[GRASS-dev] PCA question

[Michael Barton]

The constant (i.e., the band mean) was in the pca primer that someone sent me a link to in this discussion. Using the Eigenvectors resulting from i.pca, I can achieve the results of i.pca using my formula below. This is essentially the same as your formula minus the constant--which doesn't really make much (of any) difference in the final result.

However, my question is about performing an *inverse pca*--getting back to the original values from the principal components maps. The idea of pca sharpening is that you perform a pca, then do an inverse pca substituting the pan band for pc1 to enhance the resolution. The equations I show below seem to work, but what I've read indicates that it is not possible to *exactly* get the original values back; you can only approximate them.

Michael


On Jun 17, 2012, at 10:48 AM, Duccio Rocchini wrote:

> Dear all,
> first, sorry for the delay...
> Here are my 2 cents to be added to the discussion. I re-took in my
> hands the John Jensen book.
> Accordingly
> 
> new brightness values1,1,1 = a1,1*BV1,1,1  +a2,1*BV1,1,2..... + an,1*BV1,1,m
> 
> where a=eigenvector and BV=original brightness value.
> 
> I found no evidence for the mean term: "- ((b1+b2+b3)/3)"
> 
> Michael: do you have a proof/reference for that?
> 
> P.S. thanks for involving me in this discussion which is really stimulating!
> 
> Duccio
> 
> 2012/6/7 Michael Barton <michael.barton at asu.edu>:
>> 
>> I think I've figured it out.
>> 
>> If (ev1-1, ev1-2, ev1-3) are the eigenvectors of the first principal component for 3 imagery bands (b1, b2, b3), the corresponding factor scores of the PC1, PC2, and PC3 maps (fs1, fs2, fs3) are calculated as:
>> 
>> fs1 = (ev1-1*b1) + (ev1-2*b2) + (ev1-3*b3) - ((b1+b2+b3)/3)
>> fs2 = (ev2-1*b1) + (ev2-2*b2) + (ev2-3*b3) - ((b1+b2+b3)/3)
>> fs3 = (ev3-1*b1) + (ev3-2*b2) + (ev3-3*b3) - ((b1+b2+b3)/3)
>> 
>> So to do an inverse PCA on the same data you need to do the following:
>> 
>> b1' = (fs1/ev1-1) + (fs2/ev2-1) + (fs3/ev3-1)
>> b2' = (fs1/ev1-2) + (fs2/ev2-2) + (fs3/ev3-2)
>> b3' = (fs1/ev1-3) + (fs2/ev2-3) + (fs3/ev3-3)
>> 
>> Adding the constant back on doesn't really seem to matter because you need to rescale b1' to b1, b2' to b2, and b3' to b3 anyway.
>> 
>> Michael
>> 
>> On Jun 7, 2012, at 1:55 AM, Markus Neteler wrote:
>> 
>>> Hi Duccio,
>>> 
>>> On Wed, Jun 6, 2012 at 11:39 PM, Michael Barton <michael.barton at asu.edu> wrote:
>>>> On Jun 6, 2012, at 2:20 PM, Markus Neteler wrote:
>>>>> On Wed, Jun 6, 2012 at 5:09 PM, Michael Barton <michael.barton at asu.edu> wrote:
>>> ...
>>>>>> I'm working on a pan sharpening script that will combine your i.fusion.brovey with options to do other pan sharpening methods. An IHS transformation is easy. I think that a PCA sharpening is doable too if I can find an equation to rotate the PCA channels back into unrotated space--since i.pca does provide the eigenvectors.
>>>>> 
>>>>> Maybe there is material in (see m.eigenvector)
>>>>> http://grass.osgeo.org/wiki/Principal_Components_Analysis
>>>> 
>>>> This has a lot of good information and ALMOST has what I need. Everything I read suggests that there is a straightforward way to get the original values from the factor scores if you have the eigenvectors. But no one I've yet found provides the equation or algorithm to do it.
>>> 
>>> @Duccio: any idea about this by chance?
>>> 
>>> thanks
>>> Markus
>> 
>> _____________________
>> C. Michael Barton
>> Visiting Scientist, Integrated Science Program
>> National Center for Atmospheric Research &
>> University Corporation for Atmospheric Research
>> 303-497-2889 (voice)
>> 
>> Director, Center for Social Dynamics & Complexity
>> Professor of Anthropology, School of Human Evolution & Social Change
>> Arizona State University
>> www: http://www.public.asu.edu/~cmbarton, http://csdc.asu.edu
>> 
> 
> 
> 
> -- 
> Duccio Rocchini, PhD
> 
> http://gis.cri.fmach.it/rocchini/
> 
> Fondazione Edmund Mach
> Research and Innovation Centre
> Department of Biodiversity and Molecular Ecology
> GIS and Remote Sensing Unit
> Via Mach 1, 38010 San Michele all'Adige (TN) - Italy
> Phone +39 0461 615 570
> ducciorocchini at gmail.com
> duccio.rocchini at fmach.it
> skype: duccio.rocchini

_____________________
C. Michael Barton
Visiting Scientist, Integrated Science Program
National Center for Atmospheric Research &
University Consortium for Atmospheric Research
303-497-2889 (voice)

Director, Center for Social Dynamics & Complexity 
Professor of Anthropology, School of Human Evolution & Social Change
Arizona State University
www: http://www.public.asu.edu/~cmbarton, http://csdc.asu.edu