Question

Principal Component Analysis is performed on a 4-band IRS satellite image. The eigenvalues \( \mathbf{E} = [\lambda_{1,1}, \lambda_{2,2}, \lambda_{3,3}, \lambda_{4,4}] \) computed from the covariance matrix are 887.60, 75.20, 37.60 and 6.73, respectively. The percentage of total variance explained by the third principal component (\( \lambda_{3,3} \)) is __________ (rounded off to 2 decimal places).

Hint:

To find the contribution of a principal component, divide its eigenvalue by the total sum of eigenvalues and multiply by 100.

Answer 1

The percentage of total variance explained by the third principal component is calculated as:
\[ \frac{\lambda_{3,3}}{\sum_{i=1}^{4} \lambda_{i,i}} \times 100 \] Given:
\[ \lambda_{3,3} = 37.60, \quad \sum \lambda = 887.60 + 75.20 + 37.60 + 6.73 = 1007.13 \]

\[ \text{Percentage} = \frac{37.60}{1007.13} \times 100 \approx 3.732\% \]

Final Answer: \(\boxed{3.73\%}\)