Exponential least-squares fitting and initial parameters

In summary, the conversation discusses the best method for performing an exponential function least-squares fitting with multiple exponents. Different suggestions are made, including looking at the data to choose initial parameters, using arbitrary starting values, or using a more sophisticated method that avoids initial values and iterative procedures. It is also noted that the least-squares method does not work for computing all parameters in this case. Further reading on specialized methods for this issue is recommended.
  • #1
rkaminski
11
0
Dear All,

I would like to do an exponential function least-squares fitting, but having two or more exponents. For example the function looks like this:

[itex]y (x) = A \exp (-x/a) + B \exp (-x/b) [/itex]

where [itex]A[/itex], [itex]a[/itex], [itex]B[/itex] and [itex]b[/itex] are the least-squares fitted parameters. My question is how to obtain the initial parameters? I could assume that that my initial function follows the one-exponential function and then get initial estimates of [itex]a[/itex] and [itex]A[/itex], but then how to get initial values of both [itex]a[/itex] and [itex]b[/itex], and other parameters? I cannot assume for example [itex]b = 0[/itex] obviously. Thanks in advance.

Best wishes,

Radek
 
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  • #2
Where does your data come from? In particular, can you look at the data to choose initial parameters?

If yes: Draw it on a logarithmic scale, see if you can identify the two exponentials there, estimate the parameters.
If no, or the method above does not help: Use arbitrary starting values (with different values for a and b), hope that the fit converges - it should do that if the model is not completely wrong.
 
  • #3
Hi !

I think that the best way is to use a method which do not requires guessed initial values of the parameters.
There is a straightforward method for doing that : the non-linear regression is transformed to a linear regression thanks to some convenient preliminary numerical intégrations.
See pages 73-75 of the paper "Régressions et équations intégrales" published on Scribd :
http://www.scribd.com/JJacquelin/documents
The theory is written in French, but you don't need it. The practical application for fitting your function is written in English (The notations are not the same of yours. It is not difficult to change the symbols).
With this method, the approximates of the parameters are generaly sufficient for the direct practical use. But, if you need a specific fitting according to some particular criteria, it's up to you to use the good values already obtained as initial values for an iterative process using a software for non-linear regression.
 
  • #4
JJacquelin said:
I think that the best way is to use a method which do not requires guessed initial values of the parameters.

like least-squares. (i didn't even think that the least-squares method had iterations or initial conditions.)
 
  • #5
rkaminski said:
My question is how to obtain the initial parameters?

In general least squares fitting amounts to trying to find values of variables that minimize a (non-linear) function. There may be different combinations of values of the variables that do equally well. Are you applying some minimization method that requires "initial" values to get it started?
 
  • #6
rbj said:
like least-squares. (i didn't even think that the least-squares method had iterations or initial conditions.)
Of course, the least-squares method doesn't require iterations or initial conditions.
But the least-squares method doesn't work in the case of the of the equation :
y = A*exp(-x/a)+B*exp(-x/b)
because the parameters a and b are in the argument of the exponential function.
The least-squares method only works for A and B.
So you cannot use the least-squares method to compute a, b, A and B.
You have to search a more sophisticated method.
 
  • #7
Hi JJacquelin,

I will have a look at the French papers you send. These are very interesting. No problem for me to understand them:)

However, I don't understand your last post. Non-linear least-squares method does require initial values of the parameters. That is why the procedure needs to be cycled and in each case we obtain better estimates of the refined parameters. If it were linear least-squares then there is no problem like this obviously.

Radek
 
  • #8
rkaminski said:
However, I don't understand your last post. Non-linear least-squares method does require initial values of the parameters. That is why the procedure needs to be cycled and in each case we obtain better estimates of the refined parameters. If it were linear least-squares then there is no problem like this obviously.

Thay is exactly what I intended to say. Sorry if my writting was not clear enough when I was talking of linear least-squares fitting.
 
  • #9
JJacquelin said:
So you cannot use the least-squares method to compute a, b, A and B.
You have to search a more sophisticated method.
You can, you just don't get analytic solutions. That's why initial values and a fitting procedure are used.
 
  • #10
mfb said:
You can, you just don't get analytic solutions. That's why initial values and a fitting procedure are used.
OK. You are right. That is what I call "sophisticated methods".
The method I propose avoids initial values and itterative procedure, because the preliminary numerical integrations leads to a linear mean squares fitting.
 
  • #11
This is an important issue in some fields of engineering, and specialized methods have been developed for it. See here for a summary and bibliography:
http://www.csrc.sdsu.edu/csrc/research_reports/CSRSR2009-04.pdf

"General purpose" optimization methods tend not to work well, because the slowest decaying exponential tends to dominate the others.
 

1. What is exponential least-squares fitting?

Exponential least-squares fitting is a statistical method used to find the best-fit curve for a set of data points that follow an exponential trend. It involves minimizing the sum of squared errors between the data points and the predicted values from the exponential model.

2. How is exponential least-squares fitting different from linear least-squares fitting?

Exponential least-squares fitting is used for data that follows an exponential trend, while linear least-squares fitting is used for data that follows a linear trend. In exponential fitting, the model is in the form y = ae^(bx), while in linear fitting, the model is in the form y = mx + b.

3. What are initial parameters in exponential least-squares fitting?

Initial parameters are the starting values used for the coefficients in the exponential model. These values are usually chosen based on prior knowledge or estimation, and the fitting process adjusts them to find the best-fit curve for the data.

4. How do you determine the quality of an exponential least-squares fit?

The quality of an exponential least-squares fit can be determined by calculating the coefficient of determination (R^2), which measures the proportion of the variation in the data that is explained by the model. A higher R^2 value indicates a better fit.

5. Can exponential least-squares fitting be used for any type of data?

No, exponential least-squares fitting is only suitable for data that follows an exponential trend. If the data does not exhibit an exponential pattern, other methods such as polynomial fitting or non-linear least squares fitting may be more appropriate.

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