[GRASS-user] i.pca vs. r.covar/m.eigensystem/r.mapcalc

[Nikos Alexandris]

I wanted to check/verify some i.pca results so I calculated 2 principal
components using r.covar, m.eigensystem and r.mapcalc. The results are
not the same with i.pca's output.

(Maybe this post http://www.nabble.com/Eigen-vectors-tt15686112.html was
about the same issue)

# i.pca performed on (3) modis surface reflectance bands (2, 6 and 7  -
500m spatial resolution)
r.info pca.2007.242.500.b267.1 -h
[...]
Eigen vectors:
   ( -0.64 -0.65 -0.42 )
   ( -0.71 0.29 0.64 )
   ( 0.29 -0.70 0.65 )
[...]

# calculating "by hand" 
## (a) based on covariance
r.covar
map=MOD2007_242_500_sur_refl_b02,MOD2007_242_500_sur_refl_b06,MOD2007_242_500_sur_refl_b07 > covariance_b67

# eigenvalues and eigenvectors
(echo 3; cat covariance_b67) | m.eigensystem
E    193252.2225840657         0.0000000000    19.81
V         0.8480879794         0.0000000000
V        -0.0973632655         0.0000000000
V        -0.5621384304         0.0000000000
N         0.8297330080         0.0000000000
N        -0.0952560549         0.0000000000
N        -0.5499721988         0.0000000000
W       364.7544643092         0.0000000000
W       -41.8750018841         0.0000000000
W      -241.7703198653         0.0000000000

E    770286.8983235050         0.0000000000    78.97
V         0.5123041585         0.0000000000
V         0.8425333994         0.0000000000
V         0.6269758062         0.0000000000
N         0.4384249584         0.0000000000
N         0.7210319581         0.0000000000
N         0.5365598486         0.0000000000
W       384.7880047100         0.0000000000
W       632.8208355692         0.0000000000
W       470.9170625750         0.0000000000

E     11834.0908624288         0.0000000000     1.21
V         0.2892235465         0.0000000000
V        -0.5746367180         0.0000000000
V         0.5358742690         0.0000000000
N         0.3454369570         0.0000000000
N        -0.6863229556         0.0000000000
N         0.6400266474         0.0000000000
W        37.5782238381         0.0000000000
W       -74.6613734450         0.0000000000
W        69.6250477264         0.0000000000

## (b) based on correlation
r.covar -r
map=MOD2007_242_500_sur_refl_b02,MOD2007_242_500_sur_refl_b06,MOD2007_242_500_sur_refl_b07 > correlation_b267

# eigenvalues and eigenvectors
(echo 3; cat correlation_b267) | m.eigensystem
E         2.2858261608         0.0000000000    76.19
V        -0.5735822577         0.0000000000
V        -0.7657536915         0.0000000000
V        -0.6794534712         0.0000000000
N        -0.4887914931         0.0000000000
N        -0.6525548605         0.0000000000
N        -0.5790121159         0.0000000000
W        -0.7390013610         0.0000000000
W        -0.9865943596         0.0000000000
W        -0.8754054597         0.0000000000

E         0.6796466529         0.0000000000    22.65
V         0.8645087622         0.0000000000
V        -0.1090761065         0.0000000000
V        -0.6068722494         0.0000000000
N         0.8141383928         0.0000000000
N        -0.1027208167         0.0000000000
N        -0.5715130018         0.0000000000
W         0.6711812672         0.0000000000
W        -0.0846837448         0.0000000000
W        -0.4711592330         0.0000000000

E         0.0345271863         0.0000000000     1.15
V         0.2515640507         0.0000000000
V        -0.6024904816         0.0000000000
V         0.4666495170         0.0000000000
N         0.3134669897         0.0000000000
N        -0.7507466869         0.0000000000
N         0.5814790267         0.0000000000
W         0.0582468451         0.0000000000
W        -0.1394999392         0.0000000000
W         0.1080474816         0.0000000000

# range for i.pca's pc 1
r.info pca.2007.242.500.b267.1 -r
min=-7713.335449
max=-21.046232

# range for i.pca's pc 2
r.info pca.2007.242.500.b267.2 -r
min=-2313.784484
max=1838.939406

# calculating PC 1 and 2 "by hand"
# pc 1
r.mapcalc testpca1="(-0.64)*MOD2007_242_500_sur_refl_b02 +
(-0.65)*MOD2007_242_500_sur_refl_b06 +
(-0.42)*MOD2007_242_500_sur_refl_b07"
# range of testpca1
min=-7743.680000
max=-87.370000

# pc 2
r.mapcalc testpca2="(-0.71)*MOD2007_242_500_sur_refl_b02 +
0.29*MOD2007_242_500_sur_refl_b06 + 0.64*MOD2007_242_500_sur_refl_b07"
# range of testpca2
min=-2283.720000
max=1866.570000

Can someone explain which values (V or N or W) correspond to the
eigenvectors reported by i.pca? I don't think they coincide. But, as far
as I understand the PCA concept, they should or they should be close
enough.

Kind regards, Nikos