[GRASS-dev] PCA question

[Markus Metz]

An inverse PCA can be regarded as the inverse of a transformation
using matrix notation. PC scores are calculated with

b = A a

with A being the transformation matrix composed of the Eigenvectors, a
being the vector of the original values and b the PC scores. What you
now need is inverse of A, A^-1. The original values can then be
retrieved with

A^-1 b = a

A^-1 is the inverse of the transformation matrix A which you can get
in R with solve(A).

For a PCA, the original values are usually shifted to the mean and
optionally scaled to stddev before computing the Eigenvectors. The
mean shift is always performed by i.pca, scaling is optional. That
means that A^-1 b gives the original values shifted to the mean and
maybe scaled, and the mean of each original band needs to be added to
get the original values used as input to i.pca. With scaling applied,
the shifted values need to be multiplied by the stddev for each
original band.

HTH,

Markus M


On Tue, Jun 19, 2012 at 12:46 AM, Michael Barton <Michael.Barton at asu.edu> wrote:
> 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
>
>
>
>
>
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