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From: Bo on 6 Jul 2010 13:13
Hi,

I am trying to fit data sets with the following nonlinear model:

Y(t) = B0 + B1*exp(-t/tau1) + B2*exp(-t/tau2) + B3*exp(-t/tau3)

For all the data sets, the parameters, B0, B1, B2, B3, tau1, tau2 and tau3 are the same.

Is there a way to fit multiple data sets at the same time?

The reason to do this is that it seems there are multiple solutions for a single data set. I am thinking fitting multiple data sets simultaneously may help reduce the variability.

Thanks a lot

Bo
From: someone on 6 Jul 2010 14:03
"Bo " <qiangbo(a)gmail.com> wrote in message <i0vo7k$988$1(a)fred.mathworks.com>...
> Hi,
>
> I am trying to fit data sets with the following nonlinear model:
>
> Y(t) = B0 + B1*exp(-t/tau1) + B2*exp(-t/tau2) + B3*exp(-t/tau3)
>
> For all the data sets, the parameters, B0, B1, B2, B3, tau1, tau2 and tau3 are the same.
>
> Is there a way to fit multiple data sets at the same time?
>
> The reason to do this is that it seems there are multiple solutions for a single data set. I am thinking fitting multiple data sets simultaneously may help reduce the variability.
>
> Thanks a lot
>
> Bo

Maybe I'm missing something. But, if you can fit the above equation
to an individual data set, why can't you "lump" all the data sets together
and fit the above equation to that combined data set?
From: Bo on 6 Jul 2010 16:18
"someone" <someone(a)somewhere.net> wrote in message <i0vr4v$kbm$1(a)fred.mathworks.com>...
> "Bo " <qiangbo(a)gmail.com> wrote in message <i0vo7k$988$1(a)fred.mathworks.com>...
> > Hi,
> >
> > I am trying to fit data sets with the following nonlinear model:
> >
> > Y(t) = B0 + B1*exp(-t/tau1) + B2*exp(-t/tau2) + B3*exp(-t/tau3)
> >
> > For all the data sets, the parameters, B0, B1, B2, B3, tau1, tau2 and tau3 are the same.
> >
> > Is there a way to fit multiple data sets at the same time?
> >
> > The reason to do this is that it seems there are multiple solutions for a single data set. I am thinking fitting multiple data sets simultaneously may help reduce the variability.
> >
> > Thanks a lot
> >
> > Bo
>
> Maybe I'm missing something. But, if you can fit the above equation
> to an individual data set, why can't you "lump" all the data sets together
> and fit the above equation to that combined data set?

Thanks for your comments.

Because they are collected at different conditions. Therefore they are not the same. The function Y(t) takes in a set of known parameters that characterize the condition. So strictly speaking, the Y(t) are not the same for each set of data but they are correlated.

Please let me know if the question is clear now.

Thanks.
From: someone on 6 Jul 2010 16:30
"Bo " <qiangbo(a)gmail.com> wrote in message <i1032d$snq$1(a)fred.mathworks.com>...
> "someone" <someone(a)somewhere.net> wrote in message <i0vr4v$kbm$1(a)fred.mathworks.com>...
> > "Bo " <qiangbo(a)gmail.com> wrote in message <i0vo7k$988$1(a)fred.mathworks.com>...
> > > Hi,
> > >
> > > I am trying to fit data sets with the following nonlinear model:
> > >
> > > Y(t) = B0 + B1*exp(-t/tau1) + B2*exp(-t/tau2) + B3*exp(-t/tau3)
> > >
> > > For all the data sets, the parameters, B0, B1, B2, B3, tau1, tau2 and tau3 are the same.
> > >
> > > Is there a way to fit multiple data sets at the same time?
> > >
> > > The reason to do this is that it seems there are multiple solutions for a single data set. I am thinking fitting multiple data sets simultaneously may help reduce the variability.
> > >
> > > Thanks a lot
> > >
> > > Bo
> >
> > Maybe I'm missing something. But, if you can fit the above equation
> > to an individual data set, why can't you "lump" all the data sets together
> > and fit the above equation to that combined data set?
>
> Thanks for your comments.
>
> Because they are collected at different conditions. Therefore they are not the same. The function Y(t) takes in a set of known parameters that characterize the condition. So strictly speaking, the Y(t) are not the same for each set of data but they are correlated.
>
> Please let me know if the question is clear now.
>
> Thanks.

I still don't understand.

Then what does "fitting multiple data sets simultaneously" mean?
From: Walter Roberson on 6 Jul 2010 17:24
Bo wrote:

> I am trying to fit data sets with the following nonlinear model:
>
> Y(t) = B0 + B1*exp(-t/tau1) + B2*exp(-t/tau2) + B3*exp(-t/tau3)
>
> For all the data sets, the parameters, B0, B1, B2, B3, tau1, tau2 and
> tau3 are the same.
>
> Is there a way to fit multiple data sets at the same time?
>
> The reason to do this is that it seems there are multiple solutions for
> a single data set. I am thinking fitting multiple data sets
> simultaneously may help reduce the variability.

Are any of those parameters known ahead of time, or are all 7 of them to be
estimated?

If all 7 of them are to be estimated, then fitting multiple data sets will not
help: your problem would inherently always have multiple solutions.


If all 7 are to be estimated, then suppose you were to let tau2 == tau3, then
B2*exp(-t/tau2) + B3*exp(-t/tau3)
would be the same as
B2*exp(-t/tau2) + B3*exp(-t/tau2)
which would be the same as
(B2+B3)*exp(-t/tau2)
and so it would only be possible to estimate the sum (B2+B3) and not possible
to find unique values for B2 and B3.

By similar logic, when tau1 == tau2, you cannot distinguish B1 from B2, and
when tau1 == tau3 then you cannot distinguish B1 from B3.

Likewise, if one of the tau parameters becomes infinity, then t divided by
that parameter would be 0 and exp(0) is 1, which would make it impossible to
separate B0 from the B corresponding to that tau.

Between these cases, I count at least 11 combinations of circumstances in
which some parameter cannot be separated from some other parameter, and as you
have not provided any constraint such as 0 < tau1 < tau2 < tau3 < infinity,
the search *will* enter that search space and promptly deteriorate. Adding
additional cases will not have the effect of prohibiting these possibilities,
so using multiple data sets will not help under these conditions.


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