Introduction
Submission: No files are to be submitted via e-mail. Correct files are to be submitted to Moodle, as given above. You must submit the following three files:
1. One PDF file containing non-code answers to all the questions that require written answers. This file should also include all your plots.
2. An R script file containing R code answers. Please make sure this is clearly commented so it is obvious which R statements are answering which questions, and the questions are answered in the order they appear in the assignment.
Please read these submission instructions carefully and take care to submit the correct files in the correct places.
4. Use the stepwise selection procedure with the BIC penalty (using direction=”both”) to prune out potentially unimportant variables. Write down the final regression equation obtained after pruning. [1 mark]
5. Imagine that you are looking for a new car to buy to replace your existing car. The characteristics of the new car that you are looking at are given by the thirty-third row of the dataset.
(a) Use your BIC model to predict the mean fuel efficiency for this new car. Provide a 95% confidence interval for this prediction. [1 mark]
(b) The current car that you own has a mean fuel efficiency of 11km/l (measured over the life time of your ownership). Does your model suggest that the new car will have better fuel efficiency than your current car? [1 mark]
Variable name Description Values
Eng.Displacement Engine Displacement (litres, l) 0.9 − 8.4
No.Cylinders Number of Cylinders 3 − 16
Aspiration Engine Aspiration (Oxygen intake) N: Naturally∗
OT: Other
SC: Supercharged
TC: Turbocharged
TS: Turbo+supercharged
No.Gears Number of Gears 1 − 10
Lockup.Torque.Converter Lockup torque converter present? N∗ and Y
Drive.Sys Drive System 4∗: 4-wheel drive
A:All-wheel
F:Front-wheel
P:Part-time 4-wheel
R:Rear-wheel
Max.Ethanol Maximum % of Ethanol allowed 10 − 85
Fuel.Type Type of Fuel G∗: Regular Unleaded
GM: Mid-grade Unleaded Recommended
GP: Premium Unleaded Recommended
GPR: Premium Unleaded Required
Comb.FE Fuel Efficiency (km/l) 4.974 − 26.224
Table 1: Fuel efficiency data dictionary. The ∗ denotes the reference category for each categorical variable.
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When answering this question, you must use the rpart package that we used in Studio 9. The wrapper function for learning a tree using cross-validation that we used in Studio 9 is contained in the file wrappers.R. Don’t forget to source this file to get access to the function.
3. For classification problems, the rpart package only labels the leaves with the most likely class. However, if you examine the tree structure in its textural representation on the console, you can determine the probabilities of having heart disease (see Question 2.3 from Studio 9 as a guide) in each leaf (terminal node). Take a screen-capture of the plot of the tree (don’t forget to use the “zoom” button to get a larger image) or save it as an image using the “Export” button in R
Studio.
Then, use the information from the textual representation of the tree available at the console and annotate the tree in your favourite image editing software; next to all the leaves in the tree, add text giving the probability of contracting heart disease. Include this annotated image in your report file. [1 mark]
4. According to your tree, which predictor combination results in the lowest probability of having heart-disease? [1 mark]
6. Write down the regression equation for the logistic regression model you found using step-wise selection. [1 mark]
7. Please describe the effect the variable CA has on heart-disease according to this logistic regression model? [1 mark]
9. Calculate the odds of having heart disease for the 10th patient in the test dataset. The odds should be calculated for both:
(a) the tree model found using cross-validation; and (b) the step-wise logistic regression model.
10. For the logistic regression model using only those predictors selected by KIC in Question 2.5, use the bootstrap procedure (use at least 5,000 bootstrap replications) to find a confidence interval for the odds of having heart disease for the 65th and 66th patients in the test data. Use the bca option when computing this confidence interval.
Variable name Description Values
AGE Age of patient in years 29 − 77
SEX Sex of patient M = Male
F = Female
CP Chest pain type Typical = Typical angina
Atypical = Atypical angina
NonAnginal = Non anginal pain
Asymptomatic = Asymptomatic pain
TRESTBPS Resting blood pressure (in mmHg) 94 − 200
CHOL Serum cholesterol in mg/dl 126 − 564
FBS Fasting blood sugar > 120mg/dl ? <120 = No
>120 = Yes
RESTECG Resting electrocardiographic results Normal = Normal
ST.T.Wave = ST wave abnormality
Hypertrophy = showing probable hypertrophy
THALACH Maximum heart rate achieved 71 − 202
EXANG Exercise induced angina? N = No
Y = Yes
OLDPEAK Exercise induced ST depression relative to rest 0 − 6.2
SLOPE Slope of the peak exercise ST segment Up = Up-sloping
Flat = Flat
Down = Down-sloping
CA Number of major vessels colored by flourosopy 0 − 3
THAL Thallium scanning results Normal = Normal
Fixed.Defect = Fixed fluid transfer defect
Reversible.Defect = Reversible fluid transfer defect
HD Presence of heart disease N = No
Y = Yes
Table 2: Heart Disease Data Dictionary. ST depression refers to a particular type of feature in an electrocardiograph (ECG) signal during periods of exercise. Thallium scanning refers to the use of radioactive Thallium to check the fluid transfer capability of the heart.
Mass/Charge (MZ)
Figure 1: Noisy measurements from a subsection of a (simulated) mass spectrometry reading. The “true” (unknown) measurements are shown in orange, and the noisy measurements are shown in blue.
Data Smoothing
Data “smoothing” is a very common problem in data science and statistics. We are often interested in examining the unknown relationship between a dependent variable (y) and an independent variable (x), under the assumption that the dependent variable has been imperfectly measured and has been contaminated by measurement noise. The model of reality that we use is
y = f(x) + ε
where f(x) is some unknown, “true”, potentially non-linear function of x, and ε ∼ N(0,σ2) is a random disturbance or error. This is called the problem of function estimation, and the process of estimating f(x) from the noisy measurements y is sometimes called “smoothing the data” (even if the resulting curve is not “smooth” in a traditional sense, it is less rough than the original data).
In this question you will use the k-nearest neighbours machine learning technique to smooth data. This technique is used frequently in practice (think for example the 14-day rolling averages used to estimate coronavirus infection numbers). This question will explore its effectiveness as a smoothing tool.
Mass Spectrometry Data Smoothing
Questions
5. Use the cross-validation functionality in the kknn package to select an estimate of the best value of k (make sure you still use the optimal kernel). What value of k does the method select?
How does it compare to the (in practice, unknown) value of k that would minimise the actual mean-squared error (as computed in Q3.1)? [1 mark]
7. An important task when processing mass spectrometry signals is to locate the peaks, as this gives information on which elements are present in the material we are analysing. From the smoothed signal produced using the value of k found in Q3.5, which value of MZ corresponds to the maximum estimated intensity? [1 mark]
