15:50 Tutorial Sheet 2 -... where a = (a1, a2, . . ., ap). i) Show that the mean of v is v = aTy [4 marks] 6PAM1036 - Multivariate Statistics -Tutorials ii) Using the results of the previous question and given that $2 = Liel (vi - v)2 n - 1 show that $2 = a Sa where S is the sample covariance matrix of the random vector y. [6 marks] [Hint: Try writing all the v variables in terms of the corresponding vectors y.] Question 3. (To be done using 'R'). In this question, you are required to import BostonHousing data set from mlbench package in 'R', create a subset of only 3 variables: . per capita crime rate by town . average number of rooms per dwelling . weighted distances to five Boston employment centres, and then calculate certain basic multivariate statistics corresponding to these 3 columns from the data set, outlined below. Once you have successfully imported your data as a data frame: i) Calculate the sample mean vector y; [2 marks] ii) Calculate the sample covariance matrix S. Explain what the (2, 3) element of this matrix describes and interpret its value; 13 marks] iii) Calculate the diagonal matrix Ds and its inverse D -1; [3 marks] iv) Calculate the sample correlation matrix R. Again, explain what the (3, 2) element of this matrix de- scribes and interpret its value; 13 marks] v) Write down the form of the (3 x 3) matrix of coefficients, denoted A, that considers the differences between each of the three variables; [Hint: Use the theory of linear combinations] [2 marks] vi) Calculate the sample mean vector v for the difference between each of the variables; [2 marks] vii) Calculate the sample covariance matrix S. for the differences of the three variables; [2 marks ] viii) Calculate the sample correlation matrix Ro for the differences of the three variables. [3 marks] [For this question, you only need to submit the solution. You do not have to submit any code.] N 2 n 60 Dashboard Calendar To-do Notifications Inbox15:50 Tutorial Sheet 2 -... 6PAM1036 - Multivariate Statistics -Tutorials Multivariate Statistics - Coursework 1/Tutorial Sheet 2 Multivariate Data (To be submitted online by 18 February) Question 1. Consider a p-dimensional response variable y, containing p variables, with n observation vectors y1, . .., yn. The sample mean vector of these observation vectors is denoted by y = (1) =1 yi. a) Given that the sample covariance matrix, S, is defined by S = n- 12 (9 - 9) (1 - 3) T, i) Show that [4 marks] ii) Using the above result, show that S can alternatively be defined as where Y is the data matrix, I is the identity matrix and J is a matrix of 1's. [6 marks] [Hint: Consider re-writing the mean vector y in terms of the data matrix (see lecture notes).] b) Consider the following (3 x 3) data matrix, containing three observations of three variables y1, y2 and y3: Y = 5 3 5 Calculate i) The sample covariance matrix S (by hand); [4 marks] ii) The sample correlation matrix R (by hand) using the equation R = D, 'SD, ', and comment on each of the three elements in the first column of this matrix. [6 marks] [Hint: Recall how to find the inverse of a diagonal matrix.] Question 2. Consider a p-dimensional response variable y, containing p variables, with n observation vectors y1, . . ., Un. The sample mean vector of these observation vectors is denoted by y = (1) >- yi Consider a linear combination v of the observation vector y, defined as v = aly/1 + a232 + . .. + apyp = a y where aT = (a1, a2, . .., ap). i) Show that the mean of vis v = aTy [4 marks] 6PAM1036 - Multivariate Statistics -Tutorials ii) Using the results of the previous question and given that n 60 0= Dashboard Calendar To-do Notifications Inbox