Thread Subject: Estimating generalized bivariate pdfs from correlated marginals

Hello,

I am new to this newsgroup so I apologize in advance for
any improper etiquette.

I am trying to estimate a joint (bivariate) pdf/cdf from
correlated marginals. I am working with sparse data
(i.e., particle filters) so I don't want to use histogram
methods because even with interpolation, histograms
introduce undesirable artifacts in the joint densities.
I am sucessfully estimating the marginals using maximum
entropy with moment constraints. I have investigated
estimaing the joint pdf/cdf using copulas, however, I have
been unsucessful because (I think) I'm having difficulty
selecting the correct family and corresponding parameters
for my generalized application. I have also investigated
estimating the joint pdf/cdf by taking the ifft of the
joint characteristic function, however, I can't seem to
make that work either. Can anyone point me in a direction
to accomplish this or provide me with some code?

Thanks in advance.
Phil Haney

Philip Haney wrote:

> I am trying to estimate a joint (bivariate) pdf/cdf from
> correlated marginals. I am working with sparse data
> (i.e., particle filters) so I don't want to use histogram
> methods because even with interpolation, histograms
> introduce undesirable artifacts in the joint densities.
> I am sucessfully estimating the marginals using maximum
> entropy with moment constraints. I have investigated
> estimaing the joint pdf/cdf using copulas, however, I have
> been unsucessful because (I think) I'm having difficulty
> selecting the correct family and corresponding parameters
> for my generalized application. I have also investigated
> estimating the joint pdf/cdf by taking the ifft of the
> joint characteristic function, however, I can't seem to
> make that work either. Can anyone point me in a direction
> to accomplish this or provide me with some code?

Phil, this sounds like an interesting problem. I may not be able to
offer anything useful. I assume that when you say, "sparse", you mean
you don't have a lot of data. Typically that means you'll want to fit a
parametric distribution, so perhaps using a kernel density estimate will
not work. I would think that trying to make estimates via
characteristic functions would require a lot of data as well.

Since you say you've fit the marginals, it sounds like you're well down
the road towards fitting a copula, so I'm surprised to hear you say that
wih very few data, you aren't able to find a copula family that fits the
data well. With few data, I'd think you have the opposite problem.

I confess that I don't understand what you mean by "generalized".

More questions than answers, sorry.

- Peter Perkins
   The MathWorks, Inc.

Peter Perkins <Peter.PerkinsRemoveThis@mathworks.com>
wrote in message <fovc9p$i32$1@fred.mathworks.com>...
> Philip Haney wrote:
>
> > I am trying to estimate a joint (bivariate) pdf/cdf
from
> > correlated marginals. I am working with sparse data
> > (i.e., particle filters) so I don't want to use
histogram
> > methods because even with interpolation, histograms
> > introduce undesirable artifacts in the joint densities.
> > I am sucessfully estimating the marginals using
maximum
> > entropy with moment constraints. I have investigated
> > estimaing the joint pdf/cdf using copulas, however, I
have
> > been unsucessful because (I think) I'm having
difficulty
> > selecting the correct family and corresponding
parameters
> > for my generalized application. I have also
investigated
> > estimating the joint pdf/cdf by taking the ifft of the
> > joint characteristic function, however, I can't seem
to
> > make that work either. Can anyone point me in a
direction
> > to accomplish this or provide me with some code?
>
> Phil, this sounds like an interesting problem. I may
not be able to
> offer anything useful. I assume that when you
say, "sparse", you mean
> you don't have a lot of data. Typically that means
you'll want to fit a
> parametric distribution, so perhaps using a kernel
density estimate will
> not work. I would think that trying to make estimates
via
> characteristic functions would require a lot of data as
well.
>
> Since you say you've fit the marginals, it sounds like
you're well down
> the road towards fitting a copula, so I'm surprised to
hear you say that
> wih very few data, you aren't able to find a copula
family that fits the
> data well. With few data, I'd think you have the
opposite problem.
>
> I confess that I don't understand what you mean
by "generalized".
>
> More questions than answers, sorry.
>
> - Peter Perkins
> The MathWorks, Inc.

Hi Peter,

Thanks for the reply.

I may be using "sparse" incorrectly, but for my
application, I have a large sample set (i.e., a large
number of particles), however, the sample set
is "sparsely" distributed over the spatial region of
interest. Consequently, when I use histogram methods
(even with interpolation), the resulting pdf/cdf is noisy
instead of smooth as desired. This is mainly why I want
to estimate the joint pdf/cdf from the marginals.

Given the marginal CDFs, I compute the copula for a given
family and corresponding parameters. When the marginals
are independent, everything seems to work fine, however,
for my application, the marginals are not independent and
the copula method doesn't produce the expected joint cdf.
As a result, this is why I'm wondering if I'm not picking
the copula family and/or corresponding parameters
correctly in order to describe the dependence between the
marginals.

I guess by "generalized" I mean that I'm looking for a
method that will allow me to compute the joint pdf/cdf
from marginals without requiring any a-priori information
or understanding relative to the nature of the marginals.
Sorry if my terminology is confusing.

Again, thanks for the reply.
Phil

Philip Haney wrote:

> This is mainly why I want
> to estimate the joint pdf/cdf from the marginals.

Perhaps I'm just misunderstanding what you mean, but as stated, this
cannot be done. The marginals do not determine the joint distribution.


> Given the marginal CDFs, I compute the copula for a given
> family and corresponding parameters. When the marginals
> are independent, everything seems to work fine, however,
> for my application, the marginals are not independent and
> the copula method doesn't produce the expected joint cdf.

I'm not sure what you mean when you say the marginals are independent or
not independent. The random variables those marginal distributions
describe are not independent, that's why you want a joint distribution.
  And that's just what copulas are for. Each marginal only involves one
of those random variables, so in that sense, they are "independent" (I
would say "separate"), and with a copula model, you can estimate them
separately.

Again I may just be misunderstanding your terminology.

Peter Perkins <Peter.PerkinsRemoveThis@mathworks.com>
wrote in message <fp1ko8$3a$1@fred.mathworks.com>...
> Philip Haney wrote:
>
> > This is mainly why I want
> > to estimate the joint pdf/cdf from the marginals.
>
> Perhaps I'm just misunderstanding what you mean, but as
stated, this
> cannot be done. The marginals do not determine the
joint distribution.
>
>
> > Given the marginal CDFs, I compute the copula for a
given
> > family and corresponding parameters. When the
marginals
> > are independent, everything seems to work fine,
however,
> > for my application, the marginals are not independent
and
> > the copula method doesn't produce the expected joint
cdf.
>
> I'm not sure what you mean when you say the marginals
are independent or
> not independent. The random variables those marginal
distributions
> describe are not independent, that's why you want a
joint distribution.
> And that's just what copulas are for. Each marginal
only involves one
> of those random variables, so in that sense, they
are "independent" (I
> would say "separate"), and with a copula model, you can
estimate them
> separately.
>
> Again I may just be misunderstanding your terminology.

My understanding of a copula is that it is a function that
provides the ability to combine univariate distributions
in order to obtain a joint distribution with a particular
dependence structure. If the univariate distributions
(i.e., marginals) are generated from independent random
variables, then I wouldn't need a copula anymore in order
to define the joint distribution.

Respected Sir

I am also very new to copula. I can use copularnd function
to generate the copula data for gaussion,t,frank,
gunbel,clypton familes only.

BUT i want to generate the copula data for exponential
distribution. how can i do that.

Kindly advise.
Thank you very much



"Philip Haney" <philip.j.haney@baesystems.com> wrote in
message <fov2qu$3h0$1@fred.mathworks.com>...
> Hello,
>
> I am new to this newsgroup so I apologize in advance for
> any improper etiquette.
>
> I am trying to estimate a joint (bivariate) pdf/cdf from
> correlated marginals. I am working with sparse data
> (i.e., particle filters) so I don't want to use histogram
> methods because even with interpolation, histograms
> introduce undesirable artifacts in the joint densities.
> I am sucessfully estimating the marginals using maximum
> entropy with moment constraints. I have investigated
> estimaing the joint pdf/cdf using copulas, however, I have
> been unsucessful because (I think) I'm having difficulty
> selecting the correct family and corresponding parameters
> for my generalized application. I have also investigated
> estimating the joint pdf/cdf by taking the ifft of the
> joint characteristic function, however, I can't seem to
> make that work either. Can anyone point me in a direction
> to accomplish this or provide me with some code?
>
> Thanks in advance.
> Phil Haney
>

Tanuj Puri wrote:

> BUT i want to generate the copula data for exponential
> distribution. how can i do that.

Tanuj, I assume you mean the exponential copula, not the univariate exponential
distribution. the exponential copula (a.k.a. Marshal-Olkin) is not one of the
copulas that coplarnd currently supports. I've made a note to look into adding
it to the Statistics Toolbox at some point in the future.

In the absence of that, a google search turns up several papers describing this
copula; at least one describes ann algorithm for generating random values.

Hope this helps.

- Peter Perkins
   The MathWorks, Inc.