MAST90085 – Semester 2, 2019. Assignment #1
Instructions:
• The assignment contains 2 problems worth a total of 20 marks which will count towards 10% of
the final mark for the course.
• A PAPER copy of your assignment must be turned in by 5pm Wednesday 11 September, 2019. You
must complete the online plagiarism form on LMS by 5pm Wednesday 11 September, 2019. The
assignment must be return in the MAST90085 pigeonhole assigned in the School of Mathematics
and Statistics. Make sure you place your assignment in the right pigeonhole (there are three
pigeonholes, one per tutorial and you should use only the pigeonhole corresponding to your tutorial).
• [1 mark] Your assignment should clearly show your name and student ID number, your tutor’s
name and the time and day of your tutorial class. Your answers must be clearly numbered and in
the same order as the assignment questions. Your answers must be easy to read (marks may be
deducted for illegible handwriting). Include all of your working out in your answers. All R outputs,
including graphs and tables, must be accompanied by your concise and clearly written R code used
MAST90085作业代写、代做R编程设计作业、代写R课程设计作业、代做LMS课程作业
to produce it. Any graph, table or R code must be accompanied by clear and concise comments.
• Use tables, graphs and concise text explanations to support your answers. All tables and graphs
must be clearly commented and identified.
• All R code should be clearly written and commented. Uncommented R code is not acceptable.
• Comments should be brief and concise: marks will be awarded for clarity.
• Late assignments will only be accepted under exceptional circumstances with a written application
for submitting late and/or a medical certificate. A late penalty may be imposed.
• Your lecturer may not help you directly with assignment questions, but may provide some appropriate
guidance.
Data: In the assignment you will analyse some wheat data. The dataset is available in .txt format on
the LMS web page within the Assignments menu. The data come from three different varieties of wheat
denoted by 1 to 3 in the dataset. Each row of the dataset corresponds to a different wheat kernel. Seven
numerical characteristics were measured on the data: X1: area, X2: perimeter X3: compactness X4:
length of kernel, X5: width of kernel, X6: asymmetry coefficient X7: length of kernel groove. whereas
the eighth variable X8 contains values 1, 2 or 3 dependent on the variety of wheat the kernel comes from.
Problem 1 [8 marks]
(a) [3 marks] Give all possible values that a and b can take in order for the following matrix to be a
covariance matrix. Give arguments that justify your answer:
(b) [3 marks] Compute explicitly and without using R, all the eigenvectors and the eigenvalues of the
Deduce from there two orthogonal eigenvectors of norm 1 of that matrix. Give explicitly an
orthogonal matrix Γ and a diagonal matrix Λ such that we can write
Σ = ΓΛΓT
1
(c) [1 mark] Read the wheat data in R and create a data matrix X of size n × p, where n = 210
and p = 7 which contains the seven attributes X1 to X7 described above from all n kernels. Then
create a vector of length n which contains, for each kernel, the wheat variety it comes from, coded
1 to 3 as described above. If you use the menus in R studio to read your data, please print out the
corresponding instructions (they are given by R studio).
(d) [1 mark] Using R, for the covariance matrix S of X at (c), give explicitly an orthogonal matrix Γ
and a diagonal matrix Λ such that we can write
Problem 2 [11 marks]
(a) [2 marks] Perform a principal component analysis of the wheat data. Store the eigenvalues of the
covariance matrix in a vector called lambda and the eigenvectors in a matrix called gamma. What
percentage of the variability of the data does each principal component explain? Also compute the
cumulative percentages of variance ψ1, . . . ψ7 defined in class and draw a screeplot for these data.
How many principal components does this suggest we should keep?
(b) [3 mark] Give explicitly the linear combinations of the original data used in this example to create
the first and second principal components and give an interpretation of these linear combinations,
describing which variables play the biggest roles in the construction of those two PCs.
(c) [3 mark] Draw scatterplots of the principal components, using colours to identify different groups
of data. Describe what you can extract from those graphs. Which groups are visible on the graph?
What do they correspond to? How do the original variables contribute to those groups?
(d) [3 marks] Using the formula given in class, but replacing each population quantity by its empirical
estimator, compute the correlation matrix that contains the correlations between each principal
component and each original variable. Draw the correlation graph showing the correlations between
of the original variables X1 to X6 and the first two PCs. For each of the six original variables, use
an arrow to represent the correlations with the first two principal components as in the correlation
picture shown in class in week 5, and indicate the names of the variables near each arrow as done in
the example shown in class. Add to your graph a circle of radius 1 centered at the origin. Use this
and the other results of your PC analysis to describe further the results of the principal component
analysis, explicitly discussing the original variables, the groups of individuals, and the connection
between these two.
Hints: 1) To draw an arrow in R, use the command arrows 2) To add some text to a graph in R,
used the command text(x,y,yourtext) where x and y are the x and y coordinates of where to
write your text and yourtext is the text you want to write there. 3) To add a circle to a graph, use
radius <- 1
theta <- seq(0, 2 * pi, length = 200)
lines(x = radius * cos(theta), y = radius * sin(theta))
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