def estimate_i_alpha(y, co):
""" Estimate i_alpha = \int p^{\alpha}(y)dy.
The Renyi and Tsallis entropies are simple functions of this quantity.
Parameters
----------
y : (number of samples, dimension)-ndarray
One row of y corresponds to one sample.
co : cost object; details below.
co.knn_method : str
kNN computation method; 'cKDTree' or 'KDTree'.
co.k : int, >= 1
k-nearest neighbors.
co.eps : float, >= 0
the k^th returned value is guaranteed to be no further than
(1+eps) times the distance to the real kNN.
co.alpha : float
alpha in the definition of i_alpha
Returns
-------
i_alpha : float
Estimated i_alpha value.
Examples
--------
i_alpha = estimate_i_alpha(y,co)
"""
num_of_samples, dim = y.shape
distances_yy = knn_distances(y, y, True, co.knn_method, co.k, co.eps,
2)[0]
v = volume_of_the_unit_ball(dim)
# Solution-1 (normal k):
c = (gamma(co.k)/gamma(co.k + 1 - co.alpha))**(1 / (1 - co.alpha))
# Solution-2 (if k is 'extreme large', say self.k=180 [ =>
# gamma(self.k)=inf], then use this alternative form of
# 'c', after importing gammaln). Note: we used the
# 'gamma(a) / gamma(b) = exp(gammaln(a) - gammaln(b))'
# identity.
# c = exp(gammaln(co.k) - gammaln(co.k+1-co.alpha))**(1 / (1-co.alpha))
s = sum(distances_yy[:, co.k-1]**(dim * (1 - co.alpha)))
i_alpha = \
(num_of_samples - 1) / num_of_samples * v**(1 - co.alpha) * \
c**(1 - co.alpha) * s / (num_of_samples - 1)**co.alpha
return i_alpha
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