def plot(self, y="return", x="learning_steps", save=False):
"""Plots the performance of the experiment
This function has only limited capabilities.
For more advanced plotting of results consider
:py:class:`Tools.Merger.Merger`.
"""
labels = rlpy.Tools.results.default_labels
performance_fig = plt.figure("Performance")
res = self.result
plt.plot(res[x], res[y], '-bo', lw=3, markersize=10)
plt.xlim(0, res[x][-1] * 1.01)
y_arr = np.array(res[y])
m = y_arr.min()
M = y_arr.max()
delta = M - m
if delta > 0:
plt.ylim(m - .1 * delta - .1, M + .1 * delta + .1)
xlabel = labels[x] if x in labels else x
ylabel = labels[y] if y in labels else y
plt.xlabel(xlabel, fontsize=16)
plt.ylabel(ylabel, fontsize=16)
if save:
path = os.path.join(
self.full_path,
"{:3}-performance.pdf".format(self.exp_id))
performance_fig.savefig(path, transparent=True, pad_inches=.1)
plt.ioff()
plt.show()
def plot(self, y="return", x="learning_steps", save=False):
"""Plots the performance of the experiment
This function has only limited capabilities.
For more advanced plotting of results consider
:py:class:`Tools.Merger.Merger`.
"""
labels = rlpy.Tools.results.default_labels
performance_fig = plt.figure("Performance")
res = self.result
plt.plot(res[x], res[y], '-bo', lw=3, markersize=10)
plt.xlim(0, res[x][-1] * 1.01)
y_arr = np.array(res[y])
m = y_arr.min()
M = y_arr.max()
delta = M - m
if delta > 0:
plt.ylim(m - .1 * delta - .1, M + .1 * delta + .1)
xlabel = labels[x] if x in labels else x
ylabel = labels[y] if y in labels else y
plt.xlabel(xlabel, fontsize=16)
plt.ylabel(ylabel, fontsize=16)
if save:
path = os.path.join(self.full_path,
"{:3}-performance.pdf".format(self.exp_id))
performance_fig.savefig(path, transparent=True, pad_inches=.1)
plt.ioff()
plt.show()
def showDomain(self, a=None):
if a is not None:
a = self.actions[a]
T = np.empty((self.d, 2))
T[:, 0] = np.cos(self.theta)
T[:, 1] = np.sin(self.theta)
R = np.dot(self.P, T)
R1 = R - .5 * self.lengths[:, None] * T
R2 = R + .5 * self.lengths[:, None] * T
Rx = np.hstack([R1[:, 0], R2[:, 0]]) + self.pos_cm[0]
Ry = np.hstack([R1[:, 1], R2[:, 1]]) + self.pos_cm[1]
print(Rx)
print(Ry)
f = plt.figure("Swimmer Domain")
if not hasattr(self, "swimmer_lines"):
plt.plot(0., 0., "ro")
self.swimmer_lines = plt.plot(Rx, Ry)[0]
self.action_text = plt.text(-2, -8, str(a))
plt.xlim(-5, 15)
plt.ylim(-10, 10)
else:
self.swimmer_lines.set_data(Rx, Ry)
self.action_text.set_text(str(a))
plt.figure("Swimmer Domain").canvas.draw()
plt.figure("Swimmer Domain").canvas.flush_events()
def showDomain(self, a):
s = self.state
# Draw the environment
if self.domain_fig is None:
self.domain_fig = plt.imshow(self.map,
cmap='IntruderMonitoring',
interpolation='nearest',
vmin=0,
vmax=3)
plt.xticks(np.arange(self.COLS), fontsize=FONTSIZE)
plt.yticks(np.arange(self.ROWS), fontsize=FONTSIZE)
plt.show()
if self.ally_fig is not None:
self.ally_fig.pop(0).remove()
self.intruder_fig.pop(0).remove()
s_ally = s[0:self.NUMBER_OF_AGENTS * 2].reshape((-1, 2))
s_intruder = s[self.NUMBER_OF_AGENTS * 2:].reshape((-1, 2))
self.ally_fig = plt.plot(s_ally[:, 1],
s_ally[:, 0],
'bo',
markersize=30.0,
alpha=.7,
markeredgecolor='k',
markeredgewidth=2)
self.intruder_fig = plt.plot(s_intruder[:, 1],
s_intruder[:, 0],
'g>',
color='gray',
markersize=30.0,
alpha=.7,
markeredgecolor='k',
markeredgewidth=2)
plt.draw()
def show_one_variable(variable, vrange, map1="9x9-2PathR1.txt"): x = vrange ctrl = param_ranges(variable, vrange) y = param_ranges(variable, vrange, cur_map=map1) try: plt.title(variable + " vs AUC") plt.ioff() plt.plot(x, ctrl) plt.plot(x, y) plt.legend() plt.show() finally: return ctrl, y
def showDomain(self, a):
if self.gcf is None:
self.gcf = plt.gcf()
s = self.state
# Plot the car
x, y, speed, heading = s
car_xmin = x - self.REAR_WHEEL_RELATIVE_LOC
car_ymin = y - self.CAR_WIDTH / 2.
if self.domain_fig is None: # Need to initialize the figure
self.domain_fig = plt.figure()
# Goal
plt.gca(
).add_patch(
plt.Circle(
self.GOAL,
radius=self.GOAL_RADIUS,
color='g',
alpha=.4))
plt.xlim([self.XMIN, self.XMAX])
plt.ylim([self.YMIN, self.YMAX])
plt.gca().set_aspect('1')
# Car
if self.car_fig is not None:
plt.gca().patches.remove(self.car_fig)
if self.slips:
slip_x, slip_y = zip(*self.slips)
try:
line = plt.axes().lines[0]
if len(line.get_xdata()) != len(slip_x): # if plot has discrepancy from data
line.set_xdata(slip_x)
line.set_ydata(slip_y)
except IndexError:
plt.plot(slip_x, slip_y, 'x', color='b')
self.car_fig = mpatches.Rectangle(
[car_xmin,
car_ymin],
self.CAR_LENGTH,
self.CAR_WIDTH,
alpha=.4)
rotation = mpl.transforms.Affine2D().rotate_deg_around(
x, y, heading * 180 / np.pi) + plt.gca().transData
self.car_fig.set_transform(rotation)
plt.gca().add_patch(self.car_fig)
plt.draw()
# self.gcf.canvas.draw()
plt.pause(0.001)
def showDomain(self, a=0):
s = self.state
# Draw the environment
if self.domain_fig is None:
self.move_fig = plt.subplot(111)
s = s.reshape((self.BOARD_SIZE, self.BOARD_SIZE))
self.domain_fig = plt.imshow(
s,
cmap='FlipBoard',
interpolation='nearest',
vmin=0,
vmax=1)
plt.xticks(np.arange(self.BOARD_SIZE), fontsize=FONTSIZE)
plt.yticks(np.arange(self.BOARD_SIZE), fontsize=FONTSIZE)
# pl.tight_layout()
a_row, a_col = id2vec(a, [self.BOARD_SIZE, self.BOARD_SIZE])
self.move_fig = self.move_fig.plot(
a_col,
a_row,
'kx',
markersize=30.0)
plt.show()
a_row, a_col = id2vec(a, [self.BOARD_SIZE, self.BOARD_SIZE])
self.move_fig.pop(0).remove()
# print a_row,a_col
# Instead of '>' you can use 'D', 'o'
self.move_fig = plt.plot(a_col, a_row, 'kx', markersize=30.0)
s = s.reshape((self.BOARD_SIZE, self.BOARD_SIZE))
self.domain_fig.set_data(s)
plt.draw()
def showDomain(self, a=0):
s = self.state
# Draw the environment
if self.domain_fig is None:
self.move_fig = plt.subplot(111)
s = s.reshape((self.BOARD_SIZE, self.BOARD_SIZE))
self.domain_fig = plt.imshow(s,
cmap='FlipBoard',
interpolation='nearest',
vmin=0,
vmax=1)
plt.xticks(np.arange(self.BOARD_SIZE), fontsize=FONTSIZE)
plt.yticks(np.arange(self.BOARD_SIZE), fontsize=FONTSIZE)
# pl.tight_layout()
a_row, a_col = id2vec(a, [self.BOARD_SIZE, self.BOARD_SIZE])
self.move_fig = self.move_fig.plot(a_col,
a_row,
'kx',
markersize=30.0)
plt.show()
a_row, a_col = id2vec(a, [self.BOARD_SIZE, self.BOARD_SIZE])
self.move_fig.pop(0).remove()
# print a_row,a_col
# Instead of '>' you can use 'D', 'o'
self.move_fig = plt.plot(a_col, a_row, 'kx', markersize=30.0)
s = s.reshape((self.BOARD_SIZE, self.BOARD_SIZE))
self.domain_fig.set_data(s)
plt.draw()
def showDomain(self, a):
s = self.state
# Draw the environment
if self.domain_fig is None:
plt.figure("Domain")
self.domain_fig = plt.imshow(
self.map,
cmap='IntruderMonitoring',
interpolation='nearest',
vmin=0,
vmax=3)
plt.xticks(np.arange(self.COLS), fontsize=FONTSIZE)
plt.yticks(np.arange(self.ROWS), fontsize=FONTSIZE)
plt.show()
if self.ally_fig is not None:
self.ally_fig.pop(0).remove()
self.intruder_fig.pop(0).remove()
s_ally = s[0:self.NUMBER_OF_AGENTS * 2].reshape((-1, 2))
s_intruder = s[self.NUMBER_OF_AGENTS * 2:].reshape((-1, 2))
self.ally_fig = plt.plot(
s_ally[:,
1],
s_ally[:,
0],
'bo',
markersize=30.0,
alpha=.7,
markeredgecolor='k',
markeredgewidth=2)
self.intruder_fig = plt.plot(
s_intruder[:,
1],
s_intruder[:,
0],
'g>',
color='gray',
markersize=30.0,
alpha=.7,
markeredgecolor='k',
markeredgewidth=2)
plt.figure("Domain").canvas.draw()
plt.figure("Domain").canvas.flush_events()
def plot_trials(self,
y="eps_return",
x="learning_steps",
average=10,
save=False):
"""Plots the performance of the experiment
This function has only limited capabilities.
For more advanced plotting of results consider
:py:class:`Tools.Merger.Merger`.
"""
def movingaverage(interval, window_size):
window = np.ones(int(window_size)) / float(window_size)
return np.convolve(interval, window, 'same')
labels = rlpy.Tools.results.default_labels
performance_fig = plt.figure("Performance")
trials = self.trials
y_arr = np.array(trials[y])
if average:
assert type(average) is int, "Filter length is not an integer!"
y_arr = movingaverage(y_arr, average)
plt.plot(trials[x], y_arr, '-bo', lw=3, markersize=10)
plt.xlim(0, trials[x][-1] * 1.01)
m = y_arr.min()
M = y_arr.max()
delta = M - m
if delta > 0:
plt.ylim(m - .1 * delta - .1, M + .1 * delta + .1)
xlabel = labels[x] if x in labels else x
ylabel = labels[y] if y in labels else y
plt.xlabel(xlabel, fontsize=16)
plt.ylabel(ylabel, fontsize=16)
if save:
path = os.path.join(self.full_path,
"{:3}-trials.pdf".format(self.exp_id))
performance_fig.savefig(path, transparent=True, pad_inches=.1)
plt.ioff()
plt.show()
def batchDiscover(self, td_errors, phi, states):
# Discovers features using iFDD in batch setting.
# TD_Error: p-by-1 (How much error observed for each sample)
# phi: n-by-p features corresponding to all samples (each column corresponds to one sample)
# self.batchThreshold is the minimum relevance value for the feature to
# be expanded
SHOW_PLOT = 0 # Shows the histogram of relevances
maxDiscovery = self.maxBatchDiscovery
n = self.features_num # number of features
p = len(td_errors) # Number of samples
counts = np.zeros((n, n))
relevances = np.zeros((n, n))
for i in xrange(p):
phiphiT = np.outer(phi[i, :], phi[i, :])
if self.iFDDPlus:
relevances += phiphiT * td_errors[i]
else:
relevances += phiphiT * abs(td_errors[i])
counts += phiphiT
# Remove Diagonal and upper part of the relevances as they are useless
relevances = np.triu(relevances, 1)
non_zero_index = np.nonzero(relevances)
if self.iFDDPlus:
# Calculate relevances based on theoretical results of ICML 2013
# potential submission
relevances[non_zero_index] = np.divide(
np.abs(relevances[non_zero_index]),
np.sqrt(counts[non_zero_index]))
else:
# Based on Geramifard11_ICML Paper
relevances[non_zero_index] = relevances[non_zero_index]
# Find indexes to non-zero excited pairs
# F1 and F2 are the parents of the potentials
(F1, F2) = relevances.nonzero()
relevances = relevances[F1, F2]
if len(relevances) == 0:
# No feature to add
self.logger.debug("iFDD Batch: Max Relevance = 0")
return False
if SHOW_PLOT:
e_vec = relevances.flatten()
e_vec = e_vec[e_vec != 0]
e_vec = np.sort(e_vec)
plt.ioff()
plt.plot(e_vec, linewidth=3)
plt.show()
# Sort based on relevances
# We want high to low hence the reverse: [::-1]
sortedIndices = np.argsort(relevances)[::-1]
max_relevance = relevances[sortedIndices[0]]
# Add top <maxDiscovery> features
self.logger.debug(
"iFDD Batch: Max Relevance = {0:g}".format(max_relevance))
added_feature = False
new_features = 0
for j in xrange(len(relevances)):
if new_features >= maxDiscovery:
break
max_index = sortedIndices[j]
f1 = F1[max_index]
f2 = F2[max_index]
relevance = relevances[max_index]
if relevance > self.batchThreshold:
# print "Inspecting",
# f1,f2,'=>',self.getStrFeatureSet(f1),self.getStrFeatureSet(f2)
if self.inspectPair(f1, f2, np.inf):
self.logger.debug(
'New Feature %d: %s, Relevance = %0.3f' %
(self.features_num - 1,
self.getStrFeatureSet(self.features_num - 1),
relevances[max_index]))
new_features += 1
added_feature = True
else:
# Because the list is sorted, there is no use to look at the
# others
break
return (
# A signal to see if the representation has been expanded or not
added_feature)
def showDomain(self, a):
s = self.state
# Plot the car and an arrow indicating the direction of accelaration
# Parts of this code was adopted from Jose Antonio Martin H.
# <*****@*****.**> online source code
pos, vel = s
if self.domain_fig is None: # Need to initialize the figure
self.domain_fig = plt.figure("Mountain Car Domain")
# plot mountain
mountain_x = np.linspace(self.XMIN, self.XMAX, 1000)
mountain_y = np.sin(3 * mountain_x)
plt.gca(
).fill_between(mountain_x,
min(mountain_y) - self.CAR_HEIGHT * 2,
mountain_y,
color='g')
plt.xlim([self.XMIN - .2, self.XMAX])
plt.ylim(
[min(mountain_y) - self.CAR_HEIGHT * 2,
max(mountain_y) + self.CAR_HEIGHT * 2])
# plot car
self.car = lines.Line2D([], [], linewidth=20, color='b', alpha=.8)
plt.gca().add_line(self.car)
# Goal
plt.plot(self.GOAL, np.sin(3 * self.GOAL), 'yd', markersize=10.0)
plt.axis('off')
plt.gca().set_aspect('1')
self.domain_fig = plt.figure("Mountain Car Domain")
#pos = 0
#a = 0
car_middle_x = pos
car_middle_y = np.sin(3 * pos)
slope = np.arctan(3 * np.cos(3 * pos))
car_back_x = car_middle_x - self.CAR_WIDTH * np.cos(slope) / 2.
car_front_x = car_middle_x + self.CAR_WIDTH * np.cos(slope) / 2.
car_back_y = car_middle_y - self.CAR_WIDTH * np.sin(slope) / 2.
car_front_y = car_middle_y + self.CAR_WIDTH * np.sin(slope) / 2.
self.car.set_data([car_back_x, car_front_x], [car_back_y, car_front_y])
# wheels
# plott(x(1)-0.05,sin(3*(x(1)-0.05))+0.06,'ok','markersize',12,'MarkerFaceColor',[.5 .5 .5]);
# plot(x(1)+0.05,sin(3*(x(1)+0.05))+0.06,'ok','markersize',12,'MarkerFaceColor',[.5 .5 .5]);
# Arrows
if self.actionArrow is not None:
self.actionArrow.remove()
self.actionArrow = None
if self.actions[a] > 0:
self.actionArrow = fromAtoB(
car_front_x, car_front_y,
car_front_x + self.ARROW_LENGTH *
np.cos(slope), car_front_y +
self.ARROW_LENGTH * np.sin(slope),
#car_front_x + self.CAR_WIDTH*cos(slope)/2., car_front_y + self.CAR_WIDTH*sin(slope)/2.+self.CAR_HEIGHT,
'k', "arc3,rad=0",
0, 0, 'simple'
)
if self.actions[a] < 0:
self.actionArrow = fromAtoB(
car_back_x, car_back_y,
car_back_x - self.ARROW_LENGTH *
np.cos(slope), car_back_y -
self.ARROW_LENGTH * np.sin(slope),
#car_front_x + self.CAR_WIDTH*cos(slope)/2., car_front_y + self.CAR_WIDTH*sin(slope)/2.+self.CAR_HEIGHT,
'r', "arc3,rad=0",
0, 0, 'simple'
)
plt.draw()
def batchDiscover(self, td_errors, phi, states):
"""
:param td_errors: p-by-1 vector, error associated with each state
:param phi: p-by-n matrix, vector-valued feature function evaluated at
each state.
:param states: p-by-(statedimension) matrix, each state under test.
Discovers features using OMPTD
1. Find the index of remaining features in the bag \n
2. Calculate the inner product of each feature with the TD_Error vector \n
3. Add the top maxBatchDiscovery features to the selected features \n
OUTPUT: Boolean indicating expansion of features
"""
if len(self.remainingFeatures) == 0:
# No More features to Expand
return False
SHOW_RELEVANCES = 0 # Plot the relevances
self.calculateFullPhiNormalized(states)
relevances = np.zeros(len(self.remainingFeatures))
for i, f in enumerate(self.remainingFeatures):
phi_f = self.fullphi[:, f]
relevances[i] = np.abs(np.dot(phi_f, td_errors))
if SHOW_RELEVANCES:
e_vec = relevances.flatten()
e_vec = e_vec[e_vec != 0]
e_vec = np.sort(e_vec)
plt.plot(e_vec, linewidth=3)
plt.ioff()
plt.show()
plt.ion()
# Sort based on relevances
# We want high to low hence the reverse: [::-1]
sortedIndices = np.argsort(relevances)[::-1]
max_relevance = relevances[sortedIndices[0]]
# Add top <maxDiscovery> features
self.logger.debug("OMPTD Batch: Max Relevance = %0.3f" % max_relevance)
added_feature = False
to_be_deleted = [] # Record the indices of items to be removed
for j in xrange(min(self.maxBatchDiscovery, len(relevances))):
max_index = sortedIndices[j]
f = self.remainingFeatures[max_index]
relevance = relevances[max_index]
# print "Inspecting %s" % str(list(self.iFDD.getFeature(f).f_set))
if relevance >= self.batchThreshold:
self.logger.debug(
'New Feature %d: %s, Relevance = %0.3f' %
(self.features_num, str(np.sort(list(self.iFDD.getFeature(f).f_set))), relevances[max_index]))
to_be_deleted.append(max_index)
self.selectedFeatures.append(f)
self.features_num += 1
added_feature = True
else:
# Because the list is sorted, there is no use to look at the
# others
break
self.remainingFeatures = np.delete(self.remainingFeatures, to_be_deleted)
return added_feature
kernel_args=[kernel_width],
active_threshold=active_threshold,
discover_threshold=discover_threshold,
normalization=False,
max_active_base_feat=100,
max_base_feat_sim=max_base_feat_sim)
policy = SwimmerPolicy(representation)
#policy = eGreedy(representation, epsilon=0.1)
stat_bins_per_state_dim = 20
# agent = SARSA(representation,policy,domain,initial_learn_rate=initial_learn_rate,
# lambda_=.0, learn_rate_decay_mode="boyan", boyan_N0=boyan_N0)
opt["agent"] = SARSA(
policy, representation, discount_factor=domain.discount_factor,
lambda_=lambda_, initial_learn_rate=initial_learn_rate,
learn_rate_decay_mode="boyan", boyan_N0=boyan_N0)
experiment = Experiment(**opt)
return experiment
if __name__ == '__main__':
from rlpy.Tools.run import run_profiled
# run_profiled(make_experiment)
experiment = make_experiment(1)
experiment.run(visualize_performance=1, visualize_learning=True)
# experiment.plot()
# experiment.save()
from rlpy.Tools import plt
plt.figure()
for i in range(9):
plt.plot(experiment.state_counts_learn[i], label="Dim " + str(i))
plt.legend()
def batchDiscover(self, td_errors, phi, states):
# Discovers features using iFDD in batch setting.
# TD_Error: p-by-1 (How much error observed for each sample)
# phi: n-by-p features corresponding to all samples (each column corresponds to one sample)
# self.batchThreshold is the minimum relevance value for the feature to
# be expanded
SHOW_PLOT = 0 # Shows the histogram of relevances
maxDiscovery = self.maxBatchDiscovery
n = self.features_num # number of features
p = len(td_errors) # Number of samples
counts = np.zeros((n, n))
relevances = np.zeros((n, n))
for i in xrange(p):
phiphiT = np.outer(phi[i, :], phi[i,:])
if self.iFDDPlus:
relevances += phiphiT * td_errors[i]
else:
relevances += phiphiT * abs(td_errors[i])
counts += phiphiT
# Remove Diagonal and upper part of the relevances as they are useless
relevances = np.triu(relevances, 1)
non_zero_index = np.nonzero(relevances)
if self.iFDDPlus:
# Calculate relevances based on theoretical results of ICML 2013
# potential submission
relevances[non_zero_index] = np.divide(
np.abs(relevances[non_zero_index]),
np.sqrt(counts[non_zero_index]))
else:
# Based on Geramifard11_ICML Paper
relevances[non_zero_index] = relevances[non_zero_index]
# Find indexes to non-zero excited pairs
# F1 and F2 are the parents of the potentials
(F1, F2) = relevances.nonzero()
relevances = relevances[F1, F2]
if len(relevances) == 0:
# No feature to add
self.logger.debug("iFDD Batch: Max Relevance = 0")
return False
if SHOW_PLOT:
e_vec = relevances.flatten()
e_vec = e_vec[e_vec != 0]
e_vec = np.sort(e_vec)
plt.ioff()
plt.plot(e_vec, linewidth=3)
plt.show()
# Sort based on relevances
# We want high to low hence the reverse: [::-1]
sortedIndices = np.argsort(relevances)[::-1]
max_relevance = relevances[sortedIndices[0]]
# Add top <maxDiscovery> features
self.logger.debug(
"iFDD Batch: Max Relevance = {0:g}".format(max_relevance))
added_feature = False
new_features = 0
for j in xrange(len(relevances)):
if new_features >= maxDiscovery:
break
max_index = sortedIndices[j]
f1 = F1[max_index]
f2 = F2[max_index]
relevance = relevances[max_index]
if relevance > self.batchThreshold:
# print "Inspecting",
# f1,f2,'=>',self.getStrFeatureSet(f1),self.getStrFeatureSet(f2)
if self.inspectPair(f1, f2, np.inf):
self.logger.debug(
'New Feature %d: %s, Relevance = %0.3f' %
(self.features_num - 1, self.getStrFeatureSet(self.features_num - 1), relevances[max_index]))
new_features += 1
added_feature = True
else:
# Because the list is sorted, there is no use to look at the
# others
break
return (
# A signal to see if the representation has been expanded or not
added_feature
)
kernel_args=[kernel_width],
active_threshold=active_threshold,
discover_threshold=discover_threshold,
normalization=False,
max_active_base_feat=100,
max_base_feat_sim=max_base_feat_sim)
policy = SwimmerPolicy(representation)
#policy = eGreedy(representation, epsilon=0.1)
stat_bins_per_state_dim = 20
# agent = SARSA(representation,policy,domain,initial_learn_rate=initial_learn_rate,
# lambda_=.0, learn_rate_decay_mode="boyan", boyan_N0=boyan_N0)
opt["agent"] = SARSA(
policy, representation, discount_factor=domain.discount_factor,
lambda_=lambda_, initial_learn_rate=initial_learn_rate,
learn_rate_decay_mode="boyan", boyan_N0=boyan_N0)
experiment = Experiment(**opt)
return experiment
if __name__ == '__main__':
from rlpy.Tools.run import run_profiled
# run_profiled(make_experiment)
experiment = make_experiment(1)
experiment.run(visualize_performance=1, visualize_learning=True)
# experiment.plot()
# experiment.save()
from rlpy.Tools import plt
plt.figure()
for i in range(9):
plt.plot(experiment.state_counts_learn[i], label="Dim " + str(i))
plt.legend()
