# Reporting model uncertinty

I hope I'm using the right terms here. I've generated a statistical model (PLS regression) based on LWIR (8-10.5 micrometer) spectrum from some lab samples. This model predicts the concentration of a certain substance. Now let's say an environmental authority would like to use this model for their purposes, but they would like to know what is the uncertainty level of this model. I assume this has something to do with my errors (measured vs predicted, n=567), so I've plotted the errors in a histogram: How should I proceed from here? is my model uncertainty equals to the highest error (in this case 3?) meaning that my model uncertainty is 3% concentration of that substance? On one hand, it seems not right because 3% sounds a lot for an error (my concentrations range is 0-10) and as you can see the error reaches 3% only rarely, on the other hand, the model did have an error of 3%. Should I use the RMSE (1.06%) as an indicator of my model uncertainty? On one hand, it seems to represent the errors better, on the other hand, the actual error can be much higher than the RMSE in some cases. Is there some other metric or calculations I should do? Again, I apologize if my wording is not statistically accurate.

## 1 Answer

Yes, RMSE rather than observed maximum deviation would be a typical way to report verification or validation results.

RMSE behaves similarly to a standard deviation: std is a typical and sensible way to summarize the spread of a distribution, but ± 1 std is not expected to cover the full typical range of variation. The same is true for RMSE. If you like, there's nothing wrong in also reporting, say, the 95th percentile of observed error, or to construct, say, prediction intervals (however, note that this is far more complicated for PLSR than for ordinary or inverse least squares models).

Observed 95th percentile (and even more observed maximum) however do have the drawback that they are less certain ("more noisy"). OTOH, they do a much better job of giving an intuitive grasp of the magnitude of uncertainty in the predictions.

The more important point is whether your test cases are actually sufficiently representative for the cases the model will predict, and that they are truly independent of the cases used for model building (and of each other - unless you report what structure/dependence the data have and discuss its consequences).

it seems not right because 3% sounds a lot for an error (my concentrations range is 0-10)

(Assuming that this is concentration will be between 0 and 10 %)

Of course, you cannot really* conclude from the magnitude of observed error whether your error determination is correct - you'd typically want to make sure error determination/verification/validaton is correct and then conclude whether your method is fit for purpose (the application at hand).

• The most important question now is how your error compares to the expectation you had, so: What error did you expect/what is needed for the application?
• RMSE of 1.06 % analyte concentration is a relative error = $$\frac{RMSE}{c}$$ of ≈ 10 % at 10 % analyte. This would be a typical requirement for the lower limit of quantitation (LLOQ or LOQ). In other words, your model would for many applications not be considered fit for quantitative prediction.
• The most commonly used definition of the limit of detection (for qualititative predictions) corresponds to relative error of $$\frac{1}{3}$$ -- which you reach at analyte concentrations of ≈ 3%.
• OTOH, LLOQ or LOD inside the calibration range is what you'd aim for if the purpose of this particular experiment/study is to establish the LLOQ or LOD of your method.
• I used approximately above because I gave back-of-the-envelope calculations that assume homoscedasticity, i.e. RMSE is independent of analyte concentration. This is very often not the case, and for establishing LOD/LLOQ one would at the very least have a look to check whether there is noticable concentration dependence.

* Sometimes it'd good to be suspicious, though. Particularly, if the error looks too good to be true, it's often not true...

Seeing this as well as the other questions you've asked here over the last months, I'd recommend that you get yourself some more education about basic concepts in analytical chemistry/chemometrics and analytical method validation.

• Danzer: Analytical Chemistry Theoretical and Metrological Fundamentals, Springer 2007. would be a relevant book
• If you are at a university or have one closeby, you could have a look whether they have a lecture on these subjects?
• There are also chemometrics industry courses (though you'd need to check carefully whether they cover what you need: many of them consider the basics we've been discussing here as prerequisites and don't touch them).
Disclaimer: I do offer such courses, so please don't ask me which competitors I recommend.
• Thank you for the elaborate answer, you gave me a lot of thinking materials – user88484 Dec 31 '18 at 12:09