and p = 6 variables was analysed to reduce its dimensionality. As part of Principal Component Analysis, the following varia 399.019 49.66 -1.793 49.66 38.529 -6.952 -1.793 -6.952 36.051 1.7 8.333 16.583 1.733 2.409 7.873 12.142 -4.986 2.841 1.7 1.733 12.142 37.132 -4.093 -0.04 8.333 2.409 -4.986 -4.093 41.277 -3.757 16.583 7.873 2.841 -0.04 -3.757 45.062, the eigenvalue ₁ of Σ. This eigenvalue is located in the position (4, 4) of the matrix A and is simultaneously the sample v variability explained by the Principal component PC4. The number you write should be between 0 and 100 and you should i ated to da, one singular eigenvalue of the data matrix X. Compute and write the value of d4. s been set at 80\%. How many principal components must you select? Write your answer.

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A centered dataset with n = 85 observations and p = 6 variables was analysed to reduce its dimensionality. As part of Principal Component Analysis, the following variance-covariance matrix Σ was
generated
399.019 49.66 -1.793 1.7 8.333 16.583
49.66 38.529 -6.952 1.733 2.409 7.873
-1.793 -6.952 36.051 12.142 -4.986 2.841
1.7 1.733 12.142 37.132 -4.093
8.333 2.409 -4.986 -4.093 41.277 -3.757
16.583 7.873 2.841 -0.04 -3.757 45.062
-0.04
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 X 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 80\%. How many principal components must you select? Write your answer.
Transcribed Image Text:A centered dataset with n = 85 observations and p = 6 variables was analysed to reduce its dimensionality. As part of Principal Component Analysis, the following variance-covariance matrix Σ was generated 399.019 49.66 -1.793 1.7 8.333 16.583 49.66 38.529 -6.952 1.733 2.409 7.873 -1.793 -6.952 36.051 12.142 -4.986 2.841 1.7 1.733 12.142 37.132 -4.093 8.333 2.409 -4.986 -4.093 41.277 -3.757 16.583 7.873 2.841 -0.04 -3.757 45.062 -0.04 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 X 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 80\%. How many principal components must you select? Write your answer.
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