def test_quantile_regression_diff_quantile(self):
X = numpy.array([[0.1], [0.2], [0.3], [0.4], [0.5], [0.6]])
Y = numpy.array([1., 1.11, 1.21, 10, 1.29, 1.39])
clqs = []
scores = []
for q in [0.25, 0.4999, 0.5, 0.5001, 0.75]:
clq = QuantileLinearRegression(verbose=False,
fit_intercept=True,
quantile=q)
clq.fit(X, Y)
clqs.append(clq)
sc = clq.score(X, Y)
scores.append(sc)
self.assertGreater(sc, 0)
self.assertLesser(abs(clqs[1].intercept_ - clqs[2].intercept_), 0.01)
self.assertLesser(abs(clqs[2].intercept_ - clqs[3].intercept_), 0.01)
self.assertLesser(abs(clqs[1].coef_[0] - clqs[2].coef_[0]), 0.01)
self.assertLesser(abs(clqs[2].coef_[0] - clqs[3].coef_[0]), 0.01)
self.assertGreater(abs(clqs[0].intercept_ - clqs[1].intercept_), 0.01)
# self.assertGreater(abs(clqs[3].intercept_ - clqs[4].intercept_), 0.01)
self.assertGreater(abs(clqs[0].coef_[0] - clqs[1].coef_[0]), 0.05)
# self.assertGreater(abs(clqs[3].coef_[0] - clqs[4].coef_[0]), 0.05)
self.assertLesser(abs(scores[1] - scores[2]), 0.01)
self.assertLesser(abs(scores[2] - scores[3]), 0.01)
def voronoi_estimation_from_lr(L,
B,
C=None,
D=None,
cl=0,
qr=True,
max_iter=None,
verbose=False):
"""
Determines a Voronoi diagram close to a convex
partition defined by a logistic regression in *n* classes.
:math:`M \\in \\mathbb{M}_{nd}` a row matrix :math:`(L_1, ..., L_n)`.
Every border between two classes *i* and *j* is defined by:
:math:`\\scal{L_i}{X} + B = \\scal{L_j}{X} + B`.
The function looks for a set of points from which the Voronoi
diagram can be inferred. It is done through a linear regression
with norm *L1*. See :ref:`l-lrvor-connection`.
@param L matrix
@param B vector
@param C additional conditions (see below)
@param D addition condictions (see below)
@param cl class on which the additional conditions applies
@param qr use quantile regression
@param max_iter number of condition to remove until convergence
@param verbose display information while training
@return matrix :math:`P \\in \\mathbb{M}_{nd}`
The function solves the linear system:
.. math::
\\begin{array}{rcl}
& \\Longrightarrow & \\left\\{\\begin{array}{l}\\scal{\\frac{L_i-L_j}{\\norm{L_i-L_j}}}{P_i + P_j} +
2 \\frac{B_i - B_j}{\\norm{L_i-L_j}} = 0 \\\\
\\scal{P_i- P_j}{u_{ij}} - \\scal{P_i - P_j}{\\frac{L_i-L_j}{\\norm{L_i-L_j}}} \\scal{\\frac{L_i-L_j}{\\norm{L_i-L_j}}}{u_{ij}}=0
\\end{array} \\right.
\\end{array}
If the number of dimension is big and
the number of classes small, the system has
multiple solution. Addition condition must be added
such as :math:`CP_i=D` where *i=cl*, :math:`P_i`
is the Voronoï point attached to class *cl*.
`Quantile regression <https://fr.wikipedia.org/wiki/R%C3%A9gression_quantile>`_
is not implemented in :epkg:`scikit-learn`.
We use `QuantileLinearRegression <http://www.xavierdupre.fr/app/mlinsights/helpsphinx/mlinsights/mlmodel/quantile_regression.html>`_.
After the first iteration, the function determines
the furthest pair of points and removes it from the list
of equations. If *max_iter* is None, the system goes until
the number of equations is equal to the number of points * 2,
otherwise it stops after *max_iter* removals. This is not the
optimal pair to remove as they could still be neighbors but it
should be a good heuristic.
"""
labels_inv = {}
nb_constraints = numpy.zeros((L.shape[0], ))
matL = []
matB = []
for i in range(0, L.shape[0]):
for j in range(i + 1, L.shape[0]):
li = L[i, :]
lj = L[j, :]
c = (li - lj)
nc = (c.T @ c)**0.5
# first condition
mat = numpy.zeros((L.shape))
mat[i, :] = c
mat[j, :] = c
d = -2 * (B[i] - B[j])
matB.append(d)
matL.append(mat.ravel())
labels_inv[i, j, 'eq1'] = len(matL) - 1
nb_constraints[i] += 1
nb_constraints[j] += 1
# condition 2 - hides multiple equation
# we pick one
coor = 0
found = False
while not found and coor < len(c):
if c[coor] == 0:
coor += 1
continue
if c[coor] == nc:
coor += 1
continue
found = True
if not found:
raise ValueError(
"Matrix L has two similar rows {0} and {1}. Problem cannot be solved."
.format(i, j))
c /= nc
c2 = c * c[coor]
mat = numpy.zeros((L.shape))
mat[i, :] = -c2
mat[j, :] = c2
mat[i, coor] += 1
mat[j, coor] -= 1
matB.append(0)
matL.append(mat.ravel())
labels_inv[i, j, 'eq2'] = len(matL) - 1
nb_constraints[i] += 1
nb_constraints[j] += 1
nbeq = (L.shape[0] * (L.shape[0] - 1)) // 2
matL = numpy.array(matL)
matB = numpy.array(matB)
if max_iter is None:
max_iter = matL.shape[0] - matL.shape[1]
if nbeq * 2 <= L.shape[0] * L.shape[1]:
if C is None and D is None:
warnings.warn(
"[voronoi_estimation_from_lr] Additional condition are required."
)
if C is not None and D is not None:
matL = numpy.vstack([matL, numpy.zeros((1, matL.shape[1]))])
a = cl * L.shape[1]
b = a + L.shape[1]
matL[-1, a:b] = C
if not isinstance(D, float):
raise TypeError("D must be a float not {0}".format(type(D)))
matB = numpy.hstack([matB, [D]])
elif C is None and D is None:
pass
else:
raise ValueError(
"C and D must be None together or not None together.")
sample_weight = numpy.ones((matL.shape[0], ))
tol = numpy.abs(matL.ravel()).max() * 1e-8 / matL.shape[0]
order_removed = []
removed = set()
for it in range(0, max(max_iter, 1)):
if qr:
clr = QuantileLinearRegression(fit_intercept=False,
max_iter=max(matL.shape))
else:
clr = LinearRegression(fit_intercept=False)
clr.fit(matL, matB, sample_weight=sample_weight)
score = clr.score(matL, matB, sample_weight)
res = clr.coef_
res = res.reshape(L.shape)
# early stopping
if score < tol:
if verbose:
print(
'[voronoi_estimation_from_lr] iter={0}/{1} score={2} tol={3}'
.format(it + 1, max_iter, score, tol))
break
# defines the best pair of points to remove
dist2 = pairwise_distances(res, res)
dist = [(d, n // dist2.shape[0], n % dist2.shape[1])
for n, d in enumerate(dist2.ravel())]
dist = [_ for _ in dist if _[1] < _[2]]
dist.sort(reverse=True)
# test equal points
if dist[-1][0] < tol:
_, i, j = dist[-1]
eq1 = labels_inv[i, j, 'eq1']
eq2 = labels_inv[i, j, 'eq2']
if sample_weight[eq1] == 0 and sample_weight[eq2] == 0:
sample_weight[eq1] = 1
sample_weight[eq2] = 1
nb_constraints[i] += 1
nb_constraints[j] += 1
else:
keep = (i, j)
pos = len(order_removed) - 1
while pos >= 0:
i, j = order_removed[pos]
if i in keep or j in keep:
eq1 = labels_inv[i, j, 'eq1']
eq2 = labels_inv[i, j, 'eq2']
if sample_weight[eq1] == 0 and sample_weight[eq2] == 0:
sample_weight[eq1] = 1
sample_weight[eq2] = 1
nb_constraints[i] += 1
nb_constraints[j] += 1
break
pos -= 1
if pos < 0:
forma = 'Two classes have been merged in a single Voronoi point (dist={0} < {1}). max_iter should be lower than {2}'
raise VoronoiEstimationError(
forma.format(dist[-1][0], tol, it))
dmax, i, j = dist[0]
pos = 0
while (
i, j
) in removed or nb_constraints[i] == 0 or nb_constraints[j] == 0:
pos += 1
if pos == len(dist):
break
dmax, i, j = dist[pos]
if pos == len(dist):
break
removed.add((i, j))
order_removed.append((i, j))
eq1 = labels_inv[i, j, 'eq1']
eq2 = labels_inv[i, j, 'eq2']
sample_weight[eq1] = 0
sample_weight[eq2] = 0
nb_constraints[i] -= 1
nb_constraints[j] -= 1
if verbose:
print(
'[voronoi_estimation_from_lr] iter={0}/{1} score={2:.3g} tol={3:.3g} del P{4},{5} d={6:.3g}'
.format(it + 1, max_iter, score, tol, i, j, dmax))
return res