Non linear curve fit - parameter accuracy

In summary, you can calculate the standard error of the i-th parameter of a given parameter set by \sigma_i=\sqrt{\chi^2(\bold{p})C_{ii}} where C is the variance-covariance matrix. It is calculated from the Jacobian F (F_{i,j}=\partial f(\bold{p},x_j)/\partial p_i) as \bold{C}=(\bold{F}^{'} \cdot \bold{F})^{-1} If you understand this, explain me, please!
  • #1
arwelbath
10
0
Hi,
Suppose i fit some data with a curve, using Levenberg-Marquardt (or equivalent). How do I estimate the accuracy of the parameters (by this i mean p plus or minus something). I've read somewhere that its done using the jacobian and covariance matricies I think, but not sure. Anyone know if there's a 'standard' way of doing this?
Cheers
 
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  • #2
arwelbath said:
Hi,
Suppose i fit some data with a curve, using Levenberg-Marquardt (or equivalent). How do I estimate the accuracy of the parameters (by this i mean p plus or minus something). I've read somewhere that its done using the jacobian and covariance matricies I think, but not sure. Anyone know if there's a 'standard' way of doing this?
Cheers

most usually you determine the difference between each point you fit and the fitted point and you square all these difference and sum them, you then divide this sum by the number of points you have summed and call this "the mean squared error" (tip: google this).
 
  • #3
arwelbath said:
Hi,
Suppose i fit some data with a curve, using Levenberg-Marquardt (or equivalent). How do I estimate the accuracy of the parameters (by this i mean p plus or minus something). I've read somewhere that its done using the jacobian and covariance matricies I think, but not sure. Anyone know if there's a 'standard' way of doing this?
Cheers

As I read in an old Origin Help, the standard error of the i-th parameter of a given parameter set p is

[tex]\sigma_i=\sqrt{\chi^2(\bold{p})C_{ii}}[/tex]

where C is the variance-covariance matrix. It is calculated from the Jacobian F ([tex]F_{i,j}=\partial f(\bold{p},x_j)/\partial p_i)[/tex] as
[tex]\bold{C}=(\bold{F}^{'} \cdot \bold{F})^{-1}[/tex]

If you understand this, explain me, please! :smile:

ehild
 
  • #4
Erm. Thanks gerben for the definition of chi squared. Not quite what I was after.
 
  • #5
The "Numerical recipes in .." books have a discussion of this. You can find the C book online at

http://www.library.cornell.edu/nr/bookcpdf.html

The second half of chapter 15 discusses non-linear fitting and uncertainty of the estimated parameters
 
Last edited by a moderator:

1. What is non-linear curve fitting?

Non-linear curve fitting is a method used in scientific research to find the best mathematical model that describes the relationship between two or more variables. It is used when the relationship between the variables is not linear, meaning that it cannot be described by a straight line. Non-linear curve fitting involves finding the optimal values of parameters in a chosen mathematical model that minimizes the difference between the model and the actual data.

2. Why is non-linear curve fitting important?

Non-linear curve fitting is important because it allows scientists to analyze complex data and make accurate predictions. It is used in many fields of science, including physics, biology, chemistry, and economics, to understand the relationships between variables and make predictions about future outcomes.

3. How is the accuracy of parameters determined in non-linear curve fitting?

The accuracy of parameters in non-linear curve fitting is determined by calculating the standard error of the fit. This is a measure of the difference between the fitted curve and the actual data points. A lower standard error indicates a better fit and therefore more accurate parameters.

4. What are some common challenges in non-linear curve fitting?

One of the main challenges in non-linear curve fitting is choosing the appropriate mathematical model. There are many different models to choose from, and the wrong choice can lead to inaccurate results. Another challenge is finding the optimal values for the parameters, which can be a time-consuming process.

5. How can I improve the accuracy of non-linear curve fitting?

There are several ways to improve the accuracy of non-linear curve fitting. First, it is important to carefully select the mathematical model and ensure that it is appropriate for the data. Additionally, increasing the amount of data and using more advanced fitting techniques, such as gradient descent, can also improve accuracy. Finally, it is important to carefully analyze the results and consider any potential sources of error that may have affected the accuracy of the fitting process.

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