def estimation(self, y, ds=None):
""" Estimate multivariate extension of Blomqvist's beta.
Parameters
----------
y : (number of samples, dimension)-ndarray
One row of y corresponds to one sample.
ds : int vector, vector of ones
ds[i] = 1 (for all i): the i^th subspace is one-dimensional.
If ds is not given (ds=None), the vector of ones [ds =
ones(y.shape[1],dtype='int')] is emulated inside the function.
Returns
-------
a : float
Estimated multivariate extension of Blomqvist's beta.
References
----------
Friedrich Schmid, Rafael Schmidt, Thomas Blumentritt, Sandra
Gaiser, and Martin Ruppert. Copula Theory and Its Applications,
Chapter Copula based Measures of Multivariate Association. Lecture
Notes in Statistics. Springer, 2010. (multidimensional case,
len(ds)>=2)
Manuel Ubeda-Flores. Multivariate versions of Blomqvist's beta and
Spearman's footrule. Annals of the Institute of Statistical
Mathematics, 57:781-788, 2005.
Nils Blomqvist. On a measure of dependence between two random
variables. The Annals of Mathematical Statistics, 21:593-600, 1950.
(2D case, statistical properties)
Frederick Mosteller. On some useful ''inefficient'' statistics.
Annals of Mathematical Statistics, 17:377--408, 1946. (2D case,
def)
Examples
--------
a = co.estimation(y,ds)
"""
if ds is None: # emulate 'ds = vector of ones'
ds = ones(y.shape[1], dtype='int')
# verification:
self.verification_compatible_subspace_dimensions(y, ds)
self.verification_one_dimensional_subspaces(ds)
num_of_samples, dim = y.shape # number of samples, dimension
u = copula_transformation(y)
h = 2**(dim - 1) / (2**(dim - 1) - 1) # h(dim)
c1 = mean(all(u <= 1 / 2, axis=1)) # C(1/2)
c2 = mean(all(u > 1 / 2, axis=1)) # \bar{C}(1/2)
a = h * (c1 + c2 - 2**(1 - dim))
return a
def estimation(self, y, ds=None):
""" Estimate the second multivariate extension of Spearman's rho.
Parameters
----------
y : (number of samples, dimension)-ndarray
One row of y corresponds to one sample.
ds : int vector, vector of ones
ds[i] = 1 (for all i): the i^th subspace is one-dimensional.
If ds is not given (ds=None), the vector of ones [ds =
ones(y.shape[1],dtype='int')] is emulated inside the function.
Returns
-------
a : float
Estimated second multivariate extension of Spearman's rho.
References
----------
Friedrich Shmid, Rafael Schmidt, Thomas Blumentritt, Sandra
Gaiser, and Martin Ruppert. Copula Theory and Its Applications,
Chapter Copula based Measures of Multivariate Association. Lecture
Notes in Statistics. Springer, 2010.
Friedrich Schmid and Rafael Schmidt. Multivariate extensions of
Spearman's rho and related statistics. Statistics & Probability
Letters, 77:407-416, 2007.
Roger B. Nelsen. Nonparametric measures of multivariate
association. Lecture Notes-Monograph Series, Distributions with
Fixed Marginals and Related Topics, 28:223-232, 1996.
Harry Joe. Multivariate concordance. Journal of Multivariate
Analysis, 35:12-30, 1990.
C. Spearman. The proof and measurement of association between two
things. The American Journal of Psychology, 15:72-101, 1904.
Examples
--------
a = co.estimation(y,ds)
"""
if ds is None: # emulate 'ds = vector of ones'
ds = ones(y.shape[1], dtype='int')
# verification:
self.verification_compatible_subspace_dimensions(y, ds)
self.verification_one_dimensional_subspaces(ds)
dim = y.shape[1] # dimension
u = copula_transformation(y)
h = (dim + 1) / (2**dim - (dim + 1)) # h_rho(dim)
a = h * (2**dim * mean(prod(u, axis=1)) - 1)
return a
def estimation(self, y, ds=None):
""" Estimate the fourth multivariate extension of Spearman's rho.
Parameters
----------
y : (number of samples, dimension)-ndarray
One row of y corresponds to one sample.
ds : int vector, vector of ones
ds[i] = 1 (for all i): the i^th subspace is one-dimensional.
If ds is not given (ds=None), the vector of ones [ds =
ones(y.shape[1],dtype='int')] is emulated inside the function.
Returns
-------
a : float
Estimated fourth multivariate extension of Spearman's rho.
References
----------
Friedrich Shmid, Rafael Schmidt, Thomas Blumentritt, Sandra
Gaiser, and Martin Ruppert. Copula Theory and Its Applications,
Chapter Copula based Measures of Multivariate Association. Lecture
Notes in Statistics. Springer, 2010.
Friedrich Schmid and Rafael Schmidt. Multivariate extensions of
Spearman's rho and related statistics. Statistics & Probability
Letters, 77:407-416, 2007.
Maurice G. Kendall. Rank correlation methods. London, Griffin,
1970.
C. Spearman. The proof and measurement of association between two
things. The American Journal of Psychology, 15:72-101, 1904.
Examples
--------
a = co.estimation(y,ds)
"""
if ds is None: # emulate 'ds = vector of ones'
ds = ones(y.shape[1], dtype='int')
# verification:
self.verification_compatible_subspace_dimensions(y, ds)
self.verification_one_dimensional_subspaces(ds)
num_of_samples, dim = y.shape # number of samples, dimension
u = copula_transformation(y)
m_triu = triu(ones((dim, dim)), 1) # upper triangular mask
b = binom(dim, 2)
a = 12 * sum(dot((1 - u).T, (1 - u)) * m_triu) /\
(b * num_of_samples) - 3
return a
def estimation(self, y, ds=None):
""" Estimate the third multivariate extension of Spearman's rho.
Parameters
----------
y : (number of samples, dimension)-ndarray
One row of y corresponds to one sample.
ds : int vector, vector of ones
ds[i] = 1 (for all i): the i^th subspace is one-dimensional.
If ds is not given (ds=None), the vector of ones [ds =
ones(y.shape[1],dtype='int')] is emulated inside the function.
Returns
-------
a : float
Estimated third multivariate extension of Spearman's rho.
References
----------
Friedrich Shmid, Rafael Schmidt, Thomas Blumentritt, Sandra
Gaiser, and Martin Ruppert. Copula Theory and Its Applications,
Chapter Copula based Measures of Multivariate Association. Lecture
Notes in Statistics. Springer, 2010.
Roger B. Nelsen. An Introduction to Copulas (Springer Series in
Statistics). Springer, 2006.
Roger B. Nelsen. Distributions with Given Marginals and
Statistical Modelling, chapter Concordance and copulas: A survey,
pages 169-178. Kluwer Academic Publishers, Dordrecht, 2002.
C. Spearman. The proof and measurement of association between two
things. The American Journal of Psychology, 15:72-101, 1904.
Examples
--------
a = co.estimation(y,ds)
"""
if ds is None: # emulate 'ds = vector of ones'
ds = ones(y.shape[1], dtype='int')
# verification:
self.verification_compatible_subspace_dimensions(y, ds)
self.verification_one_dimensional_subspaces(ds)
dim = y.shape[1] # dimension
u = copula_transformation(y)
h = (dim + 1) / (2**dim - (dim + 1)) # h_rho(d)
a1 = h * (2**dim * mean(prod(1 - u, axis=1)) - 1)
a2 = h * (2**dim * mean(prod(u, axis=1)) - 1)
a = (a1 + a2) / 2
return a
def estimation(self, y, ds=None):
""" Estimate multivariate conditional version of Spearman's rho.
Parameters
----------
y : (number of samples, dimension)-ndarray
One row of y corresponds to one sample.
ds : int vector, vector of ones
ds[i] = 1 (for all i): the i^th subspace is one-dimensional.
If ds is not given (ds=None), the vector of ones [ds =
ones(y.shape[1],dtype='int')] is emulated inside the function.
Returns
-------
a : float
Estimated multivariate conditional version of Spearman's rho.
References
----------
Friedrich Schmid and Rafael Schmidt. Multivariate conditional
versions of Spearman's rho and related measures of tail
dependence. Journal of Multivariate Analysis, 98:1123-1140, 2007.
C. Spearman. The proof and measurement of association between two
things. The American Journal of Psychology, 15:72-101, 1904.
Examples
--------
a = co.estimation(y,ds)
"""
if ds is None: # emulate 'ds = vector of ones'
ds = ones(y.shape[1], dtype='int')
# verification:
self.verification_compatible_subspace_dimensions(y, ds)
self.verification_one_dimensional_subspaces(ds)
num_of_samples, dim = y.shape # number of samples, dimension
u = copula_transformation(y)
c = mean(prod(1 - maximum(u, 1 - self.p), axis=1))
c1 = (self.p * (2 - self.p) / 2)**dim
c2 = self.p**dim * (dim + 1 - self.p * dim) / (dim + 1)
a = (c - c1) / (c2 - c1)
return a
def estimation(self, y, ds=None):
""" Estimate Renyi mutual information.
Parameters
----------
y : (number of samples, dimension)-ndarray
One row of y corresponds to one sample.
ds : int vector, vector of ones
If ds is not given (ds=None), the vector of ones [ds =
ones(y.shape[1],dtype='int')] is emulated inside the function.
Returns
-------
i : float
Estimated Renyi mutual information.
References
----------
David Pal, Barnabas Poczos, Csaba Szepesvari. Estimation of Renyi
Entropy and Mutual Information Based on Generalized
Nearest-Neighbor Graphs. Advances in Neural Information Processing
Systems (NIPS), pages 1849-1857, 2010.
Barnabas Poczos, Sergey Krishner, Csaba Szepesvari. REGO:
Rank-based Estimation of Renyi Information using Euclidean Graph
Optimization. International Conference on Artificial Intelligence
and Statistics (AISTATS), pages 605-612, 2010.
Examples
--------
i = co.estimation(y,ds)
"""
if ds is None: # emulate 'ds = vector of ones'
ds = ones(y.shape[1], dtype='int')
# verification:
self.verification_compatible_subspace_dimensions(y, ds)
self.verification_one_dimensional_subspaces(ds)
z = copula_transformation(y)
i = -self.renyi_co.estimation(z)
return i
def estimation(self, y, ds=None):
""" Estimate copula and MMD based kernel dependency.
Parameters
----------
y : (number of samples, dimension)-ndarray
One row of y corresponds to one sample.
ds : int vector, vector of ones
If ds is not given (ds=None), the vector of ones [ds =
ones(y.shape[1],dtype='int')] is emulated inside the function.
Returns
-------
i : float
Estimated copula and MMD based kernel dependency.
References
----------
Barnabas Poczos, Zoubin Ghahramani, Jeff Schneider. Copula-based
Kernel Dependency Measures. International Conference on Machine
Learning (ICML), 2012.
Examples
--------
i = co.estimation(y,ds)
"""
if ds is None: # emulate 'ds = vector of ones'
ds = ones(y.shape[1], dtype='int')
# verification:
self.verification_compatible_subspace_dimensions(y, ds)
self.verification_one_dimensional_subspaces(ds)
z = copula_transformation(y)
u = rand(z.shape[0], z.shape[1])
i = self.mmd_co.estimation(z, u)
return i
def estimation(self, y, ds):
""" Estimate multivariate version of Hoeffding's Phi.
Parameters
----------
y : (number of samples, dimension)-ndarray
One row of y corresponds to one sample.
ds : int vector
Dimensions of the individual subspaces in y; ds[i] = i^th
subspace dimension = 1 for this estimator.
Returns
-------
i : float
Estimated value of the multivariate version of Hoeffding's Phi.
References
----------
Sandra Gaiser, Martin Ruppert, Friedrich Schmid. A multivariate
version of Hoeffding's Phi-Square. Journal of Multivariate
Analysis. 101: pages 2571-2586, 2010.
Examples
--------
i = co.estimation(y,ds)
"""
# verification:
self.verification_compatible_subspace_dimensions(y, ds)
self.verification_one_dimensional_subspaces(ds)
num_of_samples, dim = y.shape
u = copula_transformation(y)
# term1:
m = 1 - maximum(u[:, 0][:, newaxis], u[:, 0])
for i in range(1, dim):
m *= 1 - maximum(u[:, i][:, newaxis], u[:, i])
term1 = mean(m)
# term2:
if self.small_sample_adjustment:
term2 = \
- mean(prod(1 - u**2 - (1 - u) / num_of_samples,
axis=1)) / \
(2**(dim - 1))
else:
term2 = - mean(prod(1 - u**2, axis=1)) / (2 ** (dim - 1))
# term3:
if self.small_sample_adjustment:
term3 = \
((num_of_samples - 1) * (2 * num_of_samples-1) /
(3 * 2 * num_of_samples**2))**dim
else:
term3 = 1 / 3**dim
i = term1 + term2 + term3
if self.mult:
if self.small_sample_adjustment:
t1 = \
sum((1 - arange(1,
num_of_samples) / num_of_samples)**dim
* (2*arange(1, num_of_samples) - 1)) \
/ num_of_samples**2
t2 = \
-2 * mean(((num_of_samples * (num_of_samples - 1) -
arange(1, num_of_samples+1) *
arange(num_of_samples)) /
(2 * num_of_samples ** 2))**dim)
t3 = term3
inv_hd = t1 + t2 + t3 # 1 / h(d, n)
else:
inv_hd = \
2 / ((dim + 1) * (dim + 2)) - factorial(dim) / \
(2 ** dim * prod(arange(dim + 1) + 1 / 2)) + \
1 / 3 ** dim # 1 / h(d)s
i /= inv_hd
i = sqrt(abs(i))
return i