def test_find():
"""
FUNCTION: find, in general.py.
"""
assert_array_equal(general.find(np.array([[True, False]])), np.array([[0]]))
assert_array_equal(general.find(np.array([[False, True, True, False]])), np.array([[1, 2]]))
assert_array_equal(general.find(np.array([[False, False]])), np.array([[]]))
def test_find():
"""
FUNCTION: find, in general.py.
"""
assert_array_equal(general.find(np.array([[True, False]])),
np.array([[0]]))
assert_array_equal(general.find(np.array([[False, True, True, False]])),
np.array([[1, 2]]))
assert_array_equal(general.find(np.array([[False, False]])),
np.array([[]]))
def minimum_spanning_tree(C1, C2):
"""
This function finds the minimum spanning tree using Prim's algorithm.
We assume that absent edges have 0 cost. To find the maximum spanning
tree, use -1*C.
We partition the nodes into those in U and those not in U.
closest[i] is the vertex in U that is closest to i in V-U.
lowcost[i] is the cost of the edge [i, closest[i]], or infinity if i has
been used.
For details see
- Aho, Hopcroft & Ullman 1983, "Data structures and algorithms",
p 237.
Parameters
----------
C1: Numpy matrix
C1[i,j] is the primary cost of connecting i to j.
C2: Numpy matrix
C2[i,j] is the (optional) secondary cost of connecting i to j, used
to break ties.
"""
n = C1.shape[0]
A = np.zeros((n,n))
closest = np.zeros((1,n))
used = np.zeros((1,n))
used[0,0] = 1
C1 = C1 + np.nan_to_num((C1 == 0) * np.Inf )
C2 = C2 + np.nan_to_num((C2 == 0) * np.Inf )
lowcost1 = C1[0,:]
lowcost2 = C2[0,:]
for i in range(1,n):
ks = find(np.array(lowcost1) == np.min(lowcost1))
k = ks[0, np.argmin(lowcost2[0, ks])]
k=int(k)
cll=int(closest[0,k])
#A[k, closest[0,k]] = 1
#A[closest[0,k], k] = 1
A[k, cll] = 1
A[cll, k] = 1
lowcost1[0,k] = np.nan_to_num(np.Inf)
lowcost2[0,k] = np.nan_to_num(np.Inf)
used[0,k] = 1
NU = find(used == 0)
for ji in range(0, NU.shape[1]):
j = NU[0, ji]
if C1[k, j] < lowcost1[0, j]:
lowcost1[0, j] = float(C1[k, j])
lowcost2[0, j] = float(C2[k, j])
closest[0, j] = float(k)
return A
def minimum_spanning_tree(C1, C2):
"""
This function finds the minimum spanning tree using Prim's algorithm.
We assume that absent edges have 0 cost. To find the maximum spanning
tree, use -1*C.
We partition the nodes into those in U and those not in U.
closest[i] is the vertex in U that is closest to i in V-U.
lowcost[i] is the cost of the edge [i, closest[i]], or infinity if i has
been used.
For details see
- Aho, Hopcroft & Ullman 1983, "Data structures and algorithms",
p 237.
Parameters
----------
C1: Numpy matrix
C1[i,j] is the primary cost of connecting i to j.
C2: Numpy matrix
C2[i,j] is the (optional) secondary cost of connecting i to j, used
to break ties.
"""
n = C1.shape[0]
A = np.zeros((n,n))
closest = np.zeros((1,n))
used = np.zeros((1,n))
used[0,0] = 1
C1 = C1 + np.nan_to_num((C1 == 0) * np.Inf )
C2 = C2 + np.nan_to_num((C2 == 0) * np.Inf )
lowcost1 = C1[0,:]
lowcost2 = C2[0,:]
for i in range(1,n):
ks = find(np.array(lowcost1) == np.min(lowcost1))
k = ks[0, np.argmin(lowcost2[0, ks])]
A[k, closest[0,k]] = 1
A[closest[0,k], k] = 1
lowcost1[0,k] = np.nan_to_num(np.Inf)
lowcost2[0,k] = np.nan_to_num(np.Inf)
used[0,k] = 1
NU = find(used == 0)
for ji in range(0, NU.shape[1]):
j = NU[0, ji]
if C1[k, j] < lowcost1[0, j]:
lowcost1[0, j] = float(C1[k, j])
lowcost2[0, j] = float(C2[k, j])
closest[0, j] = float(k)
return A
def triangulate(G, order):
"""
This function ensures that the input graph is triangulated (chordal),
i.e., every cycle of length > 3 has a chord. To find the maximal
cliques, we save each induced cluster (created by adding connecting
neighbors) that is not a subset of any previously saved cluster. (A
cluster is a complete, but not necessarily maximal, set of nodes.)
Parameters
----------
G: Numpy ndarray
G[i,j] = 1 iff there is an edge between node i and node j.
order: List
The order in which to eliminate the nodes.
"""
MG = G.copy()
"""Obtain the the number of nodes in the graph"""
n = G.shape[0]
eliminated = np.zeros((1,n))
cliques = []
for i in range(0,n):
"""Obtain the index of the next node to be eliminated"""
u = order[0,i]
U = find(eliminated == 0)
#nodes = np.intersect1d_nu(neighbours(G, u), U)#################################################################################################################################################
nodes = np.intersect1d(neighbours(G, u), U)
nodes = np.union1d(nodes, np.array([u]))
"""
Connect all uneliminated neighbours of the node to be eliminated
together.
"""
for i in nodes:
for j in nodes:
i=int(i)
j=int(j)
G[i, j] = 1
G = setdiag(G, 0)
u=int(u)
"""Mark the node as 'eliminated'"""
eliminated[0, u] = 1
"""
If the generated clique is a subset of an existing clique, then it is
not a maximal clique, so it is excluded from the list if cliques.
"""
exclude = False
for c in range(0, len(cliques)):
if issubset(nodes, np.array(cliques[c])):
exclude = True
break
if not exclude:
cliques.append(nodes)
return [G, cliques]
def triangulate(G, order):
"""
This function ensures that the input graph is triangulated (chordal),
i.e., every cycle of length > 3 has a chord. To find the maximal
cliques, we save each induced cluster (created by adding connecting
neighbors) that is not a subset of any previously saved cluster. (A
cluster is a complete, but not necessarily maximal, set of nodes.)
Parameters
----------
G: Numpy ndarray
G[i,j] = 1 iff there is an edge between node i and node j.
order: List
The order in which to eliminate the nodes.
"""
MG = G.copy()
"""Obtain the the number of nodes in the graph"""
n = G.shape[0]
eliminated = np.zeros((1,n))
cliques = []
for i in range(0,n):
"""Obtain the index of the next node to be eliminated"""
u = order[0,i]
U = find(eliminated == 0)
nodes = np.intersect1d_nu(neighbours(G, u), U)
nodes = np.union1d(nodes, np.array([u]))
"""
Connect all uneliminated neighbours of the node to be eliminated
together.
"""
for i in nodes:
for j in nodes:
G[i, j] = 1
G = setdiag(G, 0)
"""Mark the node as 'eliminated'"""
eliminated[0, u] = 1
"""
If the generated clique is a subset of an existing clique, then it is
not a maximal clique, so it is excluded from the list if cliques.
"""
exclude = False
for c in range(0, len(cliques)):
if issubset(nodes, np.array(cliques[c])):
exclude = True
break
if not exclude:
cliques.append(nodes)
return [G, cliques]
def marginalize_pot(self, onto, maximize=False):
"""
This method marginalizes (or maximizes) a discrete potential onto
a smaller domain.
Parameters
----------
onto: List
The list of nodes specifying the domain onto which to
marginalize.
maiximize: Bool
This value is false if the function must marginalize
the potential over a domain, and true if it must maximize
the potential over a domain.
"""
ns = np.zeros((1, np.max(self.domain) + 1))
ns[0, self.domain] = self.sizes
"""Marginalize the table"""
smallT = self.T
"""Determine which dimensions to sum/max over"""
sum_over = np.setdiff1d(np.array(self.domain), np.array(onto))
ndx = []
for i in sum_over:
temp = find(np.array([np.array(self.domain) == i]))
if temp.shape != (1, ):
ndx.append(temp[0, 0])
ndx = np.array(ndx)
maximizers = dict()
if maximize:
"""
Determine which variables to resulting argmax values will
be dependants on. These values are used for back tracking.
"""
dependants = np.setdiff1d(
np.array(self.domain[:]),
np.array(self.observed_domain[:])).squeeze().tolist()
if type(dependants) != list:
dependants = [dependants]
count = 0
for i in xrange(0, len(ndx)):
if ndx[i] < smallT.ndim:
"""
If this node is unobserved, save its backtracking info.
"""
if sum_over[count] not in self.observed_domain[:]:
"""Save backtracking information"""
if sum_over[count] in dependants:
dependants.remove(sum_over[count])
"""Determine which values maximized the array"""
argmax = np.argmax(smallT, ndx[i]).squeeze()
if argmax.shape == ():
argmax = np.array(argmax).tolist()
"""Save backtracking data"""
maximizers[sum_over[count]] = \
[dependants[:], argmax]
"""Maximize out the required dimensions"""
smallT = np.max(smallT, ndx[i])
"""Compensate for reduced dimensions of smallT"""
ndx = ndx - 1
count = count + 1
else:
for i in xrange(0, len(ndx)):
if ndx[i] < smallT.ndim:
"""Sum over the dimension ndx[i]"""
smallT = np.sum(smallT, ndx[i])
"""Compensate for reduced dimensions of smallT"""
ndx = ndx - 1
"""Create marginalized potential"""
smallpot = dpot(onto, ns[0, onto], smallT)
return [smallpot, maximizers]
def best_first_elim_order(G, node_sizes, stage=[]):
"""
This function greedily searches for an optimal elimination order.
Find an order in which to eliminate nodes from the graph in such a way
as to try and minimize the weight of the resulting triangulated graph.
The weight of a graph is the sum of the weights of each of its cliques;
the weight of a clique is the product of the weights of each of its
members; the weight of a node is the number of values it can take on.
Since this is an NP-hard problem, we use the following greedy heuristic:
At each step, eliminate that node which will result in the addition of
the least number of fill-in edges, breaking ties by choosing the node
that induces the lighest clique.
For details, see
- Kjaerulff, "Triangulation of graphs -- algorithms giving small total
state space", Univ. Aalborg tech report, 1990 (www.cs.auc.dk/~uk)
- C. Huang and A. Darwiche, "Inference in Belief Networks: A procedural
guide", Intl. J. Approx. Reasoning, 11, 1994
Parameters
----------
G: Numpy ndarray
G[i,j] = 1 iff there is an edge between node i and node j.
node_sizes: List
The node sizes, where ns[i] = the number of discrete values
node i can take on [1 if observed].
stage: List
stage[i] is a list of the nodes that must be eliminated at i'th
stage.
"""
"""Obtain the number of nodes in the graph"""
n = G.shape[0]
if stage == []:
stage = [range(0, n)]
MG = G.copy()
uneliminated = np.ones((1, n))
order = np.zeros((1, n))
t = 0
"""For each node in the graph"""
for i in range(0, n):
"""Find the indices of the unelminated elements"""
U = find(uneliminated == 1)
"""Find which nodes can be removed in this stage."""
#valid = np.intersect1d_nu(np.array(U), np.array([stage[t]]))###################################################################################################################################
valid = np.intersect1d(np.array(U), np.array([stage[t]]))
"""
Determine which of the valid nodes will add the least number of fill in
edges once eliminated. If 2 or more nodes add the same least number of
fill in edges, then choose the one that results in the lightest clique.
"""
min_fill = np.zeros((1, len(valid)))
min_weight = np.zeros((1, len(valid)))
"""For each node that is valid for elimination"""
for j in range(0, len(valid)):
k = valid[j]
"""Obtain the uneliminated neighbours of the node to be eliminated"""
nbrs = neighbours(G, k)
#nbrs = np.intersect1d_nu(np.array([nbrs]), np.array(U))####################################################################################################################################
nbrs = np.intersect1d(np.array([nbrs]), np.array(U))
l = len(nbrs)
M = np.zeros((l, l))
count = 0
for x in nbrs:
for y in range(0, len(nbrs)):
M[count, y] = MG[x, nbrs[y]]
count = count + 1
"""Save the number of fill-in edges required to eliminate node j"""
min_fill[0, j] = l**2 - np.sum(M)
nbrs = nbrs.tolist()
nbrs.insert(0, k)
"""Save the clique weight obtained by eliminating node j"""
min_weight[0, j] = np.prod(node_sizes[0, nbrs])
"""Determine which of the nodes create the lightest clique."""
lightest_nbrs = find(min_weight == np.min(min_weight))
"""
Determine which of nodes found in the step above, require the least
number of fill-in edges to eliminate.
"""
best_nbr_ndx = np.argmin(min_fill[0, lightest_nbrs.tolist()])
j = lightest_nbrs[0, best_nbr_ndx]
"""
Determine which of the nodes found in the step above are valid for
elimination, these are the nodes to be eliminated.
"""
k = valid[j]
uneliminated[0, k] = 0
"""Add the nodes to be eliminated to the elimination order"""
order[0, i] = k
"""Determine the nieghbours of the nodes to be eliminated"""
ns = neighbours(G, k)
#ns = np.intersect1d_nu(np.array([ns]), np.array(U))############################################################################################################################################
ns = np.intersect1d(np.array([ns]), np.array(U))
"""Eliminate the nodes"""
if len(ns) != 0:
for x in ns:
for y in ns:
G[x, y] = 1
G = setdiag(G, 0)
"""
If all the nodes valid for elimination in this stage have been
eliminated, then advance to the next stage.
"""
if np.sum(np.abs(uneliminated[0, stage[t]])) == 0:
t = t + 1
return order
def marginalize_pot(self, onto, maximize=False, evidence=[]):
"""
This method marginalizes (or maximizes) a discrete potential onto
a smaller domain.
Parameters
----------
onto: List
The list of nodes specifying the domain onto which to
marginalize.
maiximize: Bool
This value is false if the function must marginalize
the potential over a domain, and true if it must maximize
the potential over a domain.
"""
ns = np.zeros(np.max(self.domain)+1)
ns[self.domain] = self.sizes
"""Marginalize the table"""
smallT = self.T
"""Determine which dimensions to sum/max over"""
sum_over = np.setdiff1d(np.array(self.domain), np.array(onto))
ndx = []
for i in sum_over:
temp = find(np.array([np.array(self.domain) == i]))
if temp.shape != (1,):
ndx.append(temp[0,0])
ndx = np.array(ndx)
maximizers = dict()
if maximize:
"""
Determine which variables to resulting argmax values will
be dependants on. These values are used for back tracking.
"""
dependants = np.setdiff1d(np.array(self.domain[:]),
np.array(self.observed_domain[:])).squeeze().tolist()
if type(dependants) != list:
dependants = [dependants]
count = 0
for i in xrange(0, len(ndx)):
if ndx[i]<smallT.ndim:
"""
If this node is unobserved, save its backtracking info.
"""
if sum_over[count] not in self.observed_domain[:]:
"""Save backtracking information"""
if sum_over[count] in dependants:
dependants.remove(sum_over[count])
"""Determine which values maximized the array"""
argmax = np.argmax(smallT, ndx[i]).squeeze()
if argmax.shape == ():
argmax = np.array(argmax).tolist()
"""Save backtracking data"""
maximizers[sum_over[count]] = \
[dependants[:], argmax]
"""Maximize out the required dimensions"""
smallT = np.max(smallT, ndx[i])
"""Compensate for reduced dimensions of smallT"""
ndx = ndx - 1
count = count + 1
else:
for i in xrange(0, len(ndx)):
if ndx[i]<smallT.ndim:
"""Sum over the dimension ndx[i]"""
smallT = np.sum(smallT, ndx[i])
"""Compensate for reduced dimensions of smallT"""
ndx = ndx - 1
"""Create marginalized potential"""
smallpot = DiscretePotential(onto, ns[onto], smallT)
return [smallpot, maximizers]
def best_first_elim_order(G, node_sizes, stage=[]):
"""
This function greedily searches for an optimal elimination order.
Find an order in which to eliminate nodes from the graph in such a way
as to try and minimize the weight of the resulting triangulated graph.
The weight of a graph is the sum of the weights of each of its cliques;
the weight of a clique is the product of the weights of each of its
members; the weight of a node is the number of values it can take on.
Since this is an NP-hard problem, we use the following greedy heuristic:
At each step, eliminate that node which will result in the addition of
the least number of fill-in edges, breaking ties by choosing the node
that induces the lighest clique.
For details, see
- Kjaerulff, "Triangulation of graphs -- algorithms giving small total
state space", Univ. Aalborg tech report, 1990 (www.cs.auc.dk/~uk)
- C. Huang and A. Darwiche, "Inference in Belief Networks: A procedural
guide", Intl. J. Approx. Reasoning, 11, 1994
Parameters
----------
G: Numpy ndarray
G[i,j] = 1 iff there is an edge between node i and node j.
node_sizes: List
The node sizes, where ns[i] = the number of discrete values
node i can take on [1 if observed].
stage: List
stage[i] is a list of the nodes that must be eliminated at i'th
stage.
"""
"""Obtain the number of nodes in the graph"""
n = G.shape[0]
if stage == []:
stage = [range(0, n)]
MG = G.copy()
uneliminated = np.ones((1, n))
order = np.zeros((1, n))
t = 0
"""For each node in the graph"""
for i in range(0, n):
"""Find the indices of the unelminated elements"""
U = find(uneliminated == 1)
"""Find which nodes can be removed in this stage."""
valid = np.intersect1d_nu(np.array(U), np.array([stage[t]]))
"""
Determine which of the valid nodes will add the least number of fill in
edges once eliminated. If 2 or more nodes add the same least number of
fill in edges, then choose the one that results in the lightest clique.
"""
min_fill = np.zeros((1, len(valid)))
min_weight = np.zeros((1, len(valid)))
"""For each node that is valid for elimination"""
for j in range(0, len(valid)):
k = valid[j]
"""Obtain the uneliminated neighbours of the node to be eliminated"""
nbrs = neighbours(G, k)
nbrs = np.intersect1d_nu(np.array([nbrs]), np.array(U))
l = len(nbrs)
M = np.zeros((l, l))
count = 0
for x in nbrs:
for y in range(0, len(nbrs)):
M[count, y] = MG[x, nbrs[y]]
count = count + 1
"""Save the number of fill-in edges required to eliminate node j"""
min_fill[0, j] = l**2 - np.sum(M)
nbrs = nbrs.tolist()
nbrs.insert(0, k)
"""Save the clique weight obtained by eliminating node j"""
min_weight[0, j] = np.prod(node_sizes[nbrs])
"""Determine which of the nodes create the lightest clique."""
lightest_nbrs = find(min_weight == np.min(min_weight))
"""
Determine which of nodes found in the step above, require the least
number of fill-in edges to eliminate.
"""
best_nbr_ndx = np.argmin(min_fill[0, lightest_nbrs.tolist()])
j = lightest_nbrs[0, best_nbr_ndx]
"""
Determine which of the nodes found in the step above are valid for
elimination, these are the nodes to be eliminated.
"""
k = valid[j]
uneliminated[0, k] = 0
"""Add the nodes to be eliminated to the elimination order"""
order[0, i] = k
"""Determine the nieghbours of the nodes to be eliminated"""
ns = neighbours(G, k)
ns = np.intersect1d_nu(np.array([ns]), np.array(U))
"""Eliminate the nodes"""
if len(ns) != 0:
for x in ns:
for y in ns:
G[x, y] = 1
G = setdiag(G, 0)
"""
If all the nodes valid for elimination in this stage have been
eliminated, then advance to the next stage.
"""
if np.sum(np.abs(uneliminated[0, stage[t]])) == 0:
t = t + 1
return order