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       r.covar  - Outputs a covariance/correlation matrix for user-specified raster map layer(s).


       raster, statistics


       r.covar help
       r.covar [-rq] map=name[,name,...]  [--verbose]  [--quiet]

           Print correlation matrix

           Run quietly

           Verbose module output

           Quiet module output

           Name of input raster map(s)


       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.


       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).

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


       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
        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.


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


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

       Last changed: $Date: 2008-05-16 12:09:06 -0700 (Fri, 16 May 2008) $

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