# 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.