def find_equilibria(ssd, eps=0):
""" Finds the polytope that contains the equilibrium points
@param ssd: The dynamics of the switched system
@type ssd: L{SwitchedSysDyn}
@param eps: The value by which the width of all polytopes
containing equilibrium points is increased.
@type eps: float
@return param cont_props: The polytope representations of the atomic
propositions of the state space to be used in partitiong
@type cont_props: dict of polytope.Polytope
Warning: Currently, there is no condition for points where
the polytope is critically stable.
Warning: if there is a region outside the domain, then it is
unstable. It seems to ignore the regions outside the domain.
"""
def normalize(A, B):
""" Normalizes set of equations of the form Ax<=B
"""
if A.size > 0:
Anorm = np.sqrt(np.sum(A * A, 1)).flatten()
pos = np.nonzero(Anorm > 1e-10)[0]
A = A[pos, :]
B = B[pos]
Anorm = Anorm[pos]
mult = 1 / Anorm
for i in xrange(A.shape[0]):
A[i, :] = A[i, :] * mult[i]
B = B.flatten() * mult
return A, B
cont_ss = ssd.cts_ss
min_outx = cont_ss.b[0]
min_outy = cont_ss.b[1]
max_outx = min_outx + 1
max_outy = min_outy + 1
abs_tol = 1e-7
cont_props = dict()
for mode in ssd.modes:
cont_dyn = ssd.dynamics[mode].list_subsys[0]
A = cont_dyn.A
K = cont_dyn.K.T[0]
I = np.eye(len(A), dtype=float)
rank_IA = np.linalg.matrix_rank(I - A)
concat = np.hstack((I - A, K.reshape(len(A), 1)))
rank_concat = np.linalg.matrix_rank(concat)
soln = pc.Polytope()
props_sym = "eqpnt_" + str(mode[1])
if rank_IA == rank_concat:
if rank_IA == len(A):
equil = np.dot(np.linalg.inv(I - A), K)
if (
equil[0] >= (-cont_ss.b[2])
and equil[0] <= cont_ss.b[0]
and equil[1] >= (-cont_ss.b[3])
and equil[1] <= cont_ss.b[1]
):
delta = eps + equil / 100
soln = pc.box2poly(
[[equil[0] - delta[0], equil[0] + delta[0]], [equil[1] - delta[1], equil[1] + delta[1]]]
)
else:
soln = pc.box2poly([[min_outx, max_outx], [min_outy, max_outy]])
elif rank_IA < len(A):
if eps == 0:
eps = abs(min(np.amin(-K), np.amin(A - I)))
IAn, Kn = normalize(I - A, K)
soln = pc.Polytope(np.vstack((IAn, -IAn)), np.hstack((Kn + eps, -Kn + eps)))
relevantsoln = pc.intersect(soln, cont_ss, abs_tol)
if pc.is_empty(relevantsoln) & ~pc.is_empty(soln):
soln = pc.box2poly([[min_outx, max_outx], [min_outy, max_outy]])
else:
soln = relevantsoln
else:
# Assuming trajectories go to infinity as there are no
# equilibrium points
soln = pc.box2poly([[min_outx, max_outx], [min_outy, max_outy]])
print str(mode) + " trajectories go to infinity! No solution"
cont_props[props_sym] = soln
return cont_props
def infer_srtesseler_density(cells, volume_weighted=True, **kwargs):
"""
adapted from:
SR-Tesseler: a method to segment and quantify localization-based super-resolution microscopy data.
Levet, F., Hosy, E., Kechkar, A., Butler, C., Beghin, A., Choquet, C., Sibarita, J.B.
Nature Methods 2015; 12 (11); 1065-1071.
"""
points = cells.locations
is_densest_tessellation = isinstance(cells.tessellation, Voronoi) and cells.number_of_cells == len(points)
if is_densest_tessellation:
tessellation = cells.tessellation
partition = cells
else:
tessellation = Voronoi()
tessellation.tessellate(points[['x','y']])
partition = Partition(points, tessellation)
#areas = tessellation.cell_volume # the default implementation for cell_volume
# generates a non-infinite estimation for opened cells
areas = np.full(tessellation.number_of_cells, np.nan)
indices, densities, unit_polygons = [], [], []
for i, surface_area in enumerate(areas):
unknown_area = np.isnan(surface_area) or np.isinf(surface_area) or surface_area == 0
polygons_required = not is_densest_tessellation and volume_weighted
if unknown_area or polygons_required:
hull = convex_hull(partition=partition, cell_index=i)
if unknown_area:
if hull is None:
continue
density = 1./ hull.volume # should also divide by the number of frames
else:
density = 1./ surface_area
indices.append(i)
densities.append(density)
if polygons_required:
unit_polygons.append(p.Polytope(hull.equations[:,[0,1]], -hull.equations[:,2]))
unit_density = pd.Series(index=indices, data=densities)
if is_densest_tessellation:
density = unit_density
else:
indices, densities = [], []
for i in range(cells.number_of_cells):
assigned = cells.cell_index == i
if not np.any(assigned):
indices.append(i)
densities.append(0.)
continue
local_polygons, = np.nonzero(assigned)
assert 0 < local_polygons.size
if volume_weighted:
hull = convex_hull(partition=cells, cell_index=i, point_assigned=assigned)
if hull is None:
continue
polygon = p.Polytope(hull.equations[:,[0,1]], -hull.equations[:,2])
weighted_sum = 0.
total_area = 0.
for j in local_polygons:
intersection = p.intersect(unit_polygons[j], polygon)
total_area += intersection.volume
weighted_sum += intersection.volume * unit_density[j]
average_density = weighted_sum / total_area
else:
average_density = unit_density[local_polygons].mean()
indices.append(i)
densities.append(average_density)
density = pd.Series(index=indices, data=densities)
return pd.DataFrame(dict(density=density))
def compute_calP_in_probs(self, c, d, calP, t, cp1, cp2, actions_taken):
A_upper = np.array([
[-1.0, 0.0, 0.0],
[1.0, 0.0, 0.0],
[0.0, -1.0, 0.0],
[0.0, 1.0, 0.0],
[0.0, 0.0, -1.0],
[0.0, 0.0, 1.0], #surrounding box
[-cp1[0], -cp1[1], -cp1[2]]
])
b_upper = np.array(
[1.0, 1.0, 1.0, 1.0, 1.0, 1.0, -c * self.agents_lst[t].delta])
p_upper = pc.Polytope(A_upper, b_upper) #large upper polytope
A_lower = np.array([
[-1.0, 0.0, 0.0],
[1.0, 0.0, 0.0],
[0.0, -1.0, 0.0],
[0.0, 1.0, 0.0],
[0.0, 0.0, -1.0],
[0.0, 0.0, 1.0], #surrounding box
[cp2[0], cp2[1], cp2[2]]
])
b_lower = np.array(
[1.0, 1.0, 1.0, 1.0, 1.0, 1.0, -c * self.agents_lst[t].delta])
p_lower = pc.Polytope(A_lower, b_lower) #large lower polytope
calP_u = []
calP_l = []
calP_m = []
print("actual number of polytopes:")
print(len(calP))
for pol in calP:
p = pol.pol_repr
if p.volume > 0.01:
# intersections of the polytope with upper and lower of curr round
intersect_up = pc.intersect(p, p_upper)
intersect_lo = pc.intersect(p, p_lower)
# check whether the intersection is empty-ish (very small vol)
emptyish_up = self.is_empty(intersect_up) or self.has_tiny_vol(
intersect_up)
emptyish_lo = self.is_empty(intersect_lo) or self.has_tiny_vol(
intersect_lo)
if (emptyish_up and emptyish_lo):
calP_m.append(pol)
elif (emptyish_up):
diff_lo = p.diff(
p_lower
) # what is left in the diff should go in the middle space unless emptyish
if (self.is_empty(diff_lo) or self.has_tiny_vol(diff_lo)
): # intersection is exactly the polytope
# if emptyish, no need to create new polytope
pol.updated[t] = 1
calP_l.append(pol)
else:
# otherwise we need to create new polytope
upd_lst = pol.updated #store the history of updates for curr pol
upd_lst[
t] = 0 # by default assume that you won't update it
ratio = 1.0 * diff_lo.volume / p.volume # how big is the prob that you keep for new pol
g_pol1 = Grind_Polytope(diff_lo, ratio * pol.pi,
ratio * pol.weight, d, self.T,
pol.est_loss, pol.loss,
upd_lst)
calP_m.append(g_pol1)
upd_lst2 = pol.updated
upd_lst2[
t] = 1 # this part of the pol will be part of the calP_lower
g_pol2 = Grind_Polytope(intersect_lo,
(1.0 - ratio) * pol.pi,
(1.0 - ratio) * pol.weight, d,
self.T, pol.est_loss, pol.loss,
upd_lst2)
calP_l.append(g_pol2)
elif (emptyish_lo):
diff_up = p.diff(p_upper)
if (self.is_empty(diff_up) or self.has_tiny_vol(diff_up)):
pol.updated[t] = 1
calP_u.append(pol)
else:
upd_lst = pol.updated
upd_lst[t] = 0
ratio = 1.0 * diff_up.volume / p.volume
g_pol1 = Grind_Polytope(diff_up, ratio * pol.pi,
ratio * pol.weight, d, self.T,
pol.est_loss, pol.loss,
upd_lst)
calP_m.append(g_pol1)
upd_lst2 = pol.updated
upd_lst2[t] = 1
g_pol2 = Grind_Polytope(intersect_up,
(1.0 - ratio) * pol.pi,
(1.0 - ratio) * pol.weight, d,
self.T, pol.est_loss, pol.loss,
upd_lst2)
calP_u.append(g_pol2)
else:
diff_up = p.diff(p_upper)
diff_lo = p.diff(p_lower)
ratio1 = 1.0 * intersect_up.volume / p.volume
upd_lst = pol.updated
upd_lst[t] = 1
g_pol1 = Grind_Polytope(intersect_up, ratio1 * pol.pi,
ratio1 * pol.weight, d, self.T,
pol.est_loss, pol.loss, upd_lst)
calP_u.append(g_pol1)
ratio2 = 1.0 * intersect_lo.volume / p.volume
g_pol2 = Grind_Polytope(intersect_lo, ratio2 * pol.pi,
ratio2 * pol.weight, d, self.T,
pol.est_loss, pol.loss, upd_lst)
calP_l.append(g_pol2)
#if ratio1 > 0 or ratio2 > 0:
diff_uplo = pc.intersect(diff_up, diff_lo)
if (not self.is_empty(diff_uplo)
and not self.has_tiny_vol(diff_uplo)):
upd_lst[t] = 0
g_pol3 = Grind_Polytope(
diff_uplo, (1.0 - ratio1 - ratio2) * pol.pi,
(1.0 - ratio1 - ratio2) * pol.weight, d, self.T,
pol.est_loss, pol.loss, upd_lst)
calP_m.append(g_pol3)
elif (p.volume == 0):
# discard point-polytopes
print("Timestep t=%d polytope w zero vol" % t)
print(p)
print("Was it really empty?")
print pc.is_empty(p)
else:
intersect_up = pc.intersect(p, p_upper)
intersect_lo = pc.intersect(p, p_lower)
if (self.is_empty(intersect_up)
and self.is_empty(intersect_lo)):
pol.updated[t] = 0
calP_m.append(pol)
elif (self.is_empty(intersect_lo)
or intersect_lo.volume < intersect_up.volume):
# should go with upper polytopes set
pol.updated[t] = 1
calP_u.append(pol)
elif (self.is_empty(intersect_up)
or intersect_up.volume < intersect_lo.volume):
# should go with lower polytopes set
pol.updated[t] = 1
calP_l.append(pol)
for pol in calP_u:
if pol.pi <= 0.000001:
pol.pi = 0.000001
for pol in calP_m:
if pol.pi <= 0.000001:
pol.pi = 0.000001
for pol in calP_l:
if pol.pi <= 0.000001:
pol.pi = 0.000001
actions_set = []
upd = []
pi_lst = []
tot_up = 0.0
tot_lo = 0.0
incl = [
] # indicator function whether the current action belongs in an upper or lower polytope set
for pol in calP_u:
actions_set.append(pol.action)
upd.append(pol.updated) #size |\calA| x T
pi_lst.append(pol.pi)
incl.append(1)
tot_up += pol.pi
for pol in calP_m:
actions_set.append(pol.action)
upd.append(pol.updated)
incl.append(0)
pi_lst.append(pol.pi)
for pol in calP_l:
actions_set.append(pol.action)
upd.append(pol.updated)
pi_lst.append(pol.pi)
incl.append(1)
tot_lo += pol.pi
# a lower bound in the in probabilities of the actions that are to be updated is
# the tot prob of the upper and the lower polytopes sets
in_probs_est = self.compute_in_probs_regr(pi_lst, t, upd,
actions_taken, actions_set,
tot_up + tot_lo, incl)
spammer = 1 if self.agents_lst[t].type == 0 else 0
j = 0
for pol in calP_u:
pol.est_loss += (1.0 * spammer) / in_probs_est[j]
j += 1
for pol in calP_m:
j += 1
for pol in calP_l:
pol.est_loss += (1.0 - spammer) / in_probs_est[j]
j += 1
return (calP_u, calP_m, calP_l)
def merge_partition_pair(
old_regions, ab2,
cur_mode, prev_modes,
old_parents, old_ap_labeling
):
"""Merge an Abstraction with the current partition iterate.
@param old_regions: A list of C{Region} that is from either:
1. The ppp of the first (initial) L{AbstractPwa} to be merged.
2. A list of already-merged regions
@type old_regions: list of C{Region}
@param ab2: Abstracted piecewise affine dynamics to be merged into the
@type ab2: L{AbstractPwa}
@param cur_mode: mode to be merged
@type cur_mode: tuple
@param prev_modes: list of modes that have already been merged together
@type prev_modes: list of tuple
@param old_parents: dict of modes that have already been merged to dict of
indices of new regions to indices of regions
@type old_parents: dict of modes to list of region indices in list
C{old_regions} or dict of region indices to regions in original ppp for
that mode
@param old_ap_labeling: dict of states of already-merged modes to sets of
propositions for each state
@type old_ap_labeling: dict of tuples to sets
@return: the following:
- C{new_list}, list of new regions
- C{parents}, same as input param C{old_parents}, except that it
includes the mode that was just merged and for list of regions in
return value C{new_list}
- C{ap_labeling}, same as input param C{old_ap_labeling}, except that it
includes the mode that was just merged.
"""
logger.info('merging partitions')
part2 = ab2.ppp
modes = prev_modes + [cur_mode]
new_list = []
parents = {mode:dict() for mode in modes}
ap_labeling = dict()
for i in range(len(old_regions)):
for j in range(len(part2)):
isect = pc.intersect(old_regions[i],
part2[j])
rc, xc = pc.cheby_ball(isect)
# no intersection ?
if rc < 1e-5:
continue
logger.info('merging region: A' + str(i) +
', with: B' + str(j))
# if Polytope, make it Region
if len(isect) == 0:
isect = pc.Region([isect])
# label the Region with propositions
isect.props = old_regions[i].props.copy()
new_list.append(isect)
idx = new_list.index(isect)
# keep track of parents
for mode in prev_modes:
parents[mode][idx] = old_parents[mode][i]
parents[cur_mode][idx] = j
# union of AP labels from parent states
ap_label_1 = old_ap_labeling[i]
ap_label_2 = ab2.ts.states[j]['ap']
logger.debug('AP label 1: ' + str(ap_label_1))
logger.debug('AP label 2: ' + str(ap_label_2))
# original partitions may be different if pwa_partition used
# but must originate from same initial partition,
# i.e., have same continuous propositions, checked above
#
# so no two intersecting regions can have different AP labels,
# checked here
if ap_label_1 != ap_label_2:
msg = 'Inconsistent AP labels between intersecting regions\n'
msg += 'of partitions of switched system.'
raise Exception(msg)
ap_labeling[idx] = ap_label_1
return new_list, parents, ap_labeling
def find_equilibria(ssd, eps=0):
""" Finds the polytope that contains the equilibrium points
@param ssd: The dynamics of the switched system
@type ssd: L{SwitchedSysDyn}
@param eps: The value by which the width of all polytopes
containing equilibrium points is increased.
@type eps: float
@return param cont_props: The polytope representations of the atomic
propositions of the state space to be used in partitiong
@type cont_props: dict of polytope.Polytope
Warning: Currently, there is no condition for points where
the polytope is critically stable.
Warning: if there is a region outside the domain, then it is
unstable. It seems to ignore the regions outside the domain.
"""
def normalize(A, B):
""" Normalizes set of equations of the form Ax<=B
"""
if A.size > 0:
Anorm = np.sqrt(np.sum(A * A, 1)).flatten()
pos = np.nonzero(Anorm > 1e-10)[0]
A = A[pos, :]
B = B[pos]
Anorm = Anorm[pos]
mult = 1 / Anorm
for i in xrange(A.shape[0]):
A[i, :] = A[i, :] * mult[i]
B = B.flatten() * mult
return A, B
cont_ss = ssd.cts_ss
min_outx = cont_ss.b[0]
min_outy = cont_ss.b[1]
max_outx = min_outx + 1
max_outy = min_outy + 1
abs_tol = 1e-7
cont_props = dict()
for mode in ssd.modes:
cont_dyn = ssd.dynamics[mode].list_subsys[0]
A = cont_dyn.A
K = cont_dyn.K.T[0]
I = np.eye(len(A), dtype=float)
rank_IA = np.linalg.matrix_rank(I - A)
concat = np.hstack((I - A, K.reshape(len(A), 1)))
rank_concat = np.linalg.matrix_rank(concat)
soln = pc.Polytope()
props_sym = 'eqpnt_' + str(mode[1])
if (rank_IA == rank_concat):
if (rank_IA == len(A)):
equil = np.dot(np.linalg.inv(I - A), K)
if (equil[0] >= (-cont_ss.b[2]) and equil[0] <= cont_ss.b[0]
and equil[1] >= (-cont_ss.b[3])
and equil[1] <= cont_ss.b[1]):
delta = eps + equil / 100
soln = pc.box2poly(
[[equil[0] - delta[0], equil[0] + delta[0]],
[equil[1] - delta[1], equil[1] + delta[1]]])
else:
soln = pc.box2poly([[min_outx, max_outx],
[min_outy, max_outy]])
elif (rank_IA < len(A)):
if eps == 0:
eps = abs(min(np.amin(-K), np.amin(A - I)))
IAn, Kn = normalize(I - A, K)
soln = pc.Polytope(np.vstack((IAn, -IAn)),
np.hstack((Kn + eps, -Kn + eps)))
relevantsoln = pc.intersect(soln, cont_ss, abs_tol)
if (pc.is_empty(relevantsoln) & ~pc.is_empty(soln)):
soln = pc.box2poly([[min_outx, max_outx],
[min_outy, max_outy]])
else:
soln = relevantsoln
else:
#Assuming trajectories go to infinity as there are no
#equilibrium points
soln = pc.box2poly([[min_outx, max_outx], [min_outy, max_outy]])
print str(mode) + " trajectories go to infinity! No solution"
cont_props[props_sym] = soln
return cont_props
