m.eigensystem

NAME

m.eigensystem - Computes eigen values and eigen vectors for square matricies.
(GRASS Data Import/Processing Program)

SYNOPSIS

m.eigensystem < inputfile

DESCRIPTION

m.eigensystem determines the eigen values and eigen vectors for square matricies. The inputfile must have the following format: the first line contains an integer K which is the number of rows and columns in the matrix; the remainder of the file is the matrix, i.e., K lines, each containing K real numbers. For example:

          3
          462.876649   480.411218   281.758307
          480.411218   513.015646   278.914813
          281.758307   278.914813   336.326645

The output will be K groups of lines; each group will have the format:

          E   real part imaginary part   relative importance
          V   real part imaginary part
                   ... K lines ...
          N   real part imaginary part
                   ... K lines ...
          W   real part imaginary part
                   ... K lines ...

The E line is the eigen value. The V lines are the eigen vector associated with E. The N lines are the V vector normalized to have a magnitude of 1. The W lines are the N vector multiplied by the square root of the magnitude of the eigen value (E).

For the example input matrix, the output would be:

          E  1159.7452017844    0.0000000000   88.38
          V     0.6910021591    0.0000000000
          V     0.7205280412    0.0000000000
          V     0.4805108400    0.0000000000
          N     0.6236808478    0.0000000000
          N     0.6503301526    0.0000000000
          N     0.4336967751    0.0000000000
          W    21.2394712045    0.0000000000
          W    22.1470141296    0.0000000000
          W    14.7695575384    0.0000000000

          E     5.9705414972    0.0000000000    0.45
          V     0.7119385973    0.0000000000
          V    -0.6358200627    0.0000000000
          V    -0.0703936743    0.0000000000
          N     0.7438340890    0.0000000000
          N    -0.6643053754    0.0000000000
          N    -0.0735473745    0.0000000000
          W     1.8175356507    0.0000000000
          W    -1.6232096923    0.0000000000
          W    -0.1797107407    0.0000000000

          E   146.5031967184    0.0000000000   11.16
          V     0.2265837636    0.0000000000
          V     0.3474697082    0.0000000000
          V    -0.8468727535    0.0000000000
          N     0.2402770238    0.0000000000
          N     0.3684685345    0.0000000000
          N    -0.8980522763    0.0000000000
          W     2.9082771721    0.0000000000
          W     4.4598880523    0.0000000000
          W   -10.8698904856    0.0000000000

PROGRAM NOTES

The relative importance of the eigen value (E) is the ratio (percentage) of the eigen value to the sum of the eigen values. Note that the output is not sorted by relative importance.

In general, the solution to the eigen system results in complex numbers (with both real and imaginary parts). However, in the example above, since the input matrix is symmetric (i.e., inverting the rows and columns gives the same matrix) the eigen system has only real values (i.e., the imaginary part is zero). This fact makes it possible to use eigen vectors to perform principle component transformation of data sets. The covariance or correlation matrix of any data set is symmetric and thus has only real eigen values and vectors.

PRINCIPLE COMPONENTS

To perform principle component transformation on GRASS data layers, one would use r.covar to get the covariance (or correlation) matrix for a set of data layers, m.eigensystem to extract the related eigen vectors, and r.mapcalc to form the desired components. For example, to get the eigen vectors for 3 layers:

(echo 3; r.covar map.1,map.2,map.3) | m.eigensystem

Note that since m.covar only outputs the matrix, we must manually prepend the matrix size (3) using the echo command.

Then, using the W vector, new maps are created:

r.mapcalc 'pc.1 = 21.2395*map.1 + 22.1470*map.2 + 14.7696*map.3'
r.mapcalc 'pc.2 =  2.9083*map.1 +  4.4599*map.2  - 10.8699*map.3'
r.mapcalc 'pc.3 =  1.8175*map.1  -  1.6232*map.2 \-  0.1797*map.3'

NOTES

The source code for this program lives under /src.contrib/CERL/misc/m.eigensystem and requires a Fortran compiler.

AUTHOR

This code uses routines from the EISPACK system. The interface was coded by Michael Shapiro, U.S.Army Construction Engineering Research Laboratory