Principal Component Variance Explained

The variances of the principal components ($Y_1, Y_2, Y_3, Y_4$) correspond to the eigenvalues ($\lambda_i$) of the dispersion matrix $\Sigma$, typically sorted in descending order.

• Given eigenvalues: $\lambda_1 = 6, \lambda_2 = 3, \lambda_3 = 2, \lambda_4 = 1$.
• These represent the variances of $Y_1, Y_2, Y_3, Y_4$ respectively.
• The total variation in the data is the sum of the eigenvalues: $6 + 3 + 2 + 1 = 12$.

Statement A: Variation by First Two Components

Calculate the percentage of total variation explained by the first two principal components ($Y_1, Y_2$).

• Variation explained by $Y_1$ and $Y_2$ = $\lambda_1 + \lambda_2 = 6 + 3 = 9$.
• Percentage of variation = $\frac{9}{12} \times 100\% = 75\%$.
• The statement claims this percentage is $\le 95\%$. Since $75\% \le 95\%$, statement A is correct.

Statement B: Variation by First Three Components

Calculate the percentage of total variation explained by the first three principal components ($Y_1, Y_2, Y_3$).

• Variation explained by $Y_1, Y_2,$ and $Y_3$ = $\lambda_1 + \lambda_2 + \lambda_3 = 6 + 3 + 2 = 11$.
• Percentage of variation = $\frac{11}{12} \times 100\% \approx 91.67\%$.
• The statement claims this percentage is $\ge 95\%$. Since $91.67\% < 95\%$, statement B is incorrect.

Statement C: Independence of Components

Principal components ($Y_i$) derived from a Multivariate Normal distribution are uncorrelated. For a Multivariate Normal distribution, uncorrelated components are also independent.

• Therefore, $Y_1, Y_2, Y_3, Y_4$ are independent. Statement C is correct.

Statement D: Identical Distribution of Components

The principal components $Y_i$ follow Normal distributions, but their variances are the eigenvalues $\lambda_i$.

• The variances are $6, 3, 2, 1$. Since these variances are different, the components $Y_1, Y_2, Y_3, Y_4$ do not have identical distributions. Statement D is incorrect.

Summary of Correct Statements

Based on the analysis, statements A and C are correct.