Question

The variance–covariance matrix for a 3-band image is given below (bands in the order 1, 2, 3). Which of the statement(s) is/are CORRECT? \[ \Sigma = \begin{bmatrix} 9 & 2 & 4 \\ 2 & 9 & -3 \\ 4 & -3 & 9 \end{bmatrix} \]

• A. The standard deviation of all bands is the same
• B. The bands 1 and 2 are positively correlated
• C. The bands 2 and 3 are positively correlated
• D. A line fitted to the scattergram between band 1 and band 3 will have a positive slope

Hint:

Sign of covariance $⇒$ sign of correlation {and} of the regression slope ($\beta=\text{cov}/\text{var}$). Diagonal entries of the covariance matrix are the variances.

Answer 1

The Correct Option is A

- From the diagonal of \( \Sigma \), \( \operatorname{var}(1) = \operatorname{var}(2) = \operatorname{var}(3) = 9 \Rightarrow \sigma_1 = \sigma_2 = \sigma_3 = \sqrt{9} = 3 \). Hence (A) is True.
- Covariance \( \operatorname{cov}(1,2) = 2>0 \Rightarrow \) correlation between bands 1 and 2 is positive \( \Rightarrow \) (B) True.
- \( \operatorname{cov}(2,3) = -3<0 \Rightarrow \) bands 2 and 3 are negatively correlated \( \Rightarrow \) (C) False.
- The (least-squares) regression slope of band 3 on band 1 is \[ \beta_{3|1} = \frac{\operatorname{cov}(1,3)}{\operatorname{var}(1)} = \frac{4}{9}>0, \] and the slope of band 1 on band 3 is \( \beta_{1|3} = \frac{4}{9}>0 \) as well. Hence, the fitted line in the 1–3 scattergram has a positive slope \( \Rightarrow \) (D) True.