Question 1. Simple Linear Regression
(a) Consider a data set consisting ofX values (features)X1; : : : ;Xn and Y values (responses) Y1; : : : ; Yn.
Let ^ 0; ^ 1; ^ be the output of running ordinary least squares (OLS) regression on the data. Now
deﬁne the transformation:
e Xi = c(Xi + d);
for each i = 1; : : : ; n, where c = 0 and d are arbitrary real constants. Let e 0; e 1; e be the output of
OLS on the data e X1; : : : ; e Xn and Y1; : : : ; Yn. Write equations for e 0; e 1; e in terms of ^ 0; ^ 1; ^ (and
in terms of c; d), and be sure to justify your answers. Note that the estimate of error in OLS is taken
^ eT ^ e
where ^ e is the vector of residuals, i.e. with i-the element ^ ei = Yi ^ Yi, where ^ Yi is the i-th prediction
made by the model, and p is the number of features (so in this case p = 2).
(b) Suppose you have a dataset where X takes only two values while Y can take arbitrary real values.
To consider a concrete example, consider a clinical trial where Xi = 1 indicates that the i-th patient
receives a dose of a particular drug (the treatment), and Xi = 0 indicates that they did not, and
Yi is the real-valued outcome for the i-th patient, e.g. blood pressure. Let Y T and Y P indicate the
sample mean outcomes for the treatment group and non-treatment (placebo) group, respectively.
What will be the value of the OLS coefﬁcients ^ 0; ^ 1 in terms of the group means?
What to submit: For both parts of the question, present your solution neatly – photos of handwritten work or
using a tablet to write the answers is ﬁne. Please include all working and circle your ﬁnal answers.
Question 2. LASSO vs. Ridge Regression
In this problem we will consider the dataset provided in data.csv, with response variable Y , and
features X1; : : : ;X8.
(a) Use a pairs plot to study the correlations between the features. In 3-4 sentences, describe what
you see and how this might affect a linear regression model. What to submit: a single plot, some
(b) In order for LASSO and Ridge to be run properly, we often rescale the features in the dataset. First,
rescale each feature so that it has zero mean, and then rescale it so that
ij = n where n
denotes the total number of observations. What to submit: print out the sum of squared observations of
each of the 8 (transformed) features, i.e.
ij for j = 1; : : : ; 8
(c) Now we will apply ridge regression to this dataset, recall that ridge regression is deﬁned as the
solution to the optimisation:
^ = argmin
2 + kk2
Run ridge regression with = f0:01; 0:1; 0:5; 1; 1:5; 2; 5; 10; 20; 30; 50; 100; 200; 300g. Create a plot
with x-axis representing log(), and y-axis representing the value of the coefﬁcient for each feature
in each of the ﬁtted ridge models. In other words, the plot should describe what happens to each
of the coefﬁcients in your model for the different choices of . For this problem you are permitted
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