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Q7.5-5E

Expert-verifiedFound in: Page 395

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 \(75.8956\% \).

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:

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

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.

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