import numpy as np
import pandas as pd
from statsmodels.base.model import LikelihoodModelResults
[docs]class BayesGaussMI(object):
"""
Bayesian Imputation using a Gaussian model.
The approach is Bayesian. The goal is to sample from the joint
distribution of the mean vector, covariance matrix, and missing
data values given the observed data values. Conjugate priors for
the population mean and covariance matrix are used. Gibbs
sampling is used to update the mean vector, covariance matrix, and
missing data values in turn. After burn-in, the imputed complete
data sets from the Gibbs chain can be used in multiple imputation
analyses (MI).
Parameters
----------
data : ndarray
The array of data to be imputed. Values in the array equal to
NaN are imputed.
mean_prior : ndarray, optional
The covariance matrix of the Gaussian prior distribution for
the mean vector. If not provided, the identity matrix is
used.
cov_prior : ndarray, optional
The center matrix for the inverse Wishart prior distribution
for the covariance matrix. If not provided, the identity
matrix is used.
cov_prior_df : positive float
The degrees of freedom of the inverse Wishart prior
distribution for the covariance matrix. Defaults to 1.
Examples
--------
A basic example with OLS. Data is generated assuming 10% is missing at
random.
>>> import numpy as np
>>> x = np.random.standard_normal((1000, 2))
>>> x.flat[np.random.sample(2000) < 0.1] = np.nan
The imputer is used with ``MI``.
>>> import statsmodels.api as sm
>>> def model_args_fn(x):
... # Return endog, exog from x
... return x[:, 0], x[:, 1:]
>>> imp = sm.BayesGaussMI(x)
>>> mi = sm.MI(imp, sm.OLS, model_args_fn)
"""
def __init__(self, data, mean_prior=None, cov_prior=None, cov_prior_df=1):
self.exog_names = None
if type(data) is pd.DataFrame:
self.exog_names = data.columns
data = np.asarray(data)
self.data = data
self._data = data
self.mask = np.isnan(data)
self.nobs = self.mask.shape[0]
self.nvar = self.mask.shape[1]
# Identify all distinct missing data patterns
z = 1 + np.log(1 + np.arange(self.mask.shape[1]))
c = np.dot(self.mask, z)
rowmap = {}
for i, v in enumerate(c):
if v == 0:
# No missing values
continue
if v not in rowmap:
rowmap[v] = []
rowmap[v].append(i)
self.patterns = [np.asarray(v) for v in rowmap.values()]
# Simple starting values for mean and covariance
p = self._data.shape[1]
self.cov = np.eye(p)
mean = []
for i in range(p):
v = self._data[:, i]
v = v[np.isfinite(v)]
if len(v) == 0:
msg = "Column %d has no observed values" % i
raise ValueError(msg)
mean.append(v.mean())
self.mean = np.asarray(mean)
# Default covariance matrix of the (Gaussian) mean prior
if mean_prior is None:
mean_prior = np.eye(p)
self.mean_prior = mean_prior
# Default center matrix of the (inverse Wishart) covariance prior
if cov_prior is None:
cov_prior = np.eye(p)
self.cov_prior = cov_prior
# Degrees of freedom for the (inverse Wishart) covariance prior
self.cov_prior_df = cov_prior_df
[docs] def update(self):
"""
Cycle through all Gibbs updates.
"""
self.update_data()
# Need to update data first
self.update_mean()
self.update_cov()
[docs] def update_data(self):
"""
Gibbs update of the missing data values.
"""
for ix in self.patterns:
i = ix[0]
ix_miss = np.flatnonzero(self.mask[i, :])
ix_obs = np.flatnonzero(~self.mask[i, :])
mm = self.mean[ix_miss]
mo = self.mean[ix_obs]
voo = self.cov[ix_obs, :][:, ix_obs]
vmm = self.cov[ix_miss, :][:, ix_miss]
vmo = self.cov[ix_miss, :][:, ix_obs]
r = self._data[ix, :][:, ix_obs] - mo
cm = mm + np.dot(vmo, np.linalg.solve(voo, r.T)).T
cv = vmm - np.dot(vmo, np.linalg.solve(voo, vmo.T))
cs = np.linalg.cholesky(cv)
u = np.random.normal(size=(len(ix), len(ix_miss)))
self._data[np.ix_(ix, ix_miss)] = cm + np.dot(u, cs.T)
# Set the user-visible data set.
if self.exog_names is not None:
self.data = pd.DataFrame(
self._data,
columns=self.exog_names,
copy=False)
else:
self.data = self._data
[docs] def update_mean(self):
"""
Gibbs update of the mean vector.
Do not call until update_data has been called once.
"""
# https://stats.stackexchange.com/questions/28744/multivariate-normal-posterior
# Posterior covariance matrix of the mean
cm = np.linalg.solve(self.cov/self.nobs + self.mean_prior,
self.mean_prior / self.nobs)
cm = np.dot(self.cov, cm)
# Posterior mean of the mean
vm = np.linalg.solve(self.cov, self._data.sum(0))
vm = np.dot(cm, vm)
# Sample
r = np.linalg.cholesky(cm)
self.mean = vm + np.dot(r, np.random.normal(0, 1, self.nvar))
[docs] def update_cov(self):
"""
Gibbs update of the covariance matrix.
Do not call until update_data has been called once.
"""
# https://stats.stackexchange.com/questions/50844/estimating-the-covariance-posterior-distribution-of-a-multivariate-gaussian
r = self._data - self.mean
gr = np.dot(r.T, r)
a = gr + self.cov_prior
df = int(np.ceil(self.nobs + self.cov_prior_df))
r = np.linalg.cholesky(np.linalg.inv(a))
x = np.dot(np.random.normal(size=(df, self.nvar)), r.T)
ma = np.dot(x.T, x)
self.cov = np.linalg.inv(ma)
[docs]class MI(object):
"""
MI performs multiple imputation using a provided imputer object.
Parameters
----------
imp : object
An imputer class, such as BayesGaussMI.
model : model class
Any statsmodels model class.
model_args_fn : function
A function taking an imputed dataset as input and returning
endog, exog. If the model is fit using a formula, returns
a DataFrame used to build the model. Optional when a formula
is used.
model_kwds_fn : function, optional
A function taking an imputed dataset as input and returning
a dictionary of model keyword arguments.
formula : str, optional
If provided, the model is constructed using the `from_formula`
class method, otherwise the `__init__` method is used.
fit_args : list-like, optional
List of arguments to be passed to the fit method
fit_kwds : dict-like, optional
Keyword arguments to be passed to the fit method
xfunc : function mapping ndarray to ndarray
A function that is applied to the complete data matrix
prior to fitting the model
burn : int
Number of burn-in iterations
nrep : int
Number of imputed data sets to use in the analysis
skip : int
Number of Gibbs iterations to skip between successive
multiple imputation fits.
Notes
-----
The imputer object must have an 'update' method, and a 'data'
attribute that contains the current imputed dataset.
xfunc can be used to introduce domain constraints, e.g. when
imputing binary data the imputed continuous values can be rounded
to 0/1.
"""
def __init__(self, imp, model, model_args_fn=None, model_kwds_fn=None,
formula=None, fit_args=None, fit_kwds=None, xfunc=None,
burn=100, nrep=20, skip=10):
# The imputer
self.imp = imp
# The number of imputed data sets to skip between each imputed
# data set tha that is used in the analysis.
self.skip = skip
# The model class
self.model = model
self.formula = formula
if model_args_fn is None:
def f(x):
return []
model_args_fn = f
self.model_args_fn = model_args_fn
if model_kwds_fn is None:
def f(x):
return {}
model_kwds_fn = f
self.model_kwds_fn = model_kwds_fn
if fit_args is None:
def f(x):
return []
fit_args = f
self.fit_args = fit_args
if fit_kwds is None:
def f(x):
return {}
fit_kwds = f
self.fit_kwds = fit_kwds
self.xfunc = xfunc
self.nrep = nrep
self.skip = skip
# Burn-in
for k in range(burn):
imp.update()
[docs] def fit(self, results_cb=None):
"""
Impute datasets, fit models, and pool results.
Parameters
----------
results_cb : function, optional
If provided, each results instance r is passed through `results_cb`,
then appended to the `results` attribute of the MIResults object.
To save complete results, use `results_cb=lambda x: x`. The default
behavior is to save no results.
Returns
-------
A MIResults object.
"""
par, cov = [], []
all_results = []
for k in range(self.nrep):
for k in range(self.skip+1):
self.imp.update()
da = self.imp.data
if self.xfunc is not None:
da = self.xfunc(da)
if self.formula is None:
model = self.model(*self.model_args_fn(da),
**self.model_kwds_fn(da))
else:
model = self.model.from_formula(
self.formula, *self.model_args_fn(da),
**self.model_kwds_fn(da))
result = model.fit(*self.fit_args(da), **self.fit_kwds(da))
if results_cb is not None:
all_results.append(results_cb(result))
par.append(np.asarray(result.params.copy()))
cov.append(np.asarray(result.cov_params().copy()))
params, cov_params, fmi = self._combine(par, cov)
r = MIResults(self, model, params, cov_params)
r.fmi = fmi
r.results = all_results
return r
def _combine(self, par, cov):
# Helper function to apply "Rubin's combining rule"
par = np.asarray(par)
# Number of imputations
m = par.shape[0]
# Point estimate
params = par.mean(0)
# Within-imputation covariance
wcov = sum(cov) / len(cov)
# Between-imputation covariance
bcov = np.cov(par.T)
bcov = np.atleast_2d(bcov)
# Overall covariance
covp = wcov + (1 + 1/float(m))*bcov
# Fraction of missing information
fmi = (1 + 1/float(m)) * np.diag(bcov) / np.diag(covp)
return params, covp, fmi
class MIResults(LikelihoodModelResults):
"""
A results class for multiple imputation (MI).
Parameters
----------
mi : MI instance
The MI object that produced the results
model : instance of statsmodels model class
This can be any instance from the multiple imputation runs.
It is used to get class information, the specific parameter
and data values are not used.
params : array_like
The overall multiple imputation parameter estimates.
normalized_cov_params : array_like (2d)
The overall variance covariance matrix of the estimates.
"""
def __init__(self, mi, model, params, normalized_cov_params):
super(MIResults, self).__init__(model, params, normalized_cov_params)
self.mi = mi
self._model = model
def summary(self, title=None, alpha=.05):
"""
Summarize the results of running multiple imputation.
Parameters
----------
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.
"""
from statsmodels.iolib import summary2
smry = summary2.Summary()
float_format = "%8.3f"
info = {}
info["Method:"] = "MI"
info["Model:"] = self.mi.model.__name__
info["Dependent variable:"] = self._model.endog_names
info["Sample size:"] = "%d" % self.mi.imp.data.shape[0]
info["Num. imputations"] = "%d" % self.mi.nrep
smry.add_dict(info, align='l', float_format=float_format)
param = summary2.summary_params(self, alpha=alpha)
param["FMI"] = self.fmi
smry.add_df(param, float_format=float_format)
smry.add_title(title=title, results=self)
return smry