def test_margins():
a = np.array([1])
m = margins(a)
assert_equal(len(m), 1)
m0 = m[0]
assert_array_equal(m0, np.array([1]))
a = np.array([[1]])
m0, m1 = margins(a)
expected0 = np.array([[1]])
expected1 = np.array([[1]])
assert_array_equal(m0, expected0)
assert_array_equal(m1, expected1)
a = np.arange(12).reshape(2, 6)
m0, m1 = margins(a)
expected0 = np.array([[15], [51]])
expected1 = np.array([[6, 8, 10, 12, 14, 16]])
assert_array_equal(m0, expected0)
assert_array_equal(m1, expected1)
a = np.arange(24).reshape(2, 3, 4)
m0, m1, m2 = margins(a)
expected0 = np.array([[[66]], [[210]]])
expected1 = np.array([[[60], [92], [124]]])
expected2 = np.array([[[60, 66, 72, 78]]])
assert_array_equal(m0, expected0)
assert_array_equal(m1, expected1)
assert_array_equal(m2, expected2)
def test_margins():
a = np.array([1])
m = margins(a)
assert_equal(len(m), 1)
m0 = m[0]
assert_array_equal(m0, np.array([1]))
a = np.array([[1]])
m0, m1 = margins(a)
expected0 = np.array([[1]])
expected1 = np.array([[1]])
assert_array_equal(m0, expected0)
assert_array_equal(m1, expected1)
a = np.arange(12).reshape(2, 6)
m0, m1 = margins(a)
expected0 = np.array([[15], [51]])
expected1 = np.array([[6, 8, 10, 12, 14, 16]])
assert_array_equal(m0, expected0)
assert_array_equal(m1, expected1)
a = np.arange(24).reshape(2, 3, 4)
m0, m1, m2 = margins(a)
expected0 = np.array([[[66]], [[210]]])
expected1 = np.array([[[60], [92], [124]]])
expected2 = np.array([[[60, 66, 72, 78]]])
assert_array_equal(m0, expected0)
assert_array_equal(m1, expected1)
assert_array_equal(m2, expected2)
def compute_mi(cov_xy=0.5, n_bins=100):
"""Analytic computation of MI using binned
2D Gaussian
Arguments:
cov_xy (list): Off-diagonal elements of covariance
matrix
n_bins (int): Number of bins to "quantize" the
continuous 2D Gaussian
"""
cov = [[1, cov_xy], [cov_xy, 1]]
data = sample(cov=cov)
# get joint distribution samples
# perform histogram binning
joint, edge = np.histogramdd(data, bins=n_bins)
joint /= joint.sum()
eps = np.finfo(float).eps
joint[joint < eps] = eps
# compute marginal distributions
x, y = margins(joint)
xy = x * y
xy[xy < eps] = eps
# MI is P(X,Y)*log(P(X,Y)/P(X)*P(Y))
mi = joint * np.log(joint / xy)
mi = mi.sum()
print("Computed MI: %0.6f" % mi)
return mi
def marginal(dist, dim):
"""
Compute marginal of distribution dist along axis dim.
Note:
Computes all marginals and returns the one asked, so might be slow
"""
ms = margins(dist) # compute all marginals
return np.squeeze(ms[dim]) # pick out the desired one
def stdres(observed, expected):
n = observed.sum()
rsum, csum = margins(observed)
# With integers, the calculation
# csum * rsum * (n - rsum) * (n - csum)
# might overflow, so convert rsum and csum to floating point.
rsum = rsum.astype(np.float64)
csum = csum.astype(np.float64)
v = csum * rsum * (n - rsum) * (n - csum) / n**3
return (observed - expected) / np.sqrt(v)
def compute_mi(cov_xy=0.9, n_bins=100):
cov = [[1, cov_xy], [cov_xy, 1]]
data = sample(cov=cov)
joint, edge = np.histogramdd(data, bins=n_bins)
joint /= joint.sum()
eps = np.finfo(float).eps
joint[joint < eps] = eps
x, y = margins(joint)
xy = x * y
xy[xy < eps] = eps
mi = joint * np.log(joint / xy)
mi = mi.sum()
print("Computed MI:", mi)
return mi
def std_res(observed, expected):
"""
:param observed: an 2-by-n numpy array containing the observed frequencies
:param expected: an 2-by-n numpy array containing the expected frequencies under the null hypothesis
:return res: the standardized Pearson's residuals indicating which dimensions in the observed data show
the stronger deviation from the expected frequencies
"""
n = observed.sum()
rsum, csum = contingency.margins(observed)
v = csum * rsum * (n - rsum) * (n - csum) / float(n**3)
res = (observed - expected) / np.sqrt(v)
return res
def compute_mi(cov_xy=0.5, n_bins=100):
cov=[[1, cov_xy], [cov_xy, 1]]
data = sample(cov=cov)
# get joint distribution samples
# perform histogram binning
joint, edge = np.histogramdd(data, bins=n_bins)
joint /= joint.sum()
eps = np.finfo(float).eps
joint[joint<eps] = eps
# compute marginal distributions
x, y = margins(joint)
xy = x*y
xy[xy<eps] = eps
# MI is P(X,Y)*log(P(X,Y)/P(X)*P(Y))
mi = joint*np.log(joint/xy)
mi = mi.sum()
print("Computed MI: %0.6f" % mi)
return mi
def uncertainty(j):
"""
Use the Shannon entropy of a marginal distribution and Shannon entropy of the joint distribution
to calculate nonlinear dependence using our uncertainty of a probability distribution. The joint
distribution j between X and Y must be known as a 2-D array.
"""
x, y = margins(j) # x and y describe the joint probability margins (marginal distribution)
Hx = 0 # Shannon entropy of the x marginal distribution
for i in x:
Hx += i*np.log(i)
Hx = -Hx # flip the sign for entropy
Hy = 0 # mutatis mutandis for y
for i in y:
Hy += i.np.log(i)
Hy = -Hy
Hxy = 0 # Shannon entropy of the joint distribution
for i in x:
for j in y:
Hxy += (i*j)*np.log(i*j)
Hxy = -Hxy
return Hx + Hy - Hxy # This can be a measure of mutual information from a joint probability distribution
def marginal(dist, dim):
"""Compute marginal of distribution dist along axis dim"""
# Note, this computes all marginals and returns the one asked, so this
# might be slow in some cases.
ms = margins(dist) # compute all marginals
return np.squeeze(ms[dim]) # get the right one
def stdres(observed, expected):
n = observed.sum()
rsum, csum = margins(observed)
v = csum * rsum * (n - rsum) * (n - csum) / n**3
return (observed - expected) / np.sqrt(v)
