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).
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Sort eigenvalues in descending order and accumulate until the running sum reaches 99% of the total.
Step 1: Recall the PCA variance-retention rule. Each eigenvalue $\lambda_i$ represents the variance of its principal component, ranked $\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_7$. The fraction retained by the first $k$ components is $\dfrac{\sum_{i=1}^{k}\lambda_i}{\sum_{i=1}^{7}\lambda_i}$, and we need the smallest $k$ for which this is at least 0.99. Step 2: Sum and sort. Sum of all 7 eigenvalues: $0.39+3.76+13.10+0.67+101.00+0.21+0.12=119.25$. Sorted descending: $101.00>13.10>3.76>0.67>0.39>0.21>0.12$. Step 3: Find the 99% cutoff value directly. \[ 0.99\times119.25=118.0575 \] so we need the smallest number of leading eigenvalues whose sum is at least $118.0575$. Step 4: Accumulate eigenvalues until the cutoff is crossed. \[ 101.00\ (k=1) \] \[ 101.00+13.10=114.10\ (k=2) \] \[ 114.10+3.76=117.86\ (k=3),\ \text{still}<118.0575 \] \[ 117.86+0.67=118.53\ (k=4),\ \text{now}\ge118.0575 \] The cumulative sum first exceeds the cutoff exactly when the 4th largest eigenvalue (0.67) is included. Step 5: Final answer. The minimum number of principal components needed to retain at least 99% of the total variance is 4.\[ \boxed{4} \]
