Noise reconstruction

SVDEntropyTruncation

class boulderopal.noise_reconstruction.SVDEntropyTruncation(rounding_threshold=0.5)

Configuration for noise reconstruction with the singular value decomposition (SVD) method using entropy truncation.

Parameters:

rounding_threshold (float, optional) – The rounding threshold of the entropy, between 0 and 1 (inclusive). Defaults to 0.5.

Notes

The singular value decomposition (SVD) method first finds a low rank approximation of the matrix of weighted filter functions \(F^\prime\):

\[F^\prime \approx U \Sigma V ,\]

where matrices \(U\) and \(V\) satisfy that \(U^\dagger U = VV^\dagger = \mathbb{I}_{n_{\mathrm{sv}} \times n_{\mathrm{sv}}}\), and \(\Sigma\) is a diagonal matrix of \(n_{\mathrm{sv}}\) truncated singular values, which in the entropy truncation method are determined by the entropy of the singular values \(E\).

The entropy truncation method calculates the value \(2^E\) and rounds the value to an integer \(n_{\mathrm{sv}}\). When rounding the value \(2^E\), the floor of \(2^E\) plus the rounding threshold that you chose is taken. Therefore a small value leads to rounding down, while a large value leads to rounding up. The \(n_{\mathrm{sv}}\) is then used as the truncation value.

The SVD method then estimates the noise power spectral density (PSD) \(\mathbf S\) as:

\[{\mathbf S}_{\mathrm{est}} = V^\dagger\Sigma^{-1}U^\dagger{\mathbf I} .\]

This method calculates the uncertainties in estimation using error propagation if you provide measurement uncertainties.