def analytical_value_k_ejr1(distr1, distr2, u, par1, par2):
""" Analytical value of the Jensen-Renyi kernel-1.
Parameters
----------
distr1, distr2 : str-s
Names of distributions.
u : float, >0
Parameter of the Jensen-Renyi kernel-1 (alpha = 2: fixed).
par1, par2 : dictionary-s
Parameters of distributions. If (distr1, distr2) =
('normal', 'normal'), then both distributions are normal:
distr1 = N(m1,s1^2 I) with m1 = par1['mean'], s1 =
par1['std'], distr2 = N(m2,s2^2 I) with m2 =
par2['mean'], s2 = par2['std'].
References
----------
Fei Wang, Tanveer Syeda-Mahmood, Baba C. Vemuri, David Beymer, and
Anand Rangarajan. Closed-Form Jensen-Renyi Divergence for Mixture of
Gaussians and Applications to Group-Wise Shape Registration. Medical
Image Computing and Computer-Assisted Intervention, 12: 648–655, 2009.
"""
if distr1 == 'normal' and distr2 == 'normal':
m1, s1 = par1['mean'], par1['std']
m2, s2 = par2['mean'], par2['std']
w = array([1/2, 1/2])
h = compute_h2(w, (m1, m2), (s1, s2)) # quadratic Renyi entropy
k = exp(-u * h)
else:
raise Exception('Distribution=?')
return k
def analytical_value_d_jensen_renyi(distr1, distr2, w, par1, par2):
""" Analytical value of the Jensen-Renyi divergence.
Parameters
----------
distr1, distr2 : str-s
Names of distributions.
w : vector, w[i] > 0 (for all i), sum(w) = 1
Weight used in the Jensen-Renyi divergence.
par1, par2 : dictionary-s
Parameters of distributions. If (distr1, distr2) =
('normal', 'normal'), then both distributions are normal:
distr1 = N(m1,s1^2 I) with m1 = par1['mean'], s1 =
par1['std'], distr2 = N(m2,s2^2 I) with m2 =
par2['mean'], s2 = par2['std'].
Returns
-------
d : float
Analytical value of the Jensen-Renyi divergence.
References
----------
Fei Wang, Tanveer Syeda-Mahmood, Baba C. Vemuri, David Beymer, and
Anand Rangarajan. Closed-Form Jensen-Renyi Divergence for Mixture of
Gaussians and Applications to Group-Wise Shape Registration. Medical
Image Computing and Computer-Assisted Intervention, 12: 648–655, 2009.
"""
if distr1 == 'normal' and distr2 == 'normal':
m1, s1 = par1['mean'], par1['std']
m2, s2 = par2['mean'], par2['std']
term1 = compute_h2(w, (m1, m2), (s1, s2))
term2 = \
w[0] * compute_h2((1,), (m1,), (s1,)) +\
w[1] * compute_h2((1,), (m2,), (s2,))
# H2(\sum_i wi yi) - \sum_i w_i H2(yi), where H2 is the quadratic
# Renyi entropy:
d = term1 - term2
else:
raise Exception('Distribution=?')
return d
def analytical_value_k_ejt2(distr1, distr2, u, par1, par2):
""" Analytical value of the Jensen-Tsallis kernel-2.
Parameters
----------
distr1, distr2 : str-s
Names of distributions.
u : float, >0
Parameter of the Jensen-Tsallis kernel-2 (alpha = 2: fixed).
par1, par2 : dictionary-s
Parameters of distributions. If (distr1, distr2) =
('normal', 'normal'), then both distributions are normal:
distr1 = N(m1,s1^2 I) with m1 = par1['mean'], s1 =
par1['std'], distr2 = N(m2,s2^2 I) with m2 =
par2['mean'], s2 = par2['std'].
References
----------
Fei Wang, Tanveer Syeda-Mahmood, Baba C. Vemuri, David Beymer, and
Anand Rangarajan. Closed-Form Jensen-Renyi Divergence for Mixture of
Gaussians and Applications to Group-Wise Shape Registration. Medical
Image Computing and Computer-Assisted Intervention, 12: 648–655, 2009.
(analytical value of the Jensen-Renyi divergence)
"""
if distr1 == 'normal' and distr2 == 'normal':
m1, s1 = par1['mean'], par1['std']
m2, s2 = par2['mean'], par2['std']
w = array([1/2, 1/2])
# quadratic Renyi entropy -> quadratic Tsallis entropy:
term1 = 1 - exp(-compute_h2(w, (m1, m2), (s1, s2)))
term2 = \
w[0] * (1 - exp(-compute_h2((1, ), (m1, ), (s1,)))) +\
w[1] * (1 - exp(-compute_h2((1,), (m2,), (s2,))))
# H2(\sum_i wi Yi) - \sum_i w_i H2(Yi), where H2 is the quadratic
# Tsallis entropy:
d = term1 - term2
k = exp(-u * d)
else:
raise Exception('Distribution=?')
return k