# -*- coding: utf-8 -*-
"""
Generalized Additive Models
Author: Luca Puggini
Author: Josef Perktold
created on 08/07/2015
"""
from collections.abc import Iterable
import copy # check if needed when dropping python 2.7
import numpy as np
from scipy import optimize
import pandas as pd
import statsmodels.base.wrapper as wrap
from statsmodels.discrete.discrete_model import Logit
from statsmodels.genmod.generalized_linear_model import (
GLM, GLMResults, GLMResultsWrapper, _check_convergence)
import statsmodels.regression.linear_model as lm
# import statsmodels.regression._tools as reg_tools # TODO: use this for pirls
from statsmodels.tools.sm_exceptions import (PerfectSeparationError,
ValueWarning)
from statsmodels.tools.decorators import cache_readonly
from statsmodels.tools.data import _is_using_pandas
from statsmodels.tools.linalg import matrix_sqrt
from statsmodels.base._penalized import PenalizedMixin
from statsmodels.gam.gam_penalties import MultivariateGamPenalty
from statsmodels.gam.gam_cross_validation.gam_cross_validation import (
MultivariateGAMCVPath)
from statsmodels.gam.gam_cross_validation.cross_validators import KFold
def _transform_predict_exog(model, exog, design_info=None):
"""transform exog for predict using design_info
Note: this is copied from base.model.Results.predict and converted to
standalone function with additional options.
"""
is_pandas = _is_using_pandas(exog, None)
exog_index = exog.index if is_pandas else None
if design_info is None:
design_info = getattr(model.data, 'design_info', None)
if design_info is not None and (exog is not None):
from patsy import dmatrix
if isinstance(exog, pd.Series):
# we are guessing whether it should be column or row
if (hasattr(exog, 'name') and isinstance(exog.name, str) and
exog.name in design_info.describe()):
# assume we need one column
exog = pd.DataFrame(exog)
else:
# assume we need a row
exog = pd.DataFrame(exog).T
orig_exog_len = len(exog)
is_dict = isinstance(exog, dict)
exog = dmatrix(design_info, exog, return_type="dataframe")
if orig_exog_len > len(exog) and not is_dict:
import warnings
if exog_index is None:
warnings.warn('nan values have been dropped', ValueWarning)
else:
exog = exog.reindex(exog_index)
exog_index = exog.index
if exog is not None:
exog = np.asarray(exog)
if exog.ndim == 1 and (model.exog.ndim == 1 or
model.exog.shape[1] == 1):
exog = exog[:, None]
exog = np.atleast_2d(exog) # needed in count model shape[1]
return exog, exog_index
[docs]class GLMGamResults(GLMResults):
"""Results class for generalized additive models, GAM.
This inherits from GLMResults.
Warning: some inherited methods might not correctly take account of the
penalization
GLMGamResults inherits from GLMResults
All methods related to the loglikelihood function return the penalized
values.
Attributes
----------
edf
list of effective degrees of freedom for each column of the design
matrix.
hat_matrix_diag
diagonal of hat matrix
gcv
generalized cross-validation criterion computed as
``gcv = scale / (1. - hat_matrix_trace / self.nobs)**2``
cv
cross-validation criterion computed as
``cv = ((resid_pearson / (1 - hat_matrix_diag))**2).sum() / nobs``
Notes
-----
status: experimental
"""
def __init__(self, model, params, normalized_cov_params, scale, **kwds):
# this is a messy way to compute edf and update scale
# need several attributes to compute edf
self.model = model
self.params = params
self.normalized_cov_params = normalized_cov_params
self.scale = scale
edf = self.edf.sum()
self.df_model = edf - 1 # assume constant
# need to use nobs or wnobs attribute
self.df_resid = self.model.endog.shape[0] - edf
# we are setting the model df for the case when super is using it
# df in model will be incorrect state when alpha/pen_weight changes
self.model.df_model = self.df_model
self.model.df_resid = self.df_resid
mu = self.fittedvalues
self.scale = scale = self.model.estimate_scale(mu)
super(GLMGamResults, self).__init__(model, params,
normalized_cov_params, scale,
**kwds)
def _tranform_predict_exog(self, exog=None, exog_smooth=None,
transform=True):
"""Transform original explanatory variables for prediction
Parameters
----------
exog : array_like, optional
The values for the linear explanatory variables.
exog_smooth : array_like
values for the variables in the smooth terms
transform : bool, optional
If transform is False, then ``exog`` is returned unchanged and
``x`` is ignored. It is assumed that exog contains the full
design matrix for the predict observations.
If transform is True, then the basis representation of the smooth
term will be constructed from the provided ``x``.
Returns
-------
exog_transformed : ndarray
design matrix for the prediction
"""
exog_index = None
if transform is False:
# the following allows that either or both exog are not None
if exog_smooth is None:
# exog could be None or array
ex = exog
else:
if exog is None:
ex = exog_smooth
else:
ex = np.column_stack((exog, exog_smooth))
else:
# transform exog_linear if needed
if exog is not None and hasattr(self.model, 'design_info_linear'):
exog, exog_index = _transform_predict_exog(
self.model, exog, self.model.design_info_linear)
# create smooth basis
if exog_smooth is not None:
ex_smooth = self.model.smoother.transform(exog_smooth)
if exog is None:
ex = ex_smooth
else:
# TODO: there might be problems is exog_smooth is 1-D
ex = np.column_stack((exog, ex_smooth))
else:
ex = exog
return ex, exog_index
[docs] def predict(self, exog=None, exog_smooth=None, transform=True, **kwargs):
""""
compute prediction
Parameters
----------
exog : array_like, optional
The values for the linear explanatory variables
exog_smooth : array_like
values for the variables in the smooth terms
transform : bool, optional
If transform is True, then the basis representation of the smooth
term will be constructed from the provided ``exog``.
kwargs :
Some models can take additional arguments or keywords, see the
predict method of the model for the details.
Returns
-------
prediction : ndarray, pandas.Series or pandas.DataFrame
predicted values
"""
ex, exog_index = self._tranform_predict_exog(exog=exog,
exog_smooth=exog_smooth,
transform=transform)
predict_results = super(GLMGamResults, self).predict(ex,
transform=False,
**kwargs)
if exog_index is not None and not hasattr(
predict_results, 'predicted_values'):
if predict_results.ndim == 1:
return pd.Series(predict_results, index=exog_index)
else:
return pd.DataFrame(predict_results, index=exog_index)
else:
return predict_results
[docs] def get_prediction(self, exog=None, exog_smooth=None, transform=True,
**kwargs):
"""compute prediction results
Parameters
----------
exog : array_like, optional
The values for which you want to predict.
exog_smooth : array_like
values for the variables in the smooth terms
transform : bool, optional
If transform is True, then the basis representation of the smooth
term will be constructed from the provided ``x``.
kwargs :
Some models can take additional arguments or keywords, see the
predict method of the model for the details.
Returns
-------
prediction_results : generalized_linear_model.PredictionResults
The prediction results instance contains prediction and prediction
variance and can on demand calculate confidence intervals and
summary tables for the prediction of the mean and of new
observations.
"""
ex, exog_index = self._tranform_predict_exog(exog=exog,
exog_smooth=exog_smooth,
transform=transform)
return super(GLMGamResults, self).get_prediction(ex, transform=False,
**kwargs)
[docs] def partial_values(self, smooth_index, include_constant=True):
"""contribution of a smooth term to the linear prediction
Warning: This will be replaced by a predict method
Parameters
----------
smooth_index : int
index of the smooth term within list of smooth terms
include_constant : bool
If true, then the estimated intercept is added to the prediction
and its standard errors. This avoids that the confidence interval
has zero width at the imposed identification constraint, e.g.
either at a reference point or at the mean.
Returns
-------
predicted : nd_array
predicted value of linear term.
This is not the expected response if the link function is not
linear.
se_pred : nd_array
standard error of linear prediction
"""
variable = smooth_index
smoother = self.model.smoother
mask = smoother.mask[variable]
start_idx = self.model.k_exog_linear
idx = start_idx + np.nonzero(mask)[0]
# smoother has only smooth parts, not exog_linear
exog_part = smoother.basis[:, mask]
const_idx = self.model.data.const_idx
if include_constant and const_idx is not None:
idx = np.concatenate(([const_idx], idx))
exog_part = self.model.exog[:, idx]
linpred = np.dot(exog_part, self.params[idx])
# select the submatrix corresponding to a single variable
partial_cov_params = self.cov_params(column=idx)
covb = partial_cov_params
var = (exog_part * np.dot(covb, exog_part.T).T).sum(1)
se = np.sqrt(var)
return linpred, se
[docs] def plot_partial(self, smooth_index, plot_se=True, cpr=False,
include_constant=True, ax=None):
"""plot the contribution of a smooth term to the linear prediction
Parameters
----------
smooth_index : int
index of the smooth term within list of smooth terms
plot_se : book
If plot_se is true, then the confidence interval for the linear
prediction will be added to the plot.
cpr : bool
If cpr (component plus residual) is true, the a scatter plot of
the partial working residuals will be added to the plot.
include_constant : bool
If true, then the estimated intercept is added to the prediction
and its standard errors. This avoids that the confidence interval
has zero width at the imposed identification constraint, e.g.
either at a reference point or at the mean.
ax : None or matplotlib axis instance
If ax is not None, then the plot will be added to it.
Returns
-------
fig : matplotlib Figure instance
"""
from statsmodels.graphics.utils import _import_mpl, create_mpl_ax
_import_mpl()
variable = smooth_index
y_est, se = self.partial_values(variable,
include_constant=include_constant)
smoother = self.model.smoother
x = smoother.smoothers[variable].x
sort_index = np.argsort(x)
x = x[sort_index]
y_est = y_est[sort_index]
se = se[sort_index]
fig, ax = create_mpl_ax(ax)
ax.plot(x, y_est, c='blue', lw=2)
if plot_se:
ax.plot(x, y_est + 1.96 * se, '-', c='blue')
ax.plot(x, y_est - 1.96 * se, '-', c='blue')
if cpr:
# TODO: resid_response does not make sense with nonlinear link
# use resid_working ?
cpr_ = y_est + self.resid_working
ax.plot(x, cpr_, '.', lw=2)
ax.set_xlabel(smoother.smoothers[variable].variable_name)
return fig
[docs] def test_significance(self, smooth_index):
"""hypothesis test that a smooth component is zero.
This calls `wald_test` to compute the hypothesis test, but uses
effective degrees of freedom.
Parameters
----------
smooth_index : int
index of the smooth term within list of smooth terms
Returns
-------
wald_test : ContrastResults instance
the results instance created by `wald_test`
"""
variable = smooth_index
smoother = self.model.smoother
start_idx = self.model.k_exog_linear
k_params = len(self.params)
# a bit messy, we need first index plus length of smooth term
mask = smoother.mask[variable]
k_constraints = mask.sum()
idx = start_idx + np.nonzero(mask)[0][0]
constraints = np.eye(k_constraints, k_params, idx)
df_constraints = self.edf[idx: idx + k_constraints].sum()
return self.wald_test(constraints, df_constraints=df_constraints)
[docs] def get_hat_matrix_diag(self, observed=True, _axis=1):
"""
Compute the diagonal of the hat matrix
Parameters
----------
observed : bool
If true, then observed hessian is used in the hat matrix
computation. If false, then the expected hessian is used.
In the case of a canonical link function both are the same.
This is only relevant for models that implement both observed
and expected Hessian, which is currently only GLM. Other
models only use the observed Hessian.
_axis : int
This is mainly for internal use. By default it returns the usual
diagonal of the hat matrix. If _axis is zero, then the result
corresponds to the effective degrees of freedom, ``edf`` for each
column of exog.
Returns
-------
hat_matrix_diag : ndarray
The diagonal of the hat matrix computed from the observed
or expected hessian.
"""
weights = self.model.hessian_factor(self.params, scale=self.scale,
observed=observed)
wexog = np.sqrt(weights)[:, None] * self.model.exog
# we can use inverse hessian directly instead of computing it from
# WLS/IRLS as in GLM
# TODO: does `normalized_cov_params * scale` work in all cases?
# this avoids recomputing hessian, check when used for other models.
hess_inv = self.normalized_cov_params * self.scale
# this is in GLM equivalent to the more generic and direct
# hess_inv = np.linalg.inv(-self.model.hessian(self.params))
hd = (wexog * hess_inv.dot(wexog.T).T).sum(axis=_axis)
return hd
@cache_readonly
def edf(self):
return self.get_hat_matrix_diag(_axis=0)
@cache_readonly
def hat_matrix_trace(self):
return self.hat_matrix_diag.sum()
@cache_readonly
def hat_matrix_diag(self):
return self.get_hat_matrix_diag(observed=True)
@cache_readonly
def gcv(self):
return self.scale / (1. - self.hat_matrix_trace / self.nobs)**2
@cache_readonly
def cv(self):
cv_ = ((self.resid_pearson / (1. - self.hat_matrix_diag))**2).sum()
cv_ /= self.nobs
return cv_
class GLMGamResultsWrapper(GLMResultsWrapper):
pass
wrap.populate_wrapper(GLMGamResultsWrapper, GLMGamResults)
[docs]class GLMGam(PenalizedMixin, GLM):
"""
Generalized Additive Models (GAM)
This inherits from `GLM`.
Warning: Not all inherited methods might take correctly account of the
penalization. Not all options including offset and exposure have been
verified yet.
Parameters
----------
endog : array_like
The response variable.
exog : array_like or None
This explanatory variables are treated as linear. The model in this
case is a partial linear model.
smoother : instance of additive smoother class
Examples of smoother instances include Bsplines or CyclicCubicSplines.
alpha : float or list of floats
Penalization weights for smooth terms. The length of the list needs
to be the same as the number of smooth terms in the ``smoother``.
family : instance of GLM family
See GLM.
offset : None or array_like
See GLM.
exposure : None or array_like
See GLM.
missing : 'none'
Missing value handling is not supported in this class.
**kwargs
Extra keywords are used in call to the super classes.
Notes
-----
Status: experimental. This has full unit test coverage for the core
results with Gaussian and Poisson (without offset and exposure). Other
options and additional results might not be correctly supported yet.
(Binomial with counts, i.e. with n_trials, is most likely wrong in pirls.
User specified var or freq weights are most likely also not correct for
all results.)
"""
_results_class = GLMGamResults
_results_class_wrapper = GLMGamResultsWrapper
def __init__(self, endog, exog=None, smoother=None, alpha=0, family=None,
offset=None, exposure=None, missing='none', **kwargs):
# TODO: check usage of hasconst
hasconst = kwargs.get('hasconst', None)
xnames_linear = None
if hasattr(exog, 'design_info'):
self.design_info_linear = exog.design_info
xnames_linear = self.design_info_linear.column_names
is_pandas = _is_using_pandas(exog, None)
# TODO: handle data is experimental, see #5469
# This is a bit wasteful because we need to `handle_data twice`
self.data_linear = self._handle_data(endog, exog, missing, hasconst)
if xnames_linear is None:
xnames_linear = self.data_linear.xnames
if exog is not None:
exog_linear = np.asarray(exog)
k_exog_linear = exog_linear.shape[1]
else:
exog_linear = None
k_exog_linear = 0
self.k_exog_linear = k_exog_linear
# We need exog_linear for k-fold cross validation
# TODO: alternative is to take columns from combined exog
self.exog_linear = exog_linear
self.smoother = smoother
self.k_smooths = smoother.k_variables
self.alpha = self._check_alpha(alpha)
penal = MultivariateGamPenalty(smoother, alpha=self.alpha,
start_idx=k_exog_linear)
kwargs.pop('penal', None)
if exog_linear is not None:
exog = np.column_stack((exog_linear, smoother.basis))
else:
exog = smoother.basis
# TODO: check: xnames_linear will be None instead of empty list
# if no exog_linear
# can smoother be empty ? I guess not allowed.
if xnames_linear is None:
xnames_linear = []
xnames = xnames_linear + self.smoother.col_names
if is_pandas and exog_linear is not None:
# we a dataframe so we can get a PandasData instance for wrapping
exog = pd.DataFrame(exog, index=self.data_linear.row_labels,
columns=xnames)
super(GLMGam, self).__init__(endog, exog=exog, family=family,
offset=offset, exposure=exposure,
penal=penal, missing=missing, **kwargs)
if not is_pandas:
# set exog nanmes if not given by pandas DataFrame
self.exog_names[:] = xnames
# TODO: the generic data handling might attach the design_info from the
# linear part, but this is incorrect for the full model and
# causes problems in wald_test_terms
if hasattr(self.data, 'design_info'):
del self.data.design_info
# formula also might be attached which causes problems in predict
if hasattr(self, 'formula'):
self.formula_linear = self.formula
self.formula = None
del self.formula
def _check_alpha(self, alpha):
"""check and convert alpha to required list format
Parameters
----------
alpha : scalar, list or array_like
penalization weight
Returns
-------
alpha : list
penalization weight, list with length equal to the number of
smooth terms
"""
if not isinstance(alpha, Iterable):
alpha = [alpha] * len(self.smoother.smoothers)
elif not isinstance(alpha, list):
# we want alpha to be a list
alpha = list(alpha)
return alpha
def fit(self, start_params=None, maxiter=1000, method='pirls', tol=1e-8,
scale=None, cov_type='nonrobust', cov_kwds=None, use_t=None,
full_output=True, disp=False, max_start_irls=3, **kwargs):
"""estimate parameters and create instance of GLMGamResults class
Parameters
----------
most parameters are the same as for GLM
method : optimization method
The special optimization method is "pirls" which uses a penalized
version of IRLS. Other methods are gradient optimizers as used in
base.model.LikelihoodModel.
Returns
-------
res : instance of wrapped GLMGamResults
"""
# TODO: temporary hack to remove attribute
# formula also might be attached which in inherited from_formula
# causes problems in predict
if hasattr(self, 'formula'):
self.formula_linear = self.formula
del self.formula
# TODO: alpha not allowed yet, but is in `_fit_pirls`
# alpha = self._check_alpha()
if method.lower() in ['pirls', 'irls']:
res = self._fit_pirls(self.alpha, start_params=start_params,
maxiter=maxiter, tol=tol, scale=scale,
cov_type=cov_type, cov_kwds=cov_kwds,
use_t=use_t, **kwargs)
else:
if max_start_irls > 0 and (start_params is None):
res = self._fit_pirls(self.alpha, start_params=start_params,
maxiter=max_start_irls, tol=tol,
scale=scale,
cov_type=cov_type, cov_kwds=cov_kwds,
use_t=use_t, **kwargs)
start_params = res.params
del res
res = super(GLMGam, self).fit(start_params=start_params,
maxiter=maxiter, method=method,
tol=tol, scale=scale,
cov_type=cov_type, cov_kwds=cov_kwds,
use_t=use_t,
full_output=full_output, disp=disp,
max_start_irls=0,
**kwargs)
return res
# pag 165 4.3 # pag 136 PIRLS
def _fit_pirls(self, alpha, start_params=None, maxiter=100, tol=1e-8,
scale=None, cov_type='nonrobust', cov_kwds=None, use_t=None,
weights=None):
"""fit model with penalized reweighted least squares
"""
# TODO: this currently modifies several attributes
# self.scale, self.scaletype, self.mu, self.weights
# self.data_weights,
# and possibly self._offset_exposure
# several of those might not be necessary, e.g. mu and weights
# alpha = alpha * len(y) * self.scale / 100
# TODO: we need to rescale alpha
endog = self.endog
wlsexog = self.exog # smoother.basis
spl_s = self.penal.penalty_matrix(alpha=alpha)
nobs, n_columns = wlsexog.shape
# TODO what are these values?
if weights is None:
self.data_weights = np.array([1.] * nobs)
else:
self.data_weights = weights
if not hasattr(self, '_offset_exposure'):
self._offset_exposure = 0
self.scaletype = scale
# TODO: check default scale types
# self.scaletype = 'dev'
# during iteration
self.scale = 1
if start_params is None:
mu = self.family.starting_mu(endog)
lin_pred = self.family.predict(mu)
else:
lin_pred = np.dot(wlsexog, start_params) + self._offset_exposure
mu = self.family.fitted(lin_pred)
dev = self.family.deviance(endog, mu)
history = dict(params=[None, start_params], deviance=[np.inf, dev])
converged = False
criterion = history['deviance']
# This special case is used to get the likelihood for a specific
# params vector.
if maxiter == 0:
mu = self.family.fitted(lin_pred)
self.scale = self.estimate_scale(mu)
wls_results = lm.RegressionResults(self, start_params, None)
iteration = 0
for iteration in range(maxiter):
# TODO: is this equivalent to point 1 of page 136:
# w = 1 / (V(mu) * g'(mu)) ?
self.weights = self.data_weights * self.family.weights(mu)
# TODO: is this equivalent to point 1 of page 136:
# z = g(mu)(y - mu) + X beta ?
wlsendog = (lin_pred + self.family.link.deriv(mu) * (endog - mu)
- self._offset_exposure)
# this defines the augmented matrix point 2a on page 136
wls_results = penalized_wls(wlsendog, wlsexog, spl_s, self.weights)
lin_pred = np.dot(wlsexog, wls_results.params).ravel()
lin_pred += self._offset_exposure
mu = self.family.fitted(lin_pred)
# We do not need to update scale in GLM/LEF models
# We might need it in dispersion models.
# self.scale = self.estimate_scale(mu)
history = self._update_history(wls_results, mu, history)
if endog.squeeze().ndim == 1 and np.allclose(mu - endog, 0):
msg = "Perfect separation detected, results not available"
raise PerfectSeparationError(msg)
# TODO need atol, rtol
# args of _check_convergence: (criterion, iteration, atol, rtol)
converged = _check_convergence(criterion, iteration, tol, 0)
if converged:
break
self.mu = mu
self.scale = self.estimate_scale(mu)
glm_results = GLMGamResults(self, wls_results.params,
wls_results.normalized_cov_params,
self.scale,
cov_type=cov_type, cov_kwds=cov_kwds,
use_t=use_t)
glm_results.method = "PIRLS"
history['iteration'] = iteration + 1
glm_results.fit_history = history
glm_results.converged = converged
return GLMGamResultsWrapper(glm_results)
def select_penweight(self, criterion='aic', start_params=None,
start_model_params=None,
method='basinhopping', **fit_kwds):
"""find alpha by minimizing results criterion
The objective for the minimization can be results attributes like
``gcv``, ``aic`` or ``bic`` where the latter are based on effective
degrees of freedom.
Warning: In many case the optimization might converge to a local
optimum or near optimum. Different start_params or using a global
optimizer is recommended, default is basinhopping.
Parameters
----------
criterion='aic'
name of results attribute to be minimized.
Default is 'aic', other options are 'gcv', 'cv' or 'bic'.
start_params : None or array
starting parameters for alpha in the penalization weight
minimization. The parameters are internally exponentiated and
the minimization is with respect to ``exp(alpha)``
start_model_params : None or array
starting parameter for the ``model._fit_pirls``.
method : 'basinhopping', 'nm' or 'minimize'
'basinhopping' and 'nm' directly use the underlying scipy.optimize
functions `basinhopping` and `fmin`. 'minimize' provides access
to the high level interface, `scipy.optimize.minimize`.
fit_kwds : keyword arguments
additional keyword arguments will be used in the call to the
scipy optimizer. Which keywords are supported depends on the
scipy optimization function.
Returns
-------
alpha : ndarray
penalization parameter found by minimizing the criterion.
Note that this can be only a local (near) optimum.
fit_res : tuple
results returned by the scipy optimization routine. The
parameters in the optimization problem are `log(alpha)`
history : dict
history of calls to pirls and contains alpha, the fit
criterion and the parameters to which pirls converged to for the
given alpha.
Notes
-----
In the test cases Nelder-Mead and bfgs often converge to local optima,
see also https://github.com/statsmodels/statsmodels/issues/5381.
This does not use any analytical derivatives for the criterion
minimization.
Status: experimental, It is possible that defaults change if there
is a better way to find a global optimum. API (e.g. type of return)
might also change.
"""
# copy attributes that are changed, so we can reset them
scale_keep = self.scale
scaletype_keep = self.scaletype
# TODO: use .copy() method when available for all types
alpha_keep = copy.copy(self.alpha)
if start_params is None:
start_params = np.zeros(self.k_smooths)
else:
start_params = np.log(1e-20 + start_params)
history = {}
history['alpha'] = []
history['params'] = [start_model_params]
history['criterion'] = []
def fun(p):
a = np.exp(p)
res_ = self._fit_pirls(start_params=history['params'][-1],
alpha=a)
history['alpha'].append(a)
history['params'].append(np.asarray(res_.params))
return getattr(res_, criterion)
if method == 'nm':
kwds = dict(full_output=True, maxiter=1000, maxfun=2000)
kwds.update(fit_kwds)
fit_res = optimize.fmin(fun, start_params, **kwds)
opt = fit_res[0]
elif method == 'basinhopping':
kwds = dict(minimizer_kwargs={'method': 'Nelder-Mead',
'options': {'maxiter': 100, 'maxfev': 500}},
niter=10)
kwds.update(fit_kwds)
fit_res = optimize.basinhopping(fun, start_params, **kwds)
opt = fit_res.x
elif method == 'minimize':
fit_res = optimize.minimize(fun, start_params, **fit_kwds)
opt = fit_res.x
else:
raise ValueError('method not recognized')
del history['params'][0] # remove the model start_params
alpha = np.exp(opt)
# reset attributes that have or might have changed
self.scale = scale_keep
self.scaletype = scaletype_keep
self.alpha = alpha_keep
return alpha, fit_res, history
def select_penweight_kfold(self, alphas=None, cv_iterator=None, cost=None,
k_folds=5, k_grid=11):
"""find alphas by k-fold cross-validation
Warning: This estimates ``k_folds`` models for each point in the
grid of alphas.
Parameters
----------
alphas : None or list of arrays
cv_iterator : instance
instance of a cross-validation iterator, by default this is a
KFold instance
cost : function
default is mean squared error. The cost function to evaluate the
prediction error for the left out sample. This should take two
arrays as argument and return one float.
k_folds : int
number of folds if default Kfold iterator is used.
This is ignored if ``cv_iterator`` is not None.
Returns
-------
alpha_cv : list of float
Best alpha in grid according to cross-validation
res_cv : instance of MultivariateGAMCVPath
The instance was used for cross-validation and holds the results
Notes
-----
The default alphas are defined as
``alphas = [np.logspace(0, 7, k_grid) for _ in range(k_smooths)]``
"""
if cost is None:
def cost(x1, x2):
return np.linalg.norm(x1 - x2) / len(x1)
if alphas is None:
alphas = [np.logspace(0, 7, k_grid) for _ in range(self.k_smooths)]
if cv_iterator is None:
cv_iterator = KFold(k_folds=k_folds, shuffle=True)
gam_cv = MultivariateGAMCVPath(smoother=self.smoother, alphas=alphas,
gam=GLMGam, cost=cost, endog=self.endog,
exog=self.exog_linear,
cv_iterator=cv_iterator)
gam_cv_res = gam_cv.fit()
return gam_cv_res.alpha_cv, gam_cv_res
[docs]class LogitGam(PenalizedMixin, Logit):
"""Generalized Additive model for discrete Logit
This subclasses discrete_model Logit.
Warning: not all inherited methods might take correctly account of the
penalization
not verified yet.
"""
def __init__(self, endog, smoother, alpha, *args, **kwargs):
if not isinstance(alpha, Iterable):
alpha = np.array([alpha] * len(smoother.smoothers))
self.smoother = smoother
self.alpha = alpha
self.pen_weight = 1 # TODO: pen weight should not be defined here!!
penal = MultivariateGamPenalty(smoother, alpha=alpha)
super(LogitGam, self).__init__(endog, smoother.basis, penal=penal,
*args, **kwargs)
def penalized_wls(endog, exog, penalty_matrix, weights):
"""weighted least squares with quadratic penalty
Parameters
----------
endog : ndarray
response or endogenous variable
exog : ndarray
design matrix, matrix of exogenous or explanatory variables
penalty_matrix : ndarray, 2-Dim square
penality matrix for quadratic penalization. Note, the penalty_matrix
is multiplied by two to match non-pirls fitting methods.
weights : ndarray
weights for WLS
Returns
-------
results : Results instance of WLS
"""
y, x, s = endog, exog, penalty_matrix
# TODO: I do not understand why I need 2 * s
aug_y, aug_x, aug_weights = make_augmented_matrix(y, x, 2 * s, weights)
wls_results = lm.WLS(aug_y, aug_x, aug_weights).fit()
# TODO: use MinimalWLS during iterations, less overhead
# However, MinimalWLS does not return normalized_cov_params
# which we need at the end of the iterations
# call would be
# wls_results = reg_tools._MinimalWLS(aug_y, aug_x, aug_weights).fit()
wls_results.params = wls_results.params.ravel()
return wls_results
def make_augmented_matrix(endog, exog, penalty_matrix, weights):
"""augment endog, exog and weights with stochastic restriction matrix
Parameters
----------
endog : ndarray
response or endogenous variable
exog : ndarray
design matrix, matrix of exogenous or explanatory variables
penalty_matrix : ndarray, 2-Dim square
penality matrix for quadratic penalization
weights : ndarray
weights for WLS
Returns
-------
endog_aug : ndarray
augmented response variable
exog_aug : ndarray
augmented design matrix
weights_aug : ndarray
augmented weights for WLS
"""
y, x, s, = endog, exog, penalty_matrix
nobs = x.shape[0]
# TODO: needs full because of broadcasting with weights
# check what weights should be doing
rs = matrix_sqrt(s)
x1 = np.vstack([x, rs]) # augmented x
n_samp1es_x1 = x1.shape[0]
y1 = np.array([0.] * n_samp1es_x1) # augmented y
y1[:nobs] = y
id1 = np.array([1.] * rs.shape[0])
w1 = np.concatenate([weights, id1])
return y1, x1, w1