Data are not uncertain, but your inference based on data is. You would use what ever appropriate statistics to understand the uncertainty of your inference based on your data. What exactly is it that you want estimates of uncertainty for?

Thanks Dr. Dan Gwinn for your answer. I have a function for Cherenkov light density estimation. This function depends on some parameters. I have parameterized this function and got energy dependence function with some coefficients. I got a good results in comparison with experimental data. I have already estimated the Error percentage. BUT I prefer to find a best way for uncertainty estimation. I appreciate your efforts if you give me the best way for uncertainty calculation.

With my best regards.

So for clarity, you are seeking uncertainty estimates such as standard errors or confidence intervals for the parameter estimates of your function? Is this correct?

What optimization routine and/or program are you using to estimate the parameters of your function?

Many programs supply the standard errors of parameter estimates, but not all.  For example, if you are using Optim in program R, you can profile the likelihood to obtain confidence intervals of parameter estimates OR you can use the quadratic approximation (estimates the variance of parameters by inverting the hessian matrix). Alternatively, if you are using Solver in Excel to estimate parameters, you only have the likelihood profile option.

If these suggestion are not yet helpful, please provide more details and I will continue to try to help you.

best, Dan 

Dear Dr. Dan,

I don't know how can I thank you for your answer and for trying to help me. I'm using Fortran programing to estimate the Cherenkov light density (Q) with its parameters. For accuracy between the calculation and experimental data, I have used the relation [(Qcalc - Qexp)/Qexp]^2*100% for example to get the error percentage.

I am trying to find  a method for uncertainty calculation to give me +- in data results.

Kind regards,

Ahmed

This paper is an example in serology, but I think that the principle is just the same in your case (observed and expected values):

http://www.clinchem.org/content/52/3/526.full

Hope that will be useful

Best regards

Thanks Dr. Federico Nave for your help. I will read this paper carfuly. I wish this will help.

Kind regards,

Ahmed Al-Rubaiee

A.A. -

I am not familiar with your particular application, but are you interested in comparing experimental and theoretical results? Do you have paired data, where x could be the theoretical data, and y could be the observed data that should correspond to it? If so, you might expect y = bx + e, where e is an estimated residual that should be heteroscedastic. A scatterplot might be informative, and b may be of interest. If b is somewhat greater than its standard error, it may show a systematic bias.

Such a regression could be used to find the variance of the prediction error of y, assuming x is correct. But uncertainty in x increases overall uncertainty. It depends on what you are trying to study. You might look at the first attachment where an excel file is used to plot confidence intervals for the predicted y (read vertically up and down, not perpendicular to the line).

I did some z-score-type calculations found through the second attachment, but that paper may be of less interest overall.

Hopefully this relates somewhat to your question.

Cheers - Jim 

PS - The square root of the estimated variance of the prediction error is STDI in SAS PROC REG, if that helps. 

Data CRE Prediction 'Bounds' and Graphs Example for Section 4 of ...

Article Using Prediction-Oriented Software for Survey Estimation - P...

Thank you Dr. James R. Knaub for these wonderful articles. Thank you again for your help and interesting.

Accept my best regards,

Ahmed

https://www.researchgate.net/publication/258835771_Decision_making_under_uncertainty_in_energy_systems_State_of_the_art

Article Decision making under uncertainty in energy systems: State of the art