关键词 > STAT3064/STAT4067
STAT3064/STAT4067 Semester 2, 2022 Assignment 1
发布时间：2022-08-22
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
STAT3064/STAT4067
Semester 2, 2022
Assignment 1
Assignment Questions
Data required for this assignment can be found in the Data Sets folder .
1. (a) What does reproducible calculations and reproducible simulations refer to and when or why should the calculations/simulations be reproducible?
(b) When is it a sensible strategy to use the Gaussian model in a simulation? When
is it not and why? (Hint. Your answer could contain an illustrative example .)
(c) Consider a random sample of observations x = [X1﹒ . . . ﹒ Xn] . Why would we expect that, typically, the variables of the sample x are correlated, but the principal component scores obtained from these variables are not?
2. Consider the aircraft data with the logged variables as in Question 2 of Computer Lab 1 . Divide the data into two period groups consisting of the same number of observations . We are interested in comparing changes that occur over time .
(a) Show smoothed histograms of logLength and logPower separately for the two
periods . Comment on the shapes of the histograms and how the change over time affects this shape .
(b) Construct contour plots of the 2D smoothed histograms of the pairs (logPower,
logWeight) and (logSpeed, logLength) . Describe the shapes of the density plots and discuss how they change over time .
(c) For which pair of variables would you expect the largest change in correlation or shape of their density over time and why?
3. Consider the aircraft data of Q2 of this assignment .
(a) Separately for the two periods selected in Q2, carry out a principal component
analysis using prcomp based on the logged data (without scaling) .
(b) Show eigenvalue plots for each of the two periods . Interpret the results .
(c) Show score plots of the first three PCs for each period . Comment on the results .
(d) Which logged variable contributes most to PC1 for each period? Does this change across the two periods? Comment on the results .
(e) Based on your analysis, discuss the main changes that have occurred over time .
4. The data set ass2pop .csv is available in the LMS folder ‘Data sets’ . It contains the means and covariance matrices corresponding to two populations . The first and second column of ass2pop .csv are the means μ 1 and μ2 of the first and second population respectively; columns 3:22 correspond to the covariance matrix Σ 1 of the first population, and the remaining columns correspond to the covariance matrix Σ 2 of the second populations . In this question we generate random samples from these populations as described below .
(a) Read the data into R . What is the dimension of the covariance matrix Σ 1?
(b) Generate 250 random samples from the Gaussian distribution N(μ1 ﹒ Σ 1) and 150
samples from the Gaussian distribution N(μ2 ﹒ Σ2) . What is the size of the data matrix consisting of these random samples? Calculate the sample covariance matrix S of the random samples, and find eigenvalues of S . Save the vector of eigenvalues into a file for later analysis .
(c) Repeat part (b) another 49 times, so you have a total of 50 vectors of eigenvalues .
(d) Calculate the mean vector of eigenvalues over the 50 repetitions and list/print this mean vector .
(e) Display the 50 vectors of eigenvalues and their mean vector in an eigenvalue
or scree plot . How similar are these eigenvalue plots? Where does the largest deviation from the mean vector occur?
(f) Repeat parts (b) to (e) with 250 samples from the t-distribution t10 (μ1 ﹒ Σ01)
and 150 samples from t-distribution t3 (μ2 ﹒ Σ02) . (Hint. Σ0k is the scale matrix which is obtained from the covariance matrix Σ k using the following relationship Σk = 
Σ0k, with v the degree of freedom of the t-distribution and k = 1 and 2 here .)
(g) Compare the results of the two different simulations and comment on interesting
findings and differences between them . Why do we expect differences between the pairs of simulations?
