# Main principle of the CIRCE method

As already stated, CIRCE (which stands for "Calcul des Incertitudes Relatives aux 
Correlations Elementaires") is a statistical method in which the uncertainties are defined through 
random variables, mean values and standard deviations {cite}`Circe,DeCrecy01`. Usually, if one 
considers $P_i$ the unobserved parameters ($i$ being the number of these parameters, limited here 
to a certain number $q$), one can write the following equation

```{math}
P_i = p_i \times P^{\rm nom}_i
```

Each physical parameter is expressed as a function of a nominal value ($P^{\rm nom}_i$) and a 
multiplier coefficient $p_i$. A relation can be constructed between these multipliers and the 
parameters considered by CIRCE, as 

**{eq}`eq_circeparamdef`: Relation between multipliers and CIRCE parameters.**

```{math}
:label: eq_circeparamdef

p_i = 1 + \alpha_{i} \; {\rm or} \; p_i= e^{\alpha_{i}}
```

The nominal value of $\alpha_i$ is set to 0 (which implies that the nominal value of the influential 
physical model is equal to 1). The other inputs needed by the method are the observed data (or responses) 
hereafter referred to as $R^{\rm exp}_j$ ($j$ being a realisation of the SET experiment), and the 
corresponding code result $R^{\rm code}_j$. CIRCE combines the difference between the experimental 
results and the code predictions ($R^{\rm exp}_j - R^{\rm code}_j$) with the derivatives of each code response 
with respect to each parameter $\dfrac{\partial R^{\rm code}_j}{\partial \alpha_i}$. It is also 
possible to take into account the experimental uncertainties of the response, called hereafter 
$\delta R^{\rm exp}_j$. This procedure should lead to the estimation, for every $\alpha_i$ parameter, 
of its mean value $b_i$ (for bias) and its standard deviation, $\sigma_i$.

In order to perform this estimation, there are two main hypotheses done by the CIRCE method:

- the linearity between the code response and each parameter $\alpha_i$. This hypothesis is clearly 
visible since first-order derivatives are used for the estimation $\dfrac{\partial 
R^{\rm code}_j}{\partial \alpha_i}$. It is further discussed in 
[](#calibration_circe_principle_linearity_hypothesis).

- the normality of the $\alpha_i$ parameters. A hypothesis on the PDF of CIRCE parameters is indeed 
compulsory, leading to the hypothesis of normality or lognormality of the $p_i$ multiplier if 
the additive or exponential change of variable is used in {eq}`eq_circeparamdef`. 
This is further discussed in [](#calibration_circe_principle_normality_hypothesis).

```{toctree}
principle/linearity_hypothesis
principle/normality_hypothesis
```