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Module Code: MATH5745M01
Q1.
(a) Suppose that A is a square n× n matrix. What do you understand by the phrase
“matrix A is not of full rank”? Outline some properties of a not full rank matrix.
How would you check whether the matrix A is not of full rank? [8 marks]
(b) Suppose that B is a symmetric matrix such that B > 0. Explain carefully what
the notation “B > 0” tells you about the matrix B. Outline some properties of
matrix B. [8 marks]
(c) Suppose you are told that a square n×n matrix C is an orthogonal matrix. What
does this tell you about matrixC? Outline some properties of matrix C. [6 marks]
(d) The spectral decomposition theorem states: “Any symmetric (n×n) matrix S can
be written as S = GDG′ where G is the matrix of standardized eigen-vectors of
S and D is a diagonal matrix of eigen-values of S.” Discuss the importance of the
spectral decomposition theorem to Multivariate Statistics. [4 marks]
(e) Why is the multivariate normal distribution important in Multivariate Analysis?
Discuss its usefulness and its limitations. [8 marks]
(f) The figure below shows the results of a cluster analysis of some data for 28 countries
based upon the values of two variables at the end of April 2020.
Explain to a non-statistician what the plot shows (you do not need to know what
the data is) and also explain the methodology used to construct the plot. Discuss
cluster analysis plot. [11 marks]

(a) Describe the principle of linear discriminant analysis. [3 marks]
(b) Looking solely from the horizontal axis (Length) or vertical axis (Weight) in the
above figure, can you identify a clear separation between the two varieties? Explain
(c) Find the discriminant function from the above data. What can you say about the
discriminant function line? Calculate the standardised coefficients of the discrimi-
nant function. [5 marks]
(d) Suppose the expert found two new observations. The first one is a bird with length
44 cm and weight 8.7 (in units of 100 grams). The second one is a bird with length
47 cm and weight 9.0 (in units of 100 grams).
(i) Before performing any calculations, identify to which sub-species should each
new observation be classified. Explain briefly your reasoning. [3 marks]
(ii) Now write down the discriminant rule. Based on this rule, to which sub-species
should each new observation be classified? [4 marks]
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Module Code: MATH5745M01
Question Q3 continued:
(e) Consider the following two plots each showing data from two different groups.