MATH530501 Learning from Data
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Module Code: MATH530501
Module Title: Learning from Data
School of Mathematics Semester One 202122
Assignment 1
This assignment counts for 30% of the total module mark
Instructions:
• There are 3 pages to this examination.
• Extended Deadline Monday the 29th of November, noon (UK time).
• Late submission will bear a penalty of 10 marks per day for up to three days. Submissions that are more than three days late, will receive a mark of zero.
• Answer all questions.
• You must type your report using either Word or LaTex (pdf submission). Your report must be in academic style and include all information needed for someone who is com- petent but not familiar with the questions to be able to understand.
• You must show all your calculations and you must use R to solve the computational parts.
• You should include your code as an appendix.
• There is no page limit. However, unnecessarily long reports (i.e. including irrelevant content) will bear a penalty.
• 
You only need to submit an electronic copy through Minerva Submit my Work. No hard copies required.
1. In this task you will simulate data from a data generating process with one response and two features. Then you will try to explain the response Y given the two features X1 and X2.
(a) 
Simulate 1,000,000 values from each of the following standard Uniform variables X1 U (0, 1) and X2 U (0, 1). Use these as covariates to simulate 1,000,000 values from the following response Y (this is the data generating process):
yi = (⌊10 ∗ x1 + 1⌋ + ⌊10 ∗ x2 + 1⌋) mod 2,
where ⌊·⌋ is the floor function: ⌊x⌋ = max{m ∈ Z|m ≤ x}.
(b) Convert the variable y into a factor type and create a data frame with y, x1, x2.
(c) Plot the data on a scatter plot, where X1 is on the horizontal axis, X2 is on the vertical axis, and the colour of the data is based on the class Y .
(d) Can you see a pattern based on the plot of Q1.c? Discuss how the data generating process produces this pattern.
(e) Grow a classification tree using the CART algorithm to describe the relationship between Y (the response) and X1, X2 (the features). You can decide on the parameters. Can the tree learn the DGP (i.e. can it partition the feature space correctly? Explain the reasoning behind your findings.
(f) Grow a Random Forest to answer the same problem as above. Does the forest do better than the single tree in correctly partitioning the feature space? Explain the reasoning behind your findings.
(g) Now try to solve the same problem using Boosting (gradient boosting machines). Compare its performance with the single tree and the random forest.
(h) Try changing the parameters (number of trees and maximum depth of tree) for the random forest and report the effect on the accuracy of the model. Use plots to support your arguments.
2. In this question you will work with a real dataset from Kaggle Kaggle Banking Dataset. A copy of the csv file can be found in Minerva.The problem is to model the probability that a customer will sign up to a long term deposit, given the different characteristics described in 15 features. The response variable is ”y”. In all the models that you are asked to fit below, you can decide the parameters after some trial runs. You need to explain how you chose the parameters.
(a) Introduce the dataset and perform an exploratory data analysis. Try to link your analysis to the question in hand.
(b) Create a training sample of size n = 2/3N , where N is the number of instances. The remaining data will form you testing set. Explain how you selected these sets.
(c) Fit a logistic regression model to predict the probability of signing up to a long-term deposit (Y = Yes). Evaluate the fit.
(d) Fit a Random Forest model to predict the probability of signing up to a long-term deposit. Evaluate the fit.
(e) Fit a generalised additive model using cubic splines as smoothers to predict the probability of of signing up to a long-term deposit. Evaluate the fit.
(f) Compare all the above models based on their predictive performance on the testing set. Be clear on what measures you are using for comparison.
(g) Fit a logistic regression using only the 3 most important variables as indicated by the random forest model. Explain how can you get this information. Compare this logistic model to the one you fitted using all the variables. When would such an approach be beneficial?
(h) (Bonus question!) Plot a partial dependence plot of the predictions of the random forest with respect to age.
2021-11-28