Question 3

We now review k-fold cross-validation

A. Explain how k-fold cross-validation

K-fold cross-validation is a technique used to evaluate the performance of a machine learning model in a more reliable and robust way. Here’s how it works step by step:

Step 1: Split the data into k equal-sized folds (subsets). For example, if k=5, the dataset is divided into 5 parts.

Step 2: Iterate k times

  • In each iteration, one fold is used as the validation set, and the remaining k-1 folds are used as the training set.

  • The model is trained on the training set and evaluated on the validation set.

  • The evaluation metric (like accuracy, RMSE, etc.) is recorded for that fold.

Step 3: Average the results

  • After all k iterations, the k evaluation scores are averaged to give a more reliable estimate of the model’s performance.

B. What are the advantages and disadvantages of k-fold cross-validation relative to:

  1. The validation set approach?

Validation set approach: You split your data once into a training set and a validation set (commonly 70/30 or 80/20), train on one and evaluate on the other.

  • Advantages

More reliable estimates

Less bias

Efficient use of data

  • Disadvantages

Slower

More complex to implement

  1. LOOCV?

LOOCV is a special case of k-fold where k equals the number of data points. So each fold contains exactly one data point as the validation set.

  • Advantages

Faster

Lower variance in performance estimate

  • Disadvantages

Slightly more bias

Question 4

Suppose that we use some statistical learning method to make a prediction for the response Y for a particular value of the predictor X. Carefully describe how we might estimate the standard deviation of our prediction.

To estimate the standard deviation of a prediction for the response Y at a particular value of the predictor X=x0, we need to account for both the uncertainty in the model’s estimate and the inherent variability in the data. In parametric models like linear regression, this is typically done using a formula that incorporates the variance of the residuals, the size of the dataset, and the distance of x0 from the mean of the predictors. For more complex or non-parametric models where such formulas may not apply, a common approach is to use bootstrapping. This involves repeatedly resampling the data, fitting the model, making predictions at x0, and then calculating the standard deviation of these predictions to estimate the overall uncertainty.