irfpy.util.likelihood
¶
Functions for likelihood.
Code author: Yoshifumi Futaana

Return the log likelihood for normal distribution. 

Return the log likelihood for poisson statistics 
For details about likelihood, see http://en.wikipedia.org/wiki/Likelihood_function
 irfpy.util.likelihood.log_likelihood_normal(observed, predicted)[source]¶
Return the log likelihood for normal distribution.
The loglikelihood for normal distribution is calculated from
\[\log L = \sum (y_o  y_p)^2\]where yo is the observed values and yp is the predicted values. The standard deviation of the normal distribution is assumed constant over each observed value.
Normally, predicted values are derived from some parametric function, and therefore the likelihood is a function of the parameters.
 Parameters
observed – Observed value as np.array
predicted – Predicted value as np.array
 Returns
Log likelihood under normal distribution.
>>> observed = [3, 2, 3, 4, 6, 2, 1] >>> predicted = [1.8, 2.2, 3.4, 4.6, 6.7, 5.1, 3.2] >>> print('{:.3f}'.format(log_likelihood_normal(observed, predicted))) 16.940
 irfpy.util.likelihood.log_likelihood_poisson(observed, predicted)[source]¶
Return the log likelihood for poisson statistics
The log likelihood for Poisson distribution is calculated from
\[\log L = (y_o \times \ln y_p  y_p)\]where yo is the observed values (thus, integer, for Poisson distribution) and yp is the predicted values (usually, float). Normally, predicted values are derived from some parametric function, and therefore the likelihood is a function of the parameters.
 Parameters
observed – Observed value as np.array
predicted – Predicted value as np.array
 Returns
Log likelihood under Poisson distribution.
>>> observed = [3, 2, 3, 4, 6, 2, 1] >>> predicted = [1.8, 2.2, 3.4, 4.6, 6.7, 5.1, 3.2] >>> print('{:.3f}'.format(log_likelihood_poisson(observed, predicted))) 1.950
 irfpy.util.likelihood.aic(number_of_parameter, log_likelihood)[source]¶
Return the AIC value
The AIC (https://en.wikipedia.org/wiki/Akaike_information_criterion) is calculated and returned.
\[AIC = 2 k  2 \ln(L)\]If your observations are
>>> observed = [3, 2, 3, 4, 6, 2, 1]
and the best fitted values are
>>> predicted = [1.8, 2.2, 3.4, 4.6, 6.7, 5.1, 3.2]
the loglikelihood calculate from Poisson distribution is obtained as
>>> print('{:.3f}'.format(log_likelihood_poisson(observed, predicted))) 1.950
Then, if the number of parameters used for the best fitted values are 3, the AIC is simply obtained as
>>> print('{:.3f}'.format(aic(3, log_likelihood_poisson(observed, predicted)))) 2.100