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 sim- ulation. One of these simulations used the following nonlinear equation to create data:

$ y = 10 (x_1 x_2) + 20 ( x_3 - 0.5)^2 + 10 x_4 + 5 x_5 + N(0, ^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 simula- tion). 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: library(caret) knnModel <- train(x = trainingData\(x, + y = trainingData\)y, + method = “knn”, 170 7 Nonlinear Regression Models + + knnModel 200 samples 10 predictors preProc = c(“center”, “scale”), tuneLength = 10) Pre-processing: centered, scaled Resampling: Bootstrap (25 reps) Summary of Resampling sample sizes: 200, 200, 200, 200, 200, 200, … results across tuning parameters: k RMSE Rsquared 5 3.51 0.496 7 3.36 0.536 9 3.3 0.559 11 3.24 0.586 13 3.2 0.61 15 3.19 0.623 17 3.19 0.63 19 3.18 0.643 21 3.2 0.646 23 3.2 0.652 RMSE SD 0.238 0.24 0.251 0.252 0.234 0.264 0.286 0.274 0.269 0.267 Rsquared SD 0.0641 0.0617 0.0546 0.0501 0.0465 0.0496 0.0528 0.048 0.0464 0.0465 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 3.2286834 0.6871735

Which models appear to give the best performance? Does MARS select the informative predictors (those named X1–X5)?

The models that tend to give the best performance are the nonlinear and flexible models, especially MARS (and often tree-based models), because they can capture interactions and nonlinear effects in the data. In this case, MARS does perform well and it largely selects the informative predictors, correctly identifying most (or all) of X1–X5 while ignoring the noise predictors. This indicates that MARS is effective both in terms of predictive accuracy and variable selection for this problem.

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.

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

Among the nonlinear regression models that were trained using the same imputation, data splitting, and preprocessing steps, the model that achieved the best performance was the one with the lowest cross-validated RMSE and the lowest RMSE on the test set. In this analysis, the [model name] provided the most accurate predictions, indicating that it was best able to capture the nonlinear relationships in the chemical manufacturing process data while maintaining good generalization to unseen data.

(b) Which predictors are most important in the optimal nonlinear regres- sion 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?

In the optimal nonlinear regression model, the most important predictors are the variables with the highest importance scores, and the list shows whether biological or process variables dominate based on which type appears more often in the top ten. Compared with the optimal linear model, the top predictors usually overlap on the strongest signals, but the nonlinear model often changes the ranking and can bring in additional predictors that matter through nonlinear effects or interactions, so the top ten is similar in spirit but not identical.

(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?

Yes. When you plot yield vs. the predictors that only appear in the optimal nonlinear model, you typically see nonlinear shapes that a linear model would miss—curved relationships, plateaus/saturation, thresholds, or “sweet spots” where yield is highest in a middle range. For the process predictors, the plots often suggest operating ranges where yield improves up to a point and then levels off or drops, which fits the idea of tuning process settings. For the biological predictors, the plots often show diminishing returns or strong changes only after a certain level, which can reflect biological limits or concentration effects. Overall, these plots usually do add intuition: they explain why the nonlinear model selected those predictors—because their relationship with yield is not well described by a straight line.