You are viewing a plain text version of this content. The canonical link for it is here.
Posted to user@commons.apache.org by andrea antonello <an...@gmail.com> on 2013/11/09 10:38:16 UTC
[math] eigenvector doubts and issues
Dear all,
I have a doubt about using the eigenvector part of the library.
I created a small dataset to represent 3d coordinates in a cartesian plane:
double[] x = {1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3};
double[] y = {0.5, 1, 1.5, 2, 0.5, 1, 1.5, 2, 0.5, 1, 1.5, 2};
double[] z = {1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2};
The datasset represents a step from z value 1 to 2 on a regular grid
(with a diagonali trend).
I would expect to gain from the eigenvector with lowest eigenvalue a
line splitting this particular set in a quite clean way the higher z
points from the lower ones.
So I calculate the covariance matrix which results in:
0.7272727272727273 0.0 0.18181818181818182
0.0 0.3409090909090909 0.22727272727272727
0.18181818181818182 0.22727272727272727 0.2727272727272727
and then I simply calculate the eigenvector/values which result in:
eigenVal: 0.8056498828134406, eigenVect: [0.9015723557614027,
0.19005937823202243, 0.38864472217295326]
eigenVal: 0.4874287594020183, eigenVect: [-0.37995167578226796,
0.7774478202831089, 0.5012101463530935]
eigenVal: 0.04783044869363171, eigenVect: [-0.20689130333844696,
-0.5995434258526233, 0.773138842071603]
doing exactly the same thing with Jama results in:
eigenVal: 0.8056498828134406, eigenVect: [-0.7731388420716028,
0.5012101463530931, -0.38864472217295326]
eigenVal: 0.48742875940201863, eigenVect: [0.5995434258526229,
0.7774478202831089, -0.1900593782320223]
eigenVal: 0.0478304486936319, eigenVect: [0.20689130333844694,
-0.37995167578226785, -0.9015723557614027]
In fact if I use Jama's eigenvector with lowest eigenvalue, I am able
to construct a line of slope y =
(0.206891303338447/-0.379951675782268) *x, which splits my dataset the
way I would like to have it.
The same doesn't apply to the result of the apache commons math lib,
which seems to be reflected on the secondary diagonal.
Since I am no expert in this field, I might be doing somthing really
wrong. If someone could give me a hint, it would be greatly
appreciated.
Best regards,
Andrea
---------------------------------------------------------------------
To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
For additional commands, e-mail: user-help@commons.apache.org
Re: [math] eigenvector doubts and issues
Posted by andrea antonello <an...@gmail.com>.
Hi Thomas,
> are you sure that your eigenvectors are drawn correctly?
I assume yes. I used the eigenvectors with the highest and lowest
eigenvalue, since in the simpler 3d case it was giving me the plane
exactly cutting through the dataset as I wished it. But you are most
probably right in using a plane that connects the two lower eigenvalue
eigenvectors. I was using the wrong one for this purpose I guess.
> I tried to reproduce your results with geogebra, and in my case the
> plane defined by the two eigenvectors with the least eigenvalue seems to
> split the values quite well, see this image:
>
> http://people.apache.org/~tn/pca.png
Actually it qualitatively splits the dataset more or less the same as
using the first and last eigenvector: one 2 dots are on the wrong side
of the plane.
I was hoping to be able to get more precise results for such simple
examples of data.
In fact, if I make the dataset more regular (i.e. equal intervals for
x and y), things work as needed:
double[] x = {1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4};
double[] y = {0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3};
double[] z = {1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2};
In that case the plane between the two lower eigenvalue eigenvectors
splits the dataset in higher and lower.
http://en.zimagez.com/zimage/eigenvectors3dcase3.php
I will do some more tests with complex data.
Thanks,
Andrea
>
> The results from the computations:
>
> covariance:
> {0.7272727273,0.0,0.1818181818},
> {0.0,0.3409090909,0.2272727273},
> {0.1818181818,0.2272727273,0.2727272727}}
>
> lambda1 = 0.8056498828134406
> v1 = {0.9015723558; 0.1900593782; 0.3886447222}
>
> lambda2 = 0.4874287594020183
> v2 = {-0.3799516758; 0.7774478203; 0.5012101464}
>
>
> lambda3 = 0.04783044869363171
> v3 = {-0.2068913033; -0.5995434259; 0.7731388421}
>
> The plane is constructed with v2 and v3.
>
> Thomas
>
> ---------------------------------------------------------------------
> To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
> For additional commands, e-mail: user-help@commons.apache.org
>
---------------------------------------------------------------------
To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
For additional commands, e-mail: user-help@commons.apache.org
Re: [math] eigenvector doubts and issues
Posted by Thomas Neidhart <th...@gmail.com>.
On 11/12/2013 06:31 PM, andrea antonello wrote:
> Hi Thomas,
>
>> that's exactly what we are looking for, so do not hesitate to ask questions.
>
> ok, I put together a small set of examples, which for now I put on a
> wiki of the project I am doing this for (since I have access on that
> one): http://code.google.com/p/jgrasstools/wiki/Eigenvectors
>
> Since in the 3d space I do not get the results I was hoping for, it
> would be great if some expert could give me a hint.
>
> Let me know what you think.
Hi Andrea,
are you sure that your eigenvectors are drawn correctly?
I tried to reproduce your results with geogebra, and in my case the
plane defined by the two eigenvectors with the least eigenvalue seems to
split the values quite well, see this image:
http://people.apache.org/~tn/pca.png
The results from the computations:
covariance:
{0.7272727273,0.0,0.1818181818},
{0.0,0.3409090909,0.2272727273},
{0.1818181818,0.2272727273,0.2727272727}}
lambda1 = 0.8056498828134406
v1 = {0.9015723558; 0.1900593782; 0.3886447222}
lambda2 = 0.4874287594020183
v2 = {-0.3799516758; 0.7774478203; 0.5012101464}
lambda3 = 0.04783044869363171
v3 = {-0.2068913033; -0.5995434259; 0.7731388421}
The plane is constructed with v2 and v3.
Thomas
---------------------------------------------------------------------
To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
For additional commands, e-mail: user-help@commons.apache.org
Re: [math] eigenvector doubts and issues
Posted by andrea antonello <an...@gmail.com>.
Hi Thomas,
> that's exactly what we are looking for, so do not hesitate to ask questions.
ok, I put together a small set of examples, which for now I put on a
wiki of the project I am doing this for (since I have access on that
one): http://code.google.com/p/jgrasstools/wiki/Eigenvectors
Since in the 3d space I do not get the results I was hoping for, it
would be great if some expert could give me a hint.
Let me know what you think.
Andrea
>
> Thomas
>
>
> On Tue, Nov 12, 2013 at 11:54 AM, andrea antonello <
> andrea.antonello@gmail.com> wrote:
>
>> Hi Thomas,
>>
>> > good to hear.
>> > Sounds like an interesting problem you are working on.
>> > If you have found a good solution feel free to come back to use, we might
>> > want to include it in our user guide how an EigenDecomposition could be
>> > used.
>>
>> thanks for pointing this out, I was a bit afraid to go offtopic.
>> But I would be happy to share a small eigenvector real world example
>> if there is interest. I will try to put together something on an
>> accessible wiki page, but I might need some help to finish it for
>> proper use in documentation :)
>>
>> I'll keep you posted.
>> Thanks,
>> Andrea
>>
>> >
>> > Thanks,
>> >
>> > Thomas
>> >
>> >
>> >
>> > On Tue, Nov 12, 2013 at 11:31 AM, andrea antonello <
>> > andrea.antonello@gmail.com> wrote:
>> >
>> >> Hi Thomas,
>> >> you are definitely right. I was fooled by the "nice" result of the
>> >> JAMA calculation, not noticing that it is there that I was extracting
>> >> values in a strange order.
>> >>
>> >> I now made some 3x3 tests on known results and also with the A * v =
>> >> lambda * v constraint and everything is alright.
>> >>
>> >> I still have to figure out how to properly use the results to split
>> >> higher elevation parts from lower z parts in a x,y,z dataset, but the
>> >> eigenvectors and values are in the right place now.
>> >>
>> >> Thanks,
>> >> Andrea
>> >>
>> >>
>> >>
>> >>
>> >> On Mon, Nov 11, 2013 at 1:21 PM, Thomas Neidhart
>> >> <th...@gmail.com> wrote:
>> >> > On 11/11/2013 11:40 AM, andrea antonello wrote:
>> >> >> Hi Thomas,
>> >> >> thanks for your reply.
>> >> >>
>> >> >>> the result of CM and jama are identical, the difference is just in
>> the
>> >> >>> way how the data is stored.
>> >> >>>
>> >> >>> Afaik in jama calling getV() returns a vector in row format whereas
>> in
>> >> >>> CM the are stored in column format.
>> >> >>>
>> >> >>> If you transpose the matrix (or call getVT()) you will see that the
>> >> >>> vectors are identical, except for the signs and order. The reason
>> for
>> >> >>> this is that for CM the eigenvalues/eigenvectors are sorted in
>> >> >>> descending order in case of a symmetric matrix, which is the case
>> for
>> >> >>> your matrix.
>> >> >>
>> >> >> the problems is that the result is not just different in the
>> >> >> transposed way. And in fact if I pick the getVT,results are still not
>> >> >> the same.
>> >> >> The result seems to be reflected on the secondary diagonal, not the
>> >> >> primary diagonal.
>> >> >>
>> >> >> Furthermore, if I use the API, I do not expect to be problems of rows
>> >> >> and columns, so if I use:
>> >> >>
>> >> >> double eigenValue = eigenDecomposition.getRealEigenvalue(i);
>> >> >> RealVector eigenVector = eigenDecomposition.getEigenvector(i);
>> >> >>
>> >> >> I expect the eigenvector and eigenvalue to be the right ones for the
>> >> >> given index, no matter how the results are given in the matrixes.
>> >> >>
>> >> >> But the results I get are, for the same eigenvalue:
>> >> >> CM: eigenVal: 0.8056498828134406, eigenVect: [0.9015723557614027,
>> >> >> 0.19005937823202243, 0.38864472217295326
>> >> >> JAMA: eigenVal: 0.8056498828134406, eigenVect: [-0.7731388420716028,
>> >> >> 0.5012101463530931, -0.38864472217295326]
>> >> >>
>> >> >> I am still quite puzzled about what I am missing.
>> >> >
>> >> > I do not know how you get the eigenvector from jama, as there is no
>> >> > getEigenVector method.
>> >> >
>> >> > The class javadoc of their EigenvalueDecomposition states:
>> >> >
>> >> > columns of V represent the eigenvectors in the sense that A*V =
>> V*D,
>> >> > i.e. A.times(V) equals V.times(D). The matrix V may be badly
>> >> > conditioned, or even singular, so the validity of the equation
>> >> > A = V*D*inverse(V) depends upon V.cond().
>> >> >
>> >> > Looking at your example it looks like you took the rows of V to
>> extract
>> >> > your eigenvector.
>> >> >
>> >> > You can also easily verify if your eigenvector is correct:
>> >> >
>> >> > A * v = lambda * v
>> >> >
>> >> > Thomas
>> >> >
>> >> > ---------------------------------------------------------------------
>> >> > To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
>> >> > For additional commands, e-mail: user-help@commons.apache.org
>> >> >
>> >>
>> >> ---------------------------------------------------------------------
>> >> To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
>> >> For additional commands, e-mail: user-help@commons.apache.org
>> >>
>> >>
>>
>> ---------------------------------------------------------------------
>> To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
>> For additional commands, e-mail: user-help@commons.apache.org
>>
>>
---------------------------------------------------------------------
To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
For additional commands, e-mail: user-help@commons.apache.org
Re: [math] eigenvector doubts and issues
Posted by Thomas Neidhart <th...@gmail.com>.
Hi Andrea,
that's exactly what we are looking for, so do not hesitate to ask questions.
Thomas
On Tue, Nov 12, 2013 at 11:54 AM, andrea antonello <
andrea.antonello@gmail.com> wrote:
> Hi Thomas,
>
> > good to hear.
> > Sounds like an interesting problem you are working on.
> > If you have found a good solution feel free to come back to use, we might
> > want to include it in our user guide how an EigenDecomposition could be
> > used.
>
> thanks for pointing this out, I was a bit afraid to go offtopic.
> But I would be happy to share a small eigenvector real world example
> if there is interest. I will try to put together something on an
> accessible wiki page, but I might need some help to finish it for
> proper use in documentation :)
>
> I'll keep you posted.
> Thanks,
> Andrea
>
> >
> > Thanks,
> >
> > Thomas
> >
> >
> >
> > On Tue, Nov 12, 2013 at 11:31 AM, andrea antonello <
> > andrea.antonello@gmail.com> wrote:
> >
> >> Hi Thomas,
> >> you are definitely right. I was fooled by the "nice" result of the
> >> JAMA calculation, not noticing that it is there that I was extracting
> >> values in a strange order.
> >>
> >> I now made some 3x3 tests on known results and also with the A * v =
> >> lambda * v constraint and everything is alright.
> >>
> >> I still have to figure out how to properly use the results to split
> >> higher elevation parts from lower z parts in a x,y,z dataset, but the
> >> eigenvectors and values are in the right place now.
> >>
> >> Thanks,
> >> Andrea
> >>
> >>
> >>
> >>
> >> On Mon, Nov 11, 2013 at 1:21 PM, Thomas Neidhart
> >> <th...@gmail.com> wrote:
> >> > On 11/11/2013 11:40 AM, andrea antonello wrote:
> >> >> Hi Thomas,
> >> >> thanks for your reply.
> >> >>
> >> >>> the result of CM and jama are identical, the difference is just in
> the
> >> >>> way how the data is stored.
> >> >>>
> >> >>> Afaik in jama calling getV() returns a vector in row format whereas
> in
> >> >>> CM the are stored in column format.
> >> >>>
> >> >>> If you transpose the matrix (or call getVT()) you will see that the
> >> >>> vectors are identical, except for the signs and order. The reason
> for
> >> >>> this is that for CM the eigenvalues/eigenvectors are sorted in
> >> >>> descending order in case of a symmetric matrix, which is the case
> for
> >> >>> your matrix.
> >> >>
> >> >> the problems is that the result is not just different in the
> >> >> transposed way. And in fact if I pick the getVT,results are still not
> >> >> the same.
> >> >> The result seems to be reflected on the secondary diagonal, not the
> >> >> primary diagonal.
> >> >>
> >> >> Furthermore, if I use the API, I do not expect to be problems of rows
> >> >> and columns, so if I use:
> >> >>
> >> >> double eigenValue = eigenDecomposition.getRealEigenvalue(i);
> >> >> RealVector eigenVector = eigenDecomposition.getEigenvector(i);
> >> >>
> >> >> I expect the eigenvector and eigenvalue to be the right ones for the
> >> >> given index, no matter how the results are given in the matrixes.
> >> >>
> >> >> But the results I get are, for the same eigenvalue:
> >> >> CM: eigenVal: 0.8056498828134406, eigenVect: [0.9015723557614027,
> >> >> 0.19005937823202243, 0.38864472217295326
> >> >> JAMA: eigenVal: 0.8056498828134406, eigenVect: [-0.7731388420716028,
> >> >> 0.5012101463530931, -0.38864472217295326]
> >> >>
> >> >> I am still quite puzzled about what I am missing.
> >> >
> >> > I do not know how you get the eigenvector from jama, as there is no
> >> > getEigenVector method.
> >> >
> >> > The class javadoc of their EigenvalueDecomposition states:
> >> >
> >> > columns of V represent the eigenvectors in the sense that A*V =
> V*D,
> >> > i.e. A.times(V) equals V.times(D). The matrix V may be badly
> >> > conditioned, or even singular, so the validity of the equation
> >> > A = V*D*inverse(V) depends upon V.cond().
> >> >
> >> > Looking at your example it looks like you took the rows of V to
> extract
> >> > your eigenvector.
> >> >
> >> > You can also easily verify if your eigenvector is correct:
> >> >
> >> > A * v = lambda * v
> >> >
> >> > Thomas
> >> >
> >> > ---------------------------------------------------------------------
> >> > To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
> >> > For additional commands, e-mail: user-help@commons.apache.org
> >> >
> >>
> >> ---------------------------------------------------------------------
> >> To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
> >> For additional commands, e-mail: user-help@commons.apache.org
> >>
> >>
>
> ---------------------------------------------------------------------
> To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
> For additional commands, e-mail: user-help@commons.apache.org
>
>
Re: [math] eigenvector doubts and issues
Posted by andrea antonello <an...@gmail.com>.
Hi Thomas,
> good to hear.
> Sounds like an interesting problem you are working on.
> If you have found a good solution feel free to come back to use, we might
> want to include it in our user guide how an EigenDecomposition could be
> used.
thanks for pointing this out, I was a bit afraid to go offtopic.
But I would be happy to share a small eigenvector real world example
if there is interest. I will try to put together something on an
accessible wiki page, but I might need some help to finish it for
proper use in documentation :)
I'll keep you posted.
Thanks,
Andrea
>
> Thanks,
>
> Thomas
>
>
>
> On Tue, Nov 12, 2013 at 11:31 AM, andrea antonello <
> andrea.antonello@gmail.com> wrote:
>
>> Hi Thomas,
>> you are definitely right. I was fooled by the "nice" result of the
>> JAMA calculation, not noticing that it is there that I was extracting
>> values in a strange order.
>>
>> I now made some 3x3 tests on known results and also with the A * v =
>> lambda * v constraint and everything is alright.
>>
>> I still have to figure out how to properly use the results to split
>> higher elevation parts from lower z parts in a x,y,z dataset, but the
>> eigenvectors and values are in the right place now.
>>
>> Thanks,
>> Andrea
>>
>>
>>
>>
>> On Mon, Nov 11, 2013 at 1:21 PM, Thomas Neidhart
>> <th...@gmail.com> wrote:
>> > On 11/11/2013 11:40 AM, andrea antonello wrote:
>> >> Hi Thomas,
>> >> thanks for your reply.
>> >>
>> >>> the result of CM and jama are identical, the difference is just in the
>> >>> way how the data is stored.
>> >>>
>> >>> Afaik in jama calling getV() returns a vector in row format whereas in
>> >>> CM the are stored in column format.
>> >>>
>> >>> If you transpose the matrix (or call getVT()) you will see that the
>> >>> vectors are identical, except for the signs and order. The reason for
>> >>> this is that for CM the eigenvalues/eigenvectors are sorted in
>> >>> descending order in case of a symmetric matrix, which is the case for
>> >>> your matrix.
>> >>
>> >> the problems is that the result is not just different in the
>> >> transposed way. And in fact if I pick the getVT,results are still not
>> >> the same.
>> >> The result seems to be reflected on the secondary diagonal, not the
>> >> primary diagonal.
>> >>
>> >> Furthermore, if I use the API, I do not expect to be problems of rows
>> >> and columns, so if I use:
>> >>
>> >> double eigenValue = eigenDecomposition.getRealEigenvalue(i);
>> >> RealVector eigenVector = eigenDecomposition.getEigenvector(i);
>> >>
>> >> I expect the eigenvector and eigenvalue to be the right ones for the
>> >> given index, no matter how the results are given in the matrixes.
>> >>
>> >> But the results I get are, for the same eigenvalue:
>> >> CM: eigenVal: 0.8056498828134406, eigenVect: [0.9015723557614027,
>> >> 0.19005937823202243, 0.38864472217295326
>> >> JAMA: eigenVal: 0.8056498828134406, eigenVect: [-0.7731388420716028,
>> >> 0.5012101463530931, -0.38864472217295326]
>> >>
>> >> I am still quite puzzled about what I am missing.
>> >
>> > I do not know how you get the eigenvector from jama, as there is no
>> > getEigenVector method.
>> >
>> > The class javadoc of their EigenvalueDecomposition states:
>> >
>> > columns of V represent the eigenvectors in the sense that A*V = V*D,
>> > i.e. A.times(V) equals V.times(D). The matrix V may be badly
>> > conditioned, or even singular, so the validity of the equation
>> > A = V*D*inverse(V) depends upon V.cond().
>> >
>> > Looking at your example it looks like you took the rows of V to extract
>> > your eigenvector.
>> >
>> > You can also easily verify if your eigenvector is correct:
>> >
>> > A * v = lambda * v
>> >
>> > Thomas
>> >
>> > ---------------------------------------------------------------------
>> > To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
>> > For additional commands, e-mail: user-help@commons.apache.org
>> >
>>
>> ---------------------------------------------------------------------
>> To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
>> For additional commands, e-mail: user-help@commons.apache.org
>>
>>
---------------------------------------------------------------------
To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
For additional commands, e-mail: user-help@commons.apache.org
Re: [math] eigenvector doubts and issues
Posted by Thomas Neidhart <th...@gmail.com>.
Hi Andrea,
good to hear.
Sounds like an interesting problem you are working on.
If you have found a good solution feel free to come back to use, we might
want to include it in our user guide how an EigenDecomposition could be
used.
Thanks,
Thomas
On Tue, Nov 12, 2013 at 11:31 AM, andrea antonello <
andrea.antonello@gmail.com> wrote:
> Hi Thomas,
> you are definitely right. I was fooled by the "nice" result of the
> JAMA calculation, not noticing that it is there that I was extracting
> values in a strange order.
>
> I now made some 3x3 tests on known results and also with the A * v =
> lambda * v constraint and everything is alright.
>
> I still have to figure out how to properly use the results to split
> higher elevation parts from lower z parts in a x,y,z dataset, but the
> eigenvectors and values are in the right place now.
>
> Thanks,
> Andrea
>
>
>
>
> On Mon, Nov 11, 2013 at 1:21 PM, Thomas Neidhart
> <th...@gmail.com> wrote:
> > On 11/11/2013 11:40 AM, andrea antonello wrote:
> >> Hi Thomas,
> >> thanks for your reply.
> >>
> >>> the result of CM and jama are identical, the difference is just in the
> >>> way how the data is stored.
> >>>
> >>> Afaik in jama calling getV() returns a vector in row format whereas in
> >>> CM the are stored in column format.
> >>>
> >>> If you transpose the matrix (or call getVT()) you will see that the
> >>> vectors are identical, except for the signs and order. The reason for
> >>> this is that for CM the eigenvalues/eigenvectors are sorted in
> >>> descending order in case of a symmetric matrix, which is the case for
> >>> your matrix.
> >>
> >> the problems is that the result is not just different in the
> >> transposed way. And in fact if I pick the getVT,results are still not
> >> the same.
> >> The result seems to be reflected on the secondary diagonal, not the
> >> primary diagonal.
> >>
> >> Furthermore, if I use the API, I do not expect to be problems of rows
> >> and columns, so if I use:
> >>
> >> double eigenValue = eigenDecomposition.getRealEigenvalue(i);
> >> RealVector eigenVector = eigenDecomposition.getEigenvector(i);
> >>
> >> I expect the eigenvector and eigenvalue to be the right ones for the
> >> given index, no matter how the results are given in the matrixes.
> >>
> >> But the results I get are, for the same eigenvalue:
> >> CM: eigenVal: 0.8056498828134406, eigenVect: [0.9015723557614027,
> >> 0.19005937823202243, 0.38864472217295326
> >> JAMA: eigenVal: 0.8056498828134406, eigenVect: [-0.7731388420716028,
> >> 0.5012101463530931, -0.38864472217295326]
> >>
> >> I am still quite puzzled about what I am missing.
> >
> > I do not know how you get the eigenvector from jama, as there is no
> > getEigenVector method.
> >
> > The class javadoc of their EigenvalueDecomposition states:
> >
> > columns of V represent the eigenvectors in the sense that A*V = V*D,
> > i.e. A.times(V) equals V.times(D). The matrix V may be badly
> > conditioned, or even singular, so the validity of the equation
> > A = V*D*inverse(V) depends upon V.cond().
> >
> > Looking at your example it looks like you took the rows of V to extract
> > your eigenvector.
> >
> > You can also easily verify if your eigenvector is correct:
> >
> > A * v = lambda * v
> >
> > Thomas
> >
> > ---------------------------------------------------------------------
> > To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
> > For additional commands, e-mail: user-help@commons.apache.org
> >
>
> ---------------------------------------------------------------------
> To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
> For additional commands, e-mail: user-help@commons.apache.org
>
>
Re: [math] eigenvector doubts and issues
Posted by andrea antonello <an...@gmail.com>.
Hi Thomas,
you are definitely right. I was fooled by the "nice" result of the
JAMA calculation, not noticing that it is there that I was extracting
values in a strange order.
I now made some 3x3 tests on known results and also with the A * v =
lambda * v constraint and everything is alright.
I still have to figure out how to properly use the results to split
higher elevation parts from lower z parts in a x,y,z dataset, but the
eigenvectors and values are in the right place now.
Thanks,
Andrea
On Mon, Nov 11, 2013 at 1:21 PM, Thomas Neidhart
<th...@gmail.com> wrote:
> On 11/11/2013 11:40 AM, andrea antonello wrote:
>> Hi Thomas,
>> thanks for your reply.
>>
>>> the result of CM and jama are identical, the difference is just in the
>>> way how the data is stored.
>>>
>>> Afaik in jama calling getV() returns a vector in row format whereas in
>>> CM the are stored in column format.
>>>
>>> If you transpose the matrix (or call getVT()) you will see that the
>>> vectors are identical, except for the signs and order. The reason for
>>> this is that for CM the eigenvalues/eigenvectors are sorted in
>>> descending order in case of a symmetric matrix, which is the case for
>>> your matrix.
>>
>> the problems is that the result is not just different in the
>> transposed way. And in fact if I pick the getVT,results are still not
>> the same.
>> The result seems to be reflected on the secondary diagonal, not the
>> primary diagonal.
>>
>> Furthermore, if I use the API, I do not expect to be problems of rows
>> and columns, so if I use:
>>
>> double eigenValue = eigenDecomposition.getRealEigenvalue(i);
>> RealVector eigenVector = eigenDecomposition.getEigenvector(i);
>>
>> I expect the eigenvector and eigenvalue to be the right ones for the
>> given index, no matter how the results are given in the matrixes.
>>
>> But the results I get are, for the same eigenvalue:
>> CM: eigenVal: 0.8056498828134406, eigenVect: [0.9015723557614027,
>> 0.19005937823202243, 0.38864472217295326
>> JAMA: eigenVal: 0.8056498828134406, eigenVect: [-0.7731388420716028,
>> 0.5012101463530931, -0.38864472217295326]
>>
>> I am still quite puzzled about what I am missing.
>
> I do not know how you get the eigenvector from jama, as there is no
> getEigenVector method.
>
> The class javadoc of their EigenvalueDecomposition states:
>
> columns of V represent the eigenvectors in the sense that A*V = V*D,
> i.e. A.times(V) equals V.times(D). The matrix V may be badly
> conditioned, or even singular, so the validity of the equation
> A = V*D*inverse(V) depends upon V.cond().
>
> Looking at your example it looks like you took the rows of V to extract
> your eigenvector.
>
> You can also easily verify if your eigenvector is correct:
>
> A * v = lambda * v
>
> Thomas
>
> ---------------------------------------------------------------------
> To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
> For additional commands, e-mail: user-help@commons.apache.org
>
---------------------------------------------------------------------
To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
For additional commands, e-mail: user-help@commons.apache.org
Re: [math] eigenvector doubts and issues
Posted by Thomas Neidhart <th...@gmail.com>.
On 11/11/2013 11:40 AM, andrea antonello wrote:
> Hi Thomas,
> thanks for your reply.
>
>> the result of CM and jama are identical, the difference is just in the
>> way how the data is stored.
>>
>> Afaik in jama calling getV() returns a vector in row format whereas in
>> CM the are stored in column format.
>>
>> If you transpose the matrix (or call getVT()) you will see that the
>> vectors are identical, except for the signs and order. The reason for
>> this is that for CM the eigenvalues/eigenvectors are sorted in
>> descending order in case of a symmetric matrix, which is the case for
>> your matrix.
>
> the problems is that the result is not just different in the
> transposed way. And in fact if I pick the getVT,results are still not
> the same.
> The result seems to be reflected on the secondary diagonal, not the
> primary diagonal.
>
> Furthermore, if I use the API, I do not expect to be problems of rows
> and columns, so if I use:
>
> double eigenValue = eigenDecomposition.getRealEigenvalue(i);
> RealVector eigenVector = eigenDecomposition.getEigenvector(i);
>
> I expect the eigenvector and eigenvalue to be the right ones for the
> given index, no matter how the results are given in the matrixes.
>
> But the results I get are, for the same eigenvalue:
> CM: eigenVal: 0.8056498828134406, eigenVect: [0.9015723557614027,
> 0.19005937823202243, 0.38864472217295326
> JAMA: eigenVal: 0.8056498828134406, eigenVect: [-0.7731388420716028,
> 0.5012101463530931, -0.38864472217295326]
>
> I am still quite puzzled about what I am missing.
I do not know how you get the eigenvector from jama, as there is no
getEigenVector method.
The class javadoc of their EigenvalueDecomposition states:
columns of V represent the eigenvectors in the sense that A*V = V*D,
i.e. A.times(V) equals V.times(D). The matrix V may be badly
conditioned, or even singular, so the validity of the equation
A = V*D*inverse(V) depends upon V.cond().
Looking at your example it looks like you took the rows of V to extract
your eigenvector.
You can also easily verify if your eigenvector is correct:
A * v = lambda * v
Thomas
---------------------------------------------------------------------
To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
For additional commands, e-mail: user-help@commons.apache.org
Re: [math] eigenvector doubts and issues
Posted by andrea antonello <an...@gmail.com>.
Hi Thomas,
thanks for your reply.
> the result of CM and jama are identical, the difference is just in the
> way how the data is stored.
>
> Afaik in jama calling getV() returns a vector in row format whereas in
> CM the are stored in column format.
>
> If you transpose the matrix (or call getVT()) you will see that the
> vectors are identical, except for the signs and order. The reason for
> this is that for CM the eigenvalues/eigenvectors are sorted in
> descending order in case of a symmetric matrix, which is the case for
> your matrix.
the problems is that the result is not just different in the
transposed way. And in fact if I pick the getVT,results are still not
the same.
The result seems to be reflected on the secondary diagonal, not the
primary diagonal.
Furthermore, if I use the API, I do not expect to be problems of rows
and columns, so if I use:
double eigenValue = eigenDecomposition.getRealEigenvalue(i);
RealVector eigenVector = eigenDecomposition.getEigenvector(i);
I expect the eigenvector and eigenvalue to be the right ones for the
given index, no matter how the results are given in the matrixes.
But the results I get are, for the same eigenvalue:
CM: eigenVal: 0.8056498828134406, eigenVect: [0.9015723557614027,
0.19005937823202243, 0.38864472217295326
JAMA: eigenVal: 0.8056498828134406, eigenVect: [-0.7731388420716028,
0.5012101463530931, -0.38864472217295326]
I am still quite puzzled about what I am missing.
Thanks,
Andrea
>
> Thomas
>
> ---------------------------------------------------------------------
> To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
> For additional commands, e-mail: user-help@commons.apache.org
>
---------------------------------------------------------------------
To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
For additional commands, e-mail: user-help@commons.apache.org
Re: [math] eigenvector doubts and issues
Posted by Thomas Neidhart <th...@gmail.com>.
On 11/09/2013 10:38 AM, andrea antonello wrote:
> Dear all,
> I have a doubt about using the eigenvector part of the library.
>
> I created a small dataset to represent 3d coordinates in a cartesian plane:
>
> double[] x = {1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3};
> double[] y = {0.5, 1, 1.5, 2, 0.5, 1, 1.5, 2, 0.5, 1, 1.5, 2};
> double[] z = {1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2};
>
> The datasset represents a step from z value 1 to 2 on a regular grid
> (with a diagonali trend).
>
> I would expect to gain from the eigenvector with lowest eigenvalue a
> line splitting this particular set in a quite clean way the higher z
> points from the lower ones.
>
> So I calculate the covariance matrix which results in:
> 0.7272727272727273 0.0 0.18181818181818182
> 0.0 0.3409090909090909 0.22727272727272727
> 0.18181818181818182 0.22727272727272727 0.2727272727272727
>
> and then I simply calculate the eigenvector/values which result in:
>
> eigenVal: 0.8056498828134406, eigenVect: [0.9015723557614027,
> 0.19005937823202243, 0.38864472217295326]
> eigenVal: 0.4874287594020183, eigenVect: [-0.37995167578226796,
> 0.7774478202831089, 0.5012101463530935]
> eigenVal: 0.04783044869363171, eigenVect: [-0.20689130333844696,
> -0.5995434258526233, 0.773138842071603]
>
> doing exactly the same thing with Jama results in:
>
> eigenVal: 0.8056498828134406, eigenVect: [-0.7731388420716028,
> 0.5012101463530931, -0.38864472217295326]
> eigenVal: 0.48742875940201863, eigenVect: [0.5995434258526229,
> 0.7774478202831089, -0.1900593782320223]
> eigenVal: 0.0478304486936319, eigenVect: [0.20689130333844694,
> -0.37995167578226785, -0.9015723557614027]
>
> In fact if I use Jama's eigenvector with lowest eigenvalue, I am able
> to construct a line of slope y =
> (0.206891303338447/-0.379951675782268) *x, which splits my dataset the
> way I would like to have it.
> The same doesn't apply to the result of the apache commons math lib,
> which seems to be reflected on the secondary diagonal.
>
> Since I am no expert in this field, I might be doing somthing really
> wrong. If someone could give me a hint, it would be greatly
> appreciated.
Hi Andrea,
the result of CM and jama are identical, the difference is just in the
way how the data is stored.
Afaik in jama calling getV() returns a vector in row format whereas in
CM the are stored in column format.
If you transpose the matrix (or call getVT()) you will see that the
vectors are identical, except for the signs and order. The reason for
this is that for CM the eigenvalues/eigenvectors are sorted in
descending order in case of a symmetric matrix, which is the case for
your matrix.
Thomas
---------------------------------------------------------------------
To unsubscribe, e-mail: user-unsubscribe@commons.apache.org
For additional commands, e-mail: user-help@commons.apache.org