# espei.error_functions package¶

## espei.error_functions.activity_error module¶

Calculate error due to measured activities.

espei.error_functions.activity_error.calculate_activity_error(dbf, comps, phases, datasets, parameters=None, phase_models=None, callables=None, data_weight=1.0)

Return the sum of square error from activity data

Parameters: dbf (pycalphad.Database) – Database to consider comps (list) – List of active component names phases (list) – List of phases to consider datasets (espei.utils.PickleableTinyDB) – Datasets that contain single phase data parameters (dict) – Dictionary of symbols that will be overridden in pycalphad.equilibrium phase_models (dict) – Phase models to pass to pycalphad calculations callables (dict) – Callables to pass to pycalphad data_weight (float) – Weight for standard deviation of activity measurements, dimensionless. Corresponds to the standard deviation of differences in chemical potential in typical measurements of activity, in J/mol. A single float of the sum of square errors float

Notes

General procedure: 1. Get the datasets 2. For each dataset

1. Calculate reference state equilibrium
2. Calculate current chemical potentials
3. Find the target chemical potentials
4. Calculate error due to chemical potentials
espei.error_functions.activity_error.chempot_error(sample_chempots, target_chempots, std_dev=10.0)

Return the sum of square error from chemical potentials

sample_chempots : numpy.ndarray
Calculated chemical potentials
target_activity : numpy.ndarray
Chemical potentials to target
std_dev : float
Standard deviation of activity measurements in J/mol. Corresponds to the standard deviation of differences in chemical potential in typical measurements of activity.
Returns: Error due to chemical potentials float
espei.error_functions.activity_error.target_chempots_from_activity(component, target_activity, temperatures, reference_result)

Return an array of experimental chemical potentials for the component

Parameters: component (str) – Name of the component target_activity (numpy.ndarray) – Array of experimental activities temperatures (numpy.ndarray) – Ravelled array of temperatures (of same size as exp_activity). reference_result (xarray.Dataset) – Dataset of the equilibrium reference state. Should contain a singe point calculation. Array of experimental chemical potentials numpy.ndarray

## espei.error_functions.context module¶

Convenience function to create a context for the built in error functions

espei.error_functions.context.setup_context(dbf, datasets, symbols_to_fit=None, data_weights=None, make_callables=True)

Set up a context dictionary for calculating error.

Parameters: dbf (Database) – A pycalphad Database that will be fit datasets (PickleableTinyDB) – A database of single- and multi-phase data to fit symbols_to_fit (list of str) – List of symbols in the Database that will be fit. If None (default) are passed, then all parameters prefixed with VV followed by a number, e.g. VV0001 will be fit.

Notes

A copy of the Database is made and used in the context. To commit changes back to the original database, the dbf.symbols.update method should be used.

## espei.error_functions.non_equilibrium_thermochemical_error module¶

Calculate error due to thermochemical quantities: heat capacity, entropy, enthalpy.

espei.error_functions.non_equilibrium_thermochemical_error.calculate_non_equilibrium_thermochemical_probability(dbf, thermochemical_data, parameters=None)

Calculate the weighted single phase error in the Database

Parameters: dbf (pycalphad.Database) – Database to consider thermochemical_data (list) – List of thermochemical data dicts parameters (np.ndarray) – Array of parameters to calculate the error with. A single float of the residual sum of square errors float

Notes

There are different single phase values, HM_MIX, SM_FORM, CP_FORM, etc. Each of these have different units and the error cannot be compared directly. To normalize all of the errors, a normalization factor must be used. Equation 2.59 and 2.60 in Lukas, Fries, and Sundman “Computational Thermodynamics” shows how this can be considered. Each type of error will be weighted by the reciprocal of the estimated uncertainty in the measured value and conditions. The weighting factor is calculated by $p_i = (Delta L_i)^{-1}$ where $Delta L_i$ is the uncertainty in the measurement. We will neglect the uncertainty for quantities such as temperature, assuming they are small.

espei.error_functions.non_equilibrium_thermochemical_error.calculate_points_array(phase_constituents, configuration, occupancies=None)

Calculate the points array to use in pycalphad calculate calls.

Converts the configuration data (and occupancies for mixing data) into the points array by looking up the indices in the active phase constituents.

Parameters: phase_constituents (list) – List of active constituents in a phase configuration (list) – List of the sublattice configuration occupancies (list) – List of sublattice occupancies. Required for mixing sublattices, otherwise takes no effect. numpy.ndarray

Notes

Errors will be raised if components in the configuration are not in the corresponding phase constituents sublattice.

espei.error_functions.non_equilibrium_thermochemical_error.get_prop_samples(dbf, comps, phase_name, desired_data)

Return data values and the conditions to calculate them by pycalphad calculate from the datasets

Parameters: dbf (pycalphad.Database) – Database to consider comps (list) – List of active component names phase_name (str) – Name of the phase to consider from the Database desired_data (list) – List of dictionary datasets that contain the values to sample Dictionary of condition kwargs for pycalphad’s calculate and the expected values dict
espei.error_functions.non_equilibrium_thermochemical_error.get_thermochemical_data(dbf, comps, phases, datasets, weight_dict=None, symbols_to_fit=None)
Parameters: dbf (pycalphad.Database) – Database to consider comps (list) – List of active component names phases (list) – List of phases to consider datasets (espei.utils.PickleableTinyDB) – Datasets that contain single phase data weight_dict (dict) – Dictionary of weights for each data type, e.g. {‘HM’: 200, ‘SM’: 2} symbols_to_fit (list) – Parameters to fit. Used to build the models and PhaseRecords. List of data dictionaries to iterate over list

## espei.error_functions.zpf_error module¶

Calculate driving_force due to ZPF tielines.

The general approach is similar to the PanOptimizer rough search method.

1. With all phases active, calculate the chemical potentials of the tieline endpoints via equilibrium calls. Done in estimate_hyperplane.
2. Calculate the target chemical potentials, which are the average chemical potentials of all of the current chemical potentials at the tieline endpoints.
3. Calculate the current chemical potentials of the desired single phases
4. The error is the difference between these chemical potentials

There’s some special handling for tieline endpoints where we do not know the composition conditions to calculate chemical potentials at.

class espei.error_functions.zpf_error.PhaseRegion(region_phases, potential_conds, comp_conds, phase_flags, dbf, species, phases, models, phase_records)

Bases: tuple

comp_conds

Alias for field number 2

dbf

Alias for field number 4

models

Alias for field number 7

phase_flags

Alias for field number 3

phase_records

Alias for field number 8

phases

Alias for field number 6

potential_conds

Alias for field number 1

region_phases

Alias for field number 0

species

Alias for field number 5

espei.error_functions.zpf_error.calculate_zpf_error(zpf_data: Sequence[Dict[str, Any]], parameters: <Mock name='mock.ndarray' id='139944577980288'> = None, data_weight: int = 1.0, approximate_equilibrium: bool = False)

Calculate error due to phase equilibria data

zpf_data : list
Datasets that contain single phase data
phase_models : dict
Phase models to pass to pycalphad calculations
parameters : np.ndarray
Array of parameters to calculate the error with.
callables : dict
data_weight : float
Scaling factor for the standard deviation of the measurement of a tieline which has units J/mol. The standard deviation is 1000 J/mol and the scaling factor defaults to 1.0.
approximate_equilibrium : bool
Whether or not to use an approximate version of equilibrium that does not refine the solution and uses starting_point instead.
Returns: Log probability of ZPF error float

Notes

The physical picture of the standard deviation is that we’ve measured a ZPF line. That line corresponds to some equilibrium chemical potentials. The standard deviation is the standard deviation of those ‘measured’ chemical potentials.

espei.error_functions.zpf_error.driving_force_to_hyperplane(target_hyperplane_chempots: <Mock name='mock.ndarray' id='139944577980288'>, comps: Sequence[str], phase_region: espei.error_functions.zpf_error.PhaseRegion, vertex_idx: int, parameters: <Mock name='mock.ndarray' id='139944577980288'>, approximate_equilibrium: bool = False) → float

Calculate the integrated driving force between the current hyperplane and target hyperplane.

espei.error_functions.zpf_error.estimate_hyperplane(phase_region: espei.error_functions.zpf_error.PhaseRegion, parameters: <Mock name='mock.ndarray' id='139944577980288'>, approximate_equilibrium: bool = False) → <Mock name='mock.ndarray' id='139944577980288'>

Calculate the chemical potentials for the target hyperplane, one vertex at a time

Notes

This takes just one set of phase equilibria, a phase region, e.g. a dataset point of [[‘FCC_A1’, [‘CU’], [0.1]], [‘LAVES_C15’, [‘CU’], [0.3]]] and calculates the chemical potentials given all the phases possible at the given compositions. Then the average chemical potentials of each end point are taken as the target hyperplane for the given equilibria.

espei.error_functions.zpf_error.extract_conditions(all_conditions: Dict[<Mock name='mock.variables.StateVariable' id='139944577981184'>, <Mock name='mock.ndarray' id='139944577980288'>], idx: int) → Dict[<Mock name='mock.variables.StateVariable' id='139944577981184'>, float]

Conditions are either scalar or 1d arrays for the conditions in the entire dataset. This function extracts the condition corresponding to the current region, based on the index in the 1d condition array.

espei.error_functions.zpf_error.extract_phases_comps(phase_region)

Extract the phase names, phase compositions and any phase flags from each tie-line point in the phase region

espei.error_functions.zpf_error.get_zpf_data()

Return the ZPF data used in the calculation of ZPF error

Parameters: comps (list) – List of active component names phases (list) – List of phases to consider datasets (espei.utils.PickleableTinyDB) – Datasets that contain single phase data parameters (dict) – Dictionary mapping symbols to optimize to their initial values List of data dictionaries with keys weight, data_comps and phase_regions. data_comps are the components for the data in question. phase_regions are the ZPF phases, state variables and compositions. list
espei.error_functions.zpf_error.update_phase_record_parameters(phase_records: Dict[str, <Mock name='mock.PhaseRecord' id='139944578407840'>], parameters: <Mock name='mock.ndarray' id='139944577980288'>) → None

## Module contents¶

Functions for calculating error.