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hw3 - Arthur Kunkle ECE 5526 HW 3 Problem 1 The following...

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Arthur Kunkle ECE 5526 HW # 3
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Problem 1 The following commands were run to generate the 1000 values of the random variable X1 with the three covariance matrices in the problem description: >> N = 10000; >> sigma_1=[8000 0; 0 8000]; >> sigma_A=[8000 0; 0 8000]; >> sigma_B=[8000 0; 0 18500]; >> sigma_C=[8000 8400; 8400 18500]; >> X1A=randn(N,2)*sqrt(sigma_A)+repmat(mu,N,1); >> X1B=randn(N,2)*sqrt(sigma_B)+repmat(mu,N,1); >> X1C=randn(N,2)*sqrt(sigma_C)+repmat(mu,N,1); >> gausview(X1A,mu,sigma_A,'Sample X1_A'); >> gausview(X1B,mu,sigma_B,'Sample X1_B'); >> gausview(X1C,mu,sigma_C,'Sample X1_C'); Output 2-D PDF functions using “gausview”: The symmetry of the above PDF’s is a result of the values in each random process’ covariance matrix . The diagonal values of the 2x2 matrix correspond to the independent X and Y variances. When these are equal and the other values are zero, the sample values will be equally likely to occur, varying in the X and Y directions equally. The second PDF increased the value of the Y variance. This results in a wider spread of data values in the Y direction, however, the PDF contours are still symmetrical about both axes . Finally, the process with a fully populated, non-zero matrix exhibits variation that occurs along both directions. When both of the non-diagonal entries are equal, the variation appears to be along the X-Y diagonal, as show in Figure 3.
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Problem 2 The values of the third random process in Problem 1 were used to obtain the following values. The first N values of the sample data was used in the estimate calculations. The original values from the process were: Mean: [ 730 1090] Covariance: [8000 8400 ; 8400 18500] Points (N) Est. Mean Est. Covariance Mean Distance Covar. matrix norm 10000 [729.1 1087.0] [7724 8113 ; 8113 18259] 3.1336 545.8393 1000 [730.6 1085.7] [7282 7450 ; 7450 17229] 4.3337 1983.7 100 [729.8 1077.6] [7294 8236 ; 8236 17103] 12.4494 1434.2 10 [725.1 1051.9] [8241 13384 ; 13384 26446] 38.4163 10392 The most obvious and important trend is as N decreases, the distance measures tend to go up . Having a greater amount of sample data available will lead to estimated mean and covariance that will be much closer to the true values input to the process.
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Problem 3 The following are the computed joint log-likelihoods for the X_3 and N models: N_1 N_2 N_3 N_4 X3 Sigma Log- likelihood -1.2492e+005 -1.2248e+005 -1.1923e+005 -8.5911e+005 The following are the gausview outputs comparing the X_3 and N models: The above show that the N_4 does the worst job estimating the likelihood of X_3. This is due to the drastic difference in mean value.
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