""" Generalized linear models currently supports estimation using the one-parameter exponential families References ---------- Gill, Jeff. 2000. Generalized Linear Models: A Unified Approach. SAGE QASS Series. Green, PJ. 1984. "Iteratively reweighted least squares for maximum likelihood estimation, and some robust and resistant alternatives." Journal of the Royal Statistical Society, Series B, 46, 149-192. Hardin, J.W. and Hilbe, J.M. 2007. "Generalized Linear Models and Extensions." 2nd ed. Stata Press, College Station, TX. McCullagh, P. and Nelder, J.A. 1989. "Generalized Linear Models." 2nd ed. Chapman & Hall, Boca Rotan. """ from statsmodels.compat.pandas import Appender import warnings import numpy as np from numpy.linalg import LinAlgError import statsmodels.base.model as base import statsmodels.base.wrapper as wrap from statsmodels.base import _prediction_inference as pred from statsmodels.base._prediction_inference import PredictionResultsMean import statsmodels.base._parameter_inference as pinfer from statsmodels.graphics._regressionplots_doc import ( _plot_added_variable_doc, _plot_ceres_residuals_doc, _plot_partial_residuals_doc, ) import statsmodels.regression._tools as reg_tools import statsmodels.regression.linear_model as lm from statsmodels.tools.decorators import ( cache_readonly, cached_data, cached_value, ) from statsmodels.tools.docstring import Docstring from statsmodels.tools.sm_exceptions import ( DomainWarning, HessianInversionWarning, PerfectSeparationWarning, ) from statsmodels.tools.validation import float_like # need import in module instead of lazily to copy `__doc__` from . import families __all__ = ['GLM', 'PredictionResultsMean'] def _check_convergence(criterion, iteration, atol, rtol): return np.allclose(criterion[iteration], criterion[iteration + 1], atol=atol, rtol=rtol) # Remove after 0.13 when bic changes to bic llf class _ModuleVariable: _value = None @property def use_bic_llf(self): return self._value def set_use_bic_llf(self, val): if val not in (True, False, None): raise ValueError("Must be True, False or None") self._value = bool(val) if val is not None else val _use_bic_helper = _ModuleVariable() SET_USE_BIC_LLF = _use_bic_helper.set_use_bic_llf class GLM(base.LikelihoodModel): __doc__ = """ Generalized Linear Models GLM inherits from statsmodels.base.model.LikelihoodModel Parameters ---------- endog : array_like 1d array of endogenous response variable. This array can be 1d or 2d. Binomial family models accept a 2d array with two columns. If supplied, each observation is expected to be [success, failure]. exog : array_like A nobs x k array where `nobs` is the number of observations and `k` is the number of regressors. An intercept is not included by default and should be added by the user (models specified using a formula include an intercept by default). See `statsmodels.tools.add_constant`. family : family class instance The default is Gaussian. To specify the binomial distribution family = sm.family.Binomial() Each family can take a link instance as an argument. See statsmodels.family.family for more information. offset : array_like or None An offset to be included in the model. If provided, must be an array whose length is the number of rows in exog. exposure : array_like or None Log(exposure) will be added to the linear prediction in the model. Exposure is only valid if the log link is used. If provided, it must be an array with the same length as endog. freq_weights : array_like 1d array of frequency weights. The default is None. If None is selected or a blank value, then the algorithm will replace with an array of 1's with length equal to the endog. WARNING: Using weights is not verified yet for all possible options and results, see Notes. var_weights : array_like 1d array of variance (analytic) weights. The default is None. If None is selected or a blank value, then the algorithm will replace with an array of 1's with length equal to the endog. WARNING: Using weights is not verified yet for all possible options and results, see Notes. {extra_params} Attributes ---------- df_model : float Model degrees of freedom is equal to p - 1, where p is the number of regressors. Note that the intercept is not reported as a degree of freedom. df_resid : float Residual degrees of freedom is equal to the number of observation n minus the number of regressors p. endog : ndarray See Notes. Note that `endog` is a reference to the data so that if data is already an array and it is changed, then `endog` changes as well. exposure : array_like Include ln(exposure) in model with coefficient constrained to 1. Can only be used if the link is the logarithm function. exog : ndarray See Notes. Note that `exog` is a reference to the data so that if data is already an array and it is changed, then `exog` changes as well. freq_weights : ndarray See Notes. Note that `freq_weights` is a reference to the data so that if data is already an array and it is changed, then `freq_weights` changes as well. var_weights : ndarray See Notes. Note that `var_weights` is a reference to the data so that if data is already an array and it is changed, then `var_weights` changes as well. iteration : int The number of iterations that fit has run. Initialized at 0. family : family class instance The distribution family of the model. Can be any family in statsmodels.families. Default is Gaussian. mu : ndarray The mean response of the transformed variable. `mu` is the value of the inverse of the link function at lin_pred, where lin_pred is the linear predicted value of the WLS fit of the transformed variable. `mu` is only available after fit is called. See statsmodels.families.family.fitted of the distribution family for more information. n_trials : ndarray See Notes. Note that `n_trials` is a reference to the data so that if data is already an array and it is changed, then `n_trials` changes as well. `n_trials` is the number of binomial trials and only available with that distribution. See statsmodels.families.Binomial for more information. normalized_cov_params : ndarray The p x p normalized covariance of the design / exogenous data. This is approximately equal to (X.T X)^(-1) offset : array_like Include offset in model with coefficient constrained to 1. scale : float The estimate of the scale / dispersion of the model fit. Only available after fit is called. See GLM.fit and GLM.estimate_scale for more information. scaletype : str The scaling used for fitting the model. This is only available after fit is called. The default is None. See GLM.fit for more information. weights : ndarray The value of the weights after the last iteration of fit. Only available after fit is called. See statsmodels.families.family for the specific distribution weighting functions. Examples -------- >>> import statsmodels.api as sm >>> data = sm.datasets.scotland.load() >>> data.exog = sm.add_constant(data.exog) Instantiate a gamma family model with the default link function. >>> gamma_model = sm.GLM(data.endog, data.exog, ... family=sm.families.Gamma()) >>> gamma_results = gamma_model.fit() >>> gamma_results.params array([-0.01776527, 0.00004962, 0.00203442, -0.00007181, 0.00011185, -0.00000015, -0.00051868, -0.00000243]) >>> gamma_results.scale 0.0035842831734919055 >>> gamma_results.deviance 0.087388516416999198 >>> gamma_results.pearson_chi2 0.086022796163805704 >>> gamma_results.llf -83.017202161073527 See Also -------- statsmodels.genmod.families.family.Family :ref:`families` :ref:`links` Notes ----- Note: PerfectSeparationError exception has been converted to a PerfectSeparationWarning and perfect separation or perfect prediction will not raise an exception by default. (changed in version 0.14) Only the following combinations make sense for family and link: ============= ===== === ===== ====== ======= === ==== ====== ====== ==== Family ident log logit probit cloglog pow opow nbinom loglog logc ============= ===== === ===== ====== ======= === ==== ====== ====== ==== Gaussian x x x x x x x x x inv Gaussian x x x binomial x x x x x x x x x Poisson x x x neg binomial x x x x gamma x x x Tweedie x x x ============= ===== === ===== ====== ======= === ==== ====== ====== ==== Not all of these link functions are currently available. Endog and exog are references so that if the data they refer to are already arrays and these arrays are changed, endog and exog will change. statsmodels supports two separate definitions of weights: frequency weights and variance weights. Frequency weights produce the same results as repeating observations by the frequencies (if those are integers). Frequency weights will keep the number of observations consistent, but the degrees of freedom will change to reflect the new weights. Variance weights (referred to in other packages as analytic weights) are used when ``endog`` represents an an average or mean. This relies on the assumption that that the inverse variance scales proportionally to the weight--an observation that is deemed more credible should have less variance and therefore have more weight. For the ``Poisson`` family--which assumes that occurrences scale proportionally with time--a natural practice would be to use the amount of time as the variance weight and set ``endog`` to be a rate (occurrences per period of time). Similarly, using a compound Poisson family, namely ``Tweedie``, makes a similar assumption about the rate (or frequency) of occurrences having variance proportional to time. Both frequency and variance weights are verified for all basic results with nonrobust or heteroscedasticity robust ``cov_type``. Other robust covariance types have not yet been verified, and at least the small sample correction is currently not based on the correct total frequency count. Currently, all residuals are not weighted by frequency, although they may incorporate ``n_trials`` for ``Binomial`` and ``var_weights`` +---------------+----------------------------------+ | Residual Type | Applicable weights | +===============+==================================+ | Anscombe | ``var_weights`` | +---------------+----------------------------------+ | Deviance | ``var_weights`` | +---------------+----------------------------------+ | Pearson | ``var_weights`` and ``n_trials`` | +---------------+----------------------------------+ | Reponse | ``n_trials`` | +---------------+----------------------------------+ | Working | ``n_trials`` | +---------------+----------------------------------+ WARNING: Loglikelihood and deviance are not valid in models where scale is equal to 1 (i.e., ``Binomial``, ``NegativeBinomial``, and ``Poisson``). If variance weights are specified, then results such as ``loglike`` and ``deviance`` are based on a quasi-likelihood interpretation. The loglikelihood is not correctly specified in this case, and statistics based on it, such AIC or likelihood ratio tests, are not appropriate. """.format(extra_params=base._missing_param_doc) # Maximum number of endogenous variables when using a formula _formula_max_endog = 2 def __init__(self, endog, exog, family=None, offset=None, exposure=None, freq_weights=None, var_weights=None, missing='none', **kwargs): if type(self) is GLM: self._check_kwargs(kwargs, ['n_trials']) if (family is not None) and not isinstance(family.link, tuple(family.safe_links)): warnings.warn((f"The {type(family.link).__name__} link function " "does not respect the domain of the " f"{type(family).__name__} family."), DomainWarning) if exposure is not None: exposure = np.log(exposure) if offset is not None: # this should probably be done upstream offset = np.asarray(offset) if freq_weights is not None: freq_weights = np.asarray(freq_weights) if var_weights is not None: var_weights = np.asarray(var_weights) self.freq_weights = freq_weights self.var_weights = var_weights super().__init__(endog, exog, missing=missing, offset=offset, exposure=exposure, freq_weights=freq_weights, var_weights=var_weights, **kwargs) self._check_inputs(family, self.offset, self.exposure, self.endog, self.freq_weights, self.var_weights) if offset is None: delattr(self, 'offset') if exposure is None: delattr(self, 'exposure') self.nobs = self.endog.shape[0] # things to remove_data self._data_attr.extend(['weights', 'mu', 'freq_weights', 'var_weights', 'iweights', '_offset_exposure', 'n_trials']) # register kwds for __init__, offset and exposure are added by super self._init_keys.append('family') self._setup_binomial() # internal usage for recreating a model if 'n_trials' in kwargs: self.n_trials = kwargs['n_trials'] # Construct a combined offset/exposure term. Note that # exposure has already been logged if present. offset_exposure = 0. if hasattr(self, 'offset'): offset_exposure = self.offset if hasattr(self, 'exposure'): offset_exposure = offset_exposure + self.exposure self._offset_exposure = offset_exposure self.scaletype = None def initialize(self): """ Initialize a generalized linear model. """ self.df_model = np.linalg.matrix_rank(self.exog) - 1 if (self.freq_weights is not None) and \ (self.freq_weights.shape[0] == self.endog.shape[0]): self.wnobs = self.freq_weights.sum() self.df_resid = self.wnobs - self.df_model - 1 else: self.wnobs = self.exog.shape[0] self.df_resid = self.exog.shape[0] - self.df_model - 1 def _check_inputs(self, family, offset, exposure, endog, freq_weights, var_weights): # Default family is Gaussian if family is None: family = families.Gaussian() self.family = family if exposure is not None: if not isinstance(self.family.link, families.links.Log): raise ValueError("exposure can only be used with the log " "link function") elif exposure.shape[0] != endog.shape[0]: raise ValueError("exposure is not the same length as endog") if offset is not None: if offset.shape[0] != endog.shape[0]: raise ValueError("offset is not the same length as endog") if freq_weights is not None: if freq_weights.shape[0] != endog.shape[0]: raise ValueError("freq weights not the same length as endog") if len(freq_weights.shape) > 1: raise ValueError("freq weights has too many dimensions") # internal flag to store whether freq_weights were not None self._has_freq_weights = (self.freq_weights is not None) if self.freq_weights is None: self.freq_weights = np.ones(endog.shape[0]) # TODO: check do we want to keep None as sentinel for freq_weights if np.shape(self.freq_weights) == () and self.freq_weights > 1: self.freq_weights = (self.freq_weights * np.ones(endog.shape[0])) if var_weights is not None: if var_weights.shape[0] != endog.shape[0]: raise ValueError("var weights not the same length as endog") if len(var_weights.shape) > 1: raise ValueError("var weights has too many dimensions") # internal flag to store whether var_weights were not None self._has_var_weights = (var_weights is not None) if var_weights is None: self.var_weights = np.ones(endog.shape[0]) # TODO: check do we want to keep None as sentinel for var_weights self.iweights = np.asarray(self.freq_weights * self.var_weights) def _get_init_kwds(self): # this is a temporary fixup because exposure has been transformed # see #1609, copied from discrete_model.CountModel kwds = super()._get_init_kwds() if 'exposure' in kwds and kwds['exposure'] is not None: kwds['exposure'] = np.exp(kwds['exposure']) return kwds def loglike_mu(self, mu, scale=1.): """ Evaluate the log-likelihood for a generalized linear model. """ scale = float_like(scale, "scale") return self.family.loglike(self.endog, mu, self.var_weights, self.freq_weights, scale) def loglike(self, params, scale=None): """ Evaluate the log-likelihood for a generalized linear model. """ scale = float_like(scale, "scale", optional=True) lin_pred = np.dot(self.exog, params) + self._offset_exposure expval = self.family.link.inverse(lin_pred) if scale is None: scale = self.estimate_scale(expval) llf = self.family.loglike(self.endog, expval, self.var_weights, self.freq_weights, scale) return llf def score_obs(self, params, scale=None): """score first derivative of the loglikelihood for each observation. Parameters ---------- params : ndarray Parameter at which score is evaluated. scale : None or float If scale is None, then the default scale will be calculated. Default scale is defined by `self.scaletype` and set in fit. If scale is not None, then it is used as a fixed scale. Returns ------- score_obs : ndarray, 2d The first derivative of the loglikelihood function evaluated at params for each observation. """ scale = float_like(scale, "scale", optional=True) score_factor = self.score_factor(params, scale=scale) return score_factor[:, None] * self.exog def score(self, params, scale=None): """score, first derivative of the loglikelihood function Parameters ---------- params : ndarray Parameter at which score is evaluated. scale : None or float If scale is None, then the default scale will be calculated. Default scale is defined by `self.scaletype` and set in fit. If scale is not None, then it is used as a fixed scale. Returns ------- score : ndarray_1d The first derivative of the loglikelihood function calculated as the sum of `score_obs` """ scale = float_like(scale, "scale", optional=True) score_factor = self.score_factor(params, scale=scale) return np.dot(score_factor, self.exog) def score_factor(self, params, scale=None): """weights for score for each observation This can be considered as score residuals. Parameters ---------- params : ndarray parameter at which score is evaluated scale : None or float If scale is None, then the default scale will be calculated. Default scale is defined by `self.scaletype` and set in fit. If scale is not None, then it is used as a fixed scale. Returns ------- score_factor : ndarray_1d A 1d weight vector used in the calculation of the score_obs. The score_obs are obtained by `score_factor[:, None] * exog` """ scale = float_like(scale, "scale", optional=True) mu = self.predict(params) if scale is None: scale = self.estimate_scale(mu) score_factor = (self.endog - mu) / self.family.link.deriv(mu) score_factor /= self.family.variance(mu) score_factor *= self.iweights * self.n_trials if not scale == 1: score_factor /= scale return score_factor def hessian_factor(self, params, scale=None, observed=True): """Weights for calculating Hessian Parameters ---------- params : ndarray parameter at which Hessian is evaluated scale : None or float If scale is None, then the default scale will be calculated. Default scale is defined by `self.scaletype` and set in fit. If scale is not None, then it is used as a fixed scale. observed : bool If True, then the observed Hessian is returned. If false then the expected information matrix is returned. Returns ------- hessian_factor : ndarray, 1d A 1d weight vector used in the calculation of the Hessian. The hessian is obtained by `(exog.T * hessian_factor).dot(exog)` """ # calculating eim_factor mu = self.predict(params) if scale is None: scale = self.estimate_scale(mu) eim_factor = 1 / (self.family.link.deriv(mu)**2 * self.family.variance(mu)) eim_factor *= self.iweights * self.n_trials if not observed: if not scale == 1: eim_factor /= scale return eim_factor # calculating oim_factor, eim_factor is with scale=1 score_factor = self.score_factor(params, scale=1.) if eim_factor.ndim > 1 or score_factor.ndim > 1: raise RuntimeError('something wrong') tmp = self.family.variance(mu) * self.family.link.deriv2(mu) tmp += self.family.variance.deriv(mu) * self.family.link.deriv(mu) tmp = score_factor * tmp # correct for duplicatee iweights in oim_factor and score_factor tmp /= self.iweights * self.n_trials oim_factor = eim_factor * (1 + tmp) if tmp.ndim > 1: raise RuntimeError('something wrong') if not scale == 1: oim_factor /= scale return oim_factor def hessian(self, params, scale=None, observed=None): """Hessian, second derivative of loglikelihood function Parameters ---------- params : ndarray parameter at which Hessian is evaluated scale : None or float If scale is None, then the default scale will be calculated. Default scale is defined by `self.scaletype` and set in fit. If scale is not None, then it is used as a fixed scale. observed : bool If True, then the observed Hessian is returned (default). If False, then the expected information matrix is returned. Returns ------- hessian : ndarray Hessian, i.e. observed information, or expected information matrix. """ if observed is None: if getattr(self, '_optim_hessian', None) == 'eim': observed = False else: observed = True scale = float_like(scale, "scale", optional=True) tmp = getattr(self, '_tmp_like_exog', np.empty_like(self.exog, dtype=float)) factor = self.hessian_factor(params, scale=scale, observed=observed) np.multiply(self.exog.T, factor, out=tmp.T) return -tmp.T.dot(self.exog) def information(self, params, scale=None): """ Fisher information matrix. """ scale = float_like(scale, "scale", optional=True) return self.hessian(params, scale=scale, observed=False) def _derivative_exog(self, params, exog=None, transform="dydx", dummy_idx=None, count_idx=None, offset=None, exposure=None): """ Derivative of mean, expected endog with respect to the parameters """ if exog is None: exog = self.exog if (offset is not None) or (exposure is not None): raise NotImplementedError("offset and exposure not supported") lin_pred = self.predict(params, exog, which="linear", offset=offset, exposure=exposure) k_extra = getattr(self, 'k_extra', 0) params_exog = params if k_extra == 0 else params[:-k_extra] margeff = (self.family.link.inverse_deriv(lin_pred)[:, None] * params_exog) if 'ex' in transform: margeff *= exog if 'ey' in transform: mean = self.family.link.inverse(lin_pred) margeff /= mean[:,None] return self._derivative_exog_helper(margeff, params, exog, dummy_idx, count_idx, transform) def _derivative_exog_helper(self, margeff, params, exog, dummy_idx, count_idx, transform): """ Helper for _derivative_exog to wrap results appropriately """ from statsmodels.discrete.discrete_margins import ( _get_count_effects, _get_dummy_effects, ) if count_idx is not None: margeff = _get_count_effects(margeff, exog, count_idx, transform, self, params) if dummy_idx is not None: margeff = _get_dummy_effects(margeff, exog, dummy_idx, transform, self, params) return margeff def _derivative_predict(self, params, exog=None, transform='dydx', offset=None, exposure=None): """ Derivative of the expected endog with respect to the parameters. Parameters ---------- params : ndarray parameter at which score is evaluated exog : ndarray or None Explanatory variables at which derivative are computed. If None, then the estimation exog is used. offset, exposure : None Not yet implemented. Returns ------- The value of the derivative of the expected endog with respect to the parameter vector. """ # core part is same as derivative_mean_params # additionally handles exog and transform if exog is None: exog = self.exog if (offset is not None) or (exposure is not None) or ( getattr(self, 'offset', None) is not None): raise NotImplementedError("offset and exposure not supported") lin_pred = self.predict(params, exog=exog, which="linear") idl = self.family.link.inverse_deriv(lin_pred) dmat = exog * idl[:, None] if 'ey' in transform: mean = self.family.link.inverse(lin_pred) dmat /= mean[:, None] return dmat def _deriv_mean_dparams(self, params): """ Derivative of the expected endog with respect to the parameters. Parameters ---------- params : ndarray parameter at which score is evaluated Returns ------- The value of the derivative of the expected endog with respect to the parameter vector. """ lin_pred = self.predict(params, which="linear") idl = self.family.link.inverse_deriv(lin_pred) dmat = self.exog * idl[:, None] return dmat def _deriv_score_obs_dendog(self, params, scale=None): """derivative of score_obs w.r.t. endog Parameters ---------- params : ndarray parameter at which score is evaluated scale : None or float If scale is None, then the default scale will be calculated. Default scale is defined by `self.scaletype` and set in fit. If scale is not None, then it is used as a fixed scale. Returns ------- derivative : ndarray_2d The derivative of the score_obs with respect to endog. This can is given by `score_factor0[:, None] * exog` where `score_factor0` is the score_factor without the residual. """ scale = float_like(scale, "scale", optional=True) mu = self.predict(params) if scale is None: scale = self.estimate_scale(mu) score_factor = 1 / self.family.link.deriv(mu) score_factor /= self.family.variance(mu) score_factor *= self.iweights * self.n_trials if not scale == 1: score_factor /= scale return score_factor[:, None] * self.exog def score_test(self, params_constrained, k_constraints=None, exog_extra=None, observed=True): """score test for restrictions or for omitted variables The covariance matrix for the score is based on the Hessian, i.e. observed information matrix or optionally on the expected information matrix.. Parameters ---------- params_constrained : array_like estimated parameter of the restricted model. This can be the parameter estimate for the current when testing for omitted variables. k_constraints : int or None Number of constraints that were used in the estimation of params restricted relative to the number of exog in the model. This must be provided if no exog_extra are given. If exog_extra is not None, then k_constraints is assumed to be zero if it is None. exog_extra : None or array_like Explanatory variables that are jointly tested for inclusion in the model, i.e. omitted variables. observed : bool If True, then the observed Hessian is used in calculating the covariance matrix of the score. If false then the expected information matrix is used. Returns ------- chi2_stat : float chisquare statistic for the score test p-value : float P-value of the score test based on the chisquare distribution. df : int Degrees of freedom used in the p-value calculation. This is equal to the number of constraints. Notes ----- not yet verified for case with scale not equal to 1. """ if exog_extra is None: if k_constraints is None: raise ValueError('if exog_extra is None, then k_constraints' 'needs to be given') score = self.score(params_constrained) hessian = self.hessian(params_constrained, observed=observed) else: # exog_extra = np.asarray(exog_extra) if k_constraints is None: k_constraints = 0 ex = np.column_stack((self.exog, exog_extra)) k_constraints += ex.shape[1] - self.exog.shape[1] score_factor = self.score_factor(params_constrained) score = (score_factor[:, None] * ex).sum(0) hessian_factor = self.hessian_factor(params_constrained, observed=observed) hessian = -np.dot(ex.T * hessian_factor, ex) from scipy import stats # TODO check sign, why minus? chi2stat = -score.dot(np.linalg.solve(hessian, score[:, None])) pval = stats.chi2.sf(chi2stat, k_constraints) # return a stats results instance instead? Contrast? return chi2stat, pval, k_constraints def _update_history(self, tmp_result, mu, history): """ Helper method to update history during iterative fit. """ history['params'].append(tmp_result.params) history['deviance'].append(self.family.deviance(self.endog, mu, self.var_weights, self.freq_weights, self.scale)) return history def estimate_scale(self, mu): """ Estimate the dispersion/scale. Type of scale can be chose in the fit method. Parameters ---------- mu : ndarray mu is the mean response estimate Returns ------- Estimate of scale Notes ----- The default scale for Binomial, Poisson and Negative Binomial families is 1. The default for the other families is Pearson's Chi-Square estimate. See Also -------- statsmodels.genmod.generalized_linear_model.GLM.fit """ if not self.scaletype: if isinstance(self.family, (families.Binomial, families.Poisson, families.NegativeBinomial)): return 1. else: return self._estimate_x2_scale(mu) if isinstance(self.scaletype, float): return np.array(self.scaletype) if isinstance(self.scaletype, str): if self.scaletype.lower() == 'x2': return self._estimate_x2_scale(mu) elif self.scaletype.lower() == 'dev': return (self.family.deviance(self.endog, mu, self.var_weights, self.freq_weights, 1.) / (self.df_resid)) else: raise ValueError("Scale %s with type %s not understood" % (self.scaletype, type(self.scaletype))) else: raise ValueError("Scale %s with type %s not understood" % (self.scaletype, type(self.scaletype))) def _estimate_x2_scale(self, mu): resid = np.power(self.endog - mu, 2) * self.iweights return np.sum(resid / self.family.variance(mu)) / self.df_resid def estimate_tweedie_power(self, mu, method='brentq', low=1.01, high=5.): """ Tweedie specific function to estimate scale and the variance parameter. The variance parameter is also referred to as p, xi, or shape. Parameters ---------- mu : array_like Fitted mean response variable method : str, defaults to 'brentq' Scipy optimizer used to solve the Pearson equation. Only brentq currently supported. low : float, optional Low end of the bracketing interval [a,b] to be used in the search for the power. Defaults to 1.01. high : float, optional High end of the bracketing interval [a,b] to be used in the search for the power. Defaults to 5. Returns ------- power : float The estimated shape or power. """ if method == 'brentq': from scipy.optimize import brentq def psi_p(power, mu): scale = ((self.iweights * (self.endog - mu) ** 2 / (mu ** power)).sum() / self.df_resid) return (np.sum(self.iweights * ((self.endog - mu) ** 2 / (scale * (mu ** power)) - 1) * np.log(mu)) / self.freq_weights.sum()) power = brentq(psi_p, low, high, args=(mu)) else: raise NotImplementedError('Only brentq can currently be used') return power def predict(self, params, exog=None, exposure=None, offset=None, which="mean", linear=None): """ Return predicted values for a design matrix Parameters ---------- params : array_like Parameters / coefficients of a GLM. exog : array_like, optional Design / exogenous data. Is exog is None, model exog is used. exposure : array_like, optional Exposure time values, only can be used with the log link function. See notes for details. offset : array_like, optional Offset values. See notes for details. which : 'mean', 'linear', 'var'(optional) Statitistic to predict. Default is 'mean'. - 'mean' returns the conditional expectation of endog E(y | x), i.e. inverse of the model's link function of linear predictor. - 'linear' returns the linear predictor of the mean function. - 'var_unscaled' variance of endog implied by the likelihood model. This does not include scale or var_weights. linear : bool The ``linear` keyword is deprecated and will be removed, use ``which`` keyword instead. If True, returns the linear predicted values. If False or None, then the statistic specified by ``which`` will be returned. Returns ------- An array of fitted values Notes ----- Any `exposure` and `offset` provided here take precedence over the `exposure` and `offset` used in the model fit. If `exog` is passed as an argument here, then any `exposure` and `offset` values in the fit will be ignored. Exposure values must be strictly positive. """ if linear is not None: msg = 'linear keyword is deprecated, use which="linear"' warnings.warn(msg, FutureWarning) if linear is True: which = "linear" # Use fit offset if appropriate if offset is None and exog is None and hasattr(self, 'offset'): offset = self.offset elif offset is None: offset = 0. if exposure is not None and not isinstance(self.family.link, families.links.Log): raise ValueError("exposure can only be used with the log link " "function") # Use fit exposure if appropriate if exposure is None and exog is None and hasattr(self, 'exposure'): # Already logged exposure = self.exposure elif exposure is None: exposure = 0. else: exposure = np.log(np.asarray(exposure)) if exog is None: exog = self.exog linpred = np.dot(exog, params) + offset + exposure if which == "mean": return self.family.fitted(linpred) elif which == "linear": return linpred elif which == "var_unscaled": mean = self.family.fitted(linpred) var_ = self.family.variance(mean) return var_ else: raise ValueError(f'The which value "{which}" is not recognized') def get_distribution(self, params, scale=None, exog=None, exposure=None, offset=None, var_weights=1., n_trials=1.): """ Return a instance of the predictive distribution. Parameters ---------- params : array_like The model parameters. scale : scalar The scale parameter. exog : array_like The predictor variable matrix. offset : array_like or None Offset variable for predicted mean. exposure : array_like or None Log(exposure) will be added to the linear prediction. var_weights : array_like 1d array of variance (analytic) weights. The default is None. n_trials : int Number of trials for the binomial distribution. The default is 1 which corresponds to a Bernoulli random variable. Returns ------- gen Instance of a scipy frozen distribution based on estimated parameters. Use the ``rvs`` method to generate random values. Notes ----- Due to the behavior of ``scipy.stats.distributions objects``, the returned random number generator must be called with ``gen.rvs(n)`` where ``n`` is the number of observations in the data set used to fit the model. If any other value is used for ``n``, misleading results will be produced. """ scale = float_like(scale, "scale", optional=True) # use scale=1, independent of QMLE scale for discrete if isinstance(self.family, (families.Binomial, families.Poisson, families.NegativeBinomial)): scale = 1. mu = self.predict(params, exog, exposure, offset, which="mean") kwds = {} if (np.any(n_trials != 1) and isinstance(self.family, families.Binomial)): kwds["n_trials"] = n_trials distr = self.family.get_distribution(mu, scale, var_weights=var_weights, **kwds) return distr def _setup_binomial(self): # this checks what kind of data is given for Binomial. # family will need a reference to endog if this is to be removed from # preprocessing self.n_trials = np.ones(self.endog.shape[0]) # For binomial if isinstance(self.family, families.Binomial): tmp = self.family.initialize(self.endog, self.freq_weights) self.endog = tmp[0] self.n_trials = tmp[1] self._init_keys.append('n_trials') def fit(self, start_params=None, maxiter=100, method='IRLS', tol=1e-8, scale=None, cov_type='nonrobust', cov_kwds=None, use_t=None, full_output=True, disp=False, max_start_irls=3, **kwargs): """ Fits a generalized linear model for a given family. Parameters ---------- start_params : array_like, optional Initial guess of the solution for the loglikelihood maximization. The default is family-specific and is given by the ``family.starting_mu(endog)``. If start_params is given then the initial mean will be calculated as ``np.dot(exog, start_params)``. maxiter : int, optional Default is 100. method : str Default is 'IRLS' for iteratively reweighted least squares. Otherwise gradient optimization is used. tol : float Convergence tolerance. Default is 1e-8. scale : str or float, optional `scale` can be 'X2', 'dev', or a float The default value is None, which uses `X2` for Gamma, Gaussian, and Inverse Gaussian. `X2` is Pearson's chi-square divided by `df_resid`. The default is 1 for the Binomial and Poisson families. `dev` is the deviance divided by df_resid cov_type : str The type of parameter estimate covariance matrix to compute. cov_kwds : dict-like Extra arguments for calculating the covariance of the parameter estimates. use_t : bool If True, the Student t-distribution is used for inference. full_output : bool, optional Set to True to have all available output in the Results object's mle_retvals attribute. The output is dependent on the solver. See LikelihoodModelResults notes section for more information. Not used if methhod is IRLS. disp : bool, optional Set to True to print convergence messages. Not used if method is IRLS. max_start_irls : int The number of IRLS iterations used to obtain starting values for gradient optimization. Only relevant if `method` is set to something other than 'IRLS'. atol : float, optional (available with IRLS fits) The absolute tolerance criterion that must be satisfied. Defaults to ``tol``. Convergence is attained when: :math:`rtol * prior + atol > abs(current - prior)` rtol : float, optional (available with IRLS fits) The relative tolerance criterion that must be satisfied. Defaults to 0 which means ``rtol`` is not used. Convergence is attained when: :math:`rtol * prior + atol > abs(current - prior)` tol_criterion : str, optional (available with IRLS fits) Defaults to ``'deviance'``. Can optionally be ``'params'``. wls_method : str, optional (available with IRLS fits) options are 'lstsq', 'pinv' and 'qr' specifies which linear algebra function to use for the irls optimization. Default is `lstsq` which uses the same underlying svd based approach as 'pinv', but is faster during iterations. 'lstsq' and 'pinv' regularize the estimate in singular and near-singular cases by truncating small singular values based on `rcond` of the respective numpy.linalg function. 'qr' is only valid for cases that are not singular nor near-singular. optim_hessian : {'eim', 'oim'}, optional (available with scipy optimizer fits) When 'oim'--the default--the observed Hessian is used in fitting. 'eim' is the expected Hessian. This may provide more stable fits, but adds assumption that the Hessian is correctly specified. Notes ----- If method is 'IRLS', then an additional keyword 'attach_wls' is available. This is currently for internal use only and might change in future versions. If attach_wls' is true, then the final WLS instance of the IRLS iteration is attached to the results instance as `results_wls` attribute. """ if isinstance(scale, str): scale = scale.lower() if scale not in ("x2", "dev"): raise ValueError( "scale must be either X2 or dev when a string." ) elif scale is not None: # GH-6627 try: scale = float(scale) except Exception as exc: raise type(exc)( "scale must be a float if given and no a string." ) self.scaletype = scale if method.lower() == "irls": if cov_type.lower() == 'eim': cov_type = 'nonrobust' return self._fit_irls(start_params=start_params, maxiter=maxiter, tol=tol, scale=scale, cov_type=cov_type, cov_kwds=cov_kwds, use_t=use_t, **kwargs) else: self._optim_hessian = kwargs.get('optim_hessian') if self._optim_hessian is not None: del kwargs['optim_hessian'] self._tmp_like_exog = np.empty_like(self.exog, dtype=float) fit_ = self._fit_gradient(start_params=start_params, method=method, maxiter=maxiter, tol=tol, scale=scale, full_output=full_output, disp=disp, cov_type=cov_type, cov_kwds=cov_kwds, use_t=use_t, max_start_irls=max_start_irls, **kwargs) del self._optim_hessian del self._tmp_like_exog return fit_ def _fit_gradient(self, start_params=None, method="newton", maxiter=100, tol=1e-8, full_output=True, disp=True, scale=None, cov_type='nonrobust', cov_kwds=None, use_t=None, max_start_irls=3, **kwargs): """ Fits a generalized linear model for a given family iteratively using the scipy gradient optimizers. """ # fix scale during optimization, see #4616 scaletype = self.scaletype self.scaletype = 1. if (max_start_irls > 0) and (start_params is None): irls_rslt = self._fit_irls(start_params=start_params, maxiter=max_start_irls, tol=tol, scale=1., cov_type='nonrobust', cov_kwds=None, use_t=None, **kwargs) start_params = irls_rslt.params del irls_rslt rslt = super().fit(start_params=start_params, maxiter=maxiter, full_output=full_output, method=method, disp=disp, **kwargs) # reset scaletype to original self.scaletype = scaletype mu = self.predict(rslt.params) scale = self.estimate_scale(mu) if rslt.normalized_cov_params is None: cov_p = None else: cov_p = rslt.normalized_cov_params / scale if cov_type.lower() == 'eim': oim = False cov_type = 'nonrobust' else: oim = True try: cov_p = np.linalg.inv(-self.hessian(rslt.params, observed=oim)) / scale except LinAlgError: warnings.warn('Inverting hessian failed, no bse or cov_params ' 'available', HessianInversionWarning) cov_p = None results_class = getattr(self, '_results_class', GLMResults) results_class_wrapper = getattr(self, '_results_class_wrapper', GLMResultsWrapper) glm_results = results_class(self, rslt.params, cov_p, scale, cov_type=cov_type, cov_kwds=cov_kwds, use_t=use_t) # TODO: iteration count is not always available history = {'iteration': 0} if full_output: glm_results.mle_retvals = rslt.mle_retvals if 'iterations' in rslt.mle_retvals: history['iteration'] = rslt.mle_retvals['iterations'] glm_results.method = method glm_results.fit_history = history return results_class_wrapper(glm_results) def _fit_irls(self, start_params=None, maxiter=100, tol=1e-8, scale=None, cov_type='nonrobust', cov_kwds=None, use_t=None, **kwargs): """ Fits a generalized linear model for a given family using iteratively reweighted least squares (IRLS). """ attach_wls = kwargs.pop('attach_wls', False) atol = kwargs.get('atol') rtol = kwargs.get('rtol', 0.) tol_criterion = kwargs.get('tol_criterion', 'deviance') wls_method = kwargs.get('wls_method', 'lstsq') atol = tol if atol is None else atol endog = self.endog wlsexog = self.exog if start_params is None: start_params = np.zeros(self.exog.shape[1]) mu = self.family.starting_mu(self.endog) lin_pred = self.family.predict(mu) else: lin_pred = np.dot(wlsexog, start_params) + self._offset_exposure mu = self.family.fitted(lin_pred) self.scale = self.estimate_scale(mu) dev = self.family.deviance(self.endog, mu, self.var_weights, self.freq_weights, self.scale) if np.isnan(dev): raise ValueError("The first guess on the deviance function " "returned a nan. This could be a boundary " " problem and should be reported.") # first guess on the deviance is assumed to be scaled by 1. # params are none to start, so they line up with the deviance history = dict(params=[np.inf, start_params], deviance=[np.inf, dev]) converged = False criterion = history[tol_criterion] # 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): self.weights = (self.iweights * self.n_trials * self.family.weights(mu)) wlsendog = (lin_pred + self.family.link.deriv(mu) * (self.endog-mu) - self._offset_exposure) wls_mod = reg_tools._MinimalWLS(wlsendog, wlsexog, self.weights, check_endog=True, check_weights=True) wls_results = wls_mod.fit(method=wls_method) lin_pred = np.dot(self.exog, wls_results.params) lin_pred += self._offset_exposure mu = self.family.fitted(lin_pred) history = self._update_history(wls_results, mu, history) self.scale = self.estimate_scale(mu) if endog.squeeze().ndim == 1 and np.allclose(mu - endog, 0): msg = ("Perfect separation or prediction detected, " "parameter may not be identified") warnings.warn(msg, category=PerfectSeparationWarning) converged = _check_convergence(criterion, iteration + 1, atol, rtol) if converged: break self.mu = mu if maxiter > 0: # Only if iterative used wls_method2 = 'pinv' if wls_method == 'lstsq' else wls_method wls_model = lm.WLS(wlsendog, wlsexog, self.weights) wls_results = wls_model.fit(method=wls_method2) glm_results = GLMResults(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 = "IRLS" glm_results.mle_settings = {} glm_results.mle_settings['wls_method'] = wls_method glm_results.mle_settings['optimizer'] = glm_results.method if (maxiter > 0) and (attach_wls is True): glm_results.results_wls = wls_results history['iteration'] = iteration + 1 glm_results.fit_history = history glm_results.converged = converged return GLMResultsWrapper(glm_results) def fit_regularized(self, method="elastic_net", alpha=0., start_params=None, refit=False, opt_method="bfgs", **kwargs): r""" Return a regularized fit to a linear regression model. Parameters ---------- method : {'elastic_net'} Only the `elastic_net` approach is currently implemented. alpha : scalar or array_like The penalty weight. If a scalar, the same penalty weight applies to all variables in the model. If a vector, it must have the same length as `params`, and contains a penalty weight for each coefficient. start_params : array_like Starting values for `params`. refit : bool If True, the model is refit using only the variables that have non-zero coefficients in the regularized fit. The refitted model is not regularized. opt_method : string The method used for numerical optimization. **kwargs Additional keyword arguments used when fitting the model. Returns ------- GLMResults An array or a GLMResults object, same type returned by `fit`. Notes ----- The penalty is the ``elastic net`` penalty, which is a combination of L1 and L2 penalties. The function that is minimized is: .. math:: -loglike/n + alpha*((1-L1\_wt)*|params|_2^2/2 + L1\_wt*|params|_1) where :math:`|*|_1` and :math:`|*|_2` are the L1 and L2 norms. Post-estimation results are based on the same data used to select variables, hence may be subject to overfitting biases. The elastic_net method uses the following keyword arguments: maxiter : int Maximum number of iterations L1_wt : float Must be in [0, 1]. The L1 penalty has weight L1_wt and the L2 penalty has weight 1 - L1_wt. cnvrg_tol : float Convergence threshold for maximum parameter change after one sweep through all coefficients. zero_tol : float Coefficients below this threshold are treated as zero. """ if kwargs.get("L1_wt", 1) == 0: return self._fit_ridge(alpha, start_params, opt_method) from statsmodels.base.elastic_net import fit_elasticnet if method != "elastic_net": raise ValueError("method for fit_regularized must be elastic_net") defaults = {"maxiter": 50, "L1_wt": 1, "cnvrg_tol": 1e-10, "zero_tol": 1e-10} defaults.update(kwargs) llkw = kwargs.get("loglike_kwds", {}) sckw = kwargs.get("score_kwds", {}) hekw = kwargs.get("hess_kwds", {}) llkw["scale"] = 1 sckw["scale"] = 1 hekw["scale"] = 1 defaults["loglike_kwds"] = llkw defaults["score_kwds"] = sckw defaults["hess_kwds"] = hekw result = fit_elasticnet(self, method=method, alpha=alpha, start_params=start_params, refit=refit, **defaults) self.mu = self.predict(result.params) self.scale = self.estimate_scale(self.mu) if not result.converged: warnings.warn("Elastic net fitting did not converge") return result def _fit_ridge(self, alpha, start_params, method): if start_params is None: start_params = np.zeros(self.exog.shape[1]) def fun(x): return -(self.loglike(x) / self.nobs - np.sum(alpha * x**2) / 2) def grad(x): return -(self.score(x) / self.nobs - alpha * x) from scipy.optimize import minimize from statsmodels.base.elastic_net import ( RegularizedResults, RegularizedResultsWrapper, ) mr = minimize(fun, start_params, jac=grad, method=method) params = mr.x if not mr.success: ngrad = np.sqrt(np.sum(mr.jac**2)) msg = "GLM ridge optimization may have failed, |grad|=%f" % ngrad warnings.warn(msg) results = RegularizedResults(self, params) results = RegularizedResultsWrapper(results) return results def fit_constrained(self, constraints, start_params=None, **fit_kwds): """fit the model subject to linear equality constraints The constraints are of the form `R params = q` where R is the constraint_matrix and q is the vector of constraint_values. The estimation creates a new model with transformed design matrix, exog, and converts the results back to the original parameterization. Parameters ---------- constraints : formula expression or tuple If it is a tuple, then the constraint needs to be given by two arrays (constraint_matrix, constraint_value), i.e. (R, q). Otherwise, the constraints can be given as strings or list of strings. see t_test for details start_params : None or array_like starting values for the optimization. `start_params` needs to be given in the original parameter space and are internally transformed. **fit_kwds : keyword arguments fit_kwds are used in the optimization of the transformed model. Returns ------- results : Results instance """ from patsy import DesignInfo from statsmodels.base._constraints import ( LinearConstraints, fit_constrained, ) # same pattern as in base.LikelihoodModel.t_test lc = DesignInfo(self.exog_names).linear_constraint(constraints) R, q = lc.coefs, lc.constants # TODO: add start_params option, need access to tranformation # fit_constrained needs to do the transformation params, cov, res_constr = fit_constrained(self, R, q, start_params=start_params, fit_kwds=fit_kwds) # create dummy results Instance, TODO: wire up properly res = self.fit(start_params=params, maxiter=0) # we get a wrapper back res._results.params = params res._results.cov_params_default = cov cov_type = fit_kwds.get('cov_type', 'nonrobust') if cov_type != 'nonrobust': res._results.normalized_cov_params = cov / res_constr.scale else: res._results.normalized_cov_params = None res._results.scale = res_constr.scale k_constr = len(q) res._results.df_resid += k_constr res._results.df_model -= k_constr res._results.constraints = LinearConstraints.from_patsy(lc) res._results.k_constr = k_constr res._results.results_constrained = res_constr return res get_prediction_doc = Docstring(pred.get_prediction_glm.__doc__) get_prediction_doc.remove_parameters("pred_kwds") class GLMResults(base.LikelihoodModelResults): """ Class to contain GLM results. GLMResults inherits from statsmodels.LikelihoodModelResults Attributes ---------- df_model : float See GLM.df_model df_resid : float See GLM.df_resid fit_history : dict Contains information about the iterations. Its keys are `iterations`, `deviance` and `params`. model : class instance Pointer to GLM model instance that called fit. nobs : float The number of observations n. normalized_cov_params : ndarray See GLM docstring params : ndarray The coefficients of the fitted model. Note that interpretation of the coefficients often depends on the distribution family and the data. pvalues : ndarray The two-tailed p-values for the parameters. scale : float The estimate of the scale / dispersion for the model fit. See GLM.fit and GLM.estimate_scale for more information. stand_errors : ndarray The standard errors of the fitted GLM. #TODO still named bse See Also -------- statsmodels.base.model.LikelihoodModelResults """ def __init__(self, model, params, normalized_cov_params, scale, cov_type='nonrobust', cov_kwds=None, use_t=None): super().__init__( model, params, normalized_cov_params=normalized_cov_params, scale=scale) self.family = model.family self._endog = model.endog self.nobs = model.endog.shape[0] self._freq_weights = model.freq_weights self._var_weights = model.var_weights self._iweights = model.iweights if isinstance(self.family, families.Binomial): self._n_trials = self.model.n_trials else: self._n_trials = 1 self.df_resid = model.df_resid self.df_model = model.df_model self._cache = {} # are these intermediate results needed or can we just # call the model's attributes? # for remove data and pickle without large arrays self._data_attr.extend(['results_constrained', '_freq_weights', '_var_weights', '_iweights']) self._data_in_cache.extend(['null', 'mu']) self._data_attr_model = getattr(self, '_data_attr_model', []) self._data_attr_model.append('mu') # robust covariance from statsmodels.base.covtype import get_robustcov_results if use_t is None: self.use_t = False # TODO: class default else: self.use_t = use_t # temporary warning ct = (cov_type == 'nonrobust') or (cov_type.upper().startswith('HC')) if self.model._has_freq_weights and not ct: from statsmodels.tools.sm_exceptions import SpecificationWarning warnings.warn('cov_type not fully supported with freq_weights', SpecificationWarning) if self.model._has_var_weights and not ct: from statsmodels.tools.sm_exceptions import SpecificationWarning warnings.warn('cov_type not fully supported with var_weights', SpecificationWarning) if cov_type == 'nonrobust': self.cov_type = 'nonrobust' self.cov_kwds = {'description': 'Standard Errors assume that the' + ' covariance matrix of the errors is correctly ' + 'specified.'} else: if cov_kwds is None: cov_kwds = {} get_robustcov_results(self, cov_type=cov_type, use_self=True, use_t=use_t, **cov_kwds) @cached_data def resid_response(self): """ Response residuals. The response residuals are defined as `endog` - `fittedvalues` """ return self._n_trials * (self._endog-self.mu) @cached_data def resid_pearson(self): """ Pearson residuals. The Pearson residuals are defined as (`endog` - `mu`)/sqrt(VAR(`mu`)) where VAR is the distribution specific variance function. See statsmodels.families.family and statsmodels.families.varfuncs for more information. """ return (np.sqrt(self._n_trials) * (self._endog-self.mu) * np.sqrt(self._var_weights) / np.sqrt(self.family.variance(self.mu))) @cached_data def resid_working(self): """ Working residuals. The working residuals are defined as `resid_response`/link'(`mu`). See statsmodels.family.links for the derivatives of the link functions. They are defined analytically. """ # Isn't self.resid_response is already adjusted by _n_trials? val = (self.resid_response * self.family.link.deriv(self.mu)) val *= self._n_trials return val @cached_data def resid_anscombe(self): """ Anscombe residuals. See statsmodels.families.family for distribution- specific Anscombe residuals. Currently, the unscaled residuals are provided. In a future version, the scaled residuals will be provided. """ return self.resid_anscombe_scaled @cached_data def resid_anscombe_scaled(self): """ Scaled Anscombe residuals. See statsmodels.families.family for distribution-specific Anscombe residuals. """ return self.family.resid_anscombe(self._endog, self.fittedvalues, var_weights=self._var_weights, scale=self.scale) @cached_data def resid_anscombe_unscaled(self): """ Unscaled Anscombe residuals. See statsmodels.families.family for distribution-specific Anscombe residuals. """ return self.family.resid_anscombe(self._endog, self.fittedvalues, var_weights=self._var_weights, scale=1.) @cached_data def resid_deviance(self): """ Deviance residuals. See statsmodels.families.family for distribution- specific deviance residuals. """ dev = self.family.resid_dev(self._endog, self.fittedvalues, var_weights=self._var_weights, scale=1.) return dev @cached_value def pearson_chi2(self): """ Pearson's Chi-Squared statistic is defined as the sum of the squares of the Pearson residuals. """ chisq = (self._endog - self.mu)**2 / self.family.variance(self.mu) chisq *= self._iweights * self._n_trials chisqsum = np.sum(chisq) return chisqsum @cached_data def fittedvalues(self): """ The estimated mean response. This is the value of the inverse of the link function at lin_pred, where lin_pred is the linear predicted value obtained by multiplying the design matrix by the coefficient vector. """ return self.mu @cached_data def mu(self): """ See GLM docstring. """ return self.model.predict(self.params) @cache_readonly def null(self): """ Fitted values of the null model """ endog = self._endog model = self.model exog = np.ones((len(endog), 1)) kwargs = model._get_init_kwds().copy() kwargs.pop('family') for key in getattr(model, '_null_drop_keys', []): del kwargs[key] start_params = np.atleast_1d(self.family.link(endog.mean())) oe = self.model._offset_exposure if not (np.size(oe) == 1 and oe == 0): with warnings.catch_warnings(): warnings.simplefilter("ignore", DomainWarning) mod = GLM(endog, exog, family=self.family, **kwargs) fitted = mod.fit(start_params=start_params).fittedvalues else: # correct if fitted is identical across observations wls_model = lm.WLS(endog, exog, weights=self._iweights * self._n_trials) fitted = wls_model.fit().fittedvalues return fitted @cache_readonly def deviance(self): """ See statsmodels.families.family for the distribution-specific deviance functions. """ return self.family.deviance(self._endog, self.mu, self._var_weights, self._freq_weights) @cache_readonly def null_deviance(self): """The value of the deviance function for the model fit with a constant as the only regressor.""" return self.family.deviance(self._endog, self.null, self._var_weights, self._freq_weights) @cache_readonly def llnull(self): """ Log-likelihood of the model fit with a constant as the only regressor """ return self.family.loglike(self._endog, self.null, var_weights=self._var_weights, freq_weights=self._freq_weights, scale=self.scale) def llf_scaled(self, scale=None): """ Return the log-likelihood at the given scale, using the estimated scale if the provided scale is None. In the Gaussian case with linear link, the concentrated log-likelihood is returned. """ _modelfamily = self.family if scale is None: if (isinstance(self.family, families.Gaussian) and isinstance(self.family.link, families.links.Power) and (self.family.link.power == 1.)): # Scale for the concentrated Gaussian log likelihood # (profile log likelihood with the scale parameter # profiled out). scale = (np.power(self._endog - self.mu, 2) * self._iweights).sum() scale /= self.model.wnobs else: scale = self.scale val = _modelfamily.loglike(self._endog, self.mu, var_weights=self._var_weights, freq_weights=self._freq_weights, scale=scale) return val @cached_value def llf(self): """ Value of the loglikelihood function evalued at params. See statsmodels.families.family for distribution-specific loglikelihoods. The result uses the concentrated log-likelihood if the family is Gaussian and the link is linear, otherwise it uses the non-concentrated log-likelihood evaluated at the estimated scale. """ return self.llf_scaled() def pseudo_rsquared(self, kind="cs"): """ Pseudo R-squared Cox-Snell likelihood ratio pseudo R-squared is valid for both discrete and continuous data. McFadden's pseudo R-squared is only valid for discrete data. Cox & Snell's pseudo-R-squared: 1 - exp((llnull - llf)*(2/nobs)) McFadden's pseudo-R-squared: 1 - (llf / llnull) Parameters ---------- kind : P"cs", "mcf"} Type of pseudo R-square to return Returns ------- float Pseudo R-squared """ kind = kind.lower() if kind.startswith("mcf"): prsq = 1 - self.llf / self.llnull elif kind.startswith("cox") or kind in ["cs", "lr"]: prsq = 1 - np.exp((self.llnull - self.llf) * (2 / self.nobs)) else: raise ValueError("only McFadden and Cox-Snell are available") return prsq @cached_value def aic(self): """ Akaike Information Criterion -2 * `llf` + 2 * (`df_model` + 1) """ return self.info_criteria("aic") @property def bic(self): """ Bayes Information Criterion `deviance` - `df_resid` * log(`nobs`) .. warning:: The current definition is based on the deviance rather than the log-likelihood. This is not consistent with the AIC definition, and after 0.13 both will make use of the log-likelihood definition. Notes ----- The log-likelihood version is defined -2 * `llf` + (`df_model` + 1)*log(n) """ if _use_bic_helper.use_bic_llf not in (True, False): warnings.warn( "The bic value is computed using the deviance formula. After " "0.13 this will change to the log-likelihood based formula. " "This change has no impact on the relative rank of models " "compared using BIC. You can directly access the " "log-likelihood version using the `bic_llf` attribute. You " "can suppress this message by calling " "statsmodels.genmod.generalized_linear_model.SET_USE_BIC_LLF " "with True to get the LLF-based version now or False to retain" "the deviance version.", FutureWarning ) if bool(_use_bic_helper.use_bic_llf): return self.bic_llf return self.bic_deviance @cached_value def bic_deviance(self): """ Bayes Information Criterion Based on the deviance, `deviance` - `df_resid` * log(`nobs`) """ return (self.deviance - (self.model.wnobs - self.df_model - 1) * np.log(self.model.wnobs)) @cached_value def bic_llf(self): """ Bayes Information Criterion Based on the log-likelihood, -2 * `llf` + log(n) * (`df_model` + 1) """ return self.info_criteria("bic") def info_criteria(self, crit, scale=None, dk_params=0): """Return an information criterion for the model. Parameters ---------- crit : string One of 'aic', 'bic', or 'qaic'. scale : float The scale parameter estimated using the parent model, used only for qaic. dk_params : int or float Correction to the number of parameters used in the information criterion. By default, only mean parameters are included, the scale parameter is not included in the parameter count. Use ``dk_params=1`` to include scale in the parameter count. Returns ------- Value of information criterion. Notes ----- The quasi-Akaike Information criterion (qaic) is -2 * `llf`/`scale` + 2 * (`df_model` + 1). It may not give meaningful results except for Poisson and related models. The QAIC (ic_type='qaic') must be evaluated with a provided scale parameter. Two QAIC values are only comparable if they are calculated using the same scale parameter. The scale parameter should be estimated using the largest model among all models being compared. References ---------- Burnham KP, Anderson KR (2002). Model Selection and Multimodel Inference; Springer New York. """ crit = crit.lower() k_params = self.df_model + 1 + dk_params if crit == "aic": return -2 * self.llf + 2 * k_params elif crit == "bic": nobs = self.df_model + self.df_resid + 1 bic = -2*self.llf + k_params*np.log(nobs) return bic elif crit == "qaic": f = self.model.family fl = (families.Poisson, families.NegativeBinomial, families.Binomial) if not isinstance(f, fl): msg = "QAIC is only valid for Binomial, Poisson and " msg += "Negative Binomial families." warnings.warn(msg) llf = self.llf_scaled(scale=1) return -2 * llf/scale + 2 * k_params # now explicit docs, old and new behavior, copied from generic classes # @Appender(str(get_prediction_doc)) def get_prediction(self, exog=None, exposure=None, offset=None, transform=True, which=None, linear=None, average=False, agg_weights=None, row_labels=None): """ Compute prediction results for GLM compatible models. Options and return class depend on whether "which" is None or not. Parameters ---------- exog : array_like, optional The values for which you want to predict. exposure : array_like, optional Exposure time values, only can be used with the log link function. offset : array_like, optional Offset values. transform : bool, optional If the model was fit via a formula, do you want to pass exog through the formula. Default is True. E.g., if you fit a model y ~ log(x1) + log(x2), and transform is True, then you can pass a data structure that contains x1 and x2 in their original form. Otherwise, you'd need to log the data first. which : 'mean', 'linear', 'var'(optional) Statitistic to predict. Default is 'mean'. If which is None, then the deprecated keyword "linear" applies. If which is not None, then a generic Prediction results class will be returned. Some options are only available if which is not None. See notes. - 'mean' returns the conditional expectation of endog E(y | x), i.e. inverse of the model's link function of linear predictor. - 'linear' returns the linear predictor of the mean function. - 'var_unscaled' variance of endog implied by the likelihood model. This does not include scale or var_weights. linear : bool The ``linear` keyword is deprecated and will be removed, use ``which`` keyword instead. If which is None, then the linear keyword is used, otherwise it will be ignored. If True and which is None, the linear predicted values are returned. If False or None, then the statistic specified by ``which`` will be returned. average : bool Keyword is only used if ``which`` is not None. If average is True, then the mean prediction is computed, that is, predictions are computed for individual exog and then the average over observation is used. If average is False, then the results are the predictions for all observations, i.e. same length as ``exog``. agg_weights : ndarray, optional Keyword is only used if ``which`` is not None. Aggregation weights, only used if average is True. row_labels : list of str or None If row_lables are provided, then they will replace the generated labels. Returns ------- prediction_results : instance of a PredictionResults class. 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. The Results class of the return depends on the value of ``which``. See Also -------- GLM.predict GLMResults.predict Notes ----- Changes in statsmodels 0.14: The ``which`` keyword has been added. If ``which`` is None, then the behavior is the same as in previous versions, and returns the mean and linear prediction results. If the ``which`` keyword is not None, then a generic prediction results class is returned and is not backwards compatible with the old prediction results class, e.g. column names of summary_frame differs. There are more choices for the returned predicted statistic using ``which``. More choices will be added in the next release. Two additional keyword, average and agg_weights options are now also available if ``which`` is not None. In a future version ``which`` will become not None and the backwards compatible prediction results class will be removed. """ import statsmodels.regression._prediction as linpred pred_kwds = {'exposure': exposure, 'offset': offset, 'which': 'linear'} if which is None: # two calls to a get_prediction duplicates exog generation if patsy res_linpred = linpred.get_prediction(self, exog=exog, transform=transform, row_labels=row_labels, pred_kwds=pred_kwds) pred_kwds['which'] = 'mean' res = pred.get_prediction_glm(self, exog=exog, transform=transform, row_labels=row_labels, linpred=res_linpred, link=self.model.family.link, pred_kwds=pred_kwds) else: # new generic version, if 'which' is specified pred_kwds = {'exposure': exposure, 'offset': offset} # not yet, only applies to count families # y_values is explicit so we can add it to the docstring # if y_values is not None: # pred_kwds["y_values"] = y_values res = pred.get_prediction( self, exog=exog, which=which, transform=transform, row_labels=row_labels, average=average, agg_weights=agg_weights, pred_kwds=pred_kwds ) return res @Appender(pinfer.score_test.__doc__) def score_test(self, exog_extra=None, params_constrained=None, hypothesis='joint', cov_type=None, cov_kwds=None, k_constraints=None, observed=True): if self.model._has_freq_weights is True: warnings.warn("score test has not been verified with freq_weights", UserWarning) if self.model._has_var_weights is True: warnings.warn("score test has not been verified with var_weights", UserWarning) # We need to temporarily change model.df_resid for scale computation # TODO: find a nicer way. gh #7840 mod_df_resid = self.model.df_resid self.model.df_resid = self.df_resid if k_constraints is not None: self.model.df_resid += k_constraints res = pinfer.score_test(self, exog_extra=exog_extra, params_constrained=params_constrained, hypothesis=hypothesis, cov_type=cov_type, cov_kwds=cov_kwds, k_constraints=k_constraints, scale=None, observed=observed) self.model.df_resid = mod_df_resid return res def get_hat_matrix_diag(self, observed=True): """ 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. 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, observed=observed) wexog = np.sqrt(weights)[:, None] * self.model.exog hd = (wexog * np.linalg.pinv(wexog).T).sum(1) return hd def get_influence(self, observed=True): """ Get an instance of GLMInfluence with influence and outlier measures 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. Returns ------- infl : GLMInfluence instance The instance has methods to calculate the main influence and outlier measures as attributes. See Also -------- statsmodels.stats.outliers_influence.GLMInfluence """ from statsmodels.stats.outliers_influence import GLMInfluence weights = self.model.hessian_factor(self.params, observed=observed) weights_sqrt = np.sqrt(weights) wexog = weights_sqrt[:, None] * self.model.exog wendog = weights_sqrt * self.model.endog # using get_hat_matrix_diag has duplicated computation hat_matrix_diag = self.get_hat_matrix_diag(observed=observed) infl = GLMInfluence(self, endog=wendog, exog=wexog, resid=self.resid_pearson / np.sqrt(self.scale), hat_matrix_diag=hat_matrix_diag) return infl def get_distribution(self, exog=None, exposure=None, offset=None, var_weights=1., n_trials=1.): """ Return a instance of the predictive distribution. Parameters ---------- scale : scalar The scale parameter. exog : array_like The predictor variable matrix. offset : array_like or None Offset variable for predicted mean. exposure : array_like or None Log(exposure) will be added to the linear prediction. var_weights : array_like 1d array of variance (analytic) weights. The default is None. n_trials : int Number of trials for the binomial distribution. The default is 1 which corresponds to a Bernoulli random variable. Returns ------- gen Instance of a scipy frozen distribution based on estimated parameters. Use the ``rvs`` method to generate random values. Notes ----- Due to the behavior of ``scipy.stats.distributions objects``, the returned random number generator must be called with ``gen.rvs(n)`` where ``n`` is the number of observations in the data set used to fit the model. If any other value is used for ``n``, misleading results will be produced. """ # Note this is mostly a copy of GLM.get_prediction # calling here results.predict avoids the exog check and trasnform if isinstance(self.model.family, (families.Binomial, families.Poisson, families.NegativeBinomial)): # use scale=1, independent of QMLE scale for discrete scale = 1. if self.scale != 1.: msg = "using scale=1, no exess dispersion in distribution" warnings.warn(msg, UserWarning) else: scale = self.scale mu = self.predict(exog, exposure, offset, which="mean") kwds = {} if (np.any(n_trials != 1) and isinstance(self.model.family, families.Binomial)): kwds["n_trials"] = n_trials distr = self.model.family.get_distribution( mu, scale, var_weights=var_weights, **kwds) return distr def get_margeff(self, at='overall', method='dydx', atexog=None, dummy=False, count=False): """Get marginal effects of the fitted model. Warning: offset, exposure and weights (var_weights and freq_weights) are not supported by margeff. Parameters ---------- at : str, optional Options are: - 'overall', The average of the marginal effects at each observation. - 'mean', The marginal effects at the mean of each regressor. - 'median', The marginal effects at the median of each regressor. - 'zero', The marginal effects at zero for each regressor. - 'all', The marginal effects at each observation. If `at` is all only margeff will be available from the returned object. Note that if `exog` is specified, then marginal effects for all variables not specified by `exog` are calculated using the `at` option. method : str, optional Options are: - 'dydx' - dy/dx - No transformation is made and marginal effects are returned. This is the default. - 'eyex' - estimate elasticities of variables in `exog` -- d(lny)/d(lnx) - 'dyex' - estimate semi-elasticity -- dy/d(lnx) - 'eydx' - estimate semi-elasticity -- d(lny)/dx Note that tranformations are done after each observation is calculated. Semi-elasticities for binary variables are computed using the midpoint method. 'dyex' and 'eyex' do not make sense for discrete variables. For interpretations of these methods see notes below. atexog : array_like, optional Optionally, you can provide the exogenous variables over which to get the marginal effects. This should be a dictionary with the key as the zero-indexed column number and the value of the dictionary. Default is None for all independent variables less the constant. dummy : bool, optional If False, treats binary variables (if present) as continuous. This is the default. Else if True, treats binary variables as changing from 0 to 1. Note that any variable that is either 0 or 1 is treated as binary. Each binary variable is treated separately for now. count : bool, optional If False, treats count variables (if present) as continuous. This is the default. Else if True, the marginal effect is the change in probabilities when each observation is increased by one. Returns ------- DiscreteMargins : marginal effects instance Returns an object that holds the marginal effects, standard errors, confidence intervals, etc. See `statsmodels.discrete.discrete_margins.DiscreteMargins` for more information. Notes ----- Interpretations of methods: - 'dydx' - change in `endog` for a change in `exog`. - 'eyex' - proportional change in `endog` for a proportional change in `exog`. - 'dyex' - change in `endog` for a proportional change in `exog`. - 'eydx' - proportional change in `endog` for a change in `exog`. When using after Poisson, returns the expected number of events per period, assuming that the model is loglinear. Status : unsupported features offset, exposure and weights. Default handling of freq_weights for average effect "overall" might change. """ if getattr(self.model, "offset", None) is not None: raise NotImplementedError("Margins with offset are not available.") if (np.any(self.model.var_weights != 1) or np.any(self.model.freq_weights != 1)): warnings.warn("weights are not taken into account by margeff") from statsmodels.discrete.discrete_margins import DiscreteMargins return DiscreteMargins(self, (at, method, atexog, dummy, count)) @Appender(base.LikelihoodModelResults.remove_data.__doc__) def remove_data(self): # GLM has alias/reference in result instance self._data_attr.extend([i for i in self.model._data_attr if '_data.' not in i]) super(self.__class__, self).remove_data() # TODO: what are these in results? self._endog = None self._freq_weights = None self._var_weights = None self._iweights = None self._n_trials = None @Appender(_plot_added_variable_doc % {'extra_params_doc': ''}) def plot_added_variable(self, focus_exog, resid_type=None, use_glm_weights=True, fit_kwargs=None, ax=None): from statsmodels.graphics.regressionplots import plot_added_variable fig = plot_added_variable(self, focus_exog, resid_type=resid_type, use_glm_weights=use_glm_weights, fit_kwargs=fit_kwargs, ax=ax) return fig @Appender(_plot_partial_residuals_doc % {'extra_params_doc': ''}) def plot_partial_residuals(self, focus_exog, ax=None): from statsmodels.graphics.regressionplots import plot_partial_residuals return plot_partial_residuals(self, focus_exog, ax=ax) @Appender(_plot_ceres_residuals_doc % {'extra_params_doc': ''}) def plot_ceres_residuals(self, focus_exog, frac=0.66, cond_means=None, ax=None): from statsmodels.graphics.regressionplots import plot_ceres_residuals return plot_ceres_residuals(self, focus_exog, frac, cond_means=cond_means, ax=ax) def summary(self, yname=None, xname=None, title=None, alpha=.05): """ Summarize the Regression Results Parameters ---------- yname : str, optional Default is `y` xname : list[str], optional Names for the exogenous variables, default is `var_#` for ## in the number of regressors. Must match the number of parameters in the model title : str, optional Title for the top table. If not None, then this replaces the default title alpha : float significance level for the confidence intervals Returns ------- smry : Summary instance this holds the summary tables and text, which can be printed or converted to various output formats. See Also -------- statsmodels.iolib.summary.Summary : class to hold summary results """ top_left = [('Dep. Variable:', None), ('Model:', None), ('Model Family:', [self.family.__class__.__name__]), ('Link Function:', [self.family.link.__class__.__name__]), ('Method:', [self.method]), ('Date:', None), ('Time:', None), ('No. Iterations:', ["%d" % self.fit_history['iteration']]), ] try: prsquared = self.pseudo_rsquared(kind="cs") except ValueError: prsquared = np.nan top_right = [('No. Observations:', None), ('Df Residuals:', None), ('Df Model:', None), ('Scale:', ["%#8.5g" % self.scale]), ('Log-Likelihood:', None), ('Deviance:', ["%#8.5g" % self.deviance]), ('Pearson chi2:', ["%#6.3g" % self.pearson_chi2]), ('Pseudo R-squ. (CS):', ["%#6.4g" % prsquared]) ] if hasattr(self, 'cov_type'): top_left.append(('Covariance Type:', [self.cov_type])) if title is None: title = "Generalized Linear Model Regression Results" # create summary tables from statsmodels.iolib.summary import Summary smry = Summary() smry.add_table_2cols(self, gleft=top_left, gright=top_right, yname=yname, xname=xname, title=title) smry.add_table_params(self, yname=yname, xname=xname, alpha=alpha, use_t=self.use_t) if hasattr(self, 'constraints'): smry.add_extra_txt(['Model has been estimated subject to linear ' 'equality constraints.']) return smry def summary2(self, yname=None, xname=None, title=None, alpha=.05, float_format="%.4f"): """Experimental summary for regression Results Parameters ---------- yname : str Name of the dependent variable (optional) xname : list[str], optional Names for the exogenous variables, default is `var_#` for ## in the number of regressors. Must match the number of parameters in the model title : str, optional Title for the top table. If not None, then this replaces the default title alpha : float significance level for the confidence intervals float_format : str print format for floats in parameters summary Returns ------- smry : Summary instance this holds the summary tables and text, which can be printed or converted to various output formats. See Also -------- statsmodels.iolib.summary2.Summary : class to hold summary results """ self.method = 'IRLS' from statsmodels.iolib import summary2 smry = summary2.Summary() with warnings.catch_warnings(): warnings.simplefilter("ignore", FutureWarning) smry.add_base(results=self, alpha=alpha, float_format=float_format, xname=xname, yname=yname, title=title) if hasattr(self, 'constraints'): smry.add_text('Model has been estimated subject to linear ' 'equality constraints.') return smry class GLMResultsWrapper(lm.RegressionResultsWrapper): _attrs = { 'resid_anscombe': 'rows', 'resid_deviance': 'rows', 'resid_pearson': 'rows', 'resid_response': 'rows', 'resid_working': 'rows' } _wrap_attrs = wrap.union_dicts(lm.RegressionResultsWrapper._wrap_attrs, _attrs) wrap.populate_wrapper(GLMResultsWrapper, GLMResults) if __name__ == "__main__": from statsmodels.datasets import longley data = longley.load() # data.exog = add_constant(data.exog) GLMmod = GLM(data.endog, data.exog).fit() GLMT = GLMmod.summary(returns='tables') # GLMT[0].extend_right(GLMT[1]) # print(GLMT[0]) # print(GLMT[2]) GLMTp = GLMmod.summary(title='Test GLM') """ From Stata . webuse beetle . glm r i.beetle ldose, family(binomial n) link(cloglog) Iteration 0: log likelihood = -79.012269 Iteration 1: log likelihood = -76.94951 Iteration 2: log likelihood = -76.945645 Iteration 3: log likelihood = -76.945645 Generalized linear models No. of obs = 24 Optimization : ML Residual df = 20 Scale parameter = 1 Deviance = 73.76505595 (1/df) Deviance = 3.688253 Pearson = 71.8901173 (1/df) Pearson = 3.594506 Variance function: V(u) = u*(1-u/n) [Binomial] Link function : g(u) = ln(-ln(1-u/n)) [Complementary log-log] AIC = 6.74547 Log likelihood = -76.94564525 BIC = 10.20398 ------------------------------------------------------------------------------ | OIM r | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- beetle | 2 | -.0910396 .1076132 -0.85 0.398 -.3019576 .1198783 3 | -1.836058 .1307125 -14.05 0.000 -2.09225 -1.579867 | ldose | 19.41558 .9954265 19.50 0.000 17.46458 21.36658 _cons | -34.84602 1.79333 -19.43 0.000 -38.36089 -31.33116 ------------------------------------------------------------------------------ """