fips.estimators#

Inversion estimators.

This module contains various inversion estimators for solving inverse problems.

Classes

`BayesianSolver`(z, x_0, H, S_0, S_z[, c, rf])	Bayesian inversion estimator class This class implements a Bayesian inversion framework for solving inverse problems, also known as the batch method.
`Estimator`(z, x_0, H, S_0, S_z[, c])	Base inversion estimator class.
`EstimatorRegistry`	Registry for estimator classes.

class fips.estimators.Estimator(z, x_0, H, S_0, S_z, c=None)[source]#

Base inversion estimator class.

z#

Observed data.

Type:: np.ndarray

x_0#

Prior model state estimate.

Type:: np.ndarray

H#

Forward operator.

Type:: np.ndarray

S_0#

Prior error covariance.

Type:: np.ndarray

S_z#

Model-data mismatch covariance.

Type:: np.ndarray

c#

Constant data, defaults to 0.0.

Type:: np.ndarray or float, optional

n_z#

Number of observations.

Type:: int

n_x#

Number of state variables.

Type:: int

x_hat#

Posterior mean model state estimate (solution).

Type:: np.ndarray

S_hat#

Posterior error covariance.

Type:: np.ndarray

y_hat#

Posterior modeled observations.

Type:: np.ndarray

y_0#

Prior modeled observations.

Type:: np.ndarray

K#

Kalman gain.

Type:: np.ndarray

A#

Averaging kernel.

Type:: np.ndarray

chi2#

Chi-squared statistic.

Type:: float

R2#

Coefficient of determination.

Type:: float

RMSE#

Root mean square error.

Type:: float

U_red#

Reduced uncertainty.

Type:: np.ndarray

cost(x: np.ndarray) → float[source]#: Cost/loss/misfit function.

forward(x: np.ndarray) → np.ndarray[source]#: Forward model calculation.

residual(x: np.ndarray) → np.ndarray[source]#: Forward model residual.

leverage(x: np.ndarray) → np.ndarray[source]#: Calculate the leverage matrix.

__init__(z, x_0, H, S_0, S_z, c=None)[source]#

Initialize the Estimator object.

Parameters:

z (np.ndarray) – Observed data.
x_0 (np.ndarray) – Prior model state estimate.
H (np.ndarray) – Forward operator.
S_0 (np.ndarray) – Prior error covariance.
S_z (np.ndarray) – Model-data mismatch covariance.
c (np.ndarray or float, optional) – Constant data, defaults to 0.0.

forward(x)[source]#

Forward model calculation.

\[y = Hx + c\]

Parameters:: x (np.ndarray) – State vector.
Returns:: Model output (Hx + c).
Return type:: np.ndarray

residual(x)[source]#

Forward model residual.

\[r = z - (Hx + c)\]

Parameters:: x (np.ndarray) – State vector.
Returns:: Residual (z - (Hx + c)).
Return type:: np.ndarray

leverage(x)[source]#

Calculate the leverage matrix.

Which observations are likely to have more impact on the solution.

\[L = Hx ((Hx)^T (H S_0 H^T + S_z)^{-1} Hx)^{-1} (Hx)^T (H S_0 H^T + S_z)^{-1}\]

Parameters:: x (np.ndarray) – State vector.
Returns:: Leverage matrix.
Return type:: np.ndarray

abstract cost(x)[source]#

Cost/loss/misfit function.

Parameters:: x (np.ndarray) – State vector.
Returns:: Cost value.
Return type:: float

abstract property x_hat: ndarray#

Posterior mean model state estimate (solution).

Returns:: Posterior state estimate.
Return type:: np.ndarray

abstract property S_hat: ndarray#

Posterior error covariance matrix.

Returns:: Posterior error covariance matrix.
Return type:: np.ndarray

property y_hat: ndarray#

Posterior mean observation estimate.

\[\hat{y} = H \hat{x} + c\]

Returns:: Posterior observation estimate.
Return type:: np.ndarray

property y_0: ndarray#

Prior mean data estimate.

\[\hat{y}_0 = H x_0 + c\]

Returns:: Prior data estimate.
Return type:: np.ndarray

property K#

Kalman gain matrix.

\[K = (H S_0)^T (H S_0 H^T + S_z)^{-1}\]

Returns:: Kalman gain matrix.
Return type:: np.ndarray

property A#

Averaging kernel matrix.

\[A = KH = (H S_0)^T (H S_0 H^T + S_z)^{-1} H\]

Returns:: Averaging kernel matrix.
Return type:: np.ndarray

property DOFS: float#

Degrees Of Freedom for Signal (DOFS).

\[DOFS = Tr(A)\]

Returns:: Degrees of Freedom value.
Return type:: float

property reduced_chi2: float#

Reduced Chi-squared statistic. Tarantola (1987)

\[\chi^2 = \frac{1}{n_z} ((z - H\hat{x})^T S_z^{-1} (z - H\hat{x}) + (\hat{x} - x_0)^T S_0^{-1} (\hat{x} - x_0))\]

Note

I can’t find a copy of Tarantola (1987) to verify this equation, but it appears in Kunik et al. (2019) https://doi.org/10.1525/elementa.375

Returns:: Reduced Chi-squared value.
Return type:: float

property R2: float#

Coefficient of determination (R-squared).

\[R^2 = corr(z, H\hat{x})^2\]

Returns:: R-squared value.
Return type:: float

property RMSE: float#

Root mean square error (RMSE).

\[RMSE = \sqrt{\frac{(z - H\hat{x})^2}{n_z}}\]

Returns:: RMSE value.
Return type:: float

property U_red#

Uncertainty reduction metric.

\[U_{red} = 1 - \frac{\sqrt{trace(\hat{S})}}{\sqrt{trace(S_0)}}\]

Returns:: Uncertainty reduction value.
Return type:: float

class fips.estimators.EstimatorRegistry[source]#

Registry for estimator classes.

register(name)[source]#

Parameters:: name (str) – Name to register the estimator under.
Returns:: decorator – Decorator to register the class.
Return type:: function

class fips.estimators.BayesianSolver(z, x_0, H, S_0, S_z, c=None, rf=1.0)[source]#

Bayesian inversion estimator class This class implements a Bayesian inversion framework for solving inverse problems, also known as the batch method.

__init__(z, x_0, H, S_0, S_z, c=None, rf=1.0)[source]#

Initialize inversion object

Parameters:

z (np.ndarray) – Observed data
x_0 (np.ndarray) – Prior model estimate
H (np.ndarray) – Forward operator
S_0 (np.ndarray) – Prior error covariance
S_z (np.ndarray) – Model-data mismatch covariance
c (np.ndarray | float, optional) – Constant data, defaults to 0.0
rf (float, optional) – Regularization factor, by default 1.0

cost(x)[source]#: Cost function

\[J(x) = \frac{1}{2}(x - x_0)^T S_0^{-1}(x - x_0) + \frac{1}{2}(z - Hx - c)^T S_z^{-1}(z - Hx - c)\]

property x_hat#: Posterior Mean Model Estimate (solution)

\[\hat{x} = x_0 + K(z - Hx_0 - c)\]

property S_hat#: Posterior Error Covariance Matrix

\[\hat{S} = (H^T S_z^{-1} H + S_0^{-1})^{-1} = S_0 - (H S_0)^T(H S_0 H^T + S_z)^{-1}(H S_0)\]

fips.estimators#

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