DATA7202 Statistical Methods

Statistical Methods for Data Science
DATA7202
Assignment 1 (Weight: 25%)

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 ∼ Gamma(1, 1)
   for i ∈ {1, 2, 3}; here Gamma(1, 1) stands for the continuous Gamma distribution with
   both scale and shape parameters equal to 1.
   (b) In addition, the model for y is as follow:
   Y = 0.5X1 + 3X2 + 5X3 + 5X2X3 + 2X1X2X3 + W, (1)
   where W ∼ N(0, σ2
   ) where σ = 2.
   Similar to the original example, generate train and test sets of size N = 1000. Fit the linear re￾gression 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.
   Is this a good method? Do you expect to obtain the true prediction error? Explain your answer.
   Please note that no coding is required here and one paragraph general answer is sufficient.
   1
3. [5 Marks] Suppose that we observe X1, . . . , Xn ∼ F. We model F as a Gamma distribution
   with shape parameter α > 0 and rate parameter β > 0. 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. Show that the expected value of LossT (g) over
   the choice of T equals LossD(g), namely,
   ET LossT (g) = LossD(g).
5. [15 Marks (see details below)] Consider the following dataset.
   x1 y
6. 1
7. 2
8. 3
9. 2
10. 1
    Now, suppose that we would like to consider two models.
    Model1 : y = β0 + ε,
    and
    Model2 : y = β1x1 + ε,
    where ε ∼ N(0, 1). That is, we consider two linear models Model1 is the constant model and
    Model2 is a regular linear model without the intercept.
    (a) [5 Marks)] Fit these models tot the data and write the corresponding coefficients. Namely,
    fill the following table:
    Model β0 β1
    Model1 0
    Model2
    (b) [5 Marks)] Consider the squared error loss, the absolute error loss, and the L1.5 loss. Find
    the average loss for each model. Namely, fill the following table:
    Model squared error loss absolute error loss L1.5 loss
    Model1
    Model2
    (c) [5 Marks)] Draw a conclusion from the obtained results.
11. [30 Marks (see details below)] Consider the Hitters data-set (given in Hitters.csv). Our
    objective is to predict a hitter’s salary via linear models.
    (a) [5 Marks)] Load the data-set and replace all categorical values with numbers. (You can
    use the LabelEncoder object in Python).
    (b) [5 Marks)] Generally, it is better to use OneHotEncoder when dealing with categorical
    variables. Justify the usage of LabelEncoder in (a).
    2
    (c) [20 Marks)] Fit linear regression and report 10-Fold Cross-Validation mean squared error.
12. [10 Marks] Consider a function
    f(x) = 3 + x2 2sin(x) 1 6 x 6 8.
    Write a Crude Monte Carlo algorithm for the estimation of
    ` = Z 8 1 f(x) dx,
    using N = 10000 sample size. Deliver the 95% confidence interval. Compare the obtained
    estimation with the true value `. 3