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

whether alternative methods might be suitable for producing a more informative

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

briefly your reasoning. [2 marks]

(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]

Page 4 of 13 Turn the page over

Module Code: MATH5745M01

Question Q3 continued:

(e) Consider the following two plots each showing data from two different groups.