def estimation(self, y1, y2):
""" Estimate symmetric Bregman distance.
Parameters
----------
y1 : (number of samples1, dimension)-ndarray
One row of y1 corresponds to one sample.
y2 : (number of samples2, dimension)-ndarray
One row of y2 corresponds to one sample.
Returns
-------
d : float
Estimated symmetric Bregman distance.
References
----------
Nikolai Leonenko, Luc Pronzato, and Vippal Savani. A class of
Renyi information estimators for multidimensional densities.
Annals of Statistics, 36(5):2153-2182, 2008.
Imre Csiszar. Generalized projections for non-negative functions.
Acta Mathematica Hungarica, 68:161-185, 1995.
Lev M. Bregman. The relaxation method of finding the common points
of convex sets and its application to the solution of problems in
convex programming. USSR Computational Mathematics and
Mathematical Physics, 7:200-217, 1967.
Examples
--------
d = co.estimation(y1,y2)
"""
# verification:
self.verification_equal_d_subspaces(y1, y2)
i_alpha_y1 = estimate_i_alpha(y1, self)
i_alpha_y2 = estimate_i_alpha(y2, self)
d_temp3_y1y2 = estimate_d_temp3(y1, y2, self)
d_temp3_y2y1 = estimate_d_temp3(y2, y1, self)
d = (i_alpha_y1 + i_alpha_y2 - d_temp3_y1y2 - d_temp3_y2y1) /\
(self.alpha - 1)
return d
def estimation(self, y):
""" Estimate Tsallis entropy.
Parameters
----------
y : (number of samples, dimension)-ndarray
One row of y corresponds to one sample.
Returns
-------
h : float
Estimated Tsallis entropy.
References
----------
Nikolai Leonenko, Luc Pronzato, and Vippal Savani. A class of
Renyi information estimators for multidimensional densities.
Annals of Statistics, 36(5):2153-2182, 2008.
Examples
--------
h = co.estimation(y)
"""
i_alpha = estimate_i_alpha(y, self)
h = (1 - i_alpha) / (self.alpha - 1)
return h
def estimation(self, y):
""" Estimate Renyi entropy.
Parameters
----------
y : (number of samples, dimension)-ndarray
One row of y corresponds to one sample.
Returns
-------
h : float
Estimated Renyi entropy.
References
----------
Nikolai Leonenko, Luc Pronzato, and Vippal Savani. A class of
Renyi information estimators for multidimensional densities.
Annals of Statistics, 36(5):2153-2182, 2008.
Joseph E. Yukich. Probability Theory of Classical Euclidean
Optimization Problems, Lecture Notes in Mathematics, 1998, vol.
1675.
Examples
--------
h = co.estimation(y)
"""
i_alpha = estimate_i_alpha(y, self)
h = log(i_alpha) / (1 - self.alpha)
return h
def estimation(self, y):
""" Estimate Sharma-Mittal entropy.
Parameters
----------
y : (number of samples, dimension)-ndarray
One row of y corresponds to one sample.
Returns
-------
h : float
Estimated Sharma-Mittal entropy.
References
----------
Nikolai Leonenko, Luc Pronzato, and Vippal Savani. A class of
Renyi information estimators for multidimensional densities.
Annals of Statistics, 36(5):2153-2182, 2008. (i_alpha estimation)
Joseph E. Yukich. Probability Theory of Classical Euclidean
Optimization Problems, Lecture Notes in Mathematics, 1998, vol.
1675. (i_alpha estimation)
Ethem Akturk, Baris Bagci, and Ramazan Sever. Is Sharma-Mittal
entropy really a step beyond Tsallis and Renyi entropies?
Technical report, 2007. http://arxiv.org/abs/cond-mat/0703277.
(Sharma-Mittal entropy)
Bhudev D. Sharma and Dharam P. Mittal. New nonadditive measures of
inaccuracy. Journal of Mathematical Sciences, 10:122-133, 1975.
(Sharma-Mittal entropy)
Examples
--------
h = co.estimation(y)
"""
i_alpha = estimate_i_alpha(y, self)
h = (i_alpha**((1 - self.beta) /
(1 - self.alpha)) - 1) / (1 - self.beta)
return h