Question

Principal Component Analysis (PCA) is applied on reflectance values obtained from seven bands. The following are the resultant eigenvalues:
0.39, 3.76, 13.10, 0.67, 101.00, 0.21, 0.12
The minimum number of principal components required to achieve at least 99% of total variance explained by seven bands is _____ (Answer in integer).

Hint:

Sort eigenvalues in descending order and accumulate until the running sum reaches 99% of the total.

Answer 1

Correct Answer: 4

Step 1: Compute the total variance (sum of all eigenvalues). \[ \text{Total}=0.39+3.76+13.10+0.67+101.00+0.21+0.12=119.25 \]
Step 2: Sort the eigenvalues in descending order, since each principal component's variance contribution equals its eigenvalue and the components are ranked from largest to smallest eigenvalue. \[ 101.00,\ 13.10,\ 3.76,\ 0.67,\ 0.39,\ 0.21,\ 0.12 \]
Step 3: Compute the running cumulative percentage of variance explained. \[ PC_1: \dfrac{101.00}{119.25}\times100=84.70\% \] \[ PC_1+PC_2: \dfrac{114.10}{119.25}\times100=95.68\% \] \[ PC_1+PC_2+PC_3: \dfrac{117.86}{119.25}\times100=98.83\% \] \[ PC_1+PC_2+PC_3+PC_4: \dfrac{118.53}{119.25}\times100=99.40\% \]
Step 4: Compare each cumulative percentage against the 99% requirement.
After 3 components the cumulative variance explained is only 98.83%, still below 99%. After including the 4th component (eigenvalue 0.67), the cumulative variance rises to 99.40%, which is at least 99%. So 4 is the smallest number of components for which the cumulative percentage first reaches or exceeds 99%.
Step 5: Conclusion.
The minimum number of principal components required is 4.\[ \boxed{4} \]