Contenido
El significado principal del uso de la métrica de Mahalanobis
1. Términos y definiciones
2. Distancia de Mahalanobis entre dos puntos y entre un punto y una clase
2.1. Información teórica
2.2. Algoritmo para calcular la distancia entre dos puntos y entre un punto y una clase
2.3. Un ejemplo de cálculo de la distancia entre dos puntos y entre un punto y una clase
3. Distancia de Mahalanobis entre dos clases
3.1. Información teórica
3.2. Algoritmo para calcular la distancia entre dos clases
3.3. Un ejemplo de cálculo de la distancia entre dos clases
4. Distancia de Mahalanobis y método de k vecinos más cercanos
5. Distancia ponderada de Mahalanobis
6. Conclusión
Si tiene algún comentario o error, escriba a quwarm@gmail.com o en los comentarios.
El punto principal de usar la distancia de Mahalanobis
En la Figura 1, dos observaciones se representan como puntos rojos.
El centro de la clase se muestra como un punto azul.
La pregunta es ¿qué observación está más cerca del centro de la clase?
La respuesta depende de cómo se mida la distancia.
Si medimos la distancia según la métrica euclidiana , obtenemos que la distancia del centro de la clase al punto es igual a , al punto es igual , es decir , el punto está más cerca del centro de la clase.
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Python 3.6 numpy 1.19.5
import numpy as np
classes = [
np.array([[1, 1], [2, 2], [3, 3], [4, 4], [5, 5]]),
np.array([[3, 1], [4, 0], [6, 0], [6, 2], [5, 3]])
]
centroids = [class_.mean(axis=0) for class_ in classes]
standard_deviations = [class_.std(axis=0, ddof=1) for class_ in classes]
covariance_matrices = np.array([np.cov(class_, rowvar=False, ddof=1) for class_ in classes])
det_covariance_matrices = [np.linalg.det(cov) for cov in covariance_matrices]
print("Centroids:", *centroids)
print("Standard deviations:", *standard_deviations)
print("Covariance matrices:", *covariance_matrices.tolist())
print("Determinants of covariance matrices:", det_covariance_matrices)
:
Centroids: [3. 3.] [4.8 1.2] Standard deviations: [1.58113883 1.58113883] [1.30384048 1.30384048] Covariance matrices: [[2.5, 2.5], [2.5, 2.5]] [[1.7, 0.3], [0.3, 1.7]] Determinants of covariance matrices: [0.0, 2.8]
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Python 3.6 numpy 1.19.5
import numpy as np
def mahalanobis(point_from, point_to, inverse_covariance_matrix):
delta = point_from - point_to
return max(np.float64(0), np.dot(np.dot(delta, inverse_covariance_matrix), delta)) ** 0.5
test_point = np.array([4., 2.])
class_ = np.array([[1., 1.], [2., 2.], [3., 3.], [4., 4.], [5., 5.]])
covariance_matrix = np.cov(class_, rowvar=False, ddof=1)
inverse_covariance_matrix = np.linalg.inv(covariance_matrix + np.identity(covariance_matrix.shape[0]))
print(" :\n", inverse_covariance_matrix, sep='')
for point_to in [class_.mean(axis=0), *class_]:
print("d_E-M (", test_point, ", ", point_to, ", (COV+E)^(-1)) = ",
mahalanobis(test_point, point_to, inverse_covariance_matrix), sep='')
:
: [[ 0.58333333 -0.41666667] [-0.41666667 0.58333333]] d_E-M ([4. 2.], [3. 3.], (COV+E)^(-1)) = 1.414213562373095 d_E-M ([4. 2.], [1. 1.], (COV+E)^(-1)) = 1.8257418583505538 d_E-M ([4. 2.], [2. 2.], (COV+E)^(-1)) = 1.5275252316519465 d_E-M ([4. 2.], [3. 3.], (COV+E)^(-1)) = 1.414213562373095 d_E-M ([4. 2.], [4. 4.], (COV+E)^(-1)) = 1.5275252316519465 d_E-M ([4. 2.], [5. 5.], (COV+E)^(-1)) = 1.8257418583505536
— , .
2. (LedoitWolf).
Python 3.6 numpy 1.19.5 scikit-learn 0.24.1
import numpy as np
from sklearn.covariance import LedoitWolf
def mahalanobis(point_from, point_to, inverse_covariance_matrix):
delta = point_from - point_to
return max(np.float64(0), np.dot(np.dot(delta, inverse_covariance_matrix), delta)) ** 0.5
def approx(number, *, sign, epsilon=1e-4):
return number + np.sign(sign) * epsilon
test_point = np.array([4., 2.])
class_ = np.array([[1., 1.], [2., 2.], [3., 3.], [4., 4.], [5., 5.]])
lw = LedoitWolf().fit(class_)
lw_covariance_matrix = lw.covariance_
lw_lambda = lw.shrinkage_
covariance_matrix = np.cov(class_, rowvar=False, ddof=0)
mu = np.sum(np.trace(covariance_matrix)) / class_.shape[0]
T = mu * np.identity(class_.shape[1])
print("T:", *T)
print("COV(*):", *lw_covariance_matrix)
print("Lambda:", lw_lambda)
# - T
# ( : T )
# ddof=0, . . LedoitWolf empirical_covariance (. )
first_condition = (np.linalg.eig(T)[0] > approx(0., sign=+1)).all()
print("All(", np.linalg.eig(T)[0], ") > 0 ? -> ", first_condition, sep='')
# - (0, 1]
second_condition = approx(0., sign=+1) < lw_lambda <= 1
print("Lambda =", lw_lambda, "in (0, 1] ? ->", second_condition)
# - COV(*)
# lambda, T
cov_eig = min(np.linalg.eig(lw_covariance_matrix)[0])
lambda_t_eig = lw_lambda * min(np.linalg.eig(T)[0])
third_condition = cov_eig >= lambda_t_eig
print(cov_eig, ">=", lambda_t_eig, "? ->", third_condition)
conditions = [first_condition, second_condition, third_condition]
if all(conditions):
print(" ")
#
inverse_lw_covariance_matrix = np.linalg.inv(lw_covariance_matrix)
print(" :\n", inverse_lw_covariance_matrix, sep='')
for point_to in [class_.mean(axis=0), *class_]:
print("d_M(*) (", test_point, ", ", point_to, ", COV(*)) = ",
mahalanobis(test_point, point_to, inverse_lw_covariance_matrix), sep='')
else:
print(" (1-3): ", [i for i, x in enumerate(conditions, 1) if not x])
:
T: [0.8 0. ] [0. 0.8] COV(*): [2. 1.44] [1.44 2. ] Lambda: 0.27999999999999997 All([0.8 0.8]) > 0 ? -> True Lambda = 0.27999999999999997 in (0, 1] ? -> True 0.56 >= 0.22399999999999998 ? -> True : [[ 1.03820598 -0.74750831] [-0.74750831 1.03820598]] d_M(*) ([4. 2.], [3. 3.], COV(*)) = 1.889822365046136 d_M(*) ([4. 2.], [1. 1.], COV(*)) = 2.4283759936997833 d_M(*) ([4. 2.], [2. 2.], COV(*)) = 2.037847864848056 d_M(*) ([4. 2.], [3. 3.], COV(*)) = 1.889822365046136 d_M(*) ([4. 2.], [4. 4.], COV(*)) = 2.037847864848056 d_M(*) ([4. 2.], [5. 5.], COV(*)) = 2.4283759936997833
— , .
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Python 3.6 numpy 1.19.5
import numpy as np
def mahalanobis(point_from, point_to, inverse_covariance_matrix):
delta = point_from - point_to
return max(np.float64(0), np.dot(np.dot(delta, inverse_covariance_matrix), delta)) ** 0.5
test_point = np.array([4., 2.])
class_ = np.array([[1., 1.], [2., 2.], [3., 3.], [4., 4.], [5., 5.]])
covariance_matrix = np.cov(class_, rowvar=False, ddof=1)
# (Singular Value Decomposition, SVD)
#
pseudo_inverse_covariance_matrix = np.linalg.pinv(covariance_matrix)
print(" :\n", pseudo_inverse_covariance_matrix, sep='')
for point_to in [class_.mean(axis=0), *class_]:
print("d_M+ (", test_point, ", ", point_to, ", COV+) = ",
mahalanobis(test_point, point_to, pseudo_inverse_covariance_matrix), sep='')
:
: [[0.1 0.1] [0.1 0.1]] d_M+ ([4. 2.], [3. 3.], COV+) = 0.0 d_M+ ([4. 2.], [1. 1.], COV+) = 1.2649110640673513 d_M+ ([4. 2.], [2. 2.], COV+) = 0.6324555320336757 d_M+ ([4. 2.], [3. 3.], COV+) = 0.0 d_M+ ([4. 2.], [4. 4.], COV+) = 0.6324555320336757 d_M+ ([4. 2.], [5. 5.], COV+) = 1.2649110640673513
— , .
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Python 3.6 numpy 1.19.5
import numpy as np
def euclid_std(point_from, point_to, standard_deviations):
return sum(((point_from - point_to) / standard_deviations) ** 2) ** 0.5
def approx(number, *, sign, epsilon=1e-4):
return number + np.sign(sign) * epsilon
test_point = np.array([4., 2.])
class_ = np.array([[1., 1.], [2., 2.], [3., 3.], [4., 4.], [5., 5.]])
standard_deviations = class_.std(axis=0, ddof=1)
# 0
std_le_0 = standard_deviations <= approx(0., sign=+1, epsilon=1e-6)
print(" :\n", standard_deviations, sep='')
if std_le_0.any():
print(" 0: ", np.where(std_le_0)[0])
else:
for point_to in [class_.mean(axis=0), *class_]:
print("d_std (", test_point, ", ", point_to, ", sigma) = ",
euclid_std(test_point, point_to, standard_deviations), sep='')
:
: [1.58113883 1.58113883] d_std ([4. 2.], [3. 3.], sigma) = 0.8944271909999159 d_std ([4. 2.], [1. 1.], sigma) = 1.9999999999999998 d_std ([4. 2.], [2. 2.], sigma) = 1.2649110640673518 d_std ([4. 2.], [3. 3.], sigma) = 0.8944271909999159 d_std ([4. 2.], [4. 4.], sigma) = 1.2649110640673518 d_std ([4. 2.], [5. 5.], sigma) = 1.9999999999999998
— , .
5. .
Python 3.6 numpy 1.19.5
import numpy as np
def euclid_std(point_from, point_to, standard_deviations):
return sum(((point_from - point_to) / standard_deviations) ** 2) ** 0.5
def euclid_diag(point_from, point_to, standard_deviations):
return euclid_std(point_from, point_to, standard_deviations) \
* (np.prod(standard_deviations ** 2)) ** (1. / point_from.shape[0])
def approx(number, *, sign, epsilon=1e-4):
return number + np.sign(sign) * epsilon
test_point = np.array([4., 2.])
class_ = np.array([[1., 1.], [2., 2.], [3., 3.], [4., 4.], [5., 5.]])
standard_deviations = class_.std(axis=0, ddof=1)
# 0
std_le_0 = standard_deviations <= approx(0., sign=+1, epsilon=1e-6)
print(" :\n", standard_deviations, sep='')
if std_le_0.any():
print(" 0: ", np.where(std_le_0)[0])
else:
for point_to in [class_.mean(axis=0), *class_]:
print("d_diag (", test_point, ", ", point_to, ", sigma) = ",
euclid_diag(test_point, point_to, standard_deviations), sep='')
:
: [1.58113883 1.58113883] d_diag ([4. 2.], [3. 3.], sigma) = 2.2360679774997902 d_diag ([4. 2.], [1. 1.], sigma) = 5.0 d_diag ([4. 2.], [2. 2.], sigma) = 3.16227766016838 d_diag ([4. 2.], [3. 3.], sigma) = 2.2360679774997902 d_diag ([4. 2.], [4. 4.], sigma) = 3.16227766016838 d_diag ([4. 2.], [5. 5.], sigma) = 5.0
— , .
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Python 3.6 numpy 1.19.5
import numpy as np
def mahalanobis(point_from, point_to, inverse_covariance_matrix):
delta = point_from - point_to
return max(np.float64(0), np.dot(np.dot(delta, inverse_covariance_matrix), delta)) ** 0.5
def approx(number, *, sign, epsilon=1e-4):
return number + np.sign(sign) * epsilon
test_point = np.array([4., 2.])
class_ = np.array([[3., 1.], [4., 0.], [6., 0.], [6., 2.], [5., 3.]])
covariance_matrix = np.cov(class_, rowvar=False, ddof=1)
if abs(np.linalg.det(covariance_matrix)) <= approx(0., sign=+1):
print(" 0. .")
else:
inverse_covariance_matrix = np.linalg.inv(covariance_matrix)
print(" (d_M):\n", inverse_covariance_matrix, sep='')
for point_to in [class_.mean(axis=0), *class_]:
print("d_M (", test_point, ", ", point_to, ", COV^(-1)) = ",
mahalanobis(test_point, point_to, inverse_covariance_matrix), sep='')
covariance_matrix = covariance_matrix + np.identity(class_.shape[1])
inverse_covariance_matrix = np.linalg.inv(covariance_matrix)
print(" (d_E-M):\n", inverse_covariance_matrix, sep='')
for point_to in [class_.mean(axis=0), *class_]:
print("d_E-M (", test_point, ", ", point_to, ", (COV+E)^(-1)) = ",
mahalanobis(test_point, point_to, inverse_covariance_matrix), sep='')
:
(d_M): [[ 0.60714286 -0.10714286] [-0.10714286 0.60714286]] d_M ([4. 2.], [4.8 1.2], COV^(-1)) = 0.9561828874675149 d_M ([4. 2.], [3. 1.], COV^(-1)) = 1.0 d_M ([4. 2.], [4. 0.], COV^(-1)) = 1.5583874449479593 d_M ([4. 2.], [6. 0.], COV^(-1)) = 2.3904572186687876 d_M ([4. 2.], [6. 2.], COV^(-1)) = 1.5583874449479593 d_M ([4. 2.], [5. 3.], COV^(-1)) = 1.0 (d_E-M): [[ 0.375 -0.04166667] [-0.04166667 0.375 ]] d_E-M ([4. 2.], [4.8 1.2], (COV+E)^(-1)) = 0.7302967433402214 d_E-M ([4. 2.], [3. 1.], (COV+E)^(-1)) = 0.8164965809277259 d_E-M ([4. 2.], [4. 0.], (COV+E)^(-1)) = 1.224744871391589 d_E-M ([4. 2.], [6. 0.], (COV+E)^(-1)) = 1.8257418583505536 d_E-M ([4. 2.], [6. 2.], (COV+E)^(-1)) = 1.224744871391589 d_E-M ([4. 2.], [5. 3.], (COV+E)^(-1)) = 0.8164965809277259
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Python 3.6 numpy 1.19.5
#
import heapq
from collections import Counter
from operator import itemgetter
import numpy as np
class MkNN:
def __init__(self, k, classes, inv_cov_matrices):
self.n = len(classes)
self.k = k
self.classes = classes
self.inv_cov_matrices = inv_cov_matrices
@staticmethod
def mahalanobis_sqr(point_from, point_to, inverse_covariance_matrix):
delta = point_from - point_to
return max(np.float64(0), np.dot(np.dot(delta, inverse_covariance_matrix), delta))
def _get_k_smallest(self, test_point):
generator = (
(MkNN.mahalanobis_sqr(test_point, point, inv_cov), i)
for i, (class_, inv_cov) in enumerate(zip(self.classes, self.inv_cov_matrices))
for point in class_
)
return heapq.nsmallest(self.k, generator, key=itemgetter(0))
def predict(self, test_point):
return heapq.nlargest(1, Counter((i for _, i in self._get_k_smallest(test_point))).items(),
key=lambda t: (t[1], -t[0]))[0][0]
def predict_proba(self, test_point):
most_common = Counter((i for _, i in self._get_k_smallest(test_point)))
classes_proba = np.array([most_common.get(i, 0) for i in range(self.n)])
return classes_proba / classes_proba.sum()
def predict_all_max(self, test_point):
p = self.predict_proba(test_point)
return np.where(p == max(p))[0]
def main():
# , - classes
test_points = np.array([[4., 2.]])
#
classes = [
np.array([[1., 1.], [2., 2.], [3., 3.], [4., 4.], [5., 5.]]),
np.array([[3., 1.], [4., 0.], [6., 0.], [6., 2.], [5., 3.]])
]
#
n_train_points = sum(class_.shape[0] for class_ in classes)
#
cov_matrices = [np.cov(class_, rowvar=False, ddof=1) for class_ in classes]
# -- -
inv_cov_matrices = [np.linalg.inv(cov + np.identity(cov.shape[0])) for cov in cov_matrices]
for test_point in test_points:
print("Point:", test_point)
# k 1 ( )
for i in range(1, n_train_points):
classifier = MkNN(i, classes, inv_cov_matrices)
print(f"{i}nn:",
1 + classifier.predict(test_point),
classifier.predict_proba(test_point),
classifier.predict_all_max(test_point))
if __name__ == "__main__":
main()
:
"knn: [ ( 1), ] [ ] [ ( 0), ]".
Point: [4. 2.] 1nn: 2 [0. 1.] [1] 2nn: 2 [0. 1.] [1] 3nn: 2 [0. 1.] [1] 4nn: 2 [0. 1.] [1] 5nn: 2 [0.2 0.8] [1] 6nn: 2 [0.33333333 0.66666667] [1] 7nn: 2 [0.42857143 0.57142857] [1] 8nn: 1 [0.5 0.5] [0 1] 9nn: 2 [0.44444444 0.55555556] [1]
. 4.
Python 3.6
# ...
from operator import sub
import numpy as np # 1.19.5
from matplotlib import colors, pyplot as plt # 3.3.4
#
def show_data_on_mesh(k, classes, inv_cov_matrices):
#
min_ = np.min([np.min(class_, axis=0) for class_ in classes], axis=1) - 1
max_ = np.max([np.max(class_, axis=0) for class_ in classes], axis=1) + 1
min_c = min(min_[0], min_[1])
max_c = max(max_[0], max_[1])
h = 0.05
test_mesh = np.meshgrid(np.arange(min_c, max_c, h), np.arange(min_c, max_c, h))
test_points = np.c_[test_mesh[0].ravel(), test_mesh[1].ravel()]
#
classifier = MkNN(k, classes, inv_cov_matrices)
test_mesh_labels = [sub(*classifier.predict_proba(x)) for x in test_points]
#
plt.figure(figsize=(6, 5), dpi=90)
class_colormap = colors.ListedColormap(['#070648', '#480607'])
plt.pcolormesh(test_mesh[0], test_mesh[1],
np.asarray(test_mesh_labels).reshape(test_mesh[0].shape),
cmap='coolwarm', shading='nearest')
plt.colorbar()
plt.scatter([point[0] for class_ in classes for point in class_],
[point[1] for class_ in classes for point in class_],
c=[-i for i, class_ in enumerate(classes) for _ in class_],
cmap=class_colormap)
plt.axis([min_c, max_c, min_c, max_c])
plt.xlabel("X")
plt.ylabel("Y")
plt.title("k=" + str(k))
plt.show()
# ...
— .
5.
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2. «Metric Learning» (: Aurélien Bellet, Amaury Habrard, Marc Sebban; ).
3. (, k-means: 1, 2, 3).
4. ().
5. ( 1, artículo 2 , artículo 3 ).
Si tiene algún comentario o error, escriba a quwarm@gmail.com o en los comentarios.