Question-11

PCA

Let \(\Sigma\) be the covariance matrix of a dataset, and suppose \(\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_d\) are the eigenvalues of \(\Sigma\). If we keep only the first \(k\) principal components, what is the fraction of total variance preserved by these \(k\) components?

\(\cfrac{\lambda_1 + \lambda_2 + \dots + \lambda_k}{\lambda_1 + \lambda_2 + \dots + \lambda_d}\)
\(\cfrac{\lambda_1^2 + \lambda_2^2 + \dots + \lambda_k^2}{\lambda_1^2 + \lambda_2^2 + \dots + \lambda_d^2}\)
\(\cfrac{\lambda_k}{\lambda_1}\)
\(\cfrac{\lambda_k}{\lambda_1 + \lambda_2 + \dots + \lambda_d}\)

Answer

\(\cfrac{\lambda_1 + \lambda_2 + \dots + \lambda_k}{\lambda_1 + \lambda_2 + \dots + \lambda_d}\)
\(\cfrac{\lambda_1^2 + \lambda_2^2 + \dots + \lambda_k^2}{\lambda_1^2 + \lambda_2^2 + \dots + \lambda_d^2}\)
\(\cfrac{\lambda_k}{\lambda_1}\)
\(\cfrac{\lambda_k}{\lambda_1 + \lambda_2 + \dots + \lambda_d}\)

Solution