Global fitting with shared parameters

In summary, the conversation discusses the need for an algorithm that can perform simultaneous fitting on multiple curves with shared parameters. The Levenberg-Marquardt algorithm is mentioned as a possible solution for single-curve fitting, but implementing it for global curve fitting with shared parameters can be computationally challenging. The question is posed whether there are any algorithms specifically designed for multi-curve fitting or if modifications to single-curve fitting techniques are necessary. The conversation ends with a request for further information from someone who has experience in this area.
  • #1
raul_l
105
0

Homework Statement



Hi
I'm looking for an algorithm that can perform a fitting procedure on many curves simultaneously. Let's say, for example, that I have 3 exponential decay curves with 3 different decay times but they all share the same initial amplitude. So I have to find a fit for 4 parameters (in reality I'm dealing with much more complicated systems).

Homework Equations



Levenberg-Marquardt:
[tex] \delta a = \frac{1}{\bold{J}^T\bold{J}+\lambda \bold{diag}(\bold{J}^T\bold{J})} \bold{J}^T \boldsymbol{\chi} [/tex]
with the Jacobian
[tex] J_{ij} = \frac{\partial f(x_{i};\bold a)}{\partial a_{j}} [/tex]
where f is the fitting function, x are datapoints and a are the parameters to be fitted.
[tex] \boldsymbol{\chi} [/tex] is defined as
[tex] \chi_i = \frac{y_i - f(x_{i};\bold a)}{\sigma_i} [/tex]

The Attempt at a Solution



So far I've been using the Levenberg-Marquardt algorithm to find fits for single curves. But how to implement global curve fitting when some of the parameters are shared?

I guess a straightforward approach would be to construct a global function that takes the whole data set (constructed from all the curves) as argument. For example, if a single curve consists of 10 000 data points and I'm fitting for 5 parameters, I would need to build Jacobian with dimensions 10 000 x 5 for single-curve-fitting. Now assume I have 10 curves with 2 of those 5 parameters shared. Then I would need a 100 000 x 32 Jacobian which is pretty massive.

My question is this. Are there any algorithms especially designed for multi-curve-fitting or do I have to use the usual single-curve-fitting techniques with a few modifications?
I hope anyone has had experience in that area.
 
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  • #2
Hi,

I have the same problem and was wondering if you solved it by now. If so, could you please throw some light on what you did.

Thanks,
 

What is global fitting with shared parameters?

Global fitting with shared parameters is a statistical method used in scientific research to analyze data from multiple experiments or studies simultaneously. It involves fitting a single model to all the data sets, while allowing certain parameters to vary across the different data sets.

Why is global fitting with shared parameters important?

Global fitting with shared parameters allows for more accurate and efficient analysis of complex data sets. By combining multiple data sets, it can provide a more comprehensive understanding of the underlying phenomena being studied.

What are the benefits of using global fitting with shared parameters?

Some benefits of using global fitting with shared parameters include increased statistical power, improved precision of parameter estimates, and the ability to detect subtle relationships between variables that may not be apparent when analyzing individual data sets separately.

What types of data sets are suitable for global fitting with shared parameters?

Global fitting with shared parameters is suitable for any data sets that contain similar variables and can be described by the same model. This method is commonly used in fields such as biology, chemistry, and physics to analyze data from experiments with multiple conditions or treatments.

What are some potential limitations of global fitting with shared parameters?

One potential limitation of global fitting with shared parameters is the assumption that the underlying model is the same for all data sets. This may not always be the case and can lead to biased results. Additionally, this method may be more computationally intensive and require more advanced statistical techniques compared to analyzing individual data sets separately.

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