def test_rigorous_henon(depth, box, plot=False, fpath="../sandbox/henon_test"):
# our tree, mapper, enclosure
tree = Tree(box, full=True)
m = HenonMapper()
ce = CombEnc(tree, m)
for d in range(depth):
# print 'at depth', d
ce.tree.subdivide()
# print 'subdivided:', ce.tree.size, 'boxes'
ce.update()
# print 'enclosure updated'
I = graph_mis(ce.mvm)
# print 'len(I) = ', len(I)
# print ''
# now remove all boxes not in I (the maximal invariant set)
ce.tree.remove(list(set(range(ce.tree.size)) - set(I)))
if plot:
# now display the tree!
boxes = ce.tree.boxes()
gfx.show_uboxes(boxes, col="c", ecol="b")
print "done with rigorous henon @ depth", depth
print ""
fname = fpath + dot + "CE" + dot + str(depth)
utils.write_dot(ce.mvm.graph, fname + ".dot")
utils.pickle_tree(ce.tree, fname + ".pkl")
def shift_equivalent_map( self ):
"""
Graphical representation of the shift equivalence proved in
Theorem 3.1 and Lemmas 3.2-3.4.
The required sets of nodes V1, V2, and V3 are computed as
follows:
Given G = (V,E), with V = {V1,V2,V3},
1. Find strongly connected components (scc) of G. Labeled
by nodes V2.
2. Perform a forward depth-first search; then perform
backward depth-first search by reversing the orientation
on the edges. This locates generators that are not
connected to cycles. Labeled by nodes V1 \cup V3.
3. Trim self.index_map to include only those nodes in
V2 = V\(V1 \cup V3)
4. The nodes in V1 \cup V3 are stored in the attrribute
'non_mis_nodes'.
"""
# determine the maximal invariant set of generators in
# self.index_map (1 & 2 above)
mis_nodes = algorithms.graph_mis( self.index_map )
nbunch = set( self.index_map.nodes() )
# compute V2
self.non_mis_nodes = nbunch.difference( mis_nodes )
# remove nodes to form subgraph of map on generators
# containing only nodes that are part of the maximally
# invariant set.
self.index_map.remove_nodes_from( self.non_mis_nodes )
def max_invariant_set( I, transition ):
"""Restrict a region I to the maximal invariant set within that
region.
Return indices of nodes in the maximal invariant set.
"""
R = transition.subgraph( I )
S = algorithms.graph_mis( R )
return set( S )
def prepare_regions( self ):
"""
Partition the phase space into disjoint, recurrent regions.
Must have initialized self with index_map matrix and
region2gen dictionary, or used one of the load_from_* methods
to initialize these values.
"""
# returns a DiGraph
self.map_on_regions = utils.index_map_to_region_map( self.index_map,
self.region2gen )
# graph_mis from graphs.algorithms
scc_list, scc_components, recurrent_regions = graph_mis( self.map_on_regions,
return_rsccs=True )
nbunch = [ scc_components[i] for i in recurrent_regions ]
self.recurrent_subgraphs = []
# the actual partition of phase space in graph format
for n in nbunch:
# ignore disjoint regions/nodes that have self-loops => no
# entropy
if len( n ) == 1:
continue
G = DiGraph()
G.graph = self.map_on_regions.graph.subgraph( n )
self.recurrent_subgraphs.append( G )
# construct IndexMapProcessor for each disjoint region
# assign entropies of -1 to each recurrent region.
for region in self.recurrent_subgraphs:
self.phase_space.append( [ IndexMapProcessor( self.index_map,
self.region2gen,
region,
debug=self.debug,
verbose=self.verbose ),
-1 # initial entropy for region
] )
# convert nodes to integers starting at 0
H = nx.convert_node_labels_to_integers( FR.mvm.graph, first_label=0 )
# True Henon
# our tree, mapper, enclosure
tree = Tree(box,full=True)
m = HenonMapper()
ce = CombEnc(tree,m)
for d in range(depth):
print 'at depth', d
ce.tree.subdivide()
print 'subdivided:', ce.tree.size, 'boxes'
ce.update()
#print 'enclosure updated'
I = graph_mis(ce.mvm)
#print 'len(I) = ', len(I)
print ''
# now remove all boxes not in I (the maximal invariant set)
ce.tree.remove(list(set(range(ce.tree.size))-set(I)))
# now display the tree!
boxes = ce.tree.boxes()
gfx.show_uboxes(boxes, col='c', ecol='b')
print "Plotting boxes... "
fig = FR.show_boxes()
fig = FR.show_error_boxes( color='r', fig=fig )
confidence( ce.mvm, H )
def graph_mis( self ):
self.mis = alg.graph_mis( self.mvm )
if 0:
hom_matrix = utils.load_matlab_matrix( 'leslie_index.mat', 'hom_matrix' )
region2gen = utils.convert_matlab_gens( 'leslie_gens.mat' )
map_on_regions = utils.index_map_to_region_map( hom_matrix, region2gen, shift=-1)
########################
#
# APPEARS THAT BAD EDGE SETS ARE BEING THROWN INTO ONE (1) SET. CHECK THIS!!
#
########################
debug = True
verbose = False
scc_list, scc_components, recurrent_regions = algorithms.graph_mis( map_on_regions,
return_rsccs=True )
# separate nodes by recurrent component
nbunch = [ scc_components[i] for i in recurrent_regions ]
recurrent_subgraphs = []
for n in nbunch:
# ignore self-loops, no entropy
if len( n ) == 1:
continue
G = DiGraph()
G.graph = map_on_regions.graph.subgraph( n )
recurrent_subgraphs.append( G )
# compute entropy for each scc in the symbolic system
all_regions = []
for scc in recurrent_subgraphs:
all_regions.append( IndexMapProcessor( hom_matrix, region2gen,
def graph_maximal_inv_set(self):
self.mis = alg.graph_mis(self.mvm)
import networkx as nx from rads.graphs import DiGraph from rads.graphs.algorithms import graph_mis if __name__ == "__main__": E = [(0, 8), (0, 5), (1, 19), (1, 10), (1, 3), (1, 5), (2, 11), (2, 4), (2, 7), (4, 19), (6, 19), (6, 3), (6, 5), (7, 19), (7, 3), (8, 0), (9, 16), (9, 14), (9, 6), (10, 3), (10, 12), (11, 19), (11, 6), (12, 3), (12, 12), (13, 8), (13, 19), (13, 11), (13, 5), (14, 1), (14, 9), (16, 3), (17, 1), (18, 16), (18, 13), (19, 9)] G = DiGraph() G.add_edges_from(E) correct_answer = [0, 1, 6, 8, 9, 10, 12, 14, 19] my_answer = sorted(graph_mis(G)) print 'graph_mis(G) = ', my_answer print 'correct answer =', correct_answer E = [(0,1), (1,2), (2,0), (2,3), (3,4), (1,5), (5,6), (6,6), (7,0), (8,7)] G = DiGraph() G.add_edges_from(E) my_answer = sorted(graph_mis(G)) print 'graph_mis(G) = ', my_answer print 'correct answer =', [0,1,2,5,6]