Noise reconstruction

ConvexOptimization

class boulderopal.noise_reconstruction.ConvexOptimization(power_density_lower_bound, power_density_upper_bound, regularization_hyperparameter)

Configuration for noise reconstruction with the convex optimization (CVX) method.

Parameters:

• power_density_lower_bound (float) – The lower bound for the reconstructed power spectral densities. It must be greater than or equal to 0.

• power_density_upper_bound (float) – The upper bound for the reconstructed power spectral densities. It must be greater than the power_density_lower_bound.

• regularization_hyperparameter (float) – The regularization hyperparameter \(\lambda\).

Notes

The CVX method finds the estimation of the power spectral density (PSD) matrix \({\mathbf S}\) by solving the optimization problem:

\[{\mathbf S}_{\mathrm{est}} = \mathrm{argmin}_{\textbf S} (\| F'{\mathbf S} - {\mathbf I} \|_2^2 + \lambda \| L_1 {\mathbf S} \|_2^2) ,\]

where \(F^\prime\) is the matrix of weighted filter functions and \(\| \bullet \|_2\) denotes the Euclidean norm and \(L_1\) is the first-order derivative operator defined as

\[\begin{split}\begin{align} L_1 = \begin{bmatrix} -1 & 1 & & \\ & \ddots & \ddots & \\ & & -1 & 1 \\ \end{bmatrix}_{(K - 1) \times K} . \end{align}\end{split}\]

\(\lambda\) is a positive regularization hyperparameter which determines the smoothness of \({\mathbf S}_{\mathrm{est}}\). If you provide uncertainties in measurements, this method calculates the uncertainties in estimation using a Monte Carlo method.