7.6.1.1. 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 \(\sigma^{\rm exp}\).

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

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

  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).