"""The solver module contains the different solvers that are available for
Pastas.
All solvers inherit from the BaseSolver class, which contains general method
for selecting the correct time series to misfit and options to weight the
residuals or noise series.
Supported Solvers
-----------------
.. autosummary::
:nosignatures:
:toctree: generated/
LeastSquares
LmfitSolve
Examples
--------
To solve a model the following syntax can be used:
>>> ml.solve(solver=ps.LeastSquares)
"""
from logging import getLogger
import numpy as np
from pandas import DataFrame
from scipy.linalg import svd
from scipy.optimize import least_squares
logger = getLogger(__name__)
class BaseSolver:
_name = "BaseSolver"
__doc__ = """All solver instances inherit from the BaseSolver class.
Attributes
----------
model: pastas.Model instance
pcor: pandas.DataFrame
Pandas DataFrame with the correlation between the optimized parameters.
pcov: pandas.DataFrame
Pandas DataFrame with the correlation between the optimized parameters.
nfev: int
Number of times the model is called during optimization.
result: object
The object returned by the minimization method that is used. It depends
on the solver what is actually returned.
Methods
-------
misfit
get_correlations
"""
def __init__(self, ml, pcov=None, nfev=None, **kwargs):
self.ml = ml
self.pcov = pcov # Covariances of the parameters
if pcov is None:
self.pcor = None # Correlation between parameters
else:
self.pcor = self.get_correlations(pcov)
self.nfev = nfev # number of function evaluations
self.result = None # Object returned by the optimization method
def misfit(self, parameters, noise, weights=None, callback=None):
"""This method is called by all solvers to obtain a series that are
minimized in the optimization proces. It handles the application of
the weigths, a noisemodel and other optimization options.
Parameters
----------
parameters: list, numpy.ndarray
list with the parameters
noise: Boolean
weights: pandas.Series, optional
pandas Series by which the residual or noise series are
multiplied. Typically values between 0 and 1.
callback: ufunc, optional
function that is called after each iteration. the parameters are
provided to the func. E.g. "callback(parameters)"
Returns
-------
rv:
residuals series (if noise=False) or noise series (if noise=True)
"""
# Get the residuals or the noise
if noise:
rv = self.ml.noise(parameters)
else:
rv = self.ml.residuals(parameters)
# Determine if weights need to be applied
if weights is not None:
weights = weights.reindex(rv.index)
weights.fillna(1.0, inplace=True)
rv = rv.multiply(weights)
if callback:
callback(parameters)
return rv.values
def prediction_interval(self, n=1000, alpha=0.05, **kwargs):
"""Method to calculate the prediction interval for the simulation.
Returns
-------
data : Pandas.DataFrame
DataFrame of length number of observations and two columns labeled
0.025 and 0.975 (numerical values) containing the 2.5% and 97.5%
prediction interval (for alpha=0.05)
Notes
-----
Add residuals assuming a Normal distribution with standard deviation
equal to the standard deviation of the residuals.
"""
sigr = self.ml.residuals().std()
data = self.get_realizations(func=self.ml.simulate, n=n, name=None,
**kwargs)
data = data + sigr * np.random.randn(data.shape[0], data.shape[1])
q = [alpha / 2, 1 - alpha / 2]
rv = data.quantile(q, axis=1).transpose()
return rv
def ci_simulation(self, n=None, alpha=0.05, **kwargs):
"""Method to calculate the confidence interval for the simulation.
Returns
-------
Notes
-----
The confidence interval shows the uncertainty in the simulation due
to parameter uncertainty. In other words, there is a 95% probability
that the true best-fit line for the observed data lies within the
95% confidence interval.
"""
return self.get_confidence_interval(func=self.ml.simulate, n=n,
alpha=alpha, **kwargs)
def ci_block_response(self, name, n=None, alpha=0.05, **kwargs):
dt = self.ml.get_block_response(name=name).index.values
return self.get_confidence_interval(func=self.ml.get_block_response,
n=n, alpha=alpha, name=name, dt=dt,
**kwargs)
def ci_step_response(self, name, n=None, alpha=0.05, **kwargs):
dt = self.ml.get_block_response(name=name).index.values
return self.get_confidence_interval(func=self.ml.get_step_response,
n=n, alpha=alpha, name=name,
dt=dt, **kwargs)
def ci_contribution(self, name, n=None, alpha=0.05, **kwargs):
return self.get_confidence_interval(func=self.ml.get_contribution, n=n,
alpha=alpha, name=name, **kwargs)
def get_realizations(self, func, n=None, name=None, **kwargs):
"""Internal method to obtain n number of realizations."""
if name:
kwargs["name"] = name
params = self.get_parameter_sample(n=n, name=name)
data = {}
for i, param in enumerate(params):
data[i] = func(parameters=param, **kwargs)
return DataFrame.from_dict(data, orient="columns")
def get_confidence_interval(self, func, n=None, name=None, alpha=0.05,
**kwargs):
"""Internal method to obtain a confidence interval."""
q = [alpha / 2, 1 - alpha / 2]
data = self.get_realizations(func=func, n=n, name=name, **kwargs)
return data.quantile(q=q, axis=1).transpose()
def get_parameter_sample(self, name=None, n=None):
"""Internal method to obtain a parameter sets.
Parameters
----------
n: int, optional
Number of random samples drawn from the bivariate normal
distribution.
name: str, optional
Name of the stressmodel or model component to obtain the
parameters for.
Returns
-------
ndarray
Numpy array with N parameter samples.
"""
par = self.ml.get_parameters(name=name)
pcov = self.get_covariance_matrix(name=name)
if n is None:
n = 10 ** par.size
return np.random.multivariate_normal(par, pcov, n,
check_valid="ignore")
def get_covariance_matrix(self, name=None):
"""Internal method to obtain the covariance matrix from the model.
Parameters
----------
name: str, optional
Name of the stressmodel or model component to obtain the
parameters for.
Returns
-------
pcov: pandas.DataFrame
Pandas DataFrame with the covariances for the parameters.
"""
if name:
params = self.ml.parameters.loc[self.ml.parameters.loc[:,
"name"] == name].index
else:
params = self.ml.parameters.index
pcov = self.pcov.loc[params, params].fillna(0)
return pcov
@staticmethod
def get_correlations(pcov):
"""Method to obtain the parameter correlations from the covariance
matrix.
Parameters
----------
pcov: pandas.DataFrame
n x n Pandas DataFrame with the covariances.
Returns
-------
pcor: pandas.DataFrame
n x n Pandas DataFrame with the correlations.
"""
pcor = pcov.loc[pcov.index, pcov.index].copy()
for i in pcor.index:
for j in pcor.columns:
pcor.loc[i, j] = pcov.loc[i, j] / \
np.sqrt(pcov.loc[i, i] * pcov.loc[j, j])
return pcor
def to_dict(self):
data = {
"name": self._name,
"pcov": self.pcov,
"nfev": self.nfev
}
return data
class LeastSquares(BaseSolver):
_name = "LeastSquares"
def __init__(self, ml, pcov=None, nfev=None, **kwargs):
"""Solver based on Scipy's least_squares method [scipy_ref]_.
Notes
-----
This class is the default solve method called by the pastas Model solve
method. All kwargs provided to the Model.solve() method are forwarded
to the solver. From there, they are forwarded to Scipy least_squares
solver.
Examples
--------
>>> ml.solve(solver=ps.LeastSquares)
References
----------
.. [scipy_ref] https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html
"""
BaseSolver.__init__(self, ml=ml, pcov=pcov, nfev=nfev, **kwargs)
def solve(self, noise=True, weights=None, callback=None, **kwargs):
self.vary = self.ml.parameters.vary.values.astype(bool)
self.initial = self.ml.parameters.initial.values.copy()
parameters = self.ml.parameters.loc[self.vary]
# Set the boundaries
bounds = (np.where(parameters.pmin.isnull(), -np.inf, parameters.pmin),
np.where(parameters.pmax.isnull(), np.inf, parameters.pmax))
self.result = least_squares(self.objfunction, bounds=bounds,
x0=parameters.initial.values,
args=(noise, weights, callback), **kwargs)
self.pcov = DataFrame(self.get_covariances(self.result.jac,
self.result.cost),
index=parameters.index, columns=parameters.index)
self.pcor = self.get_correlations(self.pcov)
self.nfev = self.result.nfev
# Prepare return values
success = self.result.success
optimal = self.initial
optimal[self.vary] = self.result.x
stderr = np.zeros(len(optimal)) * np.nan
stderr[self.vary] = np.sqrt(np.diag(self.pcov))
return success, optimal, stderr
def objfunction(self, parameters, noise, weights, callback):
p = self.initial
p[self.vary] = parameters
return self.misfit(p, noise, weights, callback)
def get_covariances(self, jacobian, cost, absolute_sigma=False):
"""Method to get the covariance matrix from the jacobian.
Parameters
----------
jacobian: numpy.ndarray
cost: float
absolute_sigma: bool
Default is False
Returns
-------
pcov: numpy.array
numpy array with the covariance matrix.
Notes
-----
This method is copied from Scipy, please refer to:
https://github.com/scipy/scipy/blob/v1.0.0/scipy/optimize/optimize.py
"""
cost = 2 * cost # res.cost is half sum of squares!
# Do Moore-Penrose inverse discarding zero singular values.
_, s, VT = svd(jacobian, full_matrices=False)
threshold = np.finfo(float).eps * max(jacobian.shape) * s[0]
s = s[s > threshold]
VT = VT[:s.size]
pcov = np.dot(VT.T / s ** 2, VT)
n_param = self.ml.parameters.index.size
warn_cov = False
if pcov is None:
# indeterminate covariance
pcov = np.zeros((n_param, n_param), dtype=float)
pcov.fill(np.inf)
warn_cov = True
elif not absolute_sigma:
if self.ml.oseries.series.index.size > n_param:
s_sq = cost / (self.ml.oseries.series.index.size - n_param)
pcov = pcov * s_sq
else:
pcov.fill(np.inf)
warn_cov = True
if warn_cov:
logger.warning(
'Covariance of the parameters could not be estimated')
return pcov
class LmfitSolve(BaseSolver):
_name = "LmfitSolve"
def __init__(self, ml, pcov=None, nfev=None, **kwargs):
"""Solving the model using the LmFit solver [LM]_.
This is basically a wrapper around the scipy solvers, adding some
cool functionality for boundary conditions.
References
----------
.. [LM] https://github.com/lmfit/lmfit-py/
"""
try:
global lmfit
import lmfit as lmfit # Import Lmfit here, so it is no dependency
except ImportError:
msg = "lmfit not installed. Please install lmfit first."
raise ImportError(msg)
BaseSolver.__init__(self, ml=ml, pcov=pcov, nfev=nfev, **kwargs)
def solve(self, noise=True, weights=None, callback=None, method="leastsq",
**kwargs):
# Deal with the parameters
parameters = lmfit.Parameters()
p = self.ml.parameters.loc[:, ['initial', 'pmin', 'pmax', 'vary']]
for k in p.index:
pp = np.where(p.loc[k].isnull(), None, p.loc[k])
parameters.add(k, value=pp[0], min=pp[1], max=pp[2], vary=pp[3])
# Create the Minimizer object and minimize
self.mini = lmfit.Minimizer(userfcn=self.objfunction, calc_covar=True,
fcn_args=(noise, weights, callback),
params=parameters, **kwargs)
self.result = self.mini.minimize(method=method)
# Set all parameter attributes
pcov = None
if hasattr(self.result, "covar"):
if self.result.covar is not None:
pcov = self.result.covar
names = self.result.var_names
self.pcov = DataFrame(pcov, index=names, columns=names)
self.pcor = self.get_correlations(self.pcov)
# Set all optimization attributes
self.nfev = self.result.nfev
if hasattr(self.result, "success"):
success = self.result.success
else:
success = True
optimal = np.array([p.value for p in self.result.params.values()])
stderr = np.array([p.stderr for p in self.result.params.values()])
idx = None
if "is_weighted" in kwargs:
if not kwargs["is_weighted"]:
idx = -1
return success, optimal[:idx], stderr[:idx]
def objfunction(self, parameters, noise, weights, callback):
param = np.array([p.value for p in parameters.values()])
return self.misfit(param, noise, weights, callback)
