estimation
#
This module contains the functionality for managing the inputs and outputs of an estimation.
Functions#

Function to simulate observations. 

Function to compute the azimuth angle, elevation angle and range at a ground station. 

Function to propagate system covariance through time. 

Function to propagate system formal errors through time. 
Function for creating an 
 simulate_observations(simulation_settings: List[tudat::simulation_setup::ObservationSimulationSettings<double>], observation_simulators: List[tudatpy.kernel.numerical_simulation.estimation.ObservationSimulator], bodies: tudatpy.kernel.numerical_simulation.environment.SystemOfBodies) tudat::observation_models::ObservationCollection<double, double, 0> #
Function to simulate observations.
Function to simulate observations from set observation simulators and observation simulator settings. Automatically iterates over all provided observation simulators, generating the full set of simulated observations.
 Parameters:
observation_to_simulate (List[
ObservationSimulationSettings
]) – List of settings objects, each object providing the observation time settings for simulating one type of observable and link end set.observation_simulators (List[
ObservationSimulator
]) – List ofObservationSimulator
objects, each object hosting the functionality for simulating one type of observable and link end set.bodies (
SystemOfBodies
) – Object consolidating all bodies and environment models, including ground station models, that constitute the physical environment.
 Returns:
Object collecting all products of the observation simulation.
 Return type:
 compute_target_angles_and_range(bodies: tudatpy.kernel.numerical_simulation.environment.SystemOfBodies, station_id: Tuple[str, str], target_body: str, observation_times: List[float], is_station_transmitting: bool) Dict[float, numpy.ndarray[numpy.float64[m, 1]]] #
Function to compute the azimuth angle, elevation angle and range at a ground station.
Function to compute the azimuth angle, elevation angle and range at a ground station. This functions is provided as a function of convenience, to prevent users having to manually define the relevant settings for this oftenneeded functionality. This function takes an observing station and a target body as input, and provides the observed angles and current range (without correction for aberrations, with correction for light time) as observed at that station
 Parameters:
bodies (SystemOfBodies) – System of bodies that defines the full physical environment
station_id (tuple[ str, str]) – Identifier for the observing station, as a pair of strings: the body name and the station name.
target_body (str) – Name of body which is observed by ground station
observation_times (list[float]) – List of times at which the ground station observations are to be analyzed
is_station_transmitting (Bool) – Boolean defining whether the observation times define times at which the station is transmitting to, or receiving from, the ground station. This has an impact on the whether the lighttime is computed forward or backward in time from the ground station to the target
 Returns:
Dictionary with the required output. Key defines the observation time, the value is an array of size three containing entry 0  elevation angle, entry 1  azimuth angle, entry 2  range
 Return type:
dict[float,numpy.ndarray[numpy.float64[3, 1]]]
 propagate_covariance(initial_covariance: numpy.ndarray[numpy.float64[m, n]], state_transition_interface: tudatpy.kernel.numerical_simulation.estimation.CombinedStateTransitionAndSensitivityMatrixInterface, output_times: List[float]) Dict[float, numpy.ndarray[numpy.float64[m, n]]] #
Function to propagate system covariance through time.
Function to propagate the covariance of a given system through time. The system dynamics and numerical settings of the propagation are prescribed by the state_transition_interface parameter.
 Parameters:
initial_covariance (numpy.ndarray[numpy.float64[m, n]]) – System covariance matrix (symmetric and positive semidefinite) at initial time. Dimensions have to be consistent with estimatable parameters in the system (specified by state_transition_interface)
state_transition_interface (
CombinedStateTransitionAndSensitivityMatrixInterface
) – Interface to the variational equations of the system dynamics, handling the propagation of the covariance matrix through time.output_times (List[ float ]) – Times at which the propagated covariance matrix shall be reported. Note that this argument has no impact on the integration timesteps of the covariance propagation, which always adheres to the integrator settings that the state_transition_interface links to. Output times which do not coincide with integration time steps are calculated via interpolation.
 Returns:
Dictionary reporting the propagated covariances at each output time.
 Return type:
Dict[ float, numpy.ndarray[numpy.float64[m, n]] ]
 propagate_formal_errors(initial_covariance: numpy.ndarray[numpy.float64[m, n]], state_transition_interface: tudatpy.kernel.numerical_simulation.estimation.CombinedStateTransitionAndSensitivityMatrixInterface, output_times: List[float]) Dict[float, numpy.ndarray[numpy.float64[m, 1]]] #
Function to propagate system formal errors through time.
Function to propagate the formal errors of a given system through time. Note that in practice the entire covariance matrix is propagated, but only the formal errors (variances) are reported at the output times. The system dynamics and numerical settings of the propagation are prescribed by the state_transition_interface parameter.
 Parameters:
initial_covariance (numpy.ndarray[numpy.float64[m, n]]) – System covariance matrix (symmetric and positive semidefinite) at initial time. Dimensions have to be consistent with estimatable parameters in the system (specified by state_transition_interface)
state_transition_interface (
CombinedStateTransitionAndSensitivityMatrixInterface
) – Interface to the variational equations of the system dynamics, handling the propagation of the covariance matrix through time.output_times (List[ float ]) – Times at which the propagated covariance matrix shall be reported. Note that this argument has no impact on the integration timesteps of the covariance propagation, which always adheres to the integrator settings that the state_transition_interface links to. Output times which do not coincide with integration time steps are calculated via interpolation.
 Returns:
Dictionary reporting the propagated formal errors at each output time.
 Return type:
Dict[ float, numpy.ndarray[numpy.float64[m, 1]] ]
 estimation_convergence_checker(maximum_iterations: int = 5, minimum_residual_change: float = 0.0, minimum_residual: float = 0.0, number_of_iterations_without_improvement: int = 2) tudatpy.kernel.numerical_simulation.estimation.EstimationConvergenceChecker #
Function for creating an
EstimationConvergenceChecker
object.Function for creating an
EstimationConvergenceChecker
object, which is required for defining the convergence criteria of an estimation. Parameters:
maximum_iterations (int, default = 5) – Maximum number of allowed iterations for estimation.
minimum_residual_change (float, default = 0.0) – Minimum required change in residual between two iterations.
minimum_residual (float, default = 0.0) – Minimum value of observation residual below which estimation is converged.
number_of_iterations_without_improvement (int, default = 2) – Number of iterations without reduction of residual.
 Returns:
Instance of the
EstimationConvergenceChecker
class, defining the convergence criteria for an estimation. Return type:
Classes#
Class containing a consolidated set of estimatable parameters. 

Template class for observation viability calculators. 

Class hosting the functionality for simulating observations. 

Class collecting all observations and associated data for use in an estimation. 

Class collecting a single set of observations and associated data, of a given observable type, link ends, and ancilliary data. 

Class establishing an interface with the simulation's State Transition and Sensitivity Matrices. 

Class defining the convergence criteria for an estimation. 

Class for defining all specific inputs to a covariance analysis. 

Class for defining all inputs to the estimation. 

Class collecting all outputs from the covariance analysis process. 

Class collecting all outputs from the iterative estimation process. 
 class EstimatableParameterSet#
Class containing a consolidated set of estimatable parameters.
Class containing a consolidated set of estimatable parameters, linked to the environment and acceleration settings of the simulation. The user typically creates instances of this class via the
create_parameters_to_estimate()
factory function. indices_for_parameter_type(self: tudatpy.kernel.numerical_simulation.estimation.EstimatableParameterSet, parameter_type: Tuple[tudat::estimatable_parameters::EstimatebleParametersEnum, Tuple[str, str]]) List[Tuple[int, int]] #
Function to retrieve the indices of a given type of parameter.
Function to retrieve the index of all parameters of a given type from the parameter set. This function can be very useful, since the order of parameters within the parameter set does not necessarily correspond to the order in which the elements were added to the set.
 Parameters:
parameter_type (Tuple[
EstimatableParameterTypes
, Tuple[str, str] ]) – help Returns:
help
 Return type:
 property constraints_size#
readonly
Total size of linear constraint that is to be applied during estimation.
 Type:
 property initial_multi_arc_states_size#
readonly
Amount of initial state parameters in the set, which are treated in a multiarc fashion.
 Type:
 property initial_single_arc_states_size#
readonly
Amount of initial state parameters in the set, which are treated in a singlearc fashion.
 Type:
 property initial_states_size#
readonly
Amount of initial state parameters contained in the set.
 Type:
 property parameter_set_size#
readonly
Size of the parameter set, i.e. amount of estimatable parameters contained in the set.
 Type:
 property parameter_vector#
Vector containing the parameter values of all parameters in the set.
 Type:
numpy.ndarray[numpy.float64[m, 1]]
 class ObservationViabilityCalculator#
Template class for observation viability calculators.
Template class for classes which conducts viability calculations on simulated observations. Instances of the applicable ObservationViabilityCalculators are automatically created from the given
ObservationSimulationSettings
objects during the simulation of observations (simulate_observations()
). The user typically does not interact directly with this class. is_observation_viable(self: tudatpy.kernel.numerical_simulation.estimation.ObservationViabilityCalculator, link_end_states: List[numpy.ndarray[numpy.float64[6, 1]]], link_end_times: List[float]) bool #
Function to check whether an observation is viable.
Function to check whether an observation is viable. The calculation is performed based on the given times and link end states. Note, that this function is called automatically during the simulation of observations. Direct calls to this function are generally not required.
 Parameters:
link_end_states (List[ numpy.ndarray[numpy.float64[6, 1]] ]) – Vector of states of the link ends involved in the observation.
link_end_times (List[float]) – Vector of times at the link ends involved in the observation.
 Returns:
True if observation is viable, false if not.
 Return type:
 class ObservationSimulator#
Class hosting the functionality for simulating observations.
Class hosting the functionality for simulating a given observable over a defined link geometry. Instances of this class are automatically created from the given
ObservationSettings
objects upon instantiation of theEstimator
class.
 class ObservationCollection#
Class collecting all observations and associated data for use in an estimation.
Class containing the full set of observations and associated data, typically for input into the estimation. When using simulated data, this class is instantiated via a call to the
simulate_observations()
function. More information is provided on the user guide get_single_link_and_type_observations(self: tudatpy.kernel.numerical_simulation.estimation.ObservationCollection, observable_type: tudat::observation_models::ObservableType, link_definition: tudat::observation_models::LinkDefinition) List[tudat::observation_models::SingleObservationSet<double, double, 0>] #
Function to get all observation sets for a given observable type and link definition.
 Parameters:
observable_type (
ObservableType
) – Observable type of which observations are to be simulated.link_ends (LinkDefinition) – Link ends for which observations are to be simulated.
 Returns:
List of observation sets for given observable type and link definition.
 Return type:
 property concatenated_link_definition_ids#
readonly
Vector containing concatenated indices identifying the link ends. Each set of link ends is assigned a unique integer identifier (for a given instance of this class). The definition of a given integer identifier with the link ends is given by this class’
link_definition_ids()
function. See user guide for details on storage order of the present vector. Type:
numpy.ndarray[ int ]
 property concatenated_observations#
readonly
Vector containing concatenated observable values. See user guide for details on storage order
 Type:
numpy.ndarray[numpy.float64[m, 1]]
 property concatenated_times#
readonly
Vector containing concatenated observation times. See user guide for details on storage order
 Type:
numpy.ndarray[numpy.float64[m, 1]]
 property link_definition_ids#
readonly
Dictionaty mapping a link end integer identifier to the specific link ends
 Type:
dict[ int, dict[ LinkEndType, LinkEndId ] ]
 property observable_type_start_index_and_size#
readonly
Dictionary defining per obervable type (dict key), the index in the full observation vector (
concatenated_observations()
) where the given observable type starts, and the number of subsequent entries in this vector containing a value of an observable of this type Type:
dict[ ObservableType, [ int, int ] ]
 property observation_set_start_index_and_size#
readonly
The nested dictionary/list returned by this property mirrors the structure of the
sorted_observation_sets()
property of this class. The present function provides the start index and size of the observables in the full observation vector that come from the correspoding SingleObservationSet in thesorted_observation_sets()
Consequently, the present property returns a nested dictionary defining per obervable type, link end identifier, and SingleObservationSet index (for the given observable type and link end identifier), where the observables in the given SingleObservationSet starts, and the number of subsequent entries in this vector containing data from it.
 property sorted_observation_sets#
readonly
The nested dictionary/list contains the list of SingleObservationSet objects, in the same method as they are stored internally in the present class. Specifics on the storage order are given in the user guide
 Type:
dict[ ObservableType, dict[ int, list[ SingleObservationSet ] ] ]
 class SingleObservationSet#
Class collecting a single set of observations and associated data, of a given observable type, link ends, and ancilliary data.
 property ancilliary_settings#
readonly
Ancilliary settings all stored observations
 property concatenated_observations#
readonly
Concatenated vector of all stored observations
 Type:
numpy.ndarray[numpy.float64[m, 1]]
 property link_definition#
readonly
Definition of the link ends for which the object stores observations
 Type:
 property list_of_observations#
readonly
List of separate stored observations. Each entry of this list is a vector containing a single observation. In cases where the observation is singlevalued (range, Doppler), the vector is size 1, but for multivalued observations such as angular position, each vector in the list will have size >1
 Type:
list[ numpy.ndarray[numpy.float64[m, 1]] ]
 property observable_type#
readonly
Type of observable for which the object stores observations
 Type:
 property observation_times#
readonly
Reference time for each of the observations in
list_of_observations
 property observations_history#
readonly
Dictionary of observations sorted by time. Created by making a dictionaty with
observation_times
as keys andlist_of_observations
as values Type:
dict[ float, numpy.ndarray[numpy.float64[m, 1]] ]
 property reference_link_end#
readonly
Reference link end for all stored observations
 Type:
 class CombinedStateTransitionAndSensitivityMatrixInterface#
Class establishing an interface with the simulation’s State Transition and Sensitivity Matrices.
Class establishing an interface to the State Transition and Sensitivity Matrices. Instances of this class are instantiated automatically upon creation of
Estimator
objects, using the simulation information in the observation, propagation and integration settings that theEstimator
instance is linked to. full_state_transition_sensitivity_at_epoch(self: tudatpy.kernel.numerical_simulation.estimation.CombinedStateTransitionAndSensitivityMatrixInterface, time: float, arc_defining_bodies: List[str] = []) numpy.ndarray[numpy.float64[m, n]] #
 Parameters:
time (float) – Time at which full concatenated state transition and sensitivity matrix are to be retrieved.
 Returns:
Full concatenated state transition and sensitivity matrix at a given time.
 Return type:
numpy.ndarray[numpy.float64[m, n]]
 state_transition_sensitivity_at_epoch(self: tudatpy.kernel.numerical_simulation.estimation.CombinedStateTransitionAndSensitivityMatrixInterface, time: float, arc_defining_bodies: List[str] = []) numpy.ndarray[numpy.float64[m, n]] #
Function to get the concatenated state transition and sensitivity matrix at a given time.
Function to get the concatenated state transition and sensitivity matrix at a given time. Entries corresponding to parameters which are not active at the current arc are omitted.
 Parameters:
time (float) – Time at which concatenated state transition and sensitivity matrix are to be retrieved.
 Returns:
Concatenated state transition and sensitivity matrix at a given time.
 Return type:
numpy.ndarray[numpy.float64[m, n]]
 property full_parameter_size#
readonly
Full amount of parameters w.r.t. which partials have been set up via State Transition and Sensitivity Matrices.
 Type:
 class EstimationConvergenceChecker#
Class defining the convergence criteria for an estimation.
Class defining the convergence criteria for an estimation. The user typically creates instances of this class via the
estimation_convergence_checker()
factory function.
 class CovarianceAnalysisInput#
Class for defining all specific inputs to a covariance analysis.
 __init__(self: tudatpy.kernel.numerical_simulation.estimation.CovarianceAnalysisInput, observations_and_times: tudatpy.kernel.numerical_simulation.estimation.ObservationCollection, inverse_apriori_covariance: numpy.ndarray[numpy.float64[m, n]] = array([], shape=(0, 0), dtype=float64)) None #
Class constructor.
Constructor through which the user can create instances of this class. Note that the weight are all initiated as 1.0, and the default settings of
define_covariance_settings
are used. Parameters:
observations_and_times (ObservationCollection) – Total data structure of observations and associated times/link ends/type/etc.
inverse_apriori_covariance (numpy.ndarray[numpy.float64[m, n]], default = [ ]) – A priori covariance matrix (unnormalized) of estimated parameters. This should be either a size 0x0 matrix (no a priori information), or a square matrix with the same size as the number of parameters that are considered
 Returns:
Instance of the
CovarianceAnalysisInput
class, defining the data and other settings to be used for the covariance analysis. Return type:
 define_covariance_settings(self: tudatpy.kernel.numerical_simulation.estimation.CovarianceAnalysisInput, reintegrate_equations_on_first_iteration: bool = True, reintegrate_variational_equations: bool = True, save_design_matrix: bool = True, print_output_to_terminal: bool = True) None #
Function to define specific settings for covariance analysis process
Function to define specific settings for covariance analysis process
 Parameters:
reintegrate_equations (bool, default = True) – Boolean denoting whether the dynamics and variational equations are to be reintegrated or if existing values are to be used to perform first iteration.
reintegrate_variational_equations (bool, default = True) – Boolean denoting whether the variational equations are to be reintegrated during estimation (if this is set to False, and
reintegrate_equations
to true, only the dynamics are reintegrated)save_design_matrix (bool, default = True) – Boolean denoting whether to save the partials matrix (also called design matrix) \(\mathbf{H}\) in the output. Setting this to false makes the \(\mathbf{H}\) matrix unavailable to the user, with the advantage of lower RAM usage.
print_output_to_terminal (bool, default = True) – Boolean denoting whether to print covarianceanalysisspecific output to the terminal when running the estimation.
 Returns:
Function modifies the object inplace.
 Return type:
None
 set_constant_weight(self: tudatpy.kernel.numerical_simulation.estimation.CovarianceAnalysisInput, weight: float) None #
Function to set a constant weight matrix for all observables.
Function to set a constant weight matrix for all observables. The weights are applied to all observations managed by the given PodInput object.
 Parameters:
constant_weight (float) – Constant weight factor that is to be applied to all observations.
 Returns:
Function modifies the object inplace.
 Return type:
None
 set_constant_weight_per_observable(self: tudatpy.kernel.numerical_simulation.estimation.CovarianceAnalysisInput, weight_per_observable: Dict[tudat::observation_models::ObservableType, float]) None #
Function to set a constant weight matrix for a given type of observable.
Function to set a constant weight matrix for a given type of observable. The weights are applied to all observations of the observable type specified by the weight_per_observable parameter.
 Parameters:
constant_weight (Dict[
ObservableType
, float ]) – Constant weight factor that is to be applied to all observations. Returns:
Function modifies the object inplace.
 Return type:
None
 property weight_matrix_diagonal#
readonly
Complete diagonal of the weights matrix that is to be used
 Type:
numpy.ndarray[numpy.float64[n, 1]]
 class EstimationInput#
Class for defining all inputs to the estimation.
 __init__(self: tudatpy.kernel.numerical_simulation.estimation.EstimationInput, observations_and_times: tudatpy.kernel.numerical_simulation.estimation.ObservationCollection, inverse_apriori_covariance: numpy.ndarray[numpy.float64[m, n]] = array([], shape=(0, 0), dtype=float64), convergence_checker: tudatpy.kernel.numerical_simulation.estimation.EstimationConvergenceChecker = <tudatpy.kernel.numerical_simulation.estimation.EstimationConvergenceChecker object at 0x7f3e213efaf0>) None #
Class constructor.
Constructor through which the user can create instances of this class.
 Parameters:
observations_and_times (ObservationCollection) – Total data structure of observations and associated times/link ends/type/etc.
inverse_apriori_covariance (numpy.ndarray[numpy.float64[m, n]], default = [ ]) – A priori covariance matrix (unnormalized) of estimated parameters. This should be either a size 0x0 matrix (no a priori information), or a square matrix with the same size as the number of parameters that are considered
convergence_checker (
EstimationConvergenceChecker
, default =estimation_convergence_checker()
) – Object defining when the estimation is converged.
 Returns:
Instance of the
EstimationInput
class, defining the data and other settings to be used for the estimation. Return type:
 define_estimation_settings(self: tudatpy.kernel.numerical_simulation.estimation.EstimationInput, reintegrate_equations_on_first_iteration: bool = True, reintegrate_variational_equations: bool = True, save_design_matrix: bool = True, print_output_to_terminal: bool = True, save_residuals_and_parameters_per_iteration: bool = True, save_state_history_per_iteration: bool = False) None #
Function to define specific settings for the estimation process
Function to define specific settings for covariance analysis process
 Parameters:
reintegrate_equations_on_first_iteration (bool, default = True) – Boolean denoting whether the dynamics and variational equations are to be reintegrated or if existing values are to be used to perform first iteration.
reintegrate_variational_equations (bool, default = True) – Boolean denoting whether the variational equations are to be reintegrated during estimation (if this is set to False, and
reintegrate_equations_on_first_iteration
to true, only the dynamics are reintegrated)save_design_matrix (bool, default = True) – Boolean denoting whether to save the partials matrix (also called design matrix) \(\mathbf{H}\) in the output. Setting this to false makes the \(\mathbf{H}\) matrix unavailable to the user, with the advantage of lower RAM usage.
print_output_to_terminal (bool, default = True) – Boolean denoting whether to print covarianceanalysisspecific output to the terminal when running the estimation.
save_residuals_and_parameters_per_iteration (bool, default = True) – Boolean denoting whether the residuals and parameters from the each iteration are to be saved.
save_state_history_per_iteration (bool, default = False) – Boolean denoting whether the state history and dependent variables are to be saved on each iteration.
 Returns:
Function modifies the object inplace.
 Return type:
None
 class CovarianceAnalysisOutput#
Class collecting all outputs from the covariance analysis process.
 property correlations#
readonly
Correlation matrix of the estimation result. Entry \(i,j\) is equal to \(P_{i,j}/(\sigma_{i}\sigma_{j})\)
 Type:
numpy.ndarray[numpy.float64[m, m]]
 property covariance#
readonly
(Unnormalized) estimation covariance matrix \(\mathbf{P}\).
 Type:
numpy.ndarray[numpy.float64[m, m]]
 property design_matrix#
readonly
Matrix of unnormalized partial derivatives \(\mathbf{H}=\frac{\partial\mathbf{h}}{\partial\mathbf{p}}\).
 Type:
numpy.ndarray[numpy.float64[m, n]]
 property formal_errors#
readonly
Formal error vector \(\boldsymbol{\sigma}\) of the estimation result (e.g. square root of diagonal entries of covariance)s
 Type:
numpy.ndarray[numpy.float64[m, 1]]s
 property inverse_covariance#
readonly
(Unnormalized) inverse estimation covariance matrix \(\mathbf{P}^{1}\).
 Type:
numpy.ndarray[numpy.float64[m, m]]
 property inverse_normalized_covariance#
readonly
Normalized inverse estimation covariance matrix \(\mathbf{\tilde{P}}^{1}\).
 Type:
numpy.ndarray[numpy.float64[m, m]]
 property normalization_terms#
readonly
Vector of normalization terms used for covariance and design matrix
 Type:
numpy.ndarray[numpy.float64[m, 1]]
 property normalized_covariance#
readonly
Normalized estimation covariance matrix \(\mathbf{\tilde{P}}\).
 Type:
numpy.ndarray[numpy.float64[m, m]]
 property normalized_design_matrix#
readonly
Matrix of normalized partial derivatives \(\tilde{\mathbf{H}}\).
 Type:
numpy.ndarray[numpy.float64[m, n]]
 property weighted_design_matrix#
readonly
Matrix of weighted partial derivatives, equal to \(\mathbf{W}^{1/2}{\mathbf{H}}\)
 Type:
numpy.ndarray[numpy.float64[m, n]]
 property weighted_normalized_design_matrix#
readonly
Matrix of weighted, normalized partial derivatives, equal to \(\mathbf{W}^{1/2}\tilde{\mathbf{H}}\)
 Type:
numpy.ndarray[numpy.float64[m, n]]
 class EstimationOutput#
Class collecting all outputs from the iterative estimation process.
 property final_residuals#
readonly
Vector of postfit observation residuals.
 Type:
numpy.ndarray[numpy.float64[m, 1]]
 property parameter_history#
readonly
Parameter vectors, concatenated per iteration into a matrix. Column 0 contains preestimation values. The \((i+1)^{th}\) column has the residuals from the \(i^{th}\) iteration.
 Type:
numpy.ndarray[numpy.float64[m, n]]
 property residual_history#
readonly
Residual vectors, concatenated per iteration into a matrix; the \(i^{th}\) column has the residuals from the \(i^{th}\) iteration.
 Type:
numpy.ndarray[numpy.float64[m, n]]
 property simulation_results_per_iteration#
readonly
List of complete numerical propagation results, with the \(i^{th}\) entry of thee list thee results of the \(i^{th}\) propagation
 Type: