r"""
Bayesian inference for generalized linear mixed models.
Currently only families without additional scale or shape parameters
are supported (binomial and Poisson).
Two estimation approaches are supported: Laplace approximation
('maximum a posteriori'), and variational Bayes (mean field
approximation to the posterior distribution).
All realizations of random effects are modeled to be mutually
independent in this implementation.
The `exog_vc` matrix is the design matrix for the random effects.
Every column of `exog_vc` corresponds to an independent realization of
a random effect. These random effects have mean zero and an unknown
standard deviation. The standard deviation parameters are constrained
to be equal within subsets of the columns. When not using formulas,
these subsets are specified through the parameter `ident`. `ident`
must have the same length as the number of columns of `exog_vc`, and
two columns whose `ident` values are equal have the same standard
deviation. When formulas are used, the columns of `exog_vc` derived
from a common formula are constrained to have the same standard
deviation.
In many applications, `exog_vc` will be sparse. A sparse matrix may
be passed when constructing a model class. If a dense matrix is
passed, it will be converted internally to a sparse matrix. There
currently is no way to avoid creating a temporary dense version of
`exog_vc` when using formulas.
Model and parameterization
--------------------------
The joint density of data and parameters factors as:
.. math::
p(y | vc, fep) p(vc | vcp) p(vcp) p(fe)
The terms :math:`p(vcp)` and :math:`p(fe)` are prior distributions
that are taken to be Gaussian (the :math:`vcp` parameters are log
standard deviations so the standard deviations have log-normal
distributions). The random effects distribution :math:`p(vc | vcp)`
is independent Gaussian (random effect realizations are independent
within and between values of the `ident` array). The model
:math:`p(y | vc, fep)` depends on the specific GLM being fit.
"""
import numpy as np
from scipy.optimize import minimize
from scipy import sparse
import statsmodels.base.model as base
from statsmodels.iolib import summary2
from statsmodels.genmod import families
import pandas as pd
import warnings
import patsy
# Gauss-Legendre weights
glw = [
[0.2955242247147529, -0.1488743389816312],
[0.2955242247147529, 0.1488743389816312],
[0.2692667193099963, -0.4333953941292472],
[0.2692667193099963, 0.4333953941292472],
[0.2190863625159820, -0.6794095682990244],
[0.2190863625159820, 0.6794095682990244],
[0.1494513491505806, -0.8650633666889845],
[0.1494513491505806, 0.8650633666889845],
[0.0666713443086881, -0.9739065285171717],
[0.0666713443086881, 0.9739065285171717],
]
_init_doc = r"""
Generalized Linear Mixed Model with Bayesian estimation
The class implements the Laplace approximation to the posterior
distribution (`fit_map`) and a variational Bayes approximation to
the posterior (`fit_vb`). See the two fit method docstrings for
more information about the fitting approaches.
Parameters
----------
endog : array_like
Vector of response values.
exog : array_like
Array of covariates for the fixed effects part of the mean
structure.
exog_vc : array_like
Array of covariates for the random part of the model. A
scipy.sparse array may be provided, or else the passed
array will be converted to sparse internally.
ident : array_like
Array of integer labels showing which random terms (columns
of `exog_vc`) have a common variance.
vcp_p : float
Prior standard deviation for variance component parameters
(the prior standard deviation of log(s) is vcp_p, where s is
the standard deviation of a random effect).
fe_p : float
Prior standard deviation for fixed effects parameters.
family : statsmodels.genmod.families instance
The GLM family.
fep_names : list[str]
The names of the fixed effects parameters (corresponding to
columns of exog). If None, default names are constructed.
vcp_names : list[str]
The names of the variance component parameters (corresponding
to distinct labels in ident). If None, default names are
constructed.
vc_names : list[str]
The names of the random effect realizations.
Returns
-------
MixedGLMResults object
Notes
-----
There are three types of values in the posterior distribution:
fixed effects parameters (fep), corresponding to the columns of
`exog`, random effects realizations (vc), corresponding to the
columns of `exog_vc`, and the standard deviations of the random
effects realizations (vcp), corresponding to the unique integer
labels in `ident`.
All random effects are modeled as being independent Gaussian
values (given the variance structure parameters). Every column of
`exog_vc` has a distinct realized random effect that is used to
form the linear predictors. The elements of `ident` determine the
distinct variance structure parameters. Two random effect
realizations that have the same value in `ident` have the same
variance. When fitting with a formula, `ident` is constructed
internally (each element of `vc_formulas` yields a distinct label
in `ident`).
The random effect standard deviation parameters (`vcp`) have
log-normal prior distributions with mean 0 and standard deviation
`vcp_p`.
Note that for some families, e.g. Binomial, the posterior mode may
be difficult to find numerically if `vcp_p` is set to too large of
a value. Setting `vcp_p` to 0.5 seems to work well.
The prior for the fixed effects parameters is Gaussian with mean 0
and standard deviation `fe_p`.
Examples
--------{example}
References
----------
Introduction to generalized linear mixed models:
https://stats.idre.ucla.edu/other/mult-pkg/introduction-to-generalized-linear-mixed-models
SAS documentation:
https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_intromix_a0000000215.htm
An assessment of estimation methods for generalized linear mixed
models with binary outcomes
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3866838/
"""
# The code in the example should be identical to what appears in
# the test_doc_examples unit test
_logit_example = """
A binomial (logistic) random effects model with random intercepts
for villages and random slopes for each year within each village:
>>> random = {"a": '0 + C(Village)', "b": '0 + C(Village)*year_cen'}
>>> model = BinomialBayesMixedGLM.from_formula(
'y ~ year_cen', random, data)
>>> result = model.fit_vb()
"""
# The code in the example should be identical to what appears in
# the test_doc_examples unit test
_poisson_example = """
A Poisson random effects model with random intercepts for villages
and random slopes for each year within each village:
>>> random = {"a": '0 + C(Village)', "b": '0 + C(Village)*year_cen'}
>>> model = PoissonBayesMixedGLM.from_formula(
'y ~ year_cen', random, data)
>>> result = model.fit_vb()
"""
class _BayesMixedGLM(base.Model):
def __init__(self,
endog,
exog,
exog_vc=None,
ident=None,
family=None,
vcp_p=1,
fe_p=2,
fep_names=None,
vcp_names=None,
vc_names=None,
**kwargs):
if exog.ndim == 1:
if isinstance(exog, np.ndarray):
exog = exog[:, None]
else:
exog = pd.DataFrame(exog)
if exog.ndim != 2:
msg = "'exog' must have one or two columns"
raise ValueError(msg)
if exog_vc.ndim == 1:
if isinstance(exog_vc, np.ndarray):
exog_vc = exog_vc[:, None]
else:
exog_vc = pd.DataFrame(exog_vc)
if exog_vc.ndim != 2:
msg = "'exog_vc' must have one or two columns"
raise ValueError(msg)
ident = np.asarray(ident)
if ident.ndim != 1:
msg = "ident must be a one-dimensional array"
raise ValueError(msg)
if len(ident) != exog_vc.shape[1]:
msg = "len(ident) should match the number of columns of exog_vc"
raise ValueError(msg)
if not np.issubdtype(ident.dtype, np.integer):
msg = "ident must have an integer dtype"
raise ValueError(msg)
# Get the fixed effects parameter names
if fep_names is None:
if hasattr(exog, "columns"):
fep_names = exog.columns.tolist()
else:
fep_names = ["FE_%d" % (k + 1) for k in range(exog.shape[1])]
# Get the variance parameter names
if vcp_names is None:
vcp_names = ["VC_%d" % (k + 1) for k in range(int(max(ident)) + 1)]
else:
if len(vcp_names) != len(set(ident)):
msg = "The lengths of vcp_names and ident should be the same"
raise ValueError(msg)
if not sparse.issparse(exog_vc):
exog_vc = sparse.csr_matrix(exog_vc)
ident = ident.astype(np.int)
vcp_p = float(vcp_p)
fe_p = float(fe_p)
# Number of fixed effects parameters
if exog is None:
k_fep = 0
else:
k_fep = exog.shape[1]
# Number of variance component structure parameters and
# variance component realizations.
if exog_vc is None:
k_vc = 0
k_vcp = 0
else:
k_vc = exog_vc.shape[1]
k_vcp = max(ident) + 1
# power might be better but not available in older scipy
exog_vc2 = exog_vc.multiply(exog_vc)
super(_BayesMixedGLM, self).__init__(endog, exog, **kwargs)
self.exog_vc = exog_vc
self.exog_vc2 = exog_vc2
self.ident = ident
self.family = family
self.k_fep = k_fep
self.k_vc = k_vc
self.k_vcp = k_vcp
self.fep_names = fep_names
self.vcp_names = vcp_names
self.vc_names = vc_names
self.fe_p = fe_p
self.vcp_p = vcp_p
self.names = fep_names + vcp_names
if vc_names is not None:
self.names += vc_names
def _unpack(self, vec):
ii = 0
# Fixed effects parameters
fep = vec[:ii + self.k_fep]
ii += self.k_fep
# Variance component structure parameters (standard
# deviations). These are on the log scale. The standard
# deviation for random effect j is exp(vcp[ident[j]]).
vcp = vec[ii:ii + self.k_vcp]
ii += self.k_vcp
# Random effect realizations
vc = vec[ii:]
return fep, vcp, vc
def logposterior(self, params):
"""
The overall log-density: log p(y, fe, vc, vcp).
This differs by an additive constant from the log posterior
log p(fe, vc, vcp | y).
"""
fep, vcp, vc = self._unpack(params)
# Contributions from p(y | x, vc)
lp = 0
if self.k_fep > 0:
lp += np.dot(self.exog, fep)
if self.k_vc > 0:
lp += self.exog_vc.dot(vc)
mu = self.family.link.inverse(lp)
ll = self.family.loglike(self.endog, mu)
if self.k_vc > 0:
# Contributions from p(vc | vcp)
vcp0 = vcp[self.ident]
s = np.exp(vcp0)
ll -= 0.5 * np.sum(vc**2 / s**2) + np.sum(vcp0)
# Contributions from p(vc)
ll -= 0.5 * np.sum(vcp**2 / self.vcp_p**2)
# Contributions from p(fep)
if self.k_fep > 0:
ll -= 0.5 * np.sum(fep**2 / self.fe_p**2)
return ll
def logposterior_grad(self, params):
"""
The gradient of the log posterior.
"""
fep, vcp, vc = self._unpack(params)
lp = 0
if self.k_fep > 0:
lp += np.dot(self.exog, fep)
if self.k_vc > 0:
lp += self.exog_vc.dot(vc)
mu = self.family.link.inverse(lp)
score_factor = (self.endog - mu) / self.family.link.deriv(mu)
score_factor /= self.family.variance(mu)
te = [None, None, None]
# Contributions from p(y | x, z, vc)
if self.k_fep > 0:
te[0] = np.dot(score_factor, self.exog)
if self.k_vc > 0:
te[2] = self.exog_vc.transpose().dot(score_factor)
if self.k_vc > 0:
# Contributions from p(vc | vcp)
# vcp0 = vcp[self.ident]
# s = np.exp(vcp0)
# ll -= 0.5 * np.sum(vc**2 / s**2) + np.sum(vcp0)
vcp0 = vcp[self.ident]
s = np.exp(vcp0)
u = vc**2 / s**2 - 1
te[1] = np.bincount(self.ident, weights=u)
te[2] -= vc / s**2
# Contributions from p(vcp)
# ll -= 0.5 * np.sum(vcp**2 / self.vcp_p**2)
te[1] -= vcp / self.vcp_p**2
# Contributions from p(fep)
if self.k_fep > 0:
te[0] -= fep / self.fe_p**2
te = [x for x in te if x is not None]
return np.concatenate(te)
def _get_start(self):
start_fep = np.zeros(self.k_fep)
start_vcp = np.ones(self.k_vcp)
start_vc = np.random.normal(size=self.k_vc)
start = np.concatenate((start_fep, start_vcp, start_vc))
return start
@classmethod
def from_formula(cls,
formula,
vc_formulas,
data,
family=None,
vcp_p=1,
fe_p=2):
"""
Fit a BayesMixedGLM using a formula.
Parameters
----------
formula : str
Formula for the endog and fixed effects terms (use ~ to
separate dependent and independent expressions).
vc_formulas : dictionary
vc_formulas[name] is a one-sided formula that creates one
collection of random effects with a common variance
parameter. If using categorical (factor) variables to
produce variance components, note that generally `0 + ...`
should be used so that an intercept is not included.
data : data frame
The data to which the formulas are applied.
family : genmod.families instance
A GLM family.
vcp_p : float
The prior standard deviation for the logarithms of the standard
deviations of the random effects.
fe_p : float
The prior standard deviation for the fixed effects parameters.
"""
ident = []
exog_vc = []
vcp_names = []
j = 0
for na, fml in vc_formulas.items():
mat = patsy.dmatrix(fml, data, return_type='dataframe')
exog_vc.append(mat)
vcp_names.append(na)
ident.append(j * np.ones(mat.shape[1], dtype=np.integer))
j += 1
exog_vc = pd.concat(exog_vc, axis=1)
vc_names = exog_vc.columns.tolist()
ident = np.concatenate(ident)
model = super(_BayesMixedGLM, cls).from_formula(
formula,
data=data,
family=family,
subset=None,
exog_vc=exog_vc,
ident=ident,
vc_names=vc_names,
vcp_names=vcp_names,
fe_p=fe_p,
vcp_p=vcp_p)
return model
def fit(self, method="BFGS", minim_opts=None):
"""
fit is equivalent to fit_map.
See fit_map for parameter information.
Use `fit_vb` to fit the model using variational Bayes.
"""
self.fit_map(method, minim_opts)
def fit_map(self, method="BFGS", minim_opts=None, scale_fe=False):
"""
Construct the Laplace approximation to the posterior distribution.
Parameters
----------
method : str
Optimization method for finding the posterior mode.
minim_opts : dict
Options passed to scipy.minimize.
scale_fe : bool
If True, the columns of the fixed effects design matrix
are centered and scaled to unit variance before fitting
the model. The results are back-transformed so that the
results are presented on the original scale.
Returns
-------
BayesMixedGLMResults instance.
"""
if scale_fe:
mn = self.exog.mean(0)
sc = self.exog.std(0)
self._exog_save = self.exog
self.exog = self.exog.copy()
ixs = np.flatnonzero(sc > 1e-8)
self.exog[:, ixs] -= mn[ixs]
self.exog[:, ixs] /= sc[ixs]
def fun(params):
return -self.logposterior(params)
def grad(params):
return -self.logposterior_grad(params)
start = self._get_start()
r = minimize(fun, start, method=method, jac=grad, options=minim_opts)
if not r.success:
msg = ("Laplace fitting did not converge, |gradient|=%.6f" %
np.sqrt(np.sum(r.jac**2)))
warnings.warn(msg)
from statsmodels.tools.numdiff import approx_fprime
hess = approx_fprime(r.x, grad)
cov = np.linalg.inv(hess)
params = r.x
if scale_fe:
self.exog = self._exog_save
del self._exog_save
params[ixs] /= sc[ixs]
cov[ixs, :][:, ixs] /= np.outer(sc[ixs], sc[ixs])
return BayesMixedGLMResults(self, params, cov, optim_retvals=r)
def predict(self, params, exog=None, linear=False):
"""
Return the fitted mean structure.
Parameters
----------
params : array_like
The parameter vector, may be the full parameter vector, or may
be truncated to include only the mean parameters.
exog : array_like
The design matrix for the mean structure. If omitted, use the
model's design matrix.
linear : bool
If True, return the linear predictor without passing through the
link function.
Returns
-------
A 1-dimensional array of predicted values
"""
if exog is None:
exog = self.exog
q = exog.shape[1]
pr = np.dot(exog, params[0:q])
if not linear:
pr = self.family.link.inverse(pr)
return pr
class _VariationalBayesMixedGLM(object):
"""
A mixin providing generic (not family-specific) methods for
variational Bayes mean field fitting.
"""
# Integration range (from -rng to +rng). The integrals are with
# respect to a standard Gaussian distribution so (-5, 5) will be
# sufficient in many cases.
rng = 5
verbose = False
# Returns the mean and variance of the linear predictor under the
# given distribution parameters.
def _lp_stats(self, fep_mean, fep_sd, vc_mean, vc_sd):
tm = np.dot(self.exog, fep_mean)
tv = np.dot(self.exog**2, fep_sd**2)
tm += self.exog_vc.dot(vc_mean)
tv += self.exog_vc2.dot(vc_sd**2)
return tm, tv
def vb_elbo_base(self, h, tm, fep_mean, vcp_mean, vc_mean, fep_sd, vcp_sd,
vc_sd):
"""
Returns the evidence lower bound (ELBO) for the model.
This function calculates the family-specific ELBO function
based on information provided from a subclass.
Parameters
----------
h : function mapping 1d vector to 1d vector
The contribution of the model to the ELBO function can be
expressed as y_i*lp_i + Eh_i(z), where y_i and lp_i are
the response and linear predictor for observation i, and z
is a standard normal random variable. This formulation
can be achieved for any GLM with a canonical link
function.
"""
# p(y | vc) contributions
iv = 0
for w in glw:
z = self.rng * w[1]
iv += w[0] * h(z) * np.exp(-z**2 / 2)
iv /= np.sqrt(2 * np.pi)
iv *= self.rng
iv += self.endog * tm
iv = iv.sum()
# p(vc | vcp) * p(vcp) * p(fep) contributions
iv += self._elbo_common(fep_mean, fep_sd, vcp_mean, vcp_sd, vc_mean,
vc_sd)
r = (iv + np.sum(np.log(fep_sd)) + np.sum(np.log(vcp_sd)) + np.sum(
np.log(vc_sd)))
return r
def vb_elbo_grad_base(self, h, tm, tv, fep_mean, vcp_mean, vc_mean, fep_sd,
vcp_sd, vc_sd):
"""
Return the gradient of the ELBO function.
See vb_elbo_base for parameters.
"""
fep_mean_grad = 0.
fep_sd_grad = 0.
vcp_mean_grad = 0.
vcp_sd_grad = 0.
vc_mean_grad = 0.
vc_sd_grad = 0.
# p(y | vc) contributions
for w in glw:
z = self.rng * w[1]
u = h(z) * np.exp(-z**2 / 2) / np.sqrt(2 * np.pi)
r = u / np.sqrt(tv)
fep_mean_grad += w[0] * np.dot(u, self.exog)
vc_mean_grad += w[0] * self.exog_vc.transpose().dot(u)
fep_sd_grad += w[0] * z * np.dot(r, self.exog**2 * fep_sd)
v = self.exog_vc2.multiply(vc_sd).transpose().dot(r)
v = np.squeeze(np.asarray(v))
vc_sd_grad += w[0] * z * v
fep_mean_grad *= self.rng
vc_mean_grad *= self.rng
fep_sd_grad *= self.rng
vc_sd_grad *= self.rng
fep_mean_grad += np.dot(self.endog, self.exog)
vc_mean_grad += self.exog_vc.transpose().dot(self.endog)
(fep_mean_grad_i, fep_sd_grad_i, vcp_mean_grad_i, vcp_sd_grad_i,
vc_mean_grad_i, vc_sd_grad_i) = self._elbo_grad_common(
fep_mean, fep_sd, vcp_mean, vcp_sd, vc_mean, vc_sd)
fep_mean_grad += fep_mean_grad_i
fep_sd_grad += fep_sd_grad_i
vcp_mean_grad += vcp_mean_grad_i
vcp_sd_grad += vcp_sd_grad_i
vc_mean_grad += vc_mean_grad_i
vc_sd_grad += vc_sd_grad_i
fep_sd_grad += 1 / fep_sd
vcp_sd_grad += 1 / vcp_sd
vc_sd_grad += 1 / vc_sd
mean_grad = np.concatenate((fep_mean_grad, vcp_mean_grad,
vc_mean_grad))
sd_grad = np.concatenate((fep_sd_grad, vcp_sd_grad, vc_sd_grad))
if self.verbose:
print(
"|G|=%f" % np.sqrt(np.sum(mean_grad**2) + np.sum(sd_grad**2)))
return mean_grad, sd_grad
def fit_vb(self,
mean=None,
sd=None,
fit_method="BFGS",
minim_opts=None,
scale_fe=False,
verbose=False):
"""
Fit a model using the variational Bayes mean field approximation.
Parameters
----------
mean : array_like
Starting value for VB mean vector
sd : array_like
Starting value for VB standard deviation vector
fit_method : str
Algorithm for scipy.minimize
minim_opts : dict
Options passed to scipy.minimize
scale_fe : bool
If true, the columns of the fixed effects design matrix
are centered and scaled to unit variance before fitting
the model. The results are back-transformed so that the
results are presented on the original scale.
verbose : bool
If True, print the gradient norm to the screen each time
it is calculated.
Notes
-----
The goal is to find a factored Gaussian approximation
q1*q2*... to the posterior distribution, approximately
minimizing the KL divergence from the factored approximation
to the actual posterior. The KL divergence, or ELBO function
has the form
E* log p(y, fe, vcp, vc) - E* log q
where E* is expectation with respect to the product of qj.
References
----------
Blei, Kucukelbir, McAuliffe (2017). Variational Inference: A
review for Statisticians
https://arxiv.org/pdf/1601.00670.pdf
"""
self.verbose = verbose
if scale_fe:
mn = self.exog.mean(0)
sc = self.exog.std(0)
self._exog_save = self.exog
self.exog = self.exog.copy()
ixs = np.flatnonzero(sc > 1e-8)
self.exog[:, ixs] -= mn[ixs]
self.exog[:, ixs] /= sc[ixs]
n = self.k_fep + self.k_vcp + self.k_vc
ml = self.k_fep + self.k_vcp + self.k_vc
if mean is None:
m = np.zeros(n)
else:
if len(mean) != ml:
raise ValueError(
"mean has incorrect length, %d != %d" % (len(mean), ml))
m = mean.copy()
if sd is None:
s = -0.5 + 0.1 * np.random.normal(size=n)
else:
if len(sd) != ml:
raise ValueError(
"sd has incorrect length, %d != %d" % (len(sd), ml))
# s is parametrized on the log-scale internally when
# optimizing the ELBO function (this is transparent to the
# caller)
s = np.log(sd)
# Do not allow the variance parameter starting mean values to
# be too small.
i1, i2 = self.k_fep, self.k_fep + self.k_vcp
m[i1:i2] = np.where(m[i1:i2] < -1, -1, m[i1:i2])
# Do not allow the posterior standard deviation starting values
# to be too small.
s = np.where(s < -1, -1, s)
def elbo(x):
n = len(x) // 2
return -self.vb_elbo(x[:n], np.exp(x[n:]))
def elbo_grad(x):
n = len(x) // 2
gm, gs = self.vb_elbo_grad(x[:n], np.exp(x[n:]))
gs *= np.exp(x[n:])
return -np.concatenate((gm, gs))
start = np.concatenate((m, s))
mm = minimize(
elbo, start, jac=elbo_grad, method=fit_method, options=minim_opts)
if not mm.success:
warnings.warn("VB fitting did not converge")
n = len(mm.x) // 2
params = mm.x[0:n]
va = np.exp(2 * mm.x[n:])
if scale_fe:
self.exog = self._exog_save
del self._exog_save
params[ixs] /= sc[ixs]
va[ixs] /= sc[ixs]**2
return BayesMixedGLMResults(self, params, va, mm)
# Handle terms in the ELBO that are common to all models.
def _elbo_common(self, fep_mean, fep_sd, vcp_mean, vcp_sd, vc_mean, vc_sd):
iv = 0
# p(vc | vcp) contributions
m = vcp_mean[self.ident]
s = vcp_sd[self.ident]
iv -= np.sum((vc_mean**2 + vc_sd**2) * np.exp(2 * (s**2 - m))) / 2
iv -= np.sum(m)
# p(vcp) contributions
iv -= 0.5 * (vcp_mean**2 + vcp_sd**2).sum() / self.vcp_p**2
# p(b) contributions
iv -= 0.5 * (fep_mean**2 + fep_sd**2).sum() / self.fe_p**2
return iv
def _elbo_grad_common(self, fep_mean, fep_sd, vcp_mean, vcp_sd, vc_mean,
vc_sd):
# p(vc | vcp) contributions
m = vcp_mean[self.ident]
s = vcp_sd[self.ident]
u = vc_mean**2 + vc_sd**2
ve = np.exp(2 * (s**2 - m))
dm = u * ve - 1
ds = -2 * u * ve * s
vcp_mean_grad = np.bincount(self.ident, weights=dm)
vcp_sd_grad = np.bincount(self.ident, weights=ds)
vc_mean_grad = -vc_mean.copy() * ve
vc_sd_grad = -vc_sd.copy() * ve
# p(vcp) contributions
vcp_mean_grad -= vcp_mean / self.vcp_p**2
vcp_sd_grad -= vcp_sd / self.vcp_p**2
# p(b) contributions
fep_mean_grad = -fep_mean.copy() / self.fe_p**2
fep_sd_grad = -fep_sd.copy() / self.fe_p**2
return (fep_mean_grad, fep_sd_grad, vcp_mean_grad, vcp_sd_grad,
vc_mean_grad, vc_sd_grad)
[docs]class BayesMixedGLMResults(object):
"""
Class to hold results from a Bayesian estimation of a Mixed GLM model.
Attributes
----------
fe_mean : array_like
Posterior mean of the fixed effects coefficients.
fe_sd : array_like
Posterior standard deviation of the fixed effects coefficients
vcp_mean : array_like
Posterior mean of the logged variance component standard
deviations.
vcp_sd : array_like
Posterior standard deviation of the logged variance component
standard deviations.
vc_mean : array_like
Posterior mean of the random coefficients
vc_sd : array_like
Posterior standard deviation of the random coefficients
"""
def __init__(self, model, params, cov_params, optim_retvals=None):
self.model = model
self.params = params
self._cov_params = cov_params
self.optim_retvals = optim_retvals
self.fe_mean, self.vcp_mean, self.vc_mean = (model._unpack(params))
if cov_params.ndim == 2:
cp = np.diag(cov_params)
else:
cp = cov_params
self.fe_sd, self.vcp_sd, self.vc_sd = model._unpack(cp)
self.fe_sd = np.sqrt(self.fe_sd)
self.vcp_sd = np.sqrt(self.vcp_sd)
self.vc_sd = np.sqrt(self.vc_sd)
[docs] def cov_params(self):
if hasattr(self.model.data, "frame"):
# Return the covariance matrix as a dataframe or series
na = (self.model.fep_names + self.model.vcp_names +
self.model.vc_names)
if self._cov_params.ndim == 2:
return pd.DataFrame(self._cov_params, index=na, columns=na)
else:
return pd.Series(self._cov_params, index=na)
# Return the covariance matrix as a ndarray
return self._cov_params
[docs] def summary(self):
df = pd.DataFrame()
m = self.model.k_fep + self.model.k_vcp
df["Type"] = (["M" for k in range(self.model.k_fep)] +
["V" for k in range(self.model.k_vcp)])
df["Post. Mean"] = self.params[0:m]
if self._cov_params.ndim == 2:
v = np.diag(self._cov_params)[0:m]
df["Post. SD"] = np.sqrt(v)
else:
df["Post. SD"] = np.sqrt(self._cov_params[0:m])
# Convert variance parameters to natural scale
df["SD"] = np.exp(df["Post. Mean"])
df["SD (LB)"] = np.exp(df["Post. Mean"] - 2 * df["Post. SD"])
df["SD (UB)"] = np.exp(df["Post. Mean"] + 2 * df["Post. SD"])
df["SD"] = ["%.3f" % x for x in df.SD]
df["SD (LB)"] = ["%.3f" % x for x in df["SD (LB)"]]
df["SD (UB)"] = ["%.3f" % x for x in df["SD (UB)"]]
df.loc[df.index < self.model.k_fep, "SD"] = ""
df.loc[df.index < self.model.k_fep, "SD (LB)"] = ""
df.loc[df.index < self.model.k_fep, "SD (UB)"] = ""
df.index = self.model.fep_names + self.model.vcp_names
summ = summary2.Summary()
summ.add_title(self.model.family.__class__.__name__ +
" Mixed GLM Results")
summ.add_df(df)
summ.add_text("Parameter types are mean structure (M) and "
"variance structure (V)")
summ.add_text("Variance parameters are modeled as log "
"standard deviations")
return summ
[docs] def random_effects(self, term=None):
"""
Posterior mean and standard deviation of random effects.
Parameters
----------
term : int or None
If None, results for all random effects are returned. If
an integer, returns results for a given set of random
effects. The value of `term` refers to an element of the
`ident` vector, or to a position in the `vc_formulas`
list.
Returns
-------
Data frame of posterior means and posterior standard
deviations of random effects.
"""
z = self.vc_mean
s = self.vc_sd
na = self.model.vc_names
if term is not None:
termix = self.model.vcp_names.index(term)
ii = np.flatnonzero(self.model.ident == termix)
z = z[ii]
s = s[ii]
na = [na[i] for i in ii]
x = pd.DataFrame({"Mean": z, "SD": s})
if na is not None:
x.index = na
return x
[docs] def predict(self, exog=None, linear=False):
"""
Return predicted values for the mean structure.
Parameters
----------
exog : array_like
The design matrix for the mean structure. If None,
use the model's design matrix.
linear : bool
If True, returns the linear predictor, otherwise
transform the linear predictor using the link function.
Returns
-------
A one-dimensional array of fitted values.
"""
return self.model.predict(self.params, exog, linear)
[docs]class BinomialBayesMixedGLM(_VariationalBayesMixedGLM, _BayesMixedGLM):
__doc__ = _init_doc.format(example=_logit_example)
def __init__(self,
endog,
exog,
exog_vc,
ident,
vcp_p=1,
fe_p=2,
fep_names=None,
vcp_names=None,
vc_names=None):
super(BinomialBayesMixedGLM, self).__init__(
endog,
exog,
exog_vc=exog_vc,
ident=ident,
vcp_p=vcp_p,
fe_p=fe_p,
family=families.Binomial(),
fep_names=fep_names,
vcp_names=vcp_names,
vc_names=vc_names)
if not np.all(np.unique(endog) == np.r_[0, 1]):
msg = "endog values must be 0 and 1, and not all identical"
raise ValueError(msg)
[docs] def vb_elbo(self, vb_mean, vb_sd):
"""
Returns the evidence lower bound (ELBO) for the model.
"""
fep_mean, vcp_mean, vc_mean = self._unpack(vb_mean)
fep_sd, vcp_sd, vc_sd = self._unpack(vb_sd)
tm, tv = self._lp_stats(fep_mean, fep_sd, vc_mean, vc_sd)
def h(z):
return -np.log(1 + np.exp(tm + np.sqrt(tv) * z))
return self.vb_elbo_base(h, tm, fep_mean, vcp_mean, vc_mean, fep_sd,
vcp_sd, vc_sd)
[docs] def vb_elbo_grad(self, vb_mean, vb_sd):
"""
Returns the gradient of the model's evidence lower bound (ELBO).
"""
fep_mean, vcp_mean, vc_mean = self._unpack(vb_mean)
fep_sd, vcp_sd, vc_sd = self._unpack(vb_sd)
tm, tv = self._lp_stats(fep_mean, fep_sd, vc_mean, vc_sd)
def h(z):
u = tm + np.sqrt(tv) * z
x = np.zeros_like(u)
ii = np.flatnonzero(u > 0)
uu = u[ii]
x[ii] = 1 / (1 + np.exp(-uu))
ii = np.flatnonzero(u <= 0)
uu = u[ii]
x[ii] = np.exp(uu) / (1 + np.exp(uu))
return -x
return self.vb_elbo_grad_base(h, tm, tv, fep_mean, vcp_mean, vc_mean,
fep_sd, vcp_sd, vc_sd)
[docs]class PoissonBayesMixedGLM(_VariationalBayesMixedGLM, _BayesMixedGLM):
__doc__ = _init_doc.format(example=_poisson_example)
def __init__(self,
endog,
exog,
exog_vc,
ident,
vcp_p=1,
fe_p=2,
fep_names=None,
vcp_names=None,
vc_names=None):
super(PoissonBayesMixedGLM, self).__init__(
endog=endog,
exog=exog,
exog_vc=exog_vc,
ident=ident,
vcp_p=vcp_p,
fe_p=fe_p,
family=families.Poisson(),
fep_names=fep_names,
vcp_names=vcp_names,
vc_names=vc_names)
[docs] def vb_elbo(self, vb_mean, vb_sd):
"""
Returns the evidence lower bound (ELBO) for the model.
"""
fep_mean, vcp_mean, vc_mean = self._unpack(vb_mean)
fep_sd, vcp_sd, vc_sd = self._unpack(vb_sd)
tm, tv = self._lp_stats(fep_mean, fep_sd, vc_mean, vc_sd)
def h(z):
return -np.exp(tm + np.sqrt(tv) * z)
return self.vb_elbo_base(h, tm, fep_mean, vcp_mean, vc_mean, fep_sd,
vcp_sd, vc_sd)
[docs] def vb_elbo_grad(self, vb_mean, vb_sd):
"""
Returns the gradient of the model's evidence lower bound (ELBO).
"""
fep_mean, vcp_mean, vc_mean = self._unpack(vb_mean)
fep_sd, vcp_sd, vc_sd = self._unpack(vb_sd)
tm, tv = self._lp_stats(fep_mean, fep_sd, vc_mean, vc_sd)
def h(z):
y = -np.exp(tm + np.sqrt(tv) * z)
return y
return self.vb_elbo_grad_base(h, tm, tv, fep_mean, vcp_mean, vc_mean,
fep_sd, vcp_sd, vc_sd)