Chi^2 test with dominating x-errors

[egon ll]

Hi,

I am using thie χ² test to fit a dataset in order to calibrate a sensor.

$\chi^{2} = \sum_{i}{{\left(y_{i} - (a_{0}+a_{1}*x_{i}+a_{2}*x_{i}^{2})\right)^2 \over \sigma_i^2}}$

The sensor delivers raw data $x_{i}$ and the reference values $y_{i}$ are measured with an instrument of far greater precision than the sensor that is calibrated.

The coefficients $a_0, a_1, a_2$ have to be determined with the test.


The uncertainty in the $x_{i}$ and the $y_{i}$ are known. Unfortunately the $x_{i}$ uncertainties cannot be neglected.

Could anyone tell me how I can calculate the $\sigma_i$ values?

Thank you very much!

 

[AndyW]

Hello,

Thank you for sharing your question and the equation you are using for your calibration process. The \sigma_i values in your equation represent the uncertainties in the measured reference values y_{i}. In order to calculate these values, you will need to have some knowledge or estimate of the uncertainty in your reference instrument. This can be determined through previous calibrations or through manufacturer specifications.

Once you have the uncertainty in your reference instrument, you can use it to calculate the \sigma_i values in your equation. This can be done by taking the square root of the sum of the squared uncertainties for each individual measurement. This will give you a value for \sigma_i for each data point, which can then be plugged into your equation for the χ2 test.

It is important to note that the accuracy of your calibration will greatly depend on the accuracy of your \sigma_i values. Therefore, it is crucial to have reliable and accurate measurements of the uncertainty in your reference instrument.

I hope this helps and good luck with your calibration process!