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(a) Determine the population principal components Y1, Y2 and Y3 for the covariance matrix (b) Calculate the proportion of the total population variance explained by the first principal component.

INSTRUCTIONS TO CANDIDATES
1. There is a covariance matrix of X as follows:

 8 0 1 Σ = [0 8 3] 1 3 5
• Determine the population principal components Y1, Y2 and Y3 for the covariance matrix
• Calculate the proportion of the total population variance explained by the first principal
• Calculation of correlation coefficient matrix of X

(d) Compute the correlations ππ1,π1, ππ1,π2, ππ2,π1

1. There is a correlation coefficient matrix of X as follows:

 1 0.63 0.45 ρ = [0.63 1 0.35] 0.45 0.35 1
• Calculate the factor
• Calculate communalities β2, π = 1, 2, 3
• What proportion of the total population variance is explained by the first common factor?

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