## DESCRIPTION

r.covar outputs a covariance/correlation matrix for user-specified raster map layer(s). The output can be printed, or saved by redirecting output into a file.

The output is an N x N symmetric covariance (correlation) matrix, where N is the number of raster map layers specified on the command line. For example,

  r.covar map=layer.1,layer.2,layer.3

would produce a 3x3 matrix (values are example only):

         1.000000  0.914922  0.889581
         0.914922  1.000000  0.939452
         0.889581  0.939452  1.000000

## PRINCIPLE COMPONENTS

This module can be used as the first step of a principle components transformation. The covariance matrix would be input into a system which determines eigen values and eigen vectors. An NxN covariance matrix would result in N real eigen values and N eigen vectors (each composed of N real numbers). In the above example, the eigen values and corresponding eigen vectors for the covariance matrix are:

    component   eigen value               eigen vector
        1       1159.745202   < 0.691002    0.720528    0.480511 >
        2          5.970541   < 0.711939   -0.635820   -0.070394 >
        3        146.503197   < 0.226584    0.347470   -0.846873 >

The component corresponding to each vector can be produced using r.mapcalc as follows:

  r.mapcalc 'pc.1 = 0.691002*layer.1 + 0.720528*layer.2 + 0.480511*layer.3'
  r.mapcalc 'pc.2 = 0.711939*layer.1 - 0.635820*layer.2 - 0.070394*layer.3'
  r.mapcalc 'pc.3 = 0.226584*layer.1 + 0.347470*layer.2 - 0.846873*layer.3'

Note that based on the relative sizes of the eigen values, pc.1 will contain about 88% of the variance in the data set, pc.2 will contain about 1% of the variance in the data set, and pc.3 will contain about 11% of the variance in the data set. Also, note that the range of values produced in pc.1, pc.2, and pc.3 will not (in general) be the same as those for layer.1, layer.2, and layer.3. It may be necessary to rescale pc.1, pc.2 and pc.3 to the desired range (e.g. 0-255). This can be done with r.rescale.

## NOTES

The module m.eigensystem in src.contrib can be compiled and used to generate the eigen values and vectors.

## SEE ALSO

i.pca, m.eigensystem, r.mapcalc, r.rescale

## AUTHOR

Michael Shapiro, U.S. Army Construction Engineering Research Laboratory

Last changed: $Date: 2003/05/06 12:57:35 $