How to Propagate Errors Across Models?

[nehajo88]

Hi folks,

I have a rather simple question on error propagation - I have 2 sets of models, where the results from model are used as variables in the next model. I need to know how to carry forward errors from one to another.

Case -

Model 1: Y = a*exp(b*X) + c

The errors on X (which is a vector of about 100 samples) and Y (a vector of same size as X) are not know. From fitting the above non-linear model to the data and examining the residuals, I can calculate Mean Absolute Error, RMSE, etc. So, in the end I get a vector of Y values and a single error estimate from the model (e.g. RMSE).

Model 2: Z = s*(Y)^t + u

Where Y is the variable obtained from the results of Model 1. Applying Model 1 to a large number of new X values, I now have Y as a vector with > 10,000 elements. Each element in vector Y should have an associated error. My question is - what error should I give each element of vector Y? My next question is, once the error on each element of vector Y is known, how do I propagate this error to each element of vector Z? Finally, how do I calculate RMSE for Model 2?

All help will be much appreciated!

Thanks,
Yaal

 

[Dale]

If your process involves nonlinearity and complicated methods then your best bet will be to use some bootstrapping technique to get an estimate of the errors in Z.

http://en.wikipedia.org/wiki/Bootstrapping_(statistics )

 

[dauto]

I fixed the broken link. http://en.wikipedia.org/wiki/Bootstrapping_(statistics)

 

[Dale]

Thanks

 

[pmacias]

Thank you for your question about error analysis and propagation. It is important to properly account for errors and uncertainties in scientific models, as they can greatly affect the accuracy and reliability of the results.

In your case, it is important to first determine the errors on the variables in Model 1, X and Y. This can be done by examining the residuals and calculating metrics such as Mean Absolute Error and RMSE. These errors can then be propagated to Model 2, where Y is used as a variable.

To determine the error on each element of vector Y in Model 2, you can use the error propagation formula, which states that the error in a function of multiple variables is equal to the square root of the sum of the squares of the individual errors. In this case, the error on each element of Y would be the square root of the sum of the squares of the errors on a, b, X, and c in Model 1.

Once the error on each element of Y is known, you can then propagate these errors to the elements of vector Z using the same error propagation formula. This will give you the error on each element of Z.

To calculate the RMSE for Model 2, you can use the formula for calculating RMSE, but instead of using the actual values of Y and Z, you would use the predicted values from your model and the corresponding errors on each element.

In summary, to properly account for errors in your models, you should calculate the errors on the variables in Model 1 and propagate them to Model 2 using the error propagation formula. Then, use the propagated errors to calculate the RMSE for Model 2. I hope this helps and please let me know if you have any further questions.