Estimators

See also

For notation and terminology (\(x_0\), \(H\), \(S_0\), DOFS, etc.), see Terminology & Notation.

Estimator (Base Class)

Inputs

Estimator.n_z	Number of observations (length of z).
Estimator.n_x	Number of state variables (length of x_0).
Estimator.z	Observed data.
Estimator.x_0	Prior model state estimate.
Estimator.H	Forward operator.
Estimator.S_0	Prior error covariance.
Estimator.S_z	Model-data mismatch covariance.
Estimator.c	Constant data, which can be a scalar or an array matching the shape of z.

Methods

Estimator.cost(x)	Cost/loss/misfit function.
Estimator.forward(x)	Forward model calculation.
Estimator.residual(x)	Forward model residual.
Estimator.leverage(x)	Calculate the leverage matrix.

Results

Estimator.x_hat	Posterior mean model state estimate (solution).
Estimator.S_hat	Posterior error covariance matrix.
Estimator.y_hat	Posterior mean observation estimate.
Estimator.y_0	Prior mean data estimate.
Estimator.K	Kalman gain matrix.
Estimator.A	Averaging kernel matrix.

Metrics

Estimator.reduced_chi2	Reduced Chi-squared statistic.
Estimator.R2	Coefficient of determination (R-squared).
Estimator.RMSE	Root mean square error (RMSE).
Estimator.uncertainty_reduction	Uncertainty reduction metric.
Estimator.U_red	Uncertainty reduction vector.

BayesianSolver

Constructor

BayesianSolver(z, x_0, H, S_0, S_z[, c, gamma])

Bayesian inversion estimator class.

Methods

BayesianSolver.cost(x)

Cost function with regularization.

Results

BayesianSolver.x_hat	Posterior Mean Model Estimate (solution).
BayesianSolver.S_hat	Posterior Error Covariance Matrix with regularization.