def cov(self, return_type="pytree"):
"""Calculate the variance-covariance matrix of the estimated parameters.
Args:
return_type (str): One of "pytree", "array" or "dataframe". Default pytree.
If "array", a 2d numpy array with the covariance is returned. If
"dataframe", a pandas DataFrame with parameter names in the
index and columns are returned.
The default is "pytree".
Returns:
Any: The covariance matrix of the estimated parameters as a block-pytree,
numpy.ndarray, or pandas.DataFrame.
"""
cov = self._internal_cov
if return_type == "dataframe":
registry = get_registry(extended=True)
names = np.array(leaf_names(self._base_outcome, registry=registry))
cov = pd.DataFrame(cov, columns=names, index=names)
elif return_type == "pytree":
cov = matrix_to_block_tree(cov, self._base_outcome,
self._base_outcome)
elif return_type != "array":
raise ValueError(
"return_type must be one of pytree, array, or dataframe, "
f"not {return_type}.")
return cov
def get_weighting_matrix(moments_cov,
method,
empirical_moments,
clip_value=1e-6,
return_type="pytree"):
"""Calculate a weighting matrix from moments_cov.
Args:
moments_cov (pandas.DataFrame or numpy.ndarray): Square DataFrame or Array
with the covariance matrix of the moment conditions for msm estimation.
method (str): One of "optimal", "diagonal".
empirical_moments (pytree): Pytree containing empirical moments. Used to get
the tree structure
clip_value (float): Bound at which diagonal elements of the moments_cov are
clipped to avoid dividing by zero.
return_type (str): One of "pytree", "array" or "pytree_and_array"
Returns:
pandas.DataFrame or numpy.ndarray: Weighting matrix with the same shape as
moments_cov.
"""
fast_path = isinstance(moments_cov, np.ndarray) and moments_cov.ndim == 2
if fast_path:
_internal_cov = moments_cov
else:
_internal_cov = block_tree_to_matrix(
moments_cov,
outer_tree=empirical_moments,
inner_tree=empirical_moments,
)
if method == "optimal":
array_weights = robust_inverse(_internal_cov)
elif method == "diagonal":
diagonal_values = 1 / np.clip(np.diagonal(_internal_cov), clip_value,
np.inf)
array_weights = np.diag(diagonal_values)
else:
raise ValueError(f"Invalid method: {method}")
if return_type == "array" or (fast_path and "_and_" not in return_type):
out = array_weights
elif fast_path:
out = (array_weights, array_weights)
else:
tree_weights = matrix_to_block_tree(
array_weights,
outer_tree=empirical_moments,
inner_tree=empirical_moments,
)
if return_type == "pytree":
out = tree_weights
else:
out = (tree_weights, array_weights)
return out
def get_moments_cov(data,
calculate_moments,
*,
moment_kwargs=None,
bootstrap_kwargs=None):
"""Bootstrap the covariance matrix of the moment conditions.
Args:
data (pandas.DataFrame): DataFrame with empirical data.
calculate_moments (callable): Function that calculates that takes data and
moment_kwargs as arguments and returns a 1d numpy array or pandas Series
with moment conditions.
moment_kwargs (dict): Additional keyword arguments for calculate_moments.
bootstrap_kwargs (dict): Additional keyword arguments that govern the
bootstrapping. Allowed arguments are "n_draws", "seed", "n_cores",
"batch_evaluator", "cluster" and "error_handling". For details see the
bootstrap function.
Returns:
pandas.DataFrame or numpy.ndarray: The covariance matrix of the moment
conditions for msm estimation.
"""
moment_kwargs = {} if moment_kwargs is None else moment_kwargs
bootstrap_kwargs = {} if bootstrap_kwargs is None else bootstrap_kwargs
valid_bs_kwargs = {
"n_cores",
"n_draws",
"seed",
"batch_evaluator",
"cluster",
"error_handling",
}
problematic = set(bootstrap_kwargs).difference(valid_bs_kwargs)
if problematic:
raise ValueError(f"Invalid bootstrap_kwargs: {problematic}")
first_eval = calculate_moments(data, **moment_kwargs)
registry = get_registry(extended=True)
@functools.wraps(calculate_moments)
def func(data, **kwargs):
raw = calculate_moments(data, **kwargs)
out = pd.Series(tree_just_flatten(
raw, registry=registry)) # xxxx won't be necessary soon!
return out
cov_arr = bootstrap(data=data, outcome=func,
outcome_kwargs=moment_kwargs).cov()
if isinstance(cov_arr, pd.DataFrame):
cov_arr = cov_arr.to_numpy() # xxxx won't be necessary soon
cov = matrix_to_block_tree(cov_arr, first_eval, first_eval)
return cov
def test_matrix_to_block_tree_single_element():
tree1 = {"a": 0}
tree2 = {"b": 1, "c": 2}
block_tree = {"a": {"b": 0, "c": 1}}
matrix = np.array([[0, 1]])
calculated = matrix_to_block_tree(matrix, tree1, tree2)
assert tree_equal(block_tree, calculated)
def transform_free_cov_to_cov(free_cov, free_params, params, return_type):
"""Fill non-free values and project to params block-tree."""
mask = free_params.free_mask
cov = np.full((len(mask), len(mask)), np.nan)
cov[np.ix_(mask, mask)] = free_cov
if return_type == "dataframe":
names = free_params.all_names
cov = pd.DataFrame(cov, columns=names, index=names)
elif return_type == "pytree":
cov = matrix_to_block_tree(cov, params, params)
elif return_type != "array":
raise ValueError(
"return_type must be one of pytree, array, or dataframe, "
f"not {return_type}.")
return cov
def test_matrix_to_block_tree_array_and_scalar():
t = {"a": 1.0, "b": np.arange(2)}
calculated = matrix_to_block_tree(np.arange(9).reshape(3, 3), t, t)
expected = {
"a": {
"a": np.array(0),
"b": np.array([1, 2])
},
"b": {
"a": np.array([3, 6]),
"b": np.array([[4, 5], [7, 8]])
},
}
assert _tree_equal_up_to_dtype(calculated, expected)
def test_matrix_to_block_tree_only_params_dfs():
tree = {
"a": pd.DataFrame(index=["a", "b"]).assign(value=[1, 2]),
"b": pd.DataFrame(index=["j", "k", "l"]).assign(value=[3, 4, 5]),
}
calculated = matrix_to_block_tree(np.arange(25).reshape(5, 5), tree, tree)
expected = {
"a": {
"a":
pd.DataFrame([[0, 1], [5, 6]],
columns=["a", "b"],
index=["a", "b"]),
"b":
pd.DataFrame([[2, 3, 4], [7, 8, 9]],
columns=["j", "k", "l"],
index=["a", "b"]),
},
"b": {
"a":
pd.DataFrame(
[[10, 11], [15, 16], [20, 21]],
index=["j", "k", "l"],
columns=["a", "b"],
),
"b":
pd.DataFrame(
[[12, 13, 14], [17, 18, 19], [22, 23, 24]],
index=["j", "k", "l"],
columns=["j", "k", "l"],
),
},
}
assert _tree_equal_up_to_dtype(calculated, expected)
def estimate_ml(
loglike,
params,
optimize_options,
*,
lower_bounds=None,
upper_bounds=None,
constraints=None,
logging=False,
log_options=None,
loglike_kwargs=None,
numdiff_options=None,
jacobian=None,
jacobian_kwargs=None,
hessian=None,
hessian_kwargs=None,
design_info=None,
):
"""Do a maximum likelihood (ml) estimation.
This is a high level interface of our lower level functions for maximization,
numerical differentiation and inference. It does the full workflow for maximum
likelihood estimation with just one function call.
While we have good defaults, you can still configure each aspect of each step
via the optional arguments of this function. If you find it easier to do the
maximization separately, you can do so and just provide the optimal parameters as
``params`` and set ``optimize_options=False``
Args:
loglike (callable): Likelihood function that takes a params (and potentially
other keyword arguments) and returns a dictionary that has at least the
entries "value" (a scalar float) and "contributions" (a 1d numpy array or
pytree) with the log likelihood contribution per individual.
params (pytree): A pytree containing the estimated or start parameters of the
likelihood model. If the supplied parameters are estimated parameters, set
optimize_options to False. Pytrees can be a numpy array, a pandas Series, a
DataFrame with "value" column, a float and any kind of (nested) dictionary
or list containing these elements. See :ref:`params` for examples.
optimize_options (dict, str or False): Keyword arguments that govern the
numerical optimization. Valid entries are all arguments of
:func:`~estimagic.optimization.optimize.minimize` except for those that are
passed explicilty to ``estimate_ml``. If you pass False as optimize_options
you signal that ``params`` are already the optimal parameters and no
numerical optimization is needed. If you pass a str as optimize_options it
is used as the ``algorithm`` option.
lower_bounds (pytree): A pytree with the same structure as params with lower
bounds for the parameters. Can be ``-np.inf`` for parameters with no lower
bound.
upper_bounds (pytree): As lower_bounds. Can be ``np.inf`` for parameters with
no upper bound.
constraints (list, dict): List with constraint dictionaries or single dict.
See :ref:`constraints`.
logging (pathlib.Path, str or False): Path to sqlite3 file (which typically has
the file extension ``.db``. If the file does not exist, it will be created.
The dashboard can only be used when logging is used.
log_options (dict): Additional keyword arguments to configure the logging.
- "fast_logging": A boolean that determines if "unsafe" settings are used
to speed up write processes to the database. This should only be used for
very short running criterion functions where the main purpose of the log
is a real-time dashboard and it would not be catastrophic to get a
corrupted database in case of a sudden system shutdown. If one evaluation
of the criterion function (and gradient if applicable) takes more than
100 ms, the logging overhead is negligible.
- "if_table_exists": (str) One of "extend", "replace", "raise". What to
do if the tables we want to write to already exist. Default "extend".
- "if_database_exists": (str): One of "extend", "replace", "raise". What to
do if the database we want to write to already exists. Default "extend".
loglike_kwargs (dict): Additional keyword arguments for loglike.
numdiff_options (dict): Keyword arguments for the calculation of numerical
derivatives for the calculation of standard errors. See
:ref:`first_derivative` for details.
jacobian (callable or None): A function that takes ``params`` and potentially
other keyword arguments and returns the jacobian of loglike["contributions"]
with respect to the params. Note that you only need to pass a Jacobian
function if you have a closed form Jacobian. If you pass None, a numerical
Jacobian will be calculated.
jacobian_kwargs (dict): Additional keyword arguments for the Jacobian function.
hessian (callable or None or False): A function that takes ``params`` and
potentially other keyword arguments and returns the Hessian of
loglike["value"] with respect to the params. If you pass None, a numerical
Hessian will be calculated. If you pass ``False``, you signal that no
Hessian should be calculated. Thus, no result that requires the Hessian will
be calculated.
hessian_kwargs (dict): Additional keyword arguments for the Hessian function.
design_info (pandas.DataFrame): DataFrame with one row per observation that
contains some or all of the variables "psu" (primary sampling unit),
"strata" and "fpc" (finite population corrector). See
:ref:`robust_likelihood_inference` for details.
Returns:
LikelihoodResult: A LikelihoodResult object.
"""
# ==================================================================================
# Check and process inputs
# ==================================================================================
is_optimized = optimize_options is False
if not is_optimized:
if isinstance(optimize_options, str):
optimize_options = {"algorithm": optimize_options}
check_optimization_options(
optimize_options,
usage="estimate_ml",
algorithm_mandatory=True,
)
jac_case = get_derivative_case(jacobian)
hess_case = get_derivative_case(hessian)
check_numdiff_options(numdiff_options, "estimate_ml")
numdiff_options = {} if numdiff_options in (None,
False) else numdiff_options
loglike_kwargs = {} if loglike_kwargs is None else loglike_kwargs
constraints = [] if constraints is None else constraints
jacobian_kwargs = {} if jacobian_kwargs is None else jacobian_kwargs
hessian_kwargs = {} if hessian_kwargs is None else hessian_kwargs
# ==================================================================================
# Calculate estimates via maximization (if necessary)
# ==================================================================================
if is_optimized:
estimates = params
opt_res = None
else:
opt_res = maximize(
criterion=loglike,
criterion_kwargs=loglike_kwargs,
params=params,
lower_bounds=lower_bounds,
upper_bounds=upper_bounds,
constraints=constraints,
logging=logging,
log_options=log_options,
**optimize_options,
)
estimates = opt_res.params
# ==================================================================================
# Do first function evaluations at estimated parameters
# ==================================================================================
try:
loglike_eval = loglike(estimates, **loglike_kwargs)
except (KeyboardInterrupt, SystemExit):
raise
except Exception as e:
msg = "Error while evaluating loglike at estimated params."
raise InvalidFunctionError(msg) from e
if callable(jacobian):
try:
jacobian_eval = jacobian(estimates, **jacobian_kwargs)
except (KeyboardInterrupt, SystemExit):
raise
except Exception as e:
msg = "Error while evaluating closed form jacobian at estimated params."
raise InvalidFunctionError(msg) from e
else:
jacobian_eval = None
if callable(hessian):
try:
hessian_eval = hessian(estimates, **hessian_kwargs)
except (KeyboardInterrupt, SystemExit):
raise
except Exception as e:
msg = "Error while evaluating closed form hessian at estimated params."
raise InvalidFunctionError(msg) from e
else:
hessian_eval = None
# ==================================================================================
# Get the converter for params and function outputs
# ==================================================================================
converter, internal_estimates = get_converter(
params=estimates,
constraints=constraints,
lower_bounds=lower_bounds,
upper_bounds=upper_bounds,
func_eval=loglike_eval,
primary_key="contributions",
scaling=False,
scaling_options=None,
derivative_eval=jacobian_eval,
)
# ==================================================================================
# Calculate internal jacobian
# ==================================================================================
if jac_case == "closed-form":
int_jac = converter.derivative_to_internal(jacobian_eval,
internal_estimates.values)
elif jac_case == "numerical":
def func(x):
p = converter.params_from_internal(x)
loglike_eval = loglike(p, **loglike_kwargs)["contributions"]
out = converter.func_to_internal(loglike_eval)
return out
jac_res = first_derivative(
func=func,
params=internal_estimates.values,
lower_bounds=internal_estimates.lower_bounds,
upper_bounds=internal_estimates.upper_bounds,
**numdiff_options,
)
int_jac = jac_res["derivative"]
else:
int_jac = None
if constraints in [None, []
] and jacobian_eval is None and int_jac is not None:
loglike_contribs = loglike_eval
if isinstance(loglike_contribs,
dict) and "contributions" in loglike_contribs:
loglike_contribs = loglike_contribs["contributions"]
jacobian_eval = matrix_to_block_tree(
int_jac,
outer_tree=loglike_contribs,
inner_tree=estimates,
)
if jacobian_eval is None:
_no_jac_reason = (
"no closed form jacobian was provided and there are constraints")
else:
_no_jac_reason = None
# ==================================================================================
# Calculate internal Hessian
# ==================================================================================
if hess_case == "skip":
int_hess = None
elif hess_case == "numerical":
def func(x):
p = converter.params_from_internal(x)
loglike_eval = loglike(p, **loglike_kwargs)["value"]
out = converter.func_to_internal(loglike_eval)
return out
hess_res = second_derivative(
func=func,
params=internal_estimates.values,
lower_bounds=internal_estimates.lower_bounds,
upper_bounds=internal_estimates.upper_bounds,
**numdiff_options,
)
int_hess = hess_res["derivative"]
elif hess_case == "closed-form" and constraints:
raise NotImplementedError(
"Closed-form Hessians are not yet compatible with constraints.")
elif hess_case == "closed-form":
int_hess = block_tree_to_matrix(
hessian_eval,
outer_tree=params,
inner_tree=params,
)
else:
raise ValueError()
if constraints in [None, []
] and hessian_eval is None and int_hess is not None:
hessian_eval = matrix_to_block_tree(
int_hess,
outer_tree=params,
inner_tree=params,
)
if hessian_eval is None:
if hess_case == "skip":
_no_hess_reason = "the hessian calculation was explicitly skipped."
else:
_no_hess_reason = (
"no closed form hessian was provided and there are constraints"
)
else:
_no_hess_reason = None
# ==================================================================================
# create a LikelihoodResult object
# ==================================================================================
free_estimates = calculate_free_estimates(estimates, internal_estimates)
res = LikelihoodResult(
_params=estimates,
_converter=converter,
_optimize_result=opt_res,
_jacobian=jacobian_eval,
_no_jacobian_reason=_no_jac_reason,
_hessian=hessian_eval,
_no_hessian_reason=_no_hess_reason,
_internal_jacobian=int_jac,
_internal_hessian=int_hess,
_design_info=design_info,
_internal_estimates=internal_estimates,
_free_estimates=free_estimates,
_has_constraints=constraints not in [None, []],
)
return res
def estimate_msm(
simulate_moments,
empirical_moments,
moments_cov,
params,
optimize_options,
*,
lower_bounds=None,
upper_bounds=None,
constraints=None,
logging=False,
log_options=None,
simulate_moments_kwargs=None,
weights="diagonal",
numdiff_options=None,
jacobian=None,
jacobian_kwargs=None,
):
"""Do a method of simulated moments or indirect inference estimation.
This is a high level interface for our lower level functions for minimization,
numerical differentiation, inference and sensitivity analysis. It does the full
workflow for MSM or indirect inference estimation with just one function call.
While we have good defaults, you can still configure each aspect of each steps
vial the optional arguments of this functions. If you find it easier to do the
minimization separately, you can do so and just provide the optimal parameters as
``params`` and set ``optimize_options=False``.
Args:
simulate_moments (callable): Function that takes params and potentially other
keyword arguments and returns a pytree with simulated moments. If the
function returns a dict containing the key ``"simulated_moments"`` we only
use the value corresponding to that key. Other entries are stored in the
log database if you use logging.
empirical_moments (pandas.Series): A pytree with the same structure as the
result of ``simulate_moments``.
moments_cov (pandas.DataFrame): A block-pytree containing the covariance
matrix of the empirical moments. This is typically calculated with
our ``get_moments_cov`` function.
params (pytree): A pytree containing the estimated or start parameters of the
model. If the supplied parameters are estimated parameters, set
optimize_options to False. Pytrees can be a numpy array, a pandas Series, a
DataFrame with "value" column, a float and any kind of (nested) dictionary
or list containing these elements. See :ref:`params` for examples.
optimize_options (dict, str or False): Keyword arguments that govern the
numerical optimization. Valid entries are all arguments of
:func:`~estimagic.optimization.optimize.minimize` except for those that can
be passed explicitly to ``estimate_msm``. If you pass False as
``optimize_options`` you signal that ``params`` are already
the optimal parameters and no numerical optimization is needed. If you pass
a str as optimize_options it is used as the ``algorithm`` option.
lower_bounds (pytree): A pytree with the same structure as params with lower
bounds for the parameters. Can be ``-np.inf`` for parameters with no lower
bound.
upper_bounds (pytree): As lower_bounds. Can be ``np.inf`` for parameters with
no upper bound.
simulate_moments_kwargs (dict): Additional keyword arguments for
``simulate_moments``.
weights (str): One of "diagonal" (default), "identity" or "optimal".
Note that "optimal" refers to the asymptotically optimal weighting matrix
and is often not a good choice due to large finite sample bias.
constraints (list, dict): List with constraint dictionaries or single dict.
See :ref:`constraints`.
logging (pathlib.Path, str or False): Path to sqlite3 file (which typically has
the file extension ``.db``. If the file does not exist, it will be created.
The dashboard can only be used when logging is used.
log_options (dict): Additional keyword arguments to configure the logging.
- "fast_logging" (bool):
A boolean that determines if "unsafe" settings are used to speed up
write processes to the database. This should only be used for very short
running criterion functions where the main purpose of the log is a
real-time dashboard and it would not be catastrophic to get a corrupted
database in case of a sudden system shutdown. If one evaluation of the
criterion function (and gradient if applicable) takes more than 100 ms,
the logging overhead is negligible.
- "if_table_exists" (str):
One of "extend", "replace", "raise". What to do if the tables we want to
write to already exist. Default "extend".
- "if_database_exists" (str):
One of "extend", "replace", "raise". What to do if the database we want
to write to already exists. Default "extend".
numdiff_options (dict): Keyword arguments for the calculation of numerical
derivatives for the calculation of standard errors. See
:ref:`first_derivative` for details. Note that by default we increase the
step_size by a factor of 2 compared to the rule of thumb for optimal
step sizes. This is because many msm criterion functions are slightly noisy.
jacobian (callable): A function that take ``params`` and
potentially other keyword arguments and returns the jacobian of
simulate_moments with respect to the params.
jacobian_kwargs (dict): Additional keyword arguments for the jacobian function.
Returns:
dict: The estimated parameters, standard errors and sensitivity measures
and covariance matrix of the parameters.
"""
# ==================================================================================
# Check and process inputs
# ==================================================================================
if weights not in ["diagonal", "optimal"]:
raise NotImplementedError(
"Custom weighting matrices are not yet implemented.")
is_optimized = optimize_options is False
if not is_optimized:
if isinstance(optimize_options, str):
optimize_options = {"algorithm": optimize_options}
check_optimization_options(
optimize_options,
usage="estimate_msm",
algorithm_mandatory=True,
)
jac_case = get_derivative_case(jacobian)
check_numdiff_options(numdiff_options, "estimate_msm")
numdiff_options = {} if numdiff_options in (
None, False) else numdiff_options.copy()
if "scaling_factor" not in numdiff_options:
numdiff_options["scaling_factor"] = 2
weights, internal_weights = get_weighting_matrix(
moments_cov=moments_cov,
method=weights,
empirical_moments=empirical_moments,
return_type="pytree_and_array",
)
internal_moments_cov = block_tree_to_matrix(
moments_cov,
outer_tree=empirical_moments,
inner_tree=empirical_moments,
)
constraints = [] if constraints is None else constraints
jacobian_kwargs = {} if jacobian_kwargs is None else jacobian_kwargs
simulate_moments_kwargs = ({} if simulate_moments_kwargs is None else
simulate_moments_kwargs)
# ==================================================================================
# Calculate estimates via minimization (if necessary)
# ==================================================================================
if is_optimized:
estimates = params
opt_res = None
else:
funcs = get_msm_optimization_functions(
simulate_moments=simulate_moments,
empirical_moments=empirical_moments,
weights=weights,
simulate_moments_kwargs=simulate_moments_kwargs,
# Always pass None because we do not support closed form jacobians during
# optimization yet. Otherwise we would get a NotImplementedError
jacobian=None,
jacobian_kwargs=jacobian_kwargs,
)
opt_res = minimize(
lower_bounds=lower_bounds,
upper_bounds=upper_bounds,
constraints=constraints,
logging=logging,
log_options=log_options,
params=params,
**funcs, # contains the criterion func and possibly more
**optimize_options,
)
estimates = opt_res.params
# ==================================================================================
# do first function evaluations
# ==================================================================================
try:
sim_mom_eval = simulate_moments(estimates, **simulate_moments_kwargs)
except (KeyboardInterrupt, SystemExit):
raise
except Exception as e:
msg = "Error while evaluating simulate_moments at estimated params."
raise InvalidFunctionError(msg) from e
if callable(jacobian):
try:
jacobian_eval = jacobian(estimates, **jacobian_kwargs)
except (KeyboardInterrupt, SystemExit):
raise
except Exception as e:
msg = "Error while evaluating derivative at estimated params."
raise InvalidFunctionError(msg) from e
else:
jacobian_eval = None
# ==================================================================================
# get converter for params and function outputs
# ==================================================================================
def helper(params):
raw = simulate_moments(params, **simulate_moments_kwargs)
if isinstance(raw, dict) and "simulated_moments" in raw:
out = {"contributions": raw["simulated_moments"]}
else:
out = {"contributions": raw}
return out
if isinstance(sim_mom_eval, dict) and "simulated_moments" in sim_mom_eval:
func_eval = {"contributions": sim_mom_eval["simulated_moments"]}
else:
func_eval = {"contributions": sim_mom_eval}
converter, internal_estimates = get_converter(
params=estimates,
constraints=constraints,
lower_bounds=lower_bounds,
upper_bounds=upper_bounds,
func_eval=func_eval,
primary_key="contributions",
scaling=False,
scaling_options=None,
derivative_eval=jacobian_eval,
)
# ==================================================================================
# Calculate internal jacobian
# ==================================================================================
if jac_case == "closed-form":
x = converter.params_to_internal(estimates)
int_jac = converter.derivative_to_internal(jacobian_eval, x)
else:
def func(x):
p = converter.params_from_internal(x)
sim_mom_eval = helper(p)
out = converter.func_to_internal(sim_mom_eval)
return out
int_jac = first_derivative(
func=func,
params=internal_estimates.values,
lower_bounds=internal_estimates.lower_bounds,
upper_bounds=internal_estimates.upper_bounds,
**numdiff_options,
)["derivative"]
# ==================================================================================
# Calculate external jac (if no constraints and not closed form )
# ==================================================================================
if constraints in [None, []
] and jacobian_eval is None and int_jac is not None:
jacobian_eval = matrix_to_block_tree(
int_jac,
outer_tree=empirical_moments,
inner_tree=estimates,
)
if jacobian_eval is None:
_no_jac_reason = (
"no closed form jacobian was provided and there are constraints")
else:
_no_jac_reason = None
# ==================================================================================
# Create MomentsResult
# ==================================================================================
free_estimates = calculate_free_estimates(estimates, internal_estimates)
res = MomentsResult(
_params=estimates,
_weights=weights,
_converter=converter,
_internal_weights=internal_weights,
_internal_moments_cov=internal_moments_cov,
_internal_jacobian=int_jac,
_jacobian=jacobian_eval,
_no_jacobian_reason=_no_jac_reason,
_empirical_moments=empirical_moments,
_internal_estimates=internal_estimates,
_free_estimates=free_estimates,
_has_constraints=constraints not in [None, []],
)
return res
raw = calculate_sensitivity_to_weighting(
jac=jac,
weights=weights,
moments_cov=moments_cov,
params_cov=params_cov,
)
else:
raise ValueError(f"Invalid kind: {kind}")
if return_type == "array":
out = raw
elif return_type == "pytree":
out = matrix_to_block_tree(
raw,
outer_tree=self._params,
inner_tree=self._empirical_moments,
)
elif return_type == "dataframe":
registry = get_registry(extended=True)
row_names = self._internal_estimates.names
col_names = leaf_names(self._empirical_moments, registry=registry)
out = pd.DataFrame(
data=raw,
index=row_names,
columns=col_names,
)
else:
msg = (
f"Invalid return type: {return_type}. Valid are 'pytree', 'array' "
"and 'dataframe'")
def first_derivative(
func,
params,
*,
func_kwargs=None,
method="central",
n_steps=1,
base_steps=None,
scaling_factor=1,
lower_bounds=None,
upper_bounds=None,
step_ratio=2,
min_steps=None,
f0=None,
n_cores=DEFAULT_N_CORES,
error_handling="continue",
batch_evaluator="joblib",
return_func_value=False,
return_info=False,
key=None,
):
"""Evaluate first derivative of func at params according to method and step options.
Internally, the function is converted such that it maps from a 1d array to a 1d
array. Then the Jacobian of that function is calculated.
The parameters and the function output can be estimagic-pytrees; for more details on
estimagi-pytrees see :ref:`eeppytrees`. By default the resulting Jacobian will be
returned as a block-pytree.
For a detailed description of all options that influence the step size as well as an
explanation of how steps are adjusted to bounds in case of a conflict, see
:func:`~estimagic.differentiation.generate_steps.generate_steps`.
Args:
func (callable): Function of which the derivative is calculated.
params (pytree): A pytree. See :ref:`params`.
func_kwargs (dict): Additional keyword arguments for func, optional.
method (str): One of ["central", "forward", "backward"], default "central".
n_steps (int): Number of steps needed. For central methods, this is
the number of steps per direction. It is 1 if no Richardson extrapolation
is used.
base_steps (numpy.ndarray, optional): 1d array of the same length as params.
base_steps * scaling_factor is the absolute value of the first (and possibly
only) step used in the finite differences approximation of the derivative.
If base_steps * scaling_factor conflicts with bounds, the actual steps will
be adjusted. If base_steps is not provided, it will be determined according
to a rule of thumb as long as this does not conflict with min_steps.
scaling_factor (numpy.ndarray or float): Scaling factor which is applied to
base_steps. If it is an numpy.ndarray, it needs to be as long as params.
scaling_factor is useful if you want to increase or decrease the base_step
relative to the rule-of-thumb or user provided base_step, for example to
benchmark the effect of the step size. Default 1.
lower_bounds (pytree): To be written.
upper_bounds (pytree): To be written.
step_ratio (float, numpy.array): Ratio between two consecutive Richardson
extrapolation steps in the same direction. default 2.0. Has to be larger
than one. The step ratio is only used if n_steps > 1.
min_steps (numpy.ndarray): Minimal possible step sizes that can be chosen to
accommodate bounds. Must have same length as params. By default min_steps is
equal to base_steps, i.e step size is not decreased beyond what is optimal
according to the rule of thumb.
f0 (numpy.ndarray): 1d numpy array with func(x), optional.
n_cores (int): Number of processes used to parallelize the function
evaluations. Default 1.
error_handling (str): One of "continue" (catch errors and continue to calculate
derivative estimates. In this case, some derivative estimates can be
missing but no errors are raised), "raise" (catch errors and continue
to calculate derivative estimates at fist but raise an error if all
evaluations for one parameter failed) and "raise_strict" (raise an error
as soon as a function evaluation fails).
batch_evaluator (str or callable): Name of a pre-implemented batch evaluator
(currently 'joblib' and 'pathos_mp') or Callable with the same interface
as the estimagic batch_evaluators.
return_func_value (bool): If True, return function value at params, stored in
output dict under "func_value". Default False. This is useful when using
first_derivative during optimization.
return_info (bool): If True, return additional information on function
evaluations and internal derivative candidates, stored in output dict under
"func_evals" and "derivative_candidates". Derivative candidates are only
returned if n_steps > 1. Default False.
key (str): If func returns a dictionary, take the derivative of
func(params)[key].
Returns:
result (dict): Result dictionary with keys:
- "derivative" (numpy.ndarray, pandas.Series or pandas.DataFrame): The
estimated first derivative of func at params. The shape of the output
depends on the dimension of params and func(params):
- f: R -> R leads to shape (1,), usually called derivative
- f: R^m -> R leads to shape (m, ), usually called Gradient
- f: R -> R^n leads to shape (n, 1), usually called Jacobian
- f: R^m -> R^n leads to shape (n, m), usually called Jacobian
- "func_value" (numpy.ndarray, pandas.Series or pandas.DataFrame): Function
value at params, returned if return_func_value is True.
- "func_evals" (pandas.DataFrame): Function evaluations produced by internal
derivative method, returned if return_info is True.
- "derivative_candidates" (pandas.DataFrame): Derivative candidates from
Richardson extrapolation, returned if return_info is True and n_steps >
1.
"""
_is_fast_params = isinstance(params, np.ndarray) and params.ndim == 1
registry = get_registry(extended=True)
lower_bounds, upper_bounds = get_bounds(params, lower_bounds, upper_bounds)
# handle keyword arguments
func_kwargs = {} if func_kwargs is None else func_kwargs
partialed_func = functools.partial(func, **func_kwargs)
# convert params to numpy
if not _is_fast_params:
x, params_treedef = tree_flatten(params, registry=registry)
x = np.array(x, dtype=np.float64)
else:
x = params.astype(float)
if np.isnan(x).any():
raise ValueError("The parameter vector must not contain NaNs.")
# generate the step array
steps = generate_steps(
x=x,
method=method,
n_steps=n_steps,
target="first_derivative",
base_steps=base_steps,
scaling_factor=scaling_factor,
lower_bounds=lower_bounds,
upper_bounds=upper_bounds,
step_ratio=step_ratio,
min_steps=min_steps,
)
# generate parameter vectors at which func has to be evaluated as numpy arrays
evaluation_points = []
for step_arr in steps:
for i, j in product(range(n_steps), range(len(x))):
if np.isnan(step_arr[i, j]):
evaluation_points.append(np.nan)
else:
point = x.copy()
point[j] += step_arr[i, j]
evaluation_points.append(point)
# convert the numpy arrays to whatever is needed by func
if not _is_fast_params:
evaluation_points = [
# entries are either a numpy.ndarray or np.nan
_unflatten_if_not_nan(p, params_treedef, registry)
for p in evaluation_points
]
# we always evaluate f0, so we can fall back to one-sided derivatives if
# two-sided derivatives fail. The extra cost is negligible in most cases.
if f0 is None:
evaluation_points.append(params)
# do the function evaluations, including error handling
batch_error_handling = "raise" if error_handling == "raise_strict" else "continue"
raw_evals = _nan_skipping_batch_evaluator(
func=partialed_func,
arguments=evaluation_points,
n_cores=n_cores,
error_handling=batch_error_handling,
batch_evaluator=batch_evaluator,
)
# extract information on exceptions that occurred during function evaluations
exc_info = "\n\n".join([val for val in raw_evals if isinstance(val, str)])
raw_evals = [
val if not isinstance(val, str) else np.nan for val in raw_evals
]
# store full function value at params as func_value and a processed version of it
# that we need to calculate derivatives as f0
if f0 is None:
f0 = raw_evals[-1]
raw_evals = raw_evals[:-1]
func_value = f0
use_key = key is not None and isinstance(f0, dict)
f0_tree = f0[key] if use_key else f0
scalar_out = np.isscalar(f0_tree)
vector_out = isinstance(f0_tree, np.ndarray) and f0_tree.ndim == 1
if scalar_out:
f0 = np.array([f0_tree], dtype=float)
elif vector_out:
f0 = f0_tree.astype(float)
else:
f0 = tree_leaves(f0_tree, registry=registry)
f0 = np.array(f0, dtype=np.float64)
# convert the raw evaluations to numpy arrays
raw_evals = _convert_evals_to_numpy(
raw_evals=raw_evals,
key=key,
registry=registry,
is_scalar_out=scalar_out,
is_vector_out=vector_out,
)
# apply finite difference formulae
evals = np.array(raw_evals).reshape(2, n_steps, len(x), -1)
evals = np.transpose(evals, axes=(0, 1, 3, 2))
evals = Evals(pos=evals[0], neg=evals[1])
jac_candidates = {}
for m in ["forward", "backward", "central"]:
jac_candidates[m] = finite_differences.jacobian(evals, steps, f0, m)
# get the best derivative estimate out of all derivative estimates that could be
# calculated, given the function evaluations.
orders = {
"central": ["central", "forward", "backward"],
"forward": ["forward", "backward"],
"backward": ["backward", "forward"],
}
if n_steps == 1:
jac = _consolidate_one_step_derivatives(jac_candidates, orders[method])
updated_candidates = None
else:
richardson_candidates = _compute_richardson_candidates(
jac_candidates, steps, n_steps)
jac, updated_candidates = _consolidate_extrapolated(
richardson_candidates)
# raise error if necessary
if error_handling in ("raise", "raise_strict") and np.isnan(jac).any():
raise Exception(exc_info)
# results processing
if _is_fast_params and vector_out:
derivative = jac
elif _is_fast_params and scalar_out:
derivative = jac.flatten()
else:
derivative = matrix_to_block_tree(jac, f0_tree, params)
result = {"derivative": derivative}
if return_func_value:
result["func_value"] = func_value
if return_info:
info = _collect_additional_info(steps,
evals,
updated_candidates,
target="first_derivative")
result = {**result, **info}
return result
