Suggested languages for you:

Americas

Europe

Q7.5-5E

Expert-verified
Found in: Page 395

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Question: [M] A Landsat image with three spectral components was made of Homestead Air Force Base in Florida (after the base was hit by Hurricane Andrew in 1992). The covariance matrix of the data is shown below. Find the first principal component of the data, and compute the percentage of the total variance that is contained in this component.$S = \left[ {\begin{array}{*{20}{c}}{164.12}&{32.73}&{81.04}\\{32.73}&{539.44}&{249.13}\\{81.04}&{246.13}&{189.11}\end{array}} \right]$

The required percentage is $$75.8956\%$$.

See the step by step solution

## Step 1: Mean Deviation form and Covariance Matrix

The Mean Deviation form of any $$p \times N$$ is given by:

$$B = \left[ {\begin{array}{*{20}{c}}{{{\hat X}_1}}&{{{\hat X}_2}}&{........}&{{{\hat X}_N}}\end{array}} \right]$$

Whose $$p \times p$$ covariance matrix is:

$$S = \frac{1}{{N - 1}}B{B^T}$$

## Step 2: The Variance

From question, the covariance matrix and the maximum eigenvalue we haveis:

$$\begin{array}{l}S = \left[ {\begin{array}{*{20}{c}}{164.12}&{32.73}&{81.04}\\{32.73}&{539.44}&{249.13}\\{81.04}&{246.13}&{189.11}\end{array}} \right]\\{\lambda _1} = 677.497\end{array}$$

The respective unit vector is:

$${u_1} = \left[ {\begin{array}{*{20}{c}}{0.129554}\\{0.874423}\\{0.467547}\end{array}} \right]$$

Now, the percentage of change in variance can be obtained as:

$\begin{array}{c}\Delta = \frac{{{\lambda _1}}}{{tr\left( S \right)}} \times 100\\ = \frac{{677.4978}}{{164.12 + 539.44 + 189.11}} \times 100\\ = 75.8956\% \end{array}$

Hence, this is the required answer.