(calibration_circe_principle_linearity_hypothesis)=
# The linearity hypothesis

For every response, the quantity of interest is $R^{\rm exp}_j - R^{\rm code}_j$ which can also be 
written as, if $R^{\rm real}_j $ denotes the real value of the response $R_j$, $R^{\rm exp}_j - 
R^{\rm code}_j = (R^{\rm exp}_j - R^{\rm real}_j) + (R^{\rm real}_j - R^{\rm code}_j) $. It is the 
sum of two independent random variables:

- ($R^{\rm exp}_j - R^{\rm real}_j$): the experimental uncertainty which follows a centered normal distribution 
of known standard deviation <inlineequation> <alt> $\sigma^{\rm exp}$.

- ($R^{\rm real}_j - R^{\rm code}_j$): which is obtained from a first-order development as

    ```{math}
    R^{\rm real}_j - R^{\rm code}_j = \sum_{i=1}^{q} \frac{\partial R^{\rm code}_j}{\partial \alpha_i} 
    (\alpha_{j,i} - \alpha_{i}^{\rm nom})
    ```

    In this definition, $\alpha_{j,i}$ is the unknown value assigned to the i-th parameter such that 
    $R_j^{\rm code}(\alpha_{j,1},\ldots,\alpha_{j,q}) = R^{\rm real}_j $ ($\alpha_{j,i}$ being 
    different for every response) and $\alpha_j^{\rm nom}$ is the nominal value of this i-th parameter 
    (generally 0).