smoofit.utils.build_1d_tunfold_matrix
- smoofit.utils.build_1d_tunfold_matrix(n_bins, bin_widths=None)[source]
Build matrix of discrete second derivatives for TUnfold-like regularization in 1 dimension
The second derivative operator of a 1D array with
nentries is the(n-2,n)matrix\[\begin{split}\begin{pmatrix} 1 & -2 & 1 & 0 & \dots & 0 \\ 0 & 1 & -2 & 1 & \dots & 0 \\ \dots & & & & & \\ 0 & \dots & 0 & 1 & -2 & 1 \end{pmatrix}\end{split}\]This assumes uniform spacing (bin width) of the entries. If that is not the case, the estimation of the derivatives should take that into account. Given the differential cross sections \(s_{i-1}\), \(s_i\), \(s_{i+1}\), and the corresponding bin widths \(w_{i-1}\), \(w_{i}\), and \(w_{i+1}\), the distances between the bin centers are therefore \(d_i = 0.5(w_{i-1} + w_i)\) and \(d_{i+1} = 0.5(w_i + w_{i+1})\), and the second derivative in \(i\) is estimated as:
\[s_{i-1} \frac{2}{d_i (d_i + d_{i+1})} + s_i \frac{4}{d_i d_{i+1} (d_i + d_{i+1})} + s_{i+1} \frac{2}{d_{i+1} (d_i + d_{i+1})}\]- Parameters
n_bins (
int) – number of elements in the vector for which the second derivative is to be computed (should be \(\geq 3\))bin_widths (
Union[List[float],ndarray,DeviceArrayBase,None]) – 1D array withn_binsentries specifying the bin widths of the distribution
- Return type
DeviceArrayBase- Returns
matrix of second derivatives