How to Propagate Errors Across Models?

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Discussion Overview

The discussion centers on the topic of error propagation across two models in a statistical context. The inquiry involves how to carry forward errors from the first model, which is nonlinear, to the second model that utilizes the output of the first. The scope includes theoretical considerations of error estimation and propagation methods.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Yaal presents a scenario involving two models and seeks guidance on how to propagate errors from the first model's output to the second model.
  • Yaal questions how to assign errors to each element of the output vector Y from Model 1 and how to propagate these errors to the output of Model 2.
  • One participant suggests using bootstrapping techniques to estimate the errors in the output of Model 2, indicating a potential method for handling nonlinearity and complexity in error propagation.
  • A later reply corrects a broken link to a resource on bootstrapping, ensuring that participants have access to the relevant information.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the best method for error propagation, and multiple approaches, including bootstrapping, are suggested without agreement on a definitive solution.

Contextual Notes

The discussion does not clarify the specific assumptions or limitations of the proposed methods, nor does it resolve the mathematical steps involved in error propagation between the two models.

nehajo88
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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
 
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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 )
 
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