Mathematical background

Sensitivity coefficients

A relative sensitivity coefficient of a response \(R\) to a parameter \(p_i\) is commonly written as

\[S_i = \frac{p_i}{R}\frac{\partial R}{\partial p_i}.\]

In a multi-group treatment, the sensitivity profile is represented by a vector whose components correspond to energy groups. Multiple nuclear-data parameters are identified by nuclide and ENDF reaction identifiers.

Uncertainty propagation

For a sensitivity vector \(\mathbf{S}\) and a relative covariance matrix \(\mathbf{C}\), first-order uncertainty propagation follows the sandwich rule

\[\sigma_R^2 = \mathbf{S}^{\mathrm{T}}\mathbf{C}\mathbf{S}.\]

For several reactions, the complete result includes diagonal contributions and cross-reaction terms

\[q_{ij} = \mathbf{S}_i^{\mathrm{T}}\mathbf{C}_{ij}\mathbf{S}_j.\]

These terms are variance contributions. For display, pyNDUS can apply the sign-preserving square-root convention

\[\widetilde{q}_{ij} = \operatorname{sign}(q_{ij})\sqrt{|q_{ij}|}.\]

In particular, a negative covariance contribution is displayed as \(-\sqrt{|q_{ij}|}\). These signed roots must not be summed to obtain the total uncertainty. The response standard deviation is instead

\[\sigma_R = \sqrt{\sum_{ij}q_{ij}},\]

so the original signed variance and covariance contributions are summed before the square root is taken.

Similarity

The similarity between sensitivity profile \(i\) from the first system and profile \(j\) from the second system is evaluated using the normalized scalar product

\[E_{ij} = \frac{\mathbf{S}_{1,i}^{\mathrm{T}}\mathbf{S}_{2,j}} {\sqrt{\mathbf{S}_{1,i}^{\mathrm{T}}\mathbf{S}_{1,i}} \sqrt{\mathbf{S}_{2,j}^{\mathrm{T}}\mathbf{S}_{2,j}}}.\]

It measures geometric alignment of the profiles but does not weight individual nuclear-data parameters by their covariance. Each coefficient is normalized with the same two vectors appearing in its numerator; norms are never accumulated globally over other MT reactions. Therefore, for non-zero vector norms, \(-1 \leq E_{ij} \leq 1\), and an identical profile compared with itself has \(E_{ii}=1\).

Representativity

The covariance-weighted representativity coefficient is

\[r = \frac{\mathbf{S}_1^{\mathrm{T}}\mathbf{C}\mathbf{S}_2} {\sqrt{\mathbf{S}_1^{\mathrm{T}}\mathbf{C}\mathbf{S}_1} \sqrt{\mathbf{S}_2^{\mathrm{T}}\mathbf{C}\mathbf{S}_2}}.\]

For identical systems and a non-zero denominator, \(r=1\). Unlike the unweighted similarity coefficient, representativity incorporates the relative importance and correlation structure of the nuclear-data uncertainties.

In pyNDUS the displayed ZA/MF/MT matrix contains additive contributions to this single globally normalized coefficient. The total representativity is

\[r_{\mathrm{total}} = \sum_{z,i,j} r_{z,ij}.\]

When the isotope sets differ, absent sensitivity profiles are treated as zero in the numerator, so only their intersection contributes there. The normalization terms remain the complete uncertainty of each system: isotopes exclusive to the first or second system contribute respectively to \(\mathbf{S}_1^{\mathrm{T}}\mathbf{C}\mathbf{S}_1\) or \(\mathbf{S}_2^{\mathrm{T}}\mathbf{C}\mathbf{S}_2\).

ERRORR MF and MT organization

pyNDUS groups reaction identifiers (MT) by ERRORR section (MF). For example, cross-section covariances are generally associated with MF=33, while neutron-multiplicity covariances are associated with MF=31. A given normalized MT is assigned to one MF section in the package mapping, avoiding ambiguous accumulation of the same parameter across multiple MF sections.