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DATA7202
Please answer the questions below. For theoretical questions, you should present rigorous proofs
and appropriate explanations. Your report should be visually appealing and all questions should
be answered in the order of their appearance. For programming questions, you should present your
analysis of data using Python, Matlab, or R, as a short report, clearly answering the objectives
and justifying the modeling (and hence statistical analysis) choices you make, as well as discussing
your conclusions. Do not include excessive amounts of output in your reports. All the code should
be copied into the appendix and the sources should be packaged separately and submitted on the
blackboard in a zipped folder with the name:
"student_last_name.student_first_name.student_id.zip".
For example, suppose that the student name is John Smith and the student ID is 123456789.
Then, the zipped file name will be John.Smith.123456789.zip.
1. [15 Marks] Repeat the advertisement exercise with the following changes.
(a) The data is generated via the following data generation mechanism. Xi ~ U(0, 1) for
i ∈ {1, 2, 3}; here U(0, 1) stands for the continuous uniform distribution over the [0, 1] set.
However, we require that X1 + X2 + X3 = 1, that is, the explanatory variables stand for
a percentage of the budget.
(b) In addition, the model for y is as follow:
Y = 0.5X1 + 3X2 + 5X3 + 5X2X3 + 2X1X2X3 + W, (1)
where W ~ U(0, 1).
Similar to the original example, generate train and test sets of size N = 1000. Fit the linear regression
and the random forest models to the data. For the linear regression, make an inference
about the coefficients, specifically, comment about the contributions of different advertisement
types to sales. Use the linear model and the RF (with 500 trees), to make a prediction (using
the test set), and report the corresponding mean squared errors.
When constructing datasets, please use “1” and “2” seeds for the train and the test sets,
respectively.
2. [10 Marks] Consider the following variant of the cross-validation procedure.
(i) Using the available data, find a subset of “good” predictors that show correlation with
the response variable.
(ii) Using these predictors, construct a model (for regression or classification).
(iii) Use cross-validation to estimate the model prediction error.
1
Is this a good method? Do you expect to obtain the true prediction error? Explain your
answer.
3. [5 Marks] Suppose that we observe X1, . . . , Xn ~ F. We model F as a normal distribution
with mean μ and standard deviation of σ. For this problem, determine the hypothesis class
H = {f(x, θ); θ ∈ Θ}.
and state explicitly what is θ and Θ.
4. [15 Marks] Let H be a class of binary classifiers over a set Z. Let D be an unknown distribution
over X , and let g be a target hypothesis in H. F Show that the expected value of LossT (g)
over the choice of T equals LossD(g), namely,
ET LossD(g) = LossD(g).
5. [15 Marks (see details below)] Consider the following dataset.
Now, suppose that we would like to consider two models.