SYS 6018 | Spring 2024 | University of Virginia

Homework #2: Resampling

Author

Your Name Here

Published

February 3, 2024

Required R packages and Directories

data_dir = 'https://mdporter.github.io/SYS6018/data/' # data directory
library(tidymodels)# for optional tidymodels solutions
library(tidyverse) # functions for data manipulation

Problem 1: Bootstrapping

Bootstrap resampling can be used to quantify the uncertainty in a fitted curve.

a. Data Generating Process

Create a set of functions to generate data from the following distributions: \[\begin{align*} X &\sim \mathcal{U}(0, 2) \qquad \text{Uniform between $0$ and $2$}\\ Y &= 1 + 2x + 5\sin(5x) + \epsilon \\ \epsilon &\sim \mathcal{N}(0,\, \sigma=2.5) \end{align*}\]

Solution

Add Solution Here

b. Simulate data

Simulate \(n=100\) realizations from these distributions. Produce a scatterplot and draw the true regression line \(f(x) = E[Y \mid X=x]\). Use set.seed(211) prior to generating the data.

Solution

Add Solution Here

c. 5th degree polynomial fit

Fit a 5th degree polynomial. Produce a scatterplot and draw the estimated regression curve.

Solution

Add Solution Here

d. Bootstrap sampling

Make 200 bootstrap samples. For each bootstrap sample, fit a 5th degree polynomial and make predictions at eval_pts = seq(0, 2, length=100)

  • Set the seed (use set.seed(212)) so your results are reproducible.
  • Produce a scatterplot with the original data and add the 200 bootstrap curves
Solution

Add Solution Here

e. Confidence Intervals

Calculate the pointwise 95% confidence intervals from the bootstrap samples. That is, for each \(x \in {\rm eval\_pts}\), calculate the upper and lower limits such that only 5% of the curves fall outside the interval at \(x\).

  • Remake the plot from part c, but add the upper and lower boundaries from the 95% confidence intervals.
Solution

Add Solution Here

Problem 2: V-Fold cross-validation with \(k\) nearest neighbors

Run 10-fold cross-validation on the data generated in part 1b to select the optimal \(k\) in a k-nearest neighbor (kNN) model. Then evaluate how well cross-validation performed by evaluating the performance on a large test set. The steps below will guide you.

a. Implement 10-fold cross-validation

Use \(10\)-fold cross-validation to find the value of \(k\) (i.e., neighborhood size) that provides the smallest cross-validated MSE using a kNN model.

  • Search over \(k=3,4,\ldots, 40\).
  • Use set.seed(221) prior to generating the folds to ensure the results are replicable.
  • Show the following:
    • the optimal \(k\) (as determined by cross-validation)
    • the corresponding estimated MSE
    • produce a plot with \(k\) on the x-axis and the estimated MSE on the y-axis (optional: add 1-standard error bars).
  • Notation: The \(k\) is the tuning paramter for the kNN model. The \(v=10\) is the number of folds in V-fold cross-validation. Don’t get yourself confused.
Solution

Add Solution Here

b. Find the optimal edf

The \(k\) (number of neighbors) in a kNN model determines the effective degrees of freedom edf. What is the optimal edf? Be sure to use the correct sample size when making this calculation. Produce a plot similar to that from part a, but use edf (effective degrees of freedom) on the x-axis.

Solution

Add Solution Here

c. Choose \(k\)

After running cross-validation, a final model fit from all of the training data needs to be produced to make predictions. What value of \(k\) would you choose? Why?

Solution

Add Solution Here

d. Evaluate actual performance

Now we will see how well cross-validation performed. Simulate a test data set of \(50000\) observations from the same distributions. Use set.seed(223) prior to generating the test data.

  • Fit a set of kNN models, using the full training data, and calculate the mean squared error (MSE) on the test data for each model. Use the same \(k\) values in a.
  • Report the optimal \(k\), the corresponding edf, and MSE based on the test set.
Solution

Add Solution Here

e. Performance plots

Plot both the cross-validation estimated and (true) error calculated from the test data on the same plot. See Figure 5.6 in ISL (pg 182) as a guide.

  • Produce two plots: one with \(k\) on the x-axis and one with edf on the x-axis.
  • Each plot should have two lines: one from part a and one from part d
Solution

Add Solution Here

f. Did cross-validation work as intended?

Based on the plots from e, does it appear that cross-validation worked as intended? How sensitive is the choice of \(k\) on the resulting test MSE?

Solution

Add Solution Here