def anal_Am(np_pmf, pmf_index, sys_index, m):
""" Analytic A_m """
m_ind = len(pmf_index[0]) - 2 - m
if m_ind < 0:
print("require seq_len > 1+m")
return None
s_ind = pmf_index[sys_index][:, [-2, -1]].flatten()
e_ind = np.delete(pmf_index, sys_index, axis=0)[:, m_ind:-1].flatten()
se_ind = np.concatenate((s_ind, e_ind))
margin_index = np.delete(pmf_index.flatten(), se_ind)
margin_pmf = prob.numpy_margin(np_pmf, margin_index)
x_ind = [list(se_ind).index(x) for x in s_ind][0:len(sys_index)]
y_ind = [list(se_ind).index(x)
for x in s_ind][len(sys_index):2 * len(sys_index)]
# print("indices")
# print(pmf_index)
# print(se_ind)
# print(total_ind)
# inf.numpy_conditional_mutual_information()
return inf.numpy_conditional_mutual_information(margin_pmf, x_ind, y_ind)
def anal_step_MI(np_pmf, pmf_index, xind, yind):
""" I(X;Y) """
xy_index = list(itt.chain(*xind, *yind))
margin_index = np.delete(pmf_index.flatten(), xy_index)
margin_pmf = prob.numpy_margin(np_pmf, margin_index)
XIND = [list(xy_index).index(x) for x in list(itt.chain(*xind))]
YIND = [list(xy_index).index(x) for x in list(itt.chain(*yind))]
return inf.numpy_mutual_information(margin_pmf, XIND, YIND)
def anal_Astar(np_pmf, pmf_index, sys_index):
""" Analytic A* """
s_ind = pmf_index[sys_index][:, [-2, -1]].flatten()
margin_index = np.delete(pmf_index.flatten(), s_ind)
margin_pmf = prob.numpy_margin(np_pmf, margin_index)
x_ind = [list(s_ind).index(x) for x in s_ind][0:len(sys_index)]
y_ind = [list(s_ind).index(x)
for x in s_ind][len(sys_index):2 * len(sys_index)]
return inf.numpy_mutual_information(margin_pmf, x_ind, y_ind)
def anal_step_cond_MI(np_pmf, pmf_index, xind, yind, zind=()):
""" I(X;Y|Z), z optional """
xy_index = list(itt.chain(*xind, *yind, *zind))
margin_index = np.delete(pmf_index.flatten(), xy_index)
margin_pmf = prob.numpy_margin(np_pmf, margin_index)
XIND = [list(xy_index).index(x) for x in list(itt.chain(*xind))]
YIND = [list(xy_index).index(x) for x in list(itt.chain(*yind))]
if zind == ():
return inf.numpy_mutual_information(margin_pmf, XIND, YIND)
return inf.numpy_conditional_mutual_information(margin_pmf, XIND, YIND)
def numpy_conditional_mutual_information(np_pmf, x_index, y_index):
""" Conditional Mutual Information I(X;Y|Z)
Parameters
----------
np_pmf : array_like
A numpy array of the form [x1, ... , xn, p(x1, ... , xn)] with length n+1
x_index : list
index for X in p(X,Y,Z) for I(X;Y|Z)
y_index : list
index for Y in p(X,Y,Z) for I(X;Y|Z)
"""
var_index = [x_index, y_index]
for i, var in enumerate(var_index):
var_index[i] = [var] if isinstance(var, int) else var
# marginalizing
pmf_z, inverse_index_z = prob.numpy_margin(np_pmf,
[*var_index[0], *var_index[1]],
return_inverse=True)
pmf_xz, inverse_index_xz = prob.numpy_margin(np_pmf,
var_index[1],
return_inverse=True)
pmf_yz, inverse_index_yz = prob.numpy_margin(np_pmf,
var_index[0],
return_inverse=True)
CMI = 0
for i, p_xyz in enumerate(np_pmf[-1]):
p_z = pmf_z[-1][inverse_index_z[i]]
p_xz = pmf_xz[-1][inverse_index_xz[i]]
p_yz = pmf_yz[-1][inverse_index_yz[i]]
CMI += p_xyz * np.log2((p_z * p_xyz) / (p_xz * p_yz))
return CMI
def numpy_mutual_information(np_pmf, x_index, y_index):
""" Mutual information I(X;Y)
Parameters
----------
np_pmf : array_like
A numpy array of the form [x1, ... , xn, p(x1, ... , xn)] with length n+1
x_index : list
index for X in p(X,Y,Z) for I(X;Y|Z)
y_index : list
index for Y in p(X,Y,Z) for I(X;Y|Z)
Note
----
.. math:: I(X;Y) = \sum_{x,y} p_{X,Y}(x,y) \log \frac{p_{X,Y}(x,y)}{p_X(x) p_Y(y)} = H(X) - H(X \mid Y)
"""
# y_index = np.delete(np.arange(len(np_pmf)-1), x_index)
var_index = [x_index, y_index]
# marginalizing
pmf_x, inverse_index_x = prob.numpy_margin(np_pmf,
var_index[1],
return_inverse=True)
pmf_y, inverse_index_y = prob.numpy_margin(np_pmf,
var_index[0],
return_inverse=True)
MI = 0
for i, p_xy in enumerate(np_pmf[-1]):
p_x = pmf_x[-1][inverse_index_x[i]]
p_y = pmf_y[-1][inverse_index_y[i]]
MI += p_xy * np.log2(p_xy / (p_x * p_y))
return MI
def anal_non_heteronomy(np_pmf, pmf_index, sys_index, m):
""" Analytic H(S_n+1 | E_n,... E_n-m) """
m_ind = len(pmf_index[0]) - 2 - m
if m_ind < 0:
print("require seq_len > 1+m")
return None
s_ind = pmf_index[sys_index][:, -1].flatten()
e_ind = np.delete(pmf_index, sys_index, axis=0)[:, m_ind:-1].flatten()
se_ind = np.concatenate((s_ind, e_ind))
margin_index = np.delete(pmf_index.flatten(), se_ind)
margin_pmf = prob.numpy_margin(np_pmf, margin_index)
env_cond_index = [list(se_ind).index(x) for x in e_ind]
return inf.numpy_conditional_entropy(margin_pmf, env_cond_index)
def anal_IF(np_pmf, pmf_index, x_index):
""" Analytic IF(X->Y) = I(Y_{n+1}:X_{n}|Y_{n}) """
pmf_index = np.array(pmf_index)
X_n = pmf_index[x_index][:, -2].flatten()
Y_n = np.delete(pmf_index, x_index, axis=0)[:, -2].flatten()
Y_n1 = np.delete(pmf_index, x_index, axis=0)[:, -1].flatten()
flat_ind = np.sort(np.concatenate((X_n, Y_n, Y_n1)))
margin_index = np.delete(pmf_index.flatten(), flat_ind)
margin_pmf = prob.numpy_margin(np_pmf, margin_index)
x_ind = [list(flat_ind).index(x) for x in X_n]
y_ind = [list(flat_ind).index(y) for y in Y_n1]
return inf.numpy_conditional_mutual_information(margin_pmf, x_ind, y_ind)