MATH3030-E1 The University of Nottingham SCHOOL OF MATHEMATICAL SCIENCES A LEVEL 3 MODULE, SPRING SEMESTER 2019-2020 MULTIVARIATE ANALYSIS Suggested time to complete: TWO Hours THIRTY Minutes Paper set: 21/05/2020 – 10:00 Paper due: 28/05/2020 – 10:00 Answer ALL questions Your solutions should be written on white paper using dark ink (not pencil), on a tablet, or typeset. Do not write close to the margins. Your solutions should include complete explanations and all intermediate derivations. Your solutions should be based on the material covered in the module and its prerequisites only. Any notation used should be consistent with that in the Lecture Notes. Guidance on the Alternative Assessment Arrangements can be found on the Faculty of Science Moodle page: https://moodle.nottingham.ac.uk/course/view.php?id=99154#section-2 Submit your answers as a single PDF with each page in the correct orientation, to the appropriate dropbox on the module’s Moodle page. Use the standard naming convention for your document: [StudentID]_[ModuleCode].pdf. Please check the box indicated on Moodle to confirm that you have read and understood the statement on academic integrity: https://moodle.nottingham.ac.uk/pluginfile.php/6288943/mod_ tabbedcontent/tabcontent/8496/FoS%20Statement%20on%20Academic%20Integrity.pdf A scan of handwritten notes is completely acceptable. Make sure your PDF is easily readable and does not require magnification. Text which is not in focus or is not legible for any other reason will be ignored. If your scan is larger than 20Mb, please see if it can easily be reduced in size (e.g. scan in black & white, use a lower dpi — but not so low that readability is compromised). Staff are not permitted to answer assessment or teaching queries during the assessment period. If you spot what you think may be an error on the exam paper, note this in your submission but answer the question as written. Where necessary, minor clarifications or general guidance may be posted on Moodle for all students to access. Students with approved accommodations are permitted an extension of 3 days. The standard University of Nottingham penalty of 5% deduction per working day will apply to any late submission. MATH3030-E1 Turn over MATH3030-E1 Academic Integrity in Alternative Assessments The alternative assessment tasks for summer 2020 are to replace exams that would have assessed your individual performance. You will work remotely on your alternative assessment tasks and they will all be undertaken in “open book” conditions. Work submitted for assessment should be entirely your own work. You must not collude with others or employ the services of others to work on your assessment. As with all assessments, you also need to avoid plagiarism. Plagiarism, collusion and false authorship are all examples of academic misconduct. They are defined in the University Academic Misconduct Policy at: https://www.nottingham.ac. uk/academicservices/qualitymanual/assessmentandawards/academic-misconduct.aspx Plagiarism: representing another person’s work or ideas as your own. You could do this by failing to correctly acknowledge others’ ideas and work as sources of information in an assignment or neglecting to use quotation marks. This also applies to the use of graphical material, calculations etc. in that plagiarism is not limited to text-based sources. There is further guidance about avoiding plagiarism on the University of Nottingham website. False Authorship: where you are not the author of the work you submit. This may include submitting the work of another student or submitting work that has been produced (in whole or in part) by a third party such as through an essay mill website. As it is the authorship of an assignment that is contested, there is no requirement to prove that the assignment has been purchased for this to be classed as false authorship. Collusion: cooperation in order to gain an unpermitted advantage. This may occur where you have consciously collaborated on a piece of work, in part or whole, and passed it off as your own individual effort or where you authorise another student to use your work, in part or whole, and to submit it as their own. Note that working with one or more other students to plan your assignment would be classed as collusion, even if you go on to complete your assignment independently after this preparatory work. Allowing someone else to copy your work and submit it as their own is also a form of collusion. Statement of Academic Integrity By submitting a piece of work for assessment you are agreeing to the following statements: 1. I confirm that I have read and understood the definitions of plagiarism, false authorship and collusion. 2. I confirm that this assessment is my own work and is not copied from any other person’s work (published or unpublished). 3. I confirm that I have not worked with others to complete this work. 4. I understand that plagiarism, false authorship, and collusion are academic offences and I may be referred to the Academic Misconduct Committee if plagiarism, false authorship or collusion is suspected. MATH3030-E1 Turn over 1 MATH3030-E1 1. (a) i) Briefly describe the method of principal components analysis and explain its main uses. ii) Describe the situations when it is most suitable to use principal components analysis of the sample correlation matrix rather than the sample covariance matrix , and when it is preferable to use rather than . [10 marks] (b) The profits (in £) of five banks ,, ,, from the United Kingdom were recorded as vectors of length 5 over 40 quarter year periods. A principal components analysis was performed on the sample covariance matrix with eigenvectors given by PC1 PC2 PC3 PC4 PC5 0.421 -0.526 0.541 -0.176 0.472 0.457 0.509 0.178 0.676 0.206 0.421 -0.435 0.385 -0.382 0.470 0.260 0.335 -0.400 -0.662 0.464 0.240 -0.612 -0.451 0.387 with corresponding eigenvalues 2.502, 1.273, 0.329, 0.201, 0.136. i) Calculate the value of (to two decimal places). ii) Draw a scree plot and suggest the number of components that are needed to describe the data adequately. iii) Provide an interpretation of these components. iv) Calculate the total percentage of variability explained by these components. [10 marks] (c) Data are available for another 7 banks from the United States over the same period. State what method could be used to investigate the linear combinations of the bank profits that are most highly correlated in the two datasets of UK and US banks, and give brief details of the technique. [10 marks] (d) In the table below, Euclidean distances are given in a matrix between four pension funds based on measurements of 12 financial variables. Fund A Fund B Fund C Fund D Fund A 0 2.1 2.0 2.4 Fund B 2.1 0 1.8 0.2 Fund C 2.0 1.8 0 2.5 Fund D 2.4 0.2 2.5 0 i) Apply the single linkagemethod to the matrix. Summarise your results graphically using a dendrogram. ii) Apply the complete linkage method to the matrix . Summarise your results graphically using a dendrogram. iii) Suppose exactly two clusters are required. What would be your clusters based on (I) single linkage and (II) complete linkage? iv) If two clusters are required then state which of these two linkage methods you prefer, and briefly give your reasons. [10 marks] MATH3030-E1 2 MATH3030-E1 2. (a) Let 1,… , be independent identically distributed(,) random variables. Denote the sample mean by ̄ = 1 ∑ =1 and the sample covariance matrix by = 1 ∑ =1 ( − ̄)( − ̄) ⊤. i) Using the result (̄ − )−1(̄ − ) ∼ 2 . describe how to obtain a confidence region for when is known. ii) State, without proof, the distribution of 2 = ( − 1)(̄ − )−1(̄ − ). iii) Explain how the result in ii) can be used to test 0 ∶ = 0 versus 1 ∶ ≠ 0. In practice which null distribution is used for carrying out the test? [10 marks] (b) The length and width measurements in for a particular species of fish with sample size = 30 have mean vector ̄1 = (81.50, 68.90) and covariance matrix 1 = [ 30.0 10.0 10.0 20.0] . It has been conjectured that the mean of the length and width
of this species of fish should equal 80 and 69 respectively, i.e. 0 = (80, 69) ⊤. Examine this claim by carrying out a suitable test, and carefully state your assumptions. [10 marks] (c) The length and width measurements for a second species of fish with sample size = 35 have mean vector ̄2 = (84. ∗ ∗, 70.60) ⊤ and covariance matrix 2 = [ 28.6 9.9 9.9 21.1] , where ∗∗ are the final two digits of your Student ID number. For example if your Student ID is 53256178 then ∗∗ is 78 and the sample mean length of the second species of fish is 84.78. Carry out a two sample test using test statistic 2 = ( + − − 1) ( + − 2) + (̄1 − ̄2) ⊤−1 (̄1 − ̄2), to investigate whether the two population means are the same or not, carefully stating your assumptions. Note that is the pooled unbiased covariance matrix estimator. [10 marks] MATH3030-E1 Turn Over 3 MATH3030-E1 (d) i) Comment on whether or not the assumptions for the test in (c) are reasonable. ii) Briefly discuss an alternative procedure for testing the hypothesis in (c) which is based on the multivariate linear model = + , where the terms in the model should be specified. There is no need to carry out this alternative test. [10 marks] MATH3030-E1 4 MATH3030-E1 3. (a) Consider the following road distances between some cities in Europe (in km): Athens Barcelona Brussels Calais Cherbourg Cologne Athens 0 3313 2963 3175 3339 2762 Barcelona 3313 0 1318 1326 1294 1498 Brussels 2963 1318 0 204 583 206 Calais 3175 1326 204 0 460 409 Cherbourg 3339 1294 583 460 0 785 Cologne 2762 1498 206 409 785 0 i) Briefly describe how to obtain the principal co-ordinates using the method of classical multidimensional scaling, making reference to how the centering matrix is used in the calculation. ii) The eigenvalues calculated from using classical multidimensional scaling for these data are (×106): (8.015, 1.423, 0.157, 0, −0.002, −0.019) State whether or not the resulting estimated two dimensional map frommultidimensional scaling is a good approximation to the spatial arrangement of the cities, giving your reasons. iii) Is the distance matrix between the cities a Euclidean distance matrix? Give your reasoning. [15 marks] (b) For = 1, …, let denote a population described by a probability density function (|). Provide a brief explanation of the sample ML discriminant rule in this situation. [5 marks] (c) i) Measurements of cranial length (1) and cranial breadth (2), both measured in millimetres, on samples of 40male and 40 female frogs led to the following statistics for the sample mean vector (̄) and sample covariance matrix (), with = 1 for male frogs and = 2 for female frogs. ̄1 = ( 24.1 24.8) ̄2 = ( 22.8 23.5) 1 = ( 6.5 3 3 6.5) 2 = ( 5.5 3 3 5.5) . Assuming the data are multivariate normal and stating any additional assumptions that youmake, derive a sample ML discriminant rule for allocating a new observation to 1 or 2 for this example. ii) Provide a suitable diagram and plot the straight line which separates the two allocation regions and label each region. iii) Would you classify a new observation = (23, 24) as male or female? Give your reasoning. iv) If the multivariate normal assumption looks suspect, e.g. the distributions have thicker tails than the multivariate normal distribution, briefly describe a possible strategy for obtaining an appropriate classification rule. [20 marks] MATH3030-E1 END
