Homework 1 Solution

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1. For each of parts (a) through (d), indicate whether we would generally expect the performance of a flexible statistical learning method to be better or worse than an inflexible method. Justify your answer.

  1. The sample size n is extremely large, and the number of predictors p is small.

  2. The number of predictors p is extremely large, and the number of observations n is small.

  3. The relationship between the predictors and response is highly non-linear.

  4. The variance of the error terms, i.e. σ2 = Var(), is extremely high.

2. We now revisit the bias-variance decomposition.

  1. Provide a sketch of typical (squared) bias, variance, training error, test error, and Bayes (or irreducible) error curves, on a single plot, as we go from less flexible statistical learning methods towards more flexible approaches. The x-axis should represent the amount of flexibility in the method, and the y-axis should represent the values for each curve. There should be five curves. Make sure to label each one.

  2. Explain why each of the five curves has the shape displayed in part (a).

3. What are the advantages and disadvantages of a very flexible (versus a less flexible) approach for regression or classification? Under what circumstances might a more flexible approach be preferred to a less flexible approach? When might a less flexible approach be preferred?

4. Describe the differences between a parametric and a non-parametric statistical learning approach. What are the advantages of a parametric approach to regression or classification (as opposed to a nonparametric approach)? What are its disadvantages?