def score_precision_beta(
endog,
exog_mean,
exog_precision,
bounded_reg_link,
param_mean,
param_precision,
):
"""
Computes the score vector with respect to the precision regression
parameters.
For more details we refer to:
Ferrari, S. L. P., Cribari-Neto, F. (2004). Beta regression
for modeling rates and proportions. J. Appl. Statist. 31, 799–815
Simas, A. B., Barreto-Souza, W., Rocha, A. V. (2010). Improved
estimators for a general class of beta regression models. Computational
Statistics and Data Analysis, 54, 348–366
:param endog (array_like): 1d array of endogenous response variable.
:param exog_mean (array_like): A nobs x k array where nobs is the number
of observations and k is the number of mean regressors. An intercept is
not included by default and should be added by the user.
:param exog_precision (array_like): A nobs x q array where nobs is the
number of observations and q is the number of precision regressors.
An intercept is not included by default and should be added by the user.
:param bounded_reg_link: An instance of BoundedRegLink. Recall that
the default mean link is 'logit' and that the default precision link
is None.
:param param_mean: 1d array of mean regression parameters.
:param param_precision: 1d array of precision regression parameters.
"""
estimated_mean = estimate_mean(exog_mean, param_mean, bounded_reg_link)
estimated_precision = estimate_precision(exog_precision, param_precision,
bounded_reg_link)
if exog_precision is None:
exog_precision = param_precision * np.ones_like(estimated_mean)
score_precision = np.matmul(
exog_precision.T,
(estimated_mean *
(np.log(endog) - np.log(1 - endog) -
digamma(estimated_mean * estimated_precision) + digamma(
(1 - estimated_mean) * estimated_precision)) +
digamma(estimated_precision) - digamma(
(1 - estimated_mean) * estimated_precision) + np.log(1 - endog)) *
bounded_reg_link.dphideta(estimated_precision),
)
return correct_dimension(score_precision)
def grad_q_precision_beta(
endog,
exog_mean,
exog_precision,
bounded_reg_link,
param_mean,
param_precision,
previous_precision,
):
"""
Computes the gradient of the Q function with respect to the precision
regression parameters.
For more details we refer to:
Barreto-Souza & Simas (2017) Improving estimation for beta regression
models via em-algorithm and related diagnostic tools, Volume 87, Pages
2847-2867
:param endog (array_like): 1d array of endogenous response variable.
:param exog_mean (array_like): A nobs x k array where nobs is the number
of observations and k is the number of mean regressors. An intercept is
not included by default and should be added by the user.
:param exog_precision (array_like): A nobs x q array where nobs is the
number of observations and q is the number of precision regressors.
An intercept is not included by default and should be added by the user.
:param bounded_reg_link: An instance of BoundedRegLink. Recall that
the default mean link is 'logit' and that the default precision link
is None.
:param param_mean: 1d array of mean regression parameters.
:param param_precision: 1d array of precision regression parameters.
:param previous_precision: 1d array of the regression parameters
related to the precision in the previous EM-step.
"""
estimated_mean = estimate_mean(exog_mean, param_mean, bounded_reg_link)
estimated_precision = estimate_precision(exog_precision, param_precision,
bounded_reg_link)
if exog_precision is None:
exog_precision = param_precision * np.ones_like(estimated_mean)
grad_precision = np.matmul(
exog_precision.T,
(estimated_mean * np.log(endog / (1 - endog)) +
digamma(previous_precision) + np.log(1 - endog) -
estimated_mean * digamma(estimated_mean * estimated_precision) -
(1 - estimated_mean) * digamma(
(1 - estimated_mean) * estimated_precision)) *
bounded_reg_link.dphideta(estimated_precision),
)
return correct_dimension(grad_precision)
def em_loop(endog,
exog_mean,
exog_precision,
bounded_reg_link,
param_mean_start,
param_precision_start,
em_optim_params={
"em_tolerance": 10**(-6),
"max_em_iterations": 5000,
"method": "BFGS",
},
**kwargs):
'''
Runs the EM procedure until convergence (or when reaching the
maximum number of iterations).
For more details we refer to:
Barreto-Souza & Simas (2017) Improving estimation for beta regression
models via em-algorithm and related diagnostic tools, Volume 87, Pages
2847-2867
:param endog (array_like): 1d array of endogenous response variable.
:param exog_mean (array_like): A nobs x k array where nobs is the number
of observations and k is the number of mean regressors. An intercept is
not included by default and should be added by the user.
:param exog_precision (array_like): A nobs x q array where nobs is the
number of observations and q is the number of precision regressors.
An intercept is not included by default and should be added by the user.
:param bounded_reg_link: An instance of BoundedRegLink. Recall that
the default mean link is 'logit' and that the default precision link
is None.
:param param_mean_start (array_like): initial guesses of the mean
regression parameters.
:param param_precision_start (array_like): initial guess of the
precision regression parameters.
:param optim_params (dict): A dictionary of parameters related
to optimization:
em_tolerance: the error tolerance for convergence in the EM procedure.
max_em_iterations: the maximum number of iterations in the EM
procedure.
**kwargs: additional parameters to be passed to the minimize function
from the scipy.optimize module.
The EM tolerance parameter is calculated as described in the above
reference.
'''
em_params = {
"endog": endog,
"exog_mean": exog_mean,
"bounded_reg_link": bounded_reg_link,
"exog_precision": exog_precision,
}
k = exog_mean.shape[1]
if exog_precision is None:
q = 1
else:
q = exog_precision.shape[1]
em_tolerance = 1
count = 0
param_start = np.concatenate([param_mean_start, param_precision_start])
previous_precision_param = param_precision_start
while (em_tolerance > em_optim_params["em_tolerance"]
and count < em_optim_params["max_em_iterations"]):
previous_precision = previous_precision_update(
previous_precision_param, exog_precision, bounded_reg_link)
fit = maximize_q_function_beta(
**em_params,
param_start=param_start,
previous_precision=previous_precision,
method=em_optim_params["method"],
**kwargs,
)
param_new = fit.x
em_tolerance = compute_tolerance_beta(
**em_params,
param_new=param_new,
param_start=param_start,
previous_precision=previous_precision,
)
param_start = param_new
previous_precision_param = correct_dimension(param_new[k:(k + q)])
count += 1
return {'estimated_parameters': param_new, 'count': count}