It is possible, but the better question is whether it is practical to do so in your situation.

A good starting point would be to start with some experimental work where you assess in detail the accuracy in a range of situations. A "full model" would need to differentiate between various sources of error. Two types of error are:

(i) between-instrument-variation .... different instruments may give different measurements simply because different instruments are used .

(ii)  within-instrument-variation: .... measurements of the same underlying value with the same instrument will not always be the same if set-up on different occasions under identical circumstances (observation/recording error).

Other possible sources of variation would include at least temperature and other environmental factors, but there are ways in which such effects might be combined into other "sources of variation".

A good experiment would measure these variations in a range of situations such that a realistic range of "sizes" and "environments" is represented. This would be followed/guided by a first-principles analysis of the experimental results.

If a stand-alone experiment is not feasible for your situation, then you might at least be able to introduce some of the above ideas by including repeated-measurements into your practical situation. 

It may be that you want an answer that is more directly related to your particular situation, but you can only get a vague answer since it all depends on the details of what you are trying to do. In general you would need to have a full statistical model relating "true values", observations and errors (which might involve a non-normal distribution). An often-applicable approach would then be to apply maximum-likelihood. Alternatively, if you are in very simple situation where you have a model in which all the parameters describing the size of errors are known, then you might be able to use a starting-level Bayesian approach, by using Bayes' Theorem to start from

(a) the conditional distribution of observations given true values;

(b) the marginal distribution of true values;

to give ....

(c) the conditional distribution of true values given the observations.

This last can be considered as fully representing the possible errors in the observations, and so replaces a linear-propagation-of-errors approach. You may find it possible to devise some approximation based on the conditional distribution to represent uncertainty at a level of detail you can use.

I did 1, 3 and 5 times repetitions for the measurements.

Measurements are accurate according to the claimed accuracy by manufacturing factory but they are not precise!

For example, as you can see in the attached picture, relationship between the measurements and true value cannot be modeled linearly nor non-linear.

Could you please suggest some ideas?

Thank you very much.

P.S: I am not sure why the device is so unstable but modelling error due to  temperature wont help.

P.S: black, red and green points represent relationship between the measurements and true values in three different environments.

I know that I can simply use some topological relationships to adjust the measurements but I am interested more on the mathematical/ statistical models capabilities!

I have no knowledge of magnetrometry, so can't give a really informed comment. No doubt you have done internet searches for information. You might try  https://en.wikipedia.org/wiki/Magnetometer  or www.iom3.org/fileproxy/322872 .

On looking at your graph, I am struck by the two variables both lying in the 0 to 1 interval, with the exception of a slightly negative measurement . If the 0 to 1 scale is somehow intrinsic to your instrument or to the thing being measured, it would be good to take that into account somehow. You have partly done this with your choice of trial functions, but a slightly more general form is: y=xa(1-(1-x)b)   ... but it is doubtful if this will produce anything noticeably better . However it may be worth thinking about the behaviour of your red points ... is this evidence that your instrument is hard-limiting the measured values to not produce measurements greater than one?  In which case, why not the same at the other end of the scale? The detail of such things should influence how you model errors.

The pdf file linked above (www.iom3.org/fileproxy/322872 ), indicates that instruments can be affected by, for instance, nearby metal poles.

Your graph perhaps indicates that you do not yet have enough trials to form any form conclusions. If you are worrying that errors may be of different sizes at different points on the scale then you might want to target this. If the 0 to 1 scale is important then it is possible that the sizes of errors would either increase or decrease near either or both of these limits.

We considered 4 intervals for the measurements (as seen in the attached picture).

Each measurements would be from 0 to 1 which is the ratio of (measurement - lower bound of the interval in which it is located) / (upper bound of the interval in which it is located - lower bound of the interval in which it is located). (for both measurements and true values).

Thank you very much for your suggestions.

 Indeed the problem get worse in environments with electrical instruments.

I do wanted to see if some academics had similar experiments.

Here are some of mathematical and statistical models that we used.

https://www.researchgate.net/publication/319165010_Uncertainty_modelling_based_on_Homotopy_continuation

Presentation Uncertainty modelling based on Homotopy continuation

You must consider that ANY quantity in Nature is noisy (i.e., subject to fluctuations and irreproducibilities of various kinds). So, even with an ideal instrument, physical measurementes MUST be noisy.

The first thing you should do is to devise ways of separating the noise due to the mesuring device from the actual noise in the measured quantity (magnetic field in this case). This is normally approached in two ways: (i) comparing the measurements obtained by 2 or more instruments and (ii) testing the precision of your instrument under known and controlled conditions (this may be more tricky and costly than what many people think; the manufacturer should be equipped to do this as part of their QC).

In any case, whatever you measure, never give it for granted that the measured quantity itself is constant - that is just physically impossible.

Dear Prof. Stan Sykora ,

Thank you very much for your comment.

Kind regards,

Ali