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distance matrix consolidation
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bird
science forum beginner


Joined: 11 Jun 2005
Posts: 14

PostPosted: Mon Jul 17, 2006 4:24 am    Post subject: Re: distance matrix consolidation Reply with quote

"Ray Koopman" wrote:
Quote:
"Evaluate the partitioning" means compute the objective function
(e.g., the sum of squared within-partition distances) for a given
partitioning. I am suggesting that you do this for each distance
matrix separately and then combine (e.g., sum) the results. I used
the more general term "combine" intentionally, to accomodate objective
functions for which the appropriate combining formula might be
something other than a simple sum.

If, as you said earlier, different distance matrices represent
different partitionings, it would make little sense to combine the
matrices themselves, and even less to optimize a partitioning based
on such a combined matrix.

You don't say whether all your distance matrices are in the same
units. If they are not then you will need to adjust for this when
you combine the function values, or the matrices with larger numbers
will have too much influence on the final solution.

It may help if you look at the configuration of points
that is implied by each distance matrix.
Let C = (-1/2)(I - uu'/n)B(I - uu'/n), where
n is the number of points,
B is the matrix of squared distances among the points,
I is the identity matrix,
u is a column vector whose elements are all 1s,
and ' denotes transposition.
If the points lie in an m-dimensional space
then C will be positive semidefinite, with rank m.
Let F be any factoring of such a C, so that FF' = C.
Then F contains the coordinates of the points
with respect to arbitrary orthogonal reference axes,
with the origin taken at the centroid of the points.

The above is a well-known result in multidimensional scaling.
In general, you might want to look at the multidimensional scaling
literature. In particular, google for INDSCAL and IDIOSCAL,
which are designed for multiple disparate sets of distances.

Your suggestion may be an approach to the solution of this problem.

I focused on MDS for a few days a couple of months ago, but I turned
to others for some reasons such as in computation and in pressure
of due date. Your suggestion reminds me to revisit this approach.
I appreciate all of your inputs.

Thank you.

-Kevin
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Ray Koopman
science forum Guru Wannabe


Joined: 25 Mar 2005
Posts: 216

PostPosted: Sun Jul 16, 2006 6:58 pm    Post subject: Re: distance matrix consolidation Reply with quote

bird wrote:
Quote:
Thank you first. Presently, I do minimize the "sum of the squared within-
partition distances". I am an engineering student. Most engineers like
such an objective function.

Could you please tell me more why I should evalute each distance matrix
separately and then combine the evaluations? and please suggest some
references or web pointors for me to read? I did some time before have
such an idea, but I did not proceed it because I do not know how to
combine the evaluations of matrices. Are you aware of any method to do
it?

What does the evaluation mean when you say the evaluation of a matrix?
Are you referring to anything other than a partitioning?

Thank you so much again,

-Kevin

"Evaluate the partitioning" means compute the objective function
(e.g., the sum of squared within-partition distances) for a given
partitioning. I am suggesting that you do this for each distance
matrix separately and then combine (e.g., sum) the results. I used
the more general term "combine" intentionally, to accomodate objective
functions for which the appropriate combining formula might be
something other than a simple sum.

If, as you said earlier, different distance matrices represent
different partitionings, it would make little sense to combine the
matrices themselves, and even less to optimize a partitioning based
on such a combined matrix.

You don't say whether all your distance matrices are in the same
units. If they are not then you will need to adjust for this when
you combine the function values, or the matrices with larger numbers
will have too much influence on the final solution.

It may help if you look at the configuration of points
that is implied by each distance matrix.
Let C = (-1/2)(I - uu'/n)B(I - uu'/n), where
n is the number of points,
B is the matrix of squared distances among the points,
I is the identity matrix,
u is a column vector whose elements are all 1s,
and ' denotes transposition.
If the points lie in an m-dimensional space
then C will be positive semidefinite, with rank m.
Let F be any factoring of such a C, so that FF' = C.
Then F contains the coordinates of the points
with respect to arbitrary orthogonal reference axes,
with the origin taken at the centroid of the points.

The above is a well-known result in multidimensional scaling.
In general, you might want to look at the multidimensional scaling
literature. In particular, google for INDSCAL and IDIOSCAL,
which are designed for multiple disparate sets of distances.
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bird
science forum beginner


Joined: 11 Jun 2005
Posts: 14

PostPosted: Sun Jul 16, 2006 2:14 pm    Post subject: Re: distance matrix consolidation Reply with quote

"Ray Koopman" <koopman@sfu.ca> wrote:
Quote:
bird wrote:
"Ray Koopman" wrote:
bird wrote:
Hi everyone,

What is a good way to consolidate many distance matrices?
Each of the matrices is an observation of a system from a
perspective. It represents a quite clear partitioning of the system.
Different matrices represent different partitionings.
My preliminary experiment seems telling me simple averaging
is not good.

Thank you for your help.

-Kevin

Exactly what are these things that you are calling distance matrices?
How did you get them?

Sorry for my unclear description. A distance matrix is an N by N square
if there are N objects in the system of concern. Any matrix element
a(i,j)
represents the distance of objects i and j, where 1 <= i, j <= N. These
distance matrices are computed from measurements.

By observation, I know each matrix is a quite clear partitioning of the
N
objects, which means these N objects can be clustered into several
groups; the distance between any two objects within a group is small
while the distance between any two objects not belonging to a group
is relatively large.

How can I consolidate all these matrices to obtain a single matrix
that represents the best partitioning of the system? By "the best" I
mean an objective function can be minimized with the flexibility in
choosing the number of partitions.

Thank you,

-kevin

I think you should evaluate any proposed partitioning on each distance
matrix separately and then combine the evaluations, rather than first
combining the matrices somehow and then evaluating the partitioning on
the combined matrix.

I'll leave it to others to suggest their favorite partition-evaluation
schemes -- there are many of them -- but I'll start by suggesting that
you partition the objects so as to minimize the sum of the squared
within-partition distances.

Thank you first. Presently, I do minimize the "sum of the squared within-

partition distances". I am an engineering student. Most engineers like
such an objective function.

Could you please tell me more why I should evalute each distance matrix
separately and then combine the evaluations? and please suggest some
references or web pointors for me to read? I did some time before have
such an idea, but I did not proceed it because I do not know how to
combine the evaluations of matrices. Are you aware of any method to do
it?

What does the evaluation mean when you say the evaluation of a matrix?
Are you referring to anything other than a partitioning?

Thank you so much again,

-Kevin
Back to top
Ray Koopman
science forum Guru Wannabe


Joined: 25 Mar 2005
Posts: 216

PostPosted: Sun Jul 16, 2006 6:36 am    Post subject: Re: distance matrix consolidation Reply with quote

bird wrote:
Quote:
"Ray Koopman" wrote:
bird wrote:
Hi everyone,

What is a good way to consolidate many distance matrices?
Each of the matrices is an observation of a system from a
perspective. It represents a quite clear partitioning of the system.
Different matrices represent different partitionings.
My preliminary experiment seems telling me simple averaging
is not good.

Thank you for your help.

-Kevin

Exactly what are these things that you are calling distance matrices?
How did you get them?

Sorry for my unclear description. A distance matrix is an N by N square
if there are N objects in the system of concern. Any matrix element a(i,j)
represents the distance of objects i and j, where 1 <= i, j <= N. These
distance matrices are computed from measurements.

By observation, I know each matrix is a quite clear partitioning of the N
objects, which means these N objects can be clustered into several
groups; the distance between any two objects within a group is small
while the distance between any two objects not belonging to a group
is relatively large.

How can I consolidate all these matrices to obtain a single matrix
that represents the best partitioning of the system? By "the best" I
mean an objective function can be minimized with the flexibility in
choosing the number of partitions.

Thank you,

-kevin

I think you should evaluate any proposed partitioning on each distance
matrix separately and then combine the evaluations, rather than first
combining the matrices somehow and then evaluating the partitioning on
the combined matrix.

I'll leave it to others to suggest their favorite partition-evaluation
schemes -- there are many of them -- but I'll start by suggesting that
you partition the objects so as to minimize the sum of the squared
within-partition distances.
Back to top
bird
science forum beginner


Joined: 11 Jun 2005
Posts: 14

PostPosted: Sun Jul 16, 2006 3:08 am    Post subject: Re: distance matrix consolidation Reply with quote

"Ray Koopman" wrote:
Quote:
bird wrote:
Hi everyone,

What is a good way to consolidate many distance matrices?
Each of the matrices is an observation of a system from a
perspective. It represents a quite clear partitioning of the system.
Different matrices represent different partitionings.
My preliminary experiment seems telling me simple averaging
is not good.

Thank you for your help.

-Kevin

Exactly what are these things that you are calling distance matrices?
How did you get them?

Sorry for my unclear description. A distance matrix is an N by N square

if there are N objects in the system of concern. Any matrix element a(i,j)
represents the distance of objects i and j, where 1 <= i, j <= N. These
distance matrices are computed from measurements.

By observation, I know each matrix is a quite clear partitioning of the N
objects, which means these N objects can be clustered into several
groups; the distance between any two objects within a group is small
while the distance between any two objects not belonging to a group
is relatively large.

How can I consolidate all these matrices to obtain a single matrix
that represents the best partitioning of the system? By "the best" I
mean an objective function can be minimized with the flexibility in
choosing the number of partitions.

Thank you,

-kevin
Back to top
Ray Koopman
science forum Guru Wannabe


Joined: 25 Mar 2005
Posts: 216

PostPosted: Sat Jul 15, 2006 11:58 pm    Post subject: Re: distance matrix consolidation Reply with quote

bird wrote:
Quote:
Hi everyone,

What is a good way to consolidate many distance matrices?
Each of the matrices is an observation of a system from a
perspective. It represents a quite clear partitioning of the system.
Different matrices represent different partitionings.
My preliminary experiment seems telling me simple averaging
is not good.

Thank you for your help.

-Kevin

Exactly what are these things that you are calling distance matrices?
How did you get them?
Back to top
bird
science forum beginner


Joined: 11 Jun 2005
Posts: 14

PostPosted: Sat Jul 15, 2006 9:05 pm    Post subject: distance matrix consolidation Reply with quote

Hi everyone,

What is a good way to consolidate many distance matrices?
Each of the matrices is an observation of a system from a
perspective. It represents a quite clear partitioning of the system.
Different matrices represent different partitionings.
My preliminary experiment seems telling me simple averaging
is not good.

Thank you for your help.

-Kevin
Back to top
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