IAI Actuarial Core Principles · Actuarial Statistics · Exploratory data analysis
A covariance matrix for two variables X and Y has Var(X)=9, Var(Y)=16 and Cov(X,Y)=0. PCA is performed on this covariance matrix. What is the first principal component's variance and its dominant loading?
The first component has variance 16 with loadings (0, 1). The matrix is diagonal, so its eigenvalues are 9 and 16, and the larger belongs to Y's axis.
- AVariance 25, loadings (0.6, 0.8)
- BVariance 16, loadings (0, 1)Correct
- CVariance 16, loadings (1, 0)
- DVariance 9, loadings (1, 0)
- Variance 12.5, loadings (0.707, 0.707)
Explanation
With zero covariance the matrix is diagonal, so the eigenvalues are 9 and 16 and the eigenvectors are the axes. The largest eigenvalue is 16 with eigenvector (0,1), i.e. Y. Variance 25 is wrongly the sum of the eigenvalues.
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