Principal Component Analysis Exercises: 1. Followings are mean vector, variance vector and sample correlation matrix for...

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Principal Component Analysis Exercises: 1. Followings are mean vector, variance vector and sample correlation matrix for four weather random variables based on observations for 11 years. x=(138.0, 17.8, 74.4, 1242.4), s' = (1973, 1.6, 1345, 52915) 1.000 -0.087 0.491 -0.2391 R = -0.087 1.000 -0.624 0.738 0.491 -0.624 1.000 -0.808 -0.239 0.738 -0.808 1.000 Eigen value and eigen vector matrix E for matrix R is '= (2.593, 0.978, 0.285, 0.143) 0.291 0.871 -0.332 -0.2141 -0.506 0.425 E = 0.742 -0.111 0.577 0.136 0.418 0.688 L-0.571 0.205 -0.405 0.685 Four weather random variables are - rainfall at the month of November and December, X2 - average temperature in July, X3 rainfall in July, and X4 - average radiation in July. Use the information given above: (a) Write down the equations for first and second principal components. (b) Determine variance for the two principal components and percentage of variation explained. (c) Determine 95% confidence interval for the two variances above. (d) Perform hypothesis test to prove that only first eigen value is different. (e) Determine the percentage of variance of the first principal component that is caused by each of the four weather random variables. Principal Component Analysis Exercises: 1. Followings are mean vector, variance vector and sample correlation matrix for four weather random variables based on observations for 11 years. x=(138.0, 17.8, 74.4, 1242.4), s' = (1973, 1.6, 1345, 52915) 1.000 -0.087 0.491 -0.2391 R = -0.087 1.000 -0.624 0.738 0.491 -0.624 1.000 -0.808 -0.239 0.738 -0.808 1.000 Eigen value and eigen vector matrix E for matrix R is '= (2.593, 0.978, 0.285, 0.143) 0.291 0.871 -0.332 -0.2141 -0.506 0.425 E = 0.742 -0.111 0.577 0.136 0.418 0.688 L-0.571 0.205 -0.405 0.685 Four weather random variables are - rainfall at the month of November and December, X2 - average temperature in July, X3 rainfall in July, and X4 - average radiation in July. Use the information given above: (a) Write down the equations for first and second principal components. (b) Determine variance for the two principal components and percentage of variation explained. (c) Determine 95% confidence interval for the two variances above. (d) Perform hypothesis test to prove that only first eigen value is different. (e) Determine the percentage of variance of the first principal component that is caused by each of the four weather random variables.