279732 0.5322559
## 1 6 0.6352661 0.6320216 0.5239903
## 1 7 0.6247490 0.6317278 0.5165607
## 1 8 0.6808449 0.5746501 0.5453058
## 1 9 0.6732789 0.5797236 0.5383759
## 1 10 0.6896934 0.5680038 0.5520418
## 2 2 0.7868154 0.4294405 0.6324954
## 2 3 0.6941249 0.5473943 0.5581296
## 2 4 0.7336219 0.5028495 0.5860609
## 2 5 0.6901801 0.5706026 0.5656028
## 2 6 0.7004739 0.5619804 0.5713038
## 2 7 0.6838762 0.5872813 0.5443752
## 2 8 0.7222177 0.5471597 0.5674303
## 2 9 0.7437381 0.5380753 0.5681492
## 2 10 0.7273905 0.5412025 0.5678327
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 7 and degree = 1.mars_pred2 <- predict(mars_model, test_data)
#postResample(pred = mars_pred2, obs = testData$y)The Mars model appears to perform better than KNN with a higher rsquared value and lower RMSE.
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(mars_model, 10)## earth variable importance
##
## Overall
## ManufacturingProcess32 100.000
## ManufacturingProcess09 59.823
## ManufacturingProcess13 17.294
## ManufacturingProcess01 2.199
## ManufacturingProcess42 0.000It returns 7 predictors with the manufacturing process variables dominating the predictors, with 32 being the highest in the model.
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?
library(corrplot)## Warning: package 'corrplot' was built under R version 4.2.3## corrplot 0.92 loadedtrain_data %>% select(c('ManufacturingProcess32','ManufacturingProcess13','ManufacturingProcess09','ManufacturingProcess13','ManufacturingProcess39','ManufacturingProcess04','ManufacturingProcess01','Yield')) %>% cor() %>% corrplot()
Manufacturing process 32 and 9 have stronger positive correlations with
yield, while 13 has a stronger negative relationship to yield. We can
see this in the plots, below, as well.
library(ggplot2)
predictors <- c('ManufacturingProcess32', 'ManufacturingProcess13', 'ManufacturingProcess09', 'ManufacturingProcess39', 'ManufacturingProcess04', 'ManufacturingProcess01')
plot_data <- lapply(predictors, function(pred) {
ggplot(train_data, aes_string(x = pred, y = "Yield")) +
geom_point() +
labs(x = pred, y = "Yield", title = paste(pred, "vs. Yield"))
})## Warning: `aes_string()` was deprecated in ggplot2 3.0.0.
## ℹ Please use tidy evaluation idioms with `aes()`.
## ℹ See also `vignette("ggplot2-in-packages")` for more information.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.library(gridExtra)## Warning: package 'gridExtra' was built under R version 4.2.3##
## Attaching package: 'gridExtra'## The following object is masked from 'package:dplyr':
##
## combinegrid.arrange(grobs = plot_data, ncol = 2)