[GRASS-user] Calculating eigen values and % varianceexplainedafter PCA analysis

[Hamish]

Nikos wrote:
> It should be the case with i.pca as well since eigen_VALUES_ (=represent
> the variances of the original dimensions that are "kept" in each
> component) are important for the interpretation of what exactly are each
> of the components. But, i.pca just does not report the eigen_VALUES_.
> 
> At some point some C-expert needs to have a look in the code (i.pca) and
> correct the "bug" which does not let the eigen_VALUES_ from being
> printed.

done in devbr6 (6.5svn) please test, I'm not a multivariate stats guru
and may have done something dumb so didn't port to other branches yet.

I changed the i.pca output to be like:

Eigen (vectors) and values:
PC1 ( -0.63 -0.65 -0.43 ) 88.07
PC2 ( 0.23 0.37 -0.90 ) 11.48
PC3 ( 0.75 -0.66 -0.08 ) 0.45

As it was previously sent to stderr via G_message() I don't feel bad about
breaking output text compatibility. I wanted to add "%" to the values but
due to the sprintf()+strcat() method in the code that was a pain, so I
didn't.


> >  If this is the case then both methods still differ significantly. Is
> > this possible, and which should I use.
> 
> Please have a look at my comments/questions in link [2].
> i.pca follows the "SVD" method. You performed the non-standartised PCA
> using the covariance matrix. Note that you can use also the
> standartised method by using the correlation matrix.

does the r.mapcalc command at the end of the m.eigensystem help page*
do that conversion, or ...? ie how can we test these against each other?
how to do the standardized method?

[*] (is "\-" there a typo or some old mapcalc syntax?)

also ISTR somebody (Dylan?) doing a comparison with the R-stats interface.


It would be nice to run tests using the Spearfish imagery dataset. After
my own tests I noticed it matched what was used in the m.eigensystem help
page.


my results follow.

Hamish


----------------------------
#Spearfish imagery sample dataset
g.region rast=spot.ms.1


# 'by-hand-method'
G65> echo "3" > test_m.eigensystem    # number of input maps
G65> r.covar map=spot.ms.1,spot.ms.2,spot.ms.3 >> test_m.eigensystem

G65> cat test_m.eigensystem
3
462.876649 480.411218 281.758307 
480.411218 513.015646 278.914813 
281.758307 278.914813 336.326645 


G65> m.eigensystem < test_m.eigensystem
-----
C The output is N sets of values. One E line and N V W lines
C
C  E   real  imaginary   percent-importance
C  V   real  imaginary
C  N   real  imaginary
C  W   real  imaginary
C      ...
C
C where E is the eigen value (and it relative importance)
C and   V are the eigenvector for this eigenvalue.
C       N are the normalized eigenvector for this eigenvalue.
C       W are the N vector multiplied by the square root of the
C         magnitude of the eigen value (E). 
-----

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


# 'all-in-one method' using r.covar+m.eigensystem:
G65> (echo 3; r.covar spot.ms.1,spot.ms.2,spot.ms.3 ) | m.eigensystem

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'

(is "\-" above a typo or some old mapcalc syntax?)

which look highly similar (but not identical) to i.pca output maps.
(after 'r.colors color=grey')



# 'automatic method'
imagery60:G6.5svn> i.pca in=spot.ms.1,spot.ms.2,spot.ms.3 out=spot_pca

Eigen (vectors) and values:
PC1 ( -0.63 -0.65 -0.43 ) 88.07
PC2 ( 0.23 0.37 -0.90 ) 11.48
PC3 ( 0.75 -0.66 -0.08 ) 0.45