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 q_function_beta(
endog,
exog_mean,
exog_precision,
bounded_reg_link,
param_mean,
param_precision,
previous_precision,
):
"""
Obtain the Q function for the EM estimation of the beta regression.
In the iterative procedure, we have to carry the previous value of
the precision parameter.
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
We return the value with the minus sign since we want to maximize
the Q function and we will utilize the minimize function to
optimizate.
: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)
return -np.sum(estimated_precision *
(estimated_mean * np.log(endog / (1 - endog)) +
digamma(previous_precision) + np.log(1 - endog)) -
gammaln(estimated_mean * estimated_precision) -
gammaln((1 - estimated_mean) * estimated_precision) -
np.log(endog * (1 - endog)) - digamma(previous_precision) -
np.log(1 - endog) - previous_precision)
def loglikelihood_function_beta(
endog,
exog_mean,
exog_precision,
bounded_reg_link,
param_mean,
param_precision,
):
"""
Obtain the log-likelihood function for the beta regression model.
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
We return the value with the minus sign since we want to maximize
the log-likelihood function and we will utilize the minimize function to
optimizate.
: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)
loglik = -np.sum(
gammaln(estimated_precision) -
gammaln(estimated_mean * estimated_precision) -
gammaln((1 - estimated_mean) * estimated_precision) +
(estimated_mean * estimated_precision - 1) * np.log(endog) +
((1 - estimated_mean) * estimated_precision - 1) * np.log(1 - endog))
return loglik