def all_info_measures(vars):
"""
"""
for stuff in islice(powerset(vars), 1, None):
others = set(vars) - set(stuff)
for part in partitions(stuff, tuples=True):
for cond in powerset(others):
yield (part, cond)
def all_info_measures(vars):
"""
"""
for stuff in islice(powerset(vars), 1, None):
others = set(vars) - set(stuff)
for part in partitions(stuff, tuples=True):
for cond in powerset(others):
yield (part , cond)
def poset_lattice(elements):
"""
Return the Hasse diagram of the lattice induced by `elements`.
"""
child = lambda a, b: a.issubset(b) and (len(b) - len(a) == 1)
lattice = DiGraph()
for a in powerset(elements):
for b in powerset(elements):
if child(set(a), set(b)):
lattice.add_edge(b, a)
return lattice
def poset_lattice(elements):
"""
Return the Hasse diagram of the lattice induced by `elements`.
"""
child = lambda a, b: a.issubset(b) and (len(b) - len(a) == 1)
lattice = DiGraph()
for a in powerset(elements):
for b in powerset(elements):
if child(set(a), set(b)):
lattice.add_edge(b, a)
return lattice
def all_dist_structures(outcome_length, alphabet_size):
"""
Return an iterator of distributions over the
2**(`alphabet_size`**`outcome_length`) possible combinations of joint
events.
Parameters
----------
outcome_length : int
The length of outcomes to consider.
alphabet_length : int
The size of the alphabet for each random variable.
Yields
------
d : Distribution
A uniform distribution over a subset of the possible joint events.
"""
alphabet = ''.join(str(i) for i in range(alphabet_size))
words = product(alphabet, repeat=outcome_length)
topologies = powerset(words)
next(topologies) # the first element is the null set
for t in topologies:
outcomes = [''.join(_) for _ in t]
yield uniform(outcomes)
def test_quartet_set_representable_tree_sets(self):
"""
Sloane sequence does not exist.
The extended sequence is [4, 41, 1586, ???].
"""
expected = [4, 41]
Ns = range(4, 4 + len(expected))
observed = []
for N in Ns:
# get all quartets
qs = list(get_quartets(N))
# get all resolved trees
ts = list(get_bifurcating_trees(N))
# map from quartet to quartet index
q_to_i = dict((q, i) for i, q in enumerate(qs))
# map tree index to set of indices of compatible quartets
ti_to_qi_set = dict((i, set(q_to_i[q] for q in resolved_to_quartets(t))) for i, t in enumerate(ts))
# begin the definition of representable tree index sets
representable_ti_sets = set()
# For each tree look at every subset of compatible quartets.
# Record the set of tree indices compatible with each subset.
for t in ts:
tqis = [q_to_i[q] for q in resolved_to_quartets(t)]
for qi_subset_tuple in iterutils.powerset(tqis):
qi_subset = set(qi_subset_tuple)
#ti_pattern = frozenset(i for i, tree in enumerate(ts) if
ti_pattern = frozenset(i for i, s in ti_to_qi_set.items() if qi_subset <= s)
representable_ti_sets.add(ti_pattern)
n = len(representable_ti_sets)
observed.append(n)
self.assertEqual(observed, expected)
def get_barrier(R):
"""
Return the subset of vertices on one side of a strong barrier.
A strong barrier is one that minimizes the ratio of the
flow across the barrier to the logical entropy of the partition.
An alternative characterization is that this barrier
minimizes the randomization rate of the corresponding
Markov-ized 2-state process.
@param R: general reversible rate matrix
@return: a vertex subset
"""
n = len(R)
v = mrate.R_to_distn(R)
best_subset = None
best_ratio = None
for A_tuple in iterutils.powerset(range(n)):
# define the vertex set and its complement
A = set(A_tuple)
B = set(range(n)) - A
Pa = sum(v[i] for i in A)
Pb = sum(v[i] for i in B)
if Pa and Pb:
flow = sum(v[i]*R[i,j] for i, j in product(A, B))
ratio = flow / (Pa * Pb)
if (best_ratio is None) or (ratio < best_ratio):
best_ratio = ratio
best_subset = A
print best_ratio
return set(best_subset)
def all_dist_structures(outcome_length, alphabet_size):
"""
Return an iterator of distributions over the
2**(`alphabet_size`**`outcome_length`) possible combinations of joint
events.
Parameters
----------
outcome_length : int
The length of outcomes to consider.
alphabet_length : int
The size of the alphabet for each random variable.
Yields
------
d : Distribution
A uniform distribution over a subset of the possible joint events.
"""
alphabet = ''.join(str(i) for i in range(alphabet_size))
words = product(alphabet, repeat=outcome_length)
topologies = powerset(words)
next(topologies) # the first element is the null set
for t in topologies:
outcomes = [''.join(_) for _ in t]
yield uniform(outcomes)
def test_quartet_set_representable_tree_sets(self):
"""
Sloane sequence does not exist.
The extended sequence is [4, 41, 1586, ???].
"""
expected = [4, 41]
Ns = range(4, 4 + len(expected))
observed = []
for N in Ns:
# get all quartets
qs = list(get_quartets(N))
# get all resolved trees
ts = list(get_bifurcating_trees(N))
# map from quartet to quartet index
q_to_i = dict((q, i) for i, q in enumerate(qs))
# map tree index to set of indices of compatible quartets
ti_to_qi_set = dict(
(i, set(q_to_i[q] for q in resolved_to_quartets(t)))
for i, t in enumerate(ts))
# begin the definition of representable tree index sets
representable_ti_sets = set()
# For each tree look at every subset of compatible quartets.
# Record the set of tree indices compatible with each subset.
for t in ts:
tqis = [q_to_i[q] for q in resolved_to_quartets(t)]
for qi_subset_tuple in iterutils.powerset(tqis):
qi_subset = set(qi_subset_tuple)
#ti_pattern = frozenset(i for i, tree in enumerate(ts) if
ti_pattern = frozenset(i for i, s in ti_to_qi_set.items()
if qi_subset <= s)
representable_ti_sets.add(ti_pattern)
n = len(representable_ti_sets)
observed.append(n)
self.assertEqual(observed, expected)
def test_join_sigalg():
""" Test join_sigalg """
outcomes = ['00', '01', '10', '11']
pmf = [1/4]*4
d = Distribution(outcomes, pmf)
sigalg = frozenset([frozenset(_) for _ in powerset(outcomes)])
joined = join_sigalg(d, [[0], [1]])
assert_equal(sigalg, joined)
def test_tse1():
""" Test identity comparing TSE to B from Olbrich's talk """
for i, j in zip(range(3, 6), range(2, 5)):
d = n_mod_m(i, j)
indices = [[k] for k in range(i)]
tse = TSE(d)
x = 1/2 * sum(B(d, rv)/nCk(i, len(rv)) for rv in powerset(indices))
yield assert_almost_equal, tse, x
def coinformation(dist, rvs=None, crvs=None, rv_names=None):
"""
Calculates the coinformation.
Parameters
----------
dist : Distribution
The distribution from which the coinformation is calculated.
rvs : list, None
The indexes of the random variable used to calculate the coinformation
between. If None, then the coinformation is calculated over all random
variables.
crvs : list, None
The indexes of the random variables to condition on. If None, then no
variables are condition on.
rv_names : bool
If `True`, then the elements of `rvs` are treated as random variable
names. If `False`, then the elements of `rvs` are treated as random
variable indexes. If `None`, then the value `True` is used if the
distribution has specified names for its random variables.
Returns
-------
I : float
The coinformation.
Raises
------
ditException
Raised if `dist` is not a joint distribution.
"""
if dist.is_joint():
if rvs is None:
# Set to entropy of entire distribution
rvs = [ [i] for i in range(dist.outcome_length()) ]
rv_names = False
if crvs is None:
crvs = []
else:
msg = "The coinformation is applicable to joint distributions."
raise ditException(msg)
def entropy(rvs, dist=dist, crvs=crvs, rv_names=rv_names):
return H(dist, set().union(*rvs), crvs, rv_names)
I = sum( (-1)**(len(Xs)+1) * entropy(Xs) for Xs in powerset(rvs) )
return I
def _get_cheeger_constant(R, v):
"""
This is also known as the second isoperimetric constant.
@param R: a reversible rate matrix
@param v: stationary distribution
@return: the second isoperimetric constant
"""
n = len(v)
I2 = None
for A_tuple in iterutils.powerset(range(n)):
# define the vertex set and its complement
A = set(A_tuple)
B = set(range(n)) - A
A_measure = sum(v[i] for i in A)
B_measure = sum(v[i] for i in B)
if A_measure and B_measure:
boundary_measure = sum(v[i]*R[i,j] for i, j in product(A, B))
A_connectivity = boundary_measure / A_measure
B_connectivity = boundary_measure / B_measure
connectivity = max(A_connectivity, B_connectivity)
if I2 is None or connectivity < I2:
I2 = connectivity
return I2
def _get_cheeger_constant(R, v):
"""
This is also known as the second isoperimetric constant.
@param R: a reversible rate matrix
@param v: stationary distribution
@return: the second isoperimetric constant
"""
n = len(v)
I2 = None
for A_tuple in iterutils.powerset(range(n)):
# define the vertex set and its complement
A = set(A_tuple)
B = set(range(n)) - A
A_measure = sum(v[i] for i in A)
B_measure = sum(v[i] for i in B)
if A_measure and B_measure:
boundary_measure = sum(v[i] * R[i, j] for i, j in product(A, B))
A_connectivity = boundary_measure / A_measure
B_connectivity = boundary_measure / B_measure
connectivity = max(A_connectivity, B_connectivity)
if I2 is None or connectivity < I2:
I2 = connectivity
return I2
def get_real_cheeger(Q):
"""
Get the real cheeger constant.
This is not a bound or an approximation.
And it is NP hard to compute.
"""
n = len(Q)
all_vertices = set(range(n))
min_ratio = None
for vertices in iterutils.powerset(range(n)):
if not (0 < len(vertices) <= n / 2):
continue
vset = set(vertices)
complement = all_vertices - vset
volume = len(vset)
boundary_size = 0
for v in vset:
for x in complement:
if Q[v, x]:
boundary_size += 1
ratio = boundary_size / float(volume)
if (min_ratio is None) or (ratio < min_ratio):
min_ratio = ratio
return min_ratio
def get_real_cheeger(Q):
"""
Get the real cheeger constant.
This is not a bound or an approximation.
And it is NP hard to compute.
"""
n = len(Q)
all_vertices = set(range(n))
min_ratio = None
for vertices in iterutils.powerset(range(n)):
if not (0 < len(vertices) <= n/2):
continue
vset = set(vertices)
complement = all_vertices - vset
volume = len(vset)
boundary_size = 0
for v in vset:
for x in complement:
if Q[v, x]:
boundary_size += 1
ratio = boundary_size / float(volume)
if (min_ratio is None) or (ratio < min_ratio):
min_ratio = ratio
return min_ratio
def coinformation(dist, rvs=None, crvs=None, rv_mode=None):
"""
Calculates the coinformation.
Parameters
----------
dist : Distribution
The distribution from which the coinformation is calculated.
rvs : list, None
The indexes of the random variable used to calculate the coinformation
between. If None, then the coinformation is calculated over all random
variables.
crvs : list, None
The indexes of the random variables to condition on. If None, then no
variables are condition on.
rv_mode : str, None
Specifies how to interpret `rvs` and `crvs`. Valid options are:
{'indices', 'names'}. If equal to 'indices', then the elements of
`crvs` and `rvs` are interpreted as random variable indices. If equal
to 'names', the the elements are interpreted as random variable names.
If `None`, then the value of `dist._rv_mode` is consulted, which
defaults to 'indices'.
Returns
-------
I : float
The coinformation.
Raises
------
ditException
Raised if `dist` is not a joint distribution or if `rvs` or `crvs`
contain non-existant random variables.
Examples
--------
Let's construct a 3-variable distribution for the XOR logic gate and name
the random variables X, Y, and Z.
>>> d = dit.example_dists.Xor()
>>> d.set_rv_names(['X', 'Y', 'Z'])
To calculate coinformations, recall that `rvs` specifies which groups of
random variables are involved. For example, the 3-way mutual information
I[X:Y:Z] is calculated as:
>>> dit.multivariate.coinformation(d, ['X', 'Y', 'Z'])
-1.0
It is a quirk of strings that each element of a string is also an iterable.
So an equivalent way to calculate the 3-way mutual information I[X:Y:Z] is:
>>> dit.multivariate.coinformation(d, 'XYZ')
-1.0
The reason this works is that list('XYZ') == ['X', 'Y', 'Z']. If we want
to use random variable indexes, we need to have explicit groupings:
>>> dit.multivariate.coinformation(d, [[0], [1], [2]], rv_mode='indexes')
-1.0
To calculate the mutual information I[X, Y : Z], we use explicit groups:
>>> dit.multivariate.coinformation(d, ['XY', 'Z'])
Using indexes, this looks like:
>>> dit.multivariate.coinformation(d, [[0, 1], [2]], rv_mode='indexes')
The mutual information I[X:Z] is given by:
>>> dit.multivariate.coinformation(d, 'XZ')
0.0
Equivalently,
>>> dit.multivariate.coinformation(d, ['X', 'Z'])
0.0
Using indexes, this becomes:
>>> dit.multivariate.coinformation(d, [[0], [2]])
0.0
Conditional mutual informations can be calculated by passing in the
conditional random variables. The conditional entropy I[X:Y|Z] is:
>>> dit.multivariate.coinformation(d, 'XY', 'Z')
1.0
Using indexes, this becomes:
>>> rvs = [[0], [1]]
>>> crvs = [[2]] # broken
>>> dit.multivariate.coinformation(d, rvs, crvs, rv_mode='indexes')
1.0
For the conditional random variables, groupings have no effect, so you
can also obtain this as:
>>> rvs = [[0], [1]]
>>> crvs = [2]
>>> dit.multivariate.coinformation(d, rvs, crvs, rv_mode='indexes')
1.0
Finally, note that entropy can also be calculated. The entropy H[Z|XY]
is obtained as:
>>> rvs = [[2]]
>>> crvs = [[0], [1]] # broken
>>> dit.multivariate.coinformation(d, rvs, crvs, rv_mode='indexes')
0.0
>>> crvs = [[0, 1]] # broken
>>> dit.multivariate.coinformation(d, rvs, crvs, rv_mode='indexes')
0.0
>>> crvs = [0, 1]
>>> dit.multivariate.coinformation(d, rvs, crvs, rv_mode='indexes')
0.0
>>> rvs = 'Z'
>>> crvs = 'XY'
>>> dit.multivariate.coinformation(d, rvs, crvs, rv_mode='indexes')
0.0
Note that [[0], [1]] says to condition on two groups. But conditioning
is a flat operation and doesn't respect the groups, so it is equal to
a single group of 2 random variables: [[0, 1]]. With random variable
names 'XY' is acceptable because list('XY') = ['X', 'Y'], which is
species two singleton groups. By the previous argument, this is will
be treated the same as ['XY'].
"""
rvs, crvs, rv_mode = normalize_rvs(dist, rvs, crvs, rv_mode)
def entropy(rvs, dist=dist, crvs=crvs, rv_mode=rv_mode):
"""
Helper function to aid in computing the entropy of subsets.
"""
return H(dist, set().union(*rvs), crvs, rv_mode=rv_mode)
I = sum((-1)**(len(Xs)+1) * entropy(Xs) for Xs in powerset(rvs))
return I
def coinformation(dist, rvs=None, crvs=None, rv_mode=None):
"""
Calculates the coinformation.
Parameters
----------
dist : Distribution
The distribution from which the coinformation is calculated.
rvs : list, None
The indexes of the random variable used to calculate the coinformation
between. If None, then the coinformation is calculated over all random
variables.
crvs : list, None
The indexes of the random variables to condition on. If None, then no
variables are condition on.
rv_mode : str, None
Specifies how to interpret `rvs` and `crvs`. Valid options are:
{'indices', 'names'}. If equal to 'indices', then the elements of
`crvs` and `rvs` are interpreted as random variable indices. If equal
to 'names', the the elements are interpreted as random variable names.
If `None`, then the value of `dist._rv_mode` is consulted, which
defaults to 'indices'.
Returns
-------
I : float
The coinformation.
Raises
------
ditException
Raised if `dist` is not a joint distribution or if `rvs` or `crvs`
contain non-existant random variables.
Examples
--------
Let's construct a 3-variable distribution for the XOR logic gate and name
the random variables X, Y, and Z.
>>> d = dit.example_dists.Xor()
>>> d.set_rv_names(['X', 'Y', 'Z'])
To calculate coinformations, recall that `rvs` specifies which groups of
random variables are involved. For example, the 3-way mutual information
I[X:Y:Z] is calculated as:
>>> dit.multivariate.coinformation(d, ['X', 'Y', 'Z'])
-1.0
It is a quirk of strings that each element of a string is also an iterable.
So an equivalent way to calculate the 3-way mutual information I[X:Y:Z] is:
>>> dit.multivariate.coinformation(d, 'XYZ')
-1.0
The reason this works is that list('XYZ') == ['X', 'Y', 'Z']. If we want
to use random variable indexes, we need to have explicit groupings:
>>> dit.multivariate.coinformation(d, [[0], [1], [2]], rv_mode='indexes')
-1.0
To calculate the mutual information I[X, Y : Z], we use explicit groups:
>>> dit.multivariate.coinformation(d, ['XY', 'Z'])
Using indexes, this looks like:
>>> dit.multivariate.coinformation(d, [[0, 1], [2]], rv_mode='indexes')
The mutual information I[X:Z] is given by:
>>> dit.multivariate.coinformation(d, 'XZ')
0.0
Equivalently,
>>> dit.multivariate.coinformation(d, ['X', 'Z'])
0.0
Using indexes, this becomes:
>>> dit.multivariate.coinformation(d, [[0], [2]])
0.0
Conditional mutual informations can be calculated by passing in the
conditional random variables. The conditional entropy I[X:Y|Z] is:
>>> dit.multivariate.coinformation(d, 'XY', 'Z')
1.0
Using indexes, this becomes:
>>> rvs = [[0], [1]]
>>> crvs = [[2]] # broken
>>> dit.multivariate.coinformation(d, rvs, crvs, rv_mode='indexes')
1.0
For the conditional random variables, groupings have no effect, so you
can also obtain this as:
>>> rvs = [[0], [1]]
>>> crvs = [2]
>>> dit.multivariate.coinformation(d, rvs, crvs, rv_mode='indexes')
1.0
Finally, note that entropy can also be calculated. The entropy H[Z|XY]
is obtained as:
>>> rvs = [[2]]
>>> crvs = [[0], [1]] # broken
>>> dit.multivariate.coinformation(d, rvs, crvs, rv_mode='indexes')
0.0
>>> crvs = [[0, 1]] # broken
>>> dit.multivariate.coinformation(d, rvs, crvs, rv_mode='indexes')
0.0
>>> crvs = [0, 1]
>>> dit.multivariate.coinformation(d, rvs, crvs, rv_mode='indexes')
0.0
>>> rvs = 'Z'
>>> crvs = 'XY'
>>> dit.multivariate.coinformation(d, rvs, crvs, rv_mode='indexes')
0.0
Note that [[0], [1]] says to condition on two groups. But conditioning
is a flat operation and doesn't respect the groups, so it is equal to
a single group of 2 random variables: [[0, 1]]. With random variable
names 'XY' is acceptable because list('XY') = ['X', 'Y'], which is
species two singleton groups. By the previous argument, this is will
be treated the same as ['XY'].
"""
rvs, crvs, rv_mode = normalize_rvs(dist, rvs, crvs, rv_mode)
def entropy(rvs, dist=dist, crvs=crvs, rv_mode=rv_mode):
"""
Helper function to aid in computing the entropy of subsets.
"""
return H(dist, set().union(*rvs), crvs, rv_mode=rv_mode)
I = sum((-1)**(len(Xs) + 1) * entropy(Xs) for Xs in powerset(rvs))
return I