ST309 – Exercise 6
This counts for 10% of the final assessment of the course.
The marks in brackets reflect marks for each question.
Please submit your solutions in a pdf file to Moodle by 5pm (UK time) on Wednesday, 14 December. Late sub-
mission entails penalties: 5 marks (out of maximum 100) will be deducted for every half-day (12 hours) within
the first 24 hours after the deadline, and 5 marks will be deducted for each further 24 hours. Submissions are
not accepted after 5pm on Monday , 19 December.
You should only submit your own work, and cannot use materials from the past and/or your classmates.
Plagiarism is a very serious offense that is quite easy to detect. It will result in instant failure (mark 0).
This exercise is on credit card fraud detection based on a data set downloaded from Kaggle Datasets at
https://www.kaggle.com/mlg-ulb/creditcardfraud
Background information on the data is available at
https://www.kaggle.com/mlg-ulb/creditcardfraud/home
Previous attempts can be found at
https://www.kaggle.com/janiobachmann/credit-fraud-dealing-with-imbalanced-datasets/versions
(All those analysis was done using Python. But you should be able to follow the ideas, understand most the
results. Especially some initial data exploration is easy to follow.)
The dataset contains 284,807 credit card transactions in two days in September 2013 by European card-
holders, of which 492 are frauds. So the data is highly unbalanced: the positive cases (i.e. frauds) account for
merely 0.172% of all transactions.
Due to the confidentiality issues, the original features for each transaction are masked via a linear transfor-
mation. The 28 transformed features are presented as V1, V2, · · · , V28. According to the above webpage,
those 28 features are the principal components of the original features. No further information on those features
is provided. In addition to those 28 variables, there are 3 untransformed variables:
Time: number of seconds elapsed between each transaction and the first transaction in the dataset
Amount: the amount of the transaction
Class: binary label with value 1 for ‘fraud’ and 0 otherwise.
The whole dataset has 284,807 rows and 31 columns. The task is to build up effective algorithms for detecting
fraudulent credit card transactions.
The data is extremely imbalanced with only 0.172% ‘positives’ (i.e. frauds). Hence the information on
frauds is overwhelmed by that on true and genuine transactions. This imbalance leads the fitted models using
the whole data predominately led by the information on ‘negatives’, and the signal on ‘positives’ is too weak
to be picked up. To balance the information used in building classifiers, we have created a more balanced but,
unfortunately, much smaller training data with 24.62% positive cases, and also a testing data set which is about
equally imbalanced as the whole data set.
creditCardTrain.csv: of size 1592×31, consisting of 1200 randomly selected non-fraudulent transactions
plus 392 randomly selected fraud transactions. The true positive rate is about 24.62%.
creditCardTest.csv: of size 57889× 31, consisting of 57789 randomly selected non-fraudulent transac-
tions plus 100 remaining fraud transactions. It has no overlaps with creditCardTrain.csv. The true
positive rate is 0.173%.
The two data sets are placed on the course Moodle page. For your information, I attach below the codes for
constructing those two data sets.
> library(readr); library(dplyr)
> CC=read_csv("creditcard.csv") # read_csv is a much faster version of read.csv
> CC1=CC[CC$Class==1,] # extract all frauds
> dim(CC1)
[1] 492 31
> train1=sample(1:492, 392)
> CC1train=CC1[train1,]
> CC1test=CC1[-train1,]
> CC0=CC[CC$Class==0,] # extract all genuine transactions
> dim(CC0)
[1] 284315 31
> train0=sample(1:284315, 58988)
> Dtrain=bind_rows(CC1train, CC0[train0[1:1200],]) # bind the rows from two data together
> dim(Dtrain)
[1] 1592 31
> Dtrain=arrange(Dtrain, Time) # re-arrange rows according to ascending order of Time
> write.csv(Dtrain, row.names=F, "creditCardTrain.csv")
> Dtest=bind_rows(CC1test, CC0[train0[1200:58988],])
> dim(Dtest)
[1] 57889 31
> Dtest=arrange(Dtest, Time)
> write.csv(Dtest, row.names=F, "creditCardTest.csv")
> rm(CC, CC0, CC1, CC1test, CC1train, Dtest, Dtrain, train0, train1) # remove those objects
Your analysis should be based on creditCardTrain.csv. creditCardTest.csv represents the true reality,
and should be used only to test the performance of your models.
1. Carry out some exploratory data analysis first. You may like to address the issues such as
are there any missing values and outliers? [5 marks]
should you apply any transformations to any variables, for example, log(Amount+ 1)? [10 marks]
is Time relevant to detecting frauds? [5 marks]
2. Suppose that the credit card company charges 2% fees for each transaction (deducted from the payment
to payee).
(a) Estimate the expected potential loss of a fraudulent transaction. [5 marks]
(b) Estimate the expected profit from a genuine transaction. [5 marks]
(c) Let α denote the probability that a transaction is fraud. What is the minimum value of α to declare
‘Fraud’ in order to ensure that the expected profit from a single transaction is non-negative?
[5 marks]
A simple illustration on how a credit card works: Suppose you purchase an item from a shop for
￡100 payed out of your credit card, the credit card company will pay ￡98 to the shop at the time. By
the end of the month, you pay back ￡￡100 to the credit card company. So the company make a profit of
￡2. But if the purchase was not made by you (i.e. a fraud), you will not pay anything to the credit card
company. The company will make a loss of ￡98.
3. Let the profit matrix be
non-Fraud Fraud
No B C
Yes 1 0
where C and B are calculated, respectively, in 2(a) and 2(b) above. The nominal value ?1 reflects cus-
tomer’s unhappiness when a genuine transaction is denied.
(a) Construct a decision tree for detecting frauds. [10 marks]
(b) Find the value of the cutting-off probability, denoted by α?, which maximizes the expected profit.
[10 marks]
(c) Test the performance of your decision tree on the testing data set, with the cutting-off probability 0.5
and α? respectively. Now you should calculate the true profits or losses according to the real amount
of each transaction in the testing data sets. [10 marks]
(d) Construct a logistic regression model for detecting frauds. You may use the same predictor selected
in the tree model above. [10 marks]
(e) Plot the ROC curves with the testing data for both the tree and the logistic regression classifiers
constructed above, and compare them using the ‘area under curve’. [15 marks]
4. In your opinion, what are the pros and cons of the above analysis? Do you have any suggestions for further
improvement? [10 marks]
Note. The strategy to build classifiers using a subset with a much higher positive rate was merged after some
initial and less successful attempts. This learning process also reflects one important principle of data analytics:
Iteration is the law of learning!