10.8 Principal Component Analysis

10.8 Principal Component Analysis

Principal component analysis is used to reduce the dimensions of measured variables (p dimension) to the representative principal components (m dimension , m

Let the measured p dimensional variables be {xi } i = 1,p, the principal components{zk } k = 1, m can be expressed as the linear combination as follows.

zk = a1k x1 + a2k x2 + ...... + apk xp

The coefficients (a1k - apk ) are determined under the following constrains.

(1) aik = 1
(2) Variance zk should be maximum
(3) zk and z k+1 should be independent of each other

The solution of the above problem can be obtained by determining the unique values and the unique vectors which correspond to the variance and vector of the principal components respectively.

The unique value represents the contribution ratio which indicates how much percentage the principal component represents of the total tendency of the variables. The accumulative contribution ratio percentage all the principal components represent of the total tendency of the variables. Using an accumulative contribution ratio of 80 - 90 percent, will indicate how many principal components should be adopted to effectively represent the major variations in the image data.

Graphically speaking, the first principal component for example in the case of two dimensional variables (see Figure 10.8.1) will be the principal axis which gives the maximum variance. The principal component analysis can be used for the following applications.

(1) Effective classification of land use with multi-band data
(2) Color representation or visual interpretation with multi-band data
(3) Change detection with multi-temporal data

In the case of multi-band data with more than four bands, all bands cannot be assigned to R, G or B at the same time. However the first three principal components can represent up to five spectral variables with little information loss.

Figure 10.8.2 show the principal components and their color composite of Landsat TM (6 bands). Generally the first principal component corresponds to the total radiance (brightness), while the second principal component represents the vegetation activity (greenness).