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Sensitivity Computation

Sensitivity matrix computation utilities for parameter estimation and consider covariance analysis.

Overview

Brahe provides both analytical and numerical sensitivity computation through a unified interface. Sensitivity matrices describe how a function's output changes with respect to consider parameters (\(\partial f/\partial p\)), which is essential for orbit determination with consider parameters and covariance analysis.

NumericalSensitivity 

NumericalSensitivity(dynamics_fn: Any)

Numerical sensitivity provider for dynamic-sized systems using finite differences.

Computes the sensitivity matrix ∂f/∂p numerically by perturbing the parameters and evaluating the dynamics function.

Example 
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import brahe as bh
import numpy as np

# Dynamics with consider parameters
def dynamics(t, state, params):
    cd_area_m = params[0]
    # ... compute derivatives using cd_area_m
    return np.array([...])

sensitivity = bh.NumericalSensitivity(dynamics)
state = np.array([7000e3, 0, 0, 0, 7.5e3, 0])
params = np.array([0.044])  # cd*A/m
sens_matrix = sensitivity.compute(0.0, state, params)

Initialize instance.

backward builtin 

backward(dynamics_fn: callable) -> NumericalSensitivity

Create with backward finite differences.

Parameters:

Name Type Description Default
dynamics_fn callable

Function with signature (t: float, state: ndarray, params: ndarray) -> ndarray

 required

Returns:

Name Type Description
NumericalSensitivity NumericalSensitivity

Sensitivity provider using backward differences

central builtin 

central(dynamics_fn: callable) -> NumericalSensitivity

Create with central finite differences.

Parameters:

Name Type Description Default
dynamics_fn callable

Function with signature (t: float, state: ndarray, params: ndarray) -> ndarray

 required

Returns:

Name Type Description
NumericalSensitivity NumericalSensitivity

Sensitivity provider using central differences

compute method descriptor 

compute(t: float, state: ndarray, params: ndarray) -> ndarray

Compute the sensitivity matrix at the given time, state, and parameters.

Parameters:

Name Type Description Default
t float

Current time

 required
state ndarray

State vector at time t

 required
params ndarray

Consider parameters

 required

Returns:

Name Type Description
ndarray ndarray

Sensitivity matrix ∂f/∂p (state_dim × param_dim)

forward builtin 

forward(dynamics_fn: callable) -> NumericalSensitivity

Create with forward finite differences.

Parameters:

Name Type Description Default
dynamics_fn callable

Function with signature (t: float, state: ndarray, params: ndarray) -> ndarray

 required

Returns:

Name Type Description
NumericalSensitivity NumericalSensitivity

Sensitivity provider using forward differences

with_adaptive method descriptor 

with_adaptive(scale_factor: float, min_value: float) -> NumericalSensitivity

Set adaptive perturbation with custom parameters.

Parameters:

Name Type Description Default
scale_factor float

Multiplier on sqrt(ε), typically 1.0

 required
min_value float

Minimum reference value

 required

Returns:

Name Type Description
NumericalSensitivity NumericalSensitivity

Self for method chaining

with_fixed_offset method descriptor 

with_fixed_offset(offset: float) -> NumericalSensitivity

Set fixed absolute perturbation for all parameters.

Parameters:

Name Type Description Default
offset float

Fixed perturbation size

 required

Returns:

Name Type Description
NumericalSensitivity NumericalSensitivity

Self for method chaining

with_method method descriptor 

with_method(method: DifferenceMethod) -> NumericalSensitivity

Set the difference method.

Parameters:

Name Type Description Default
method DifferenceMethod

Finite difference method to use

 required

Returns:

Name Type Description
NumericalSensitivity NumericalSensitivity

Self for method chaining

with_percentage method descriptor 

with_percentage(percentage: float) -> NumericalSensitivity

Set percentage-based perturbation.

Parameters:

Name Type Description Default
percentage float

Percentage of parameter value (e.g., 1e-6 for 0.0001%)

 required

Returns:

Name Type Description
NumericalSensitivity NumericalSensitivity

Self for method chaining

AnalyticSensitivity 

AnalyticSensitivity(sensitivity_fn: Any)

Analytical sensitivity provider for dynamic-sized systems.

Uses a user-provided function that directly computes the analytical sensitivity matrix.

Example 
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import brahe as bh
import numpy as np

def sensitivity_fn(t, state, params):
    # Return ∂f/∂p matrix
    sens = np.zeros((6, 1))
    # ... compute analytical sensitivity
    return sens

sensitivity = bh.AnalyticSensitivity(sensitivity_fn)
state = np.array([7000e3, 0, 0, 0, 7.5e3, 0])
params = np.array([0.044])
sens_matrix = sensitivity.compute(0.0, state, params)

Initialize instance.

compute method descriptor 

compute(t: float, state: ndarray, params: ndarray) -> ndarray

Compute the sensitivity matrix at the given time, state, and parameters.

Parameters:

Name Type Description Default
t float

Current time

 required
state ndarray

State vector at time t

 required
params ndarray

Consider parameters

 required

Returns:

Name Type Description
ndarray ndarray

Sensitivity matrix ∂f/∂p (state_dim × param_dim)

See Also