A centered dataset with n = 100 observations and p = 6 variables was analysed to reduce its dimensionality. As part of Principal Component Analysis, the following variance-covariance matrix was generated 0.138 -0.101 -0.029 1 0.156 0.156 1 -0.108 0.162 0.03 -0.072 -0.002 0.138 -0.108 1 0.164 -0.122 0.072 -0.101 -0.029 -0.072 -0.002 0.162 0.164 1 -0.089 -0.003 0.03 -0.122 -0.089 1 0.072 -0.003 -0.02 -0.02 1 A) Compute and write the numerical value of the eigenvalue 4 of Σ. This eigenvalue is located in the position (4, 4) of the matrix A and is simultaneously the sample variance of the score PC4: B) Compute and write the percentage of total variability explained by the Principal component PC4. The number you write should be between 0 and 100 and you should include decimals in your answer. C) As seen in lectures, the eigenvalue X4 is related to d4, one singular eigenvalue of the data matrix X. Compute and write the value of d4. D) A threshold of total variability explained has been set at 70%. How many principal components must you select? Write your answer.

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