def plot_features(self, plot_kind="3D", dim=None, row_size=5, show=True):
"""
This function plot the features vector of
all candidates in the given dimensions.
Parameters
----------
plot_kind : str
The kind of plot we want to show.
Can be ``'3D'`` or ``'ternary'``.
dim : list
The 3 dimensions we are using for our plot.
By default, it is set to ``[0, 1, 2]``.
row_size : int
The number of subplots by row.
By default, it is set to 5 plots by row.
show : bool
If True, plot the figure
at the end of the function.
"""
if dim is None:
dim = [0, 1, 2]
else:
if len(dim) != 3:
raise ValueError("The number of dimensions should be 3")
n_candidates = self.ratings_.n_candidates
n_rows = (n_candidates - 1) // row_size + 1
fig = plt.figure(figsize=(row_size * 5, n_rows * 5))
plot_position = [n_rows, row_size, 1]
features = self.features_
for candidate in range(n_candidates):
ax = self.embeddings_.plot_candidate(self.ratings_,
candidate,
plot_kind=plot_kind,
dim=dim,
fig=fig,
plot_position=plot_position,
show=False)
if plot_kind == "3D":
x1 = features[candidate, dim[0]]
x2 = features[candidate, dim[1]]
x3 = features[candidate, dim[2]]
ax.plot([0, x1], [0, x2], [0, x3], color='k', linewidth=2)
ax.scatter([x1], [x2], [x3], color='k', s=5)
elif plot_kind == "ternary":
x1 = features[candidate, dim[0]]
x2 = features[candidate, dim[2]]
x3 = features[candidate, dim[1]]
feature_bis = [x1, x2, x3]
feature_bis = np.maximum(feature_bis, 0)
size_features = np.linalg.norm(feature_bis)
feature_bis = normalize(feature_bis)
ax.scatter([feature_bis**2],
color='k',
s=50 * size_features + 1)
plot_position[2] += 1
if show:
plt.show() # pragma: no cover
def _plot_ternary(self, fig, dim, plot_position=None):
"""
Plot a figure of the ratings on a 2D space
representing the surface of the unit sphere
on the non-negative orthant.
Parameters
----------
fig : matplotlib figure
The figure on which we add the plot.
dim : list
The 3 dimensions we are using for our plot.
plot_position : list
The position of the plot on the figure.
Should be of the form
``[n_rows, n_columns, position]``.
Return
------
matplotlib ax
The matplotlib ax with the figure,
if you want to add something to it.
"""
tax = create_ternary_plot(fig, plot_position)
for i, v in enumerate(self):
x1 = v[dim[0]]
x2 = v[dim[2]]
x3 = v[dim[1]]
vec = [x1, x2, x3]
tax.scatter([normalize(vec)**2], color=(x1**2 * 0.8, x3**2 * 0.8, x2**2 * 0.8), alpha=0.9, s=30)
return tax
def _get_center(self):
"""
Return the center of the ratings, computed
as the center of the :attr:`n_dim`-dimensional
cube of maximal volume.
Return
------
np.ndarray
The position of the center vector. Should be of length :attr:`n_dim`.
"""
positions = np.array(self)
matrix_rank = np.linalg.matrix_rank(positions)
volume = 0
n_voters = self.n_voters
current_subset = list(np.arange(matrix_rank))
mean = np.zeros(self.n_dim)
while current_subset[0] <= n_voters - matrix_rank:
current_embeddings = positions[current_subset, ...]
new_volume = np.sqrt(np.linalg.det(np.dot(current_embeddings, current_embeddings.T)))
if new_volume > volume:
volume = new_volume
mean = normalize(current_embeddings.sum(axis=0))
x = 1
while current_subset[matrix_rank - x] == n_voters - x:
x += 1
val = current_subset[matrix_rank - x] + 1
while x > 0:
current_subset[matrix_rank - x] = val
val += 1
x -= 1
return mean
def __call__(self, polarisation=0.0):
"""
Update the parameter of the parametric embeddings
and create a new ratings.
Parameters
_________
polarisation : float
Should be between `0` and `1`.
If it is equal to `0`, then the
embeddings are uniformly distributed.
If it is equal to `1`, then each voter's
embeddings align to the dimension of its group.
Return
------
Embeddings
The embeddings generated
Examples
--------
>>> np.random.seed(42)
>>> generator = EmbeddingsGeneratorPolarized(100, 3)
>>> embs = generator(.8)
>>> embs.voter_embeddings(0)
array([0.12915167, 0.03595039, 0.99097296])
"""
if polarisation > 1 or polarisation < 0:
raise ValueError("Polarisation should be between 0 and 1")
n = len(self._thetas)
positions = np.zeros((n, self.n_dim))
for i in range(n):
p_1 = np.dot(self._orthogonal_profile[i],
self._random_profile[i]) * self._orthogonal_profile[i]
p_2 = self._random_profile[i] - p_1
e_2 = normalize(p_2)
positions[i] = self._orthogonal_profile[i] * np.cos(
self._thetas[i] *
(1 - polarisation)) + e_2 * np.sin(self._thetas[i] *
(1 - polarisation))
return Embeddings(positions)
def plot_scores_evolution(self, show=True):
"""
This function plot the evolution
of the scores of the candidates
when the moving voters' embeddings
are changing.
Parameters
----------
show : bool
If True, displays the figure
at the end of the function.
Examples
--------
>>> p = MovingVoter()(SVDNash())
>>> p.plot_scores_evolution(show=False)
"""
tab_x = np.linspace(0, 1, 50)
tab_y = []
for x in tab_x:
self.embeddings[self.moving_voter] = normalize([1 - x, x, 0])
tab_y.append(
self.rule(self.ratings_, self.embeddings).scores_float_)
tab_y = np.array(tab_y).T
name = ["Start", "End", "Orthogonal", "Consensus"]
for i in range(self.ratings_.n_candidates):
plt.plot(tab_x, tab_y[i], label=name[i])
plt.title("Evolution of the score")
plt.xlabel("X coordinate of moving voter")
plt.ylabel("Score")
plt.xlim(0, 1)
plt.legend()
if show:
plt.show() # pragma: no cover
def _build_profiles(self):
"""
This function build the two embeddings
of the parametrized embeddings (uniform and orthogonal).
Return
------
EmbeddingsGeneratorPolarized
The object itself.
"""
for i in range(self.n_voters):
new_vec = np.abs(np.random.randn(self.n_dim))
r = np.argmax(new_vec * self.prob)
new_vec = normalize(new_vec)
self._orthogonal_profile[i, r] = 1
self._random_profile[i] = new_vec
theta = np.arccos(
np.dot(self._random_profile[i], self._orthogonal_profile[i]))
self._thetas[i] = theta
return self
def _plot_scores_ternary(self, sizes, fig, plot_position, dim):
"""
Plot a figure of the ratings on a 2D space
representing the sphere in the non-negative orthant,
with the voters dots having the sizes passed
as parameters.
Parameters
----------
sizes : np.ndarray
The size of the dots.
Should be of length :attr:`n_voters`.
fig : matplotlib figure
The figure on which we add the plot.
plot_position : list
The position of the plot on the figure. Should be of the form
``[n_rows, n_columns, position]``.
dim : list
The 3 dimensions we are using for our plot.
Return
------
matplotlib ax
The matplotlib ax with the figure,
if you want to add something to it.
"""
tax = create_ternary_plot(fig, plot_position)
for i, (v, s) in enumerate(zip(np.array(self), sizes)):
x1 = v[dim[0]]
x2 = v[dim[1]]
x3 = v[dim[2]]
vec = [x1, x2, x3]
tax.scatter([normalize(vec)**2], color=(x1**2 * 0.8, x3**2 * 0.8, x2**2 * 0.8), alpha=0.7, s=max(s * 50, 1))
return tax
def plot_features_evolution(self, show=True):
"""
This function plot the evolution
of the features of the candidates
when the moving voters' embeddings
are changing. Only works for :class:`SVDMax` and :class:`FeaturesRule`.
Parameters
----------
show : bool
If True, displays the figure
at the end of the function.
Examples
--------
>>> p = MovingVoter()(SVDMax())
>>> p.plot_features_evolution(show=False)
"""
tab_x = np.linspace(0, 1, 20)
tab_y = []
for x in tab_x:
self.embeddings[self.moving_voter] = normalize([1 - x, x, 0])
tab_y.append(self.rule(self.ratings_, self.embeddings).features_)
tab_y = np.array(tab_y)
fig = plt.figure(figsize=(10, 5))
ax = create_3D_plot(fig, position=[1, 2, 1])
name = ["Start", "End", "Orthogonal", "Consensus"]
for i in range(self.ratings_.n_candidates):
vec_init = normalize(tab_y[0, i])**2
ax.plot(tab_y[::, i, 0],
tab_y[::, i, 1],
tab_y[::, i, 2],
color=(vec_init[0] * 0.8, vec_init[1] * 0.8,
vec_init[2] * 0.8),
alpha=0.5,
label=name[i])
for j, v in enumerate(tab_y[::, i]):
vec_normalized = normalize(v)
ax.plot([0, v[0]], [0, v[1]], [0, v[2]],
color=(vec_normalized[0] * 0.8,
vec_normalized[1] * 0.8,
vec_normalized[2] * 0.8),
alpha=j / 60)
ax.set_title("Evolution of the features")
plt.legend()
tax = create_ternary_plot(fig, position=[1, 2, 2])
for i in range(self.ratings_.n_candidates):
points = [normalize(x[[0, 2, 1]])**2 for x in tab_y[::, i]]
vec_init = normalize(tab_y[0, i, [0, 2, 1]])**2
tax.plot(points,
color=(vec_init[0] * 0.8, vec_init[2] * 0.8,
vec_init[1] * 0.8),
alpha=0.8)
tax.scatter([vec_init],
color=(vec_init[0] * 0.8, vec_init[2] * 0.8,
vec_init[1] * 0.8),
alpha=0.7,
s=50)
if show:
plt.show() # pragma: no cover
def plot_weights(self, plot_kind="3D", dim=None, row_size=5, verbose=True, show=True):
"""
This function plot the evolution of
the voters' weights after each step of the rule.
Parameters
----------
plot_kind : str
The kind of plot we want to show.
Can be ``3D`` or ``ternary``.
dim : list
The 3 dimensions we are using for our plot.
By default, it is set to ``[0, 1, 2]``.
row_size : int
Number of subplots by row.
By default, it is set to 5.
verbose : bool
If True, print the total weight divided by
the number of remaining candidates at
the end of each step.
show : bool
If True, displays the figure
at the end of the function.
"""
ls_weight = self._ruleResults["weights_list"]
vectors = self._ruleResults["vectors"]
n_candidates = len(ls_weight)
n_rows = (n_candidates - 1) // row_size + 1
fig = plt.figure(figsize=(5 * row_size, n_rows * 5))
plot_position = [n_rows, row_size, 1]
if dim is None:
dim = [0, 1, 2]
for i in range(n_candidates):
ax = self.embeddings.plot_scores(ls_weight[i],
plot_kind=plot_kind,
title="Step %i" % i,
dim=dim,
fig=fig,
plot_position=plot_position,
show=False)
if i < n_candidates - 1:
x1 = vectors[i][dim[0]]
x2 = vectors[i][dim[1]]
x3 = vectors[i][dim[2]]
if plot_kind == "3D":
ax.plot([0, x1], [0, x2], [0, x3], color='k', linewidth=2)
ax.scatter([x1], [x2], [x3], color='k', s=5)
elif plot_kind == "ternary":
feature_bis = [x1, x3, x2]
feature_bis = np.maximum(feature_bis, 0)
size_features = np.linalg.norm(feature_bis)
feature_bis = normalize(feature_bis)
ax.scatter([feature_bis ** 2], color='k', s=50*size_features+1)
plot_position[2] += 1
if verbose:
sum_w = [ls_weight[i].sum() / (n_candidates - i - 1) for i in range(n_candidates - 1)]
print("Weight / remaining candidate : ", sum_w)
if show:
plt.show() # pragma: no cover
def recenter(self, approx=True):
"""
Recenter the embeddings of the
voters so that they are the most
possible on the non-negative orthant.
Parameters
----------
approx : bool
If True, we compute the center of the population
with a polynomial time algorithm. If False, we use
an algorithm exponential in :attr:`n_dim`.
Return
------
Embeddings
The embeddings itself.
Examples
--------
>>> embs = Embeddings(-np.array([[.5,.9,.4],[.4,.7,.5],[.4,.2,.4]])).normalize()
>>> embs
Embeddings([[-0.45267873, -0.81482171, -0.36214298],
[-0.42163702, -0.73786479, -0.52704628],
[-0.66666667, -0.33333333, -0.66666667]])
>>> embs.recenter()
Embeddings([[0.40215359, 0.75125134, 0.52334875],
[0.56352875, 0.6747875 , 0.47654713],
[0.70288844, 0.24253193, 0.66867489]])
"""
if self.n_voters < 2:
raise ValueError("Cannot recenter a ratings with less than 2 candidates")
positions = np.array(self)
if approx:
center = normalize(positions.sum(axis=0))
else:
center = self._get_center()
target_center = np.ones(self.n_dim)
target_center = normalize(target_center)
if np.dot(center, target_center) == -1:
new_positions = - self
elif np.dot(center, target_center) == 1:
new_positions = self
else:
orthogonal_center = center - np.dot(center, target_center)*target_center
orthogonal_center = normalize(orthogonal_center)
theta = -np.arccos(np.dot(center, target_center))
rotation_matrix = np.array([[np.cos(theta), - np.sin(theta)], [np.sin(theta), np.cos(theta)]])
new_positions = np.zeros((self.n_voters, self.n_dim))
for i in range(self.n_voters):
position_i = self.voter_embeddings(i)
comp_1 = position_i.dot(target_center)
comp_2 = position_i.dot(orthogonal_center)
vector = [comp_1, comp_2]
remainder = position_i - comp_1*target_center - comp_2*orthogonal_center
new_vector = rotation_matrix.dot(vector)
new_positions[i] = new_vector[0]*target_center + new_vector[1]*orthogonal_center + remainder
return Embeddings(new_positions, False)
def dilate(self, approx=True):
"""
Dilate the embeddings of the
voters so that they take all
the space possible in the non-negative orthant.
Parameters
----------
approx : bool
If True, we compute the center of the population
with a polynomial time algorithm. If False, we use
an algorithm exponential in :attr:`n_dim`.
Return
------
Embeddings
The embeddings itself.
Examples
--------
>>> embs = Embeddings(np.array([[.5,.4,.4],[.4,.4,.5],[.4,.5,.4]])).normalize()
>>> embs
Embeddings([[0.66226618, 0.52981294, 0.52981294],
[0.52981294, 0.52981294, 0.66226618],
[0.52981294, 0.66226618, 0.52981294]])
>>> embs.dilate()
Embeddings([[0.98559856, 0.11957316, 0.11957316],
[0.11957316, 0.11957316, 0.98559856],
[0.11957316, 0.98559856, 0.11957316]])
"""
if self.n_voters < 2:
raise ValueError("Cannot dilate a ratings with less than 2 candidates")
positions = np.array(self)
if approx:
center = normalize(positions.sum(axis=0))
else:
center = self._get_center()
min_value = np.dot(self.voter_embeddings(0), center)
for i in range(self.n_voters):
val = np.dot(self.voter_embeddings(i), center)
if val < min_value:
min_value = val
theta_max = np.arccos(min_value)
k = np.pi / (4 * theta_max)
new_positions = np.zeros((self.n_voters, self.n_dim))
for i in range(self.n_voters):
v_i = self.voter_embeddings(i)
theta_i = np.arccos(np.dot(v_i, center))
if theta_i == 0:
new_positions[i] = v_i
else:
p_1 = np.dot(center, v_i) * center
p_2 = v_i - p_1
e_2 = normalize(p_2)
new_positions[i] = center * np.cos(k * theta_i) + e_2 * np.sin(k * theta_i)
return Embeddings(new_positions, False)
