def solve(a, b, overwrite_a=False, overwrite_b=False):
"""
Solve a linear system ``a x = b``
Parameters
----------
a : (N, N) array_like
left-hand-side matrix
b : (N, NRHS) array_like
right-hand-side matrix
overwrite_a : bool, optional
whether we are allowed to overwrite the matrix `a`
overwrite_b : bool, optional
whether we are allowed to overwrite the matrix `b`
Returns
-------
x : (N, NRHS) ndarray
solution matrix
"""
a1 = atleast_2d(_asarray_validated(a, check_finite=False))
b1 = atleast_1d(_asarray_validated(b, check_finite=False))
n = a1.shape[0]
overwrite_a = overwrite_a or _datacopied(a1, a)
overwrite_b = overwrite_b or _datacopied(b1, b)
if a1.shape[0] != a1.shape[1]:
raise ValueError('LHS needs to be a square matrix.')
if n != b1.shape[0]:
# Last chance to catch 1x1 scalar a and 1D b arrays
if not (n == 1 and b1.size != 0):
raise ValueError('Input b has to have same number of rows as '
'input a')
# regularize 1D b arrays to 2D
if b1.ndim == 1:
if n == 1:
b1 = b1[None, :]
else:
b1 = b1[:, None]
b_is_1D = True
else:
b_is_1D = False
gesv = get_lapack_funcs('gesv', (a1, b1))
_, _, x, info = gesv(a1,
b1,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
if info > 0:
raise LinAlgError("Singular matrix")
if info < 0:
raise ValueError('illegal value in %d-th argument of internal '
'gesv' % -info)
if b_is_1D:
return x.ravel()
return x
def py_vq2(obs, code_book, check_finite=True):
"""2nd Python version of vq algorithm.
The algorithm simply computes the euclidian distance between each
observation and every frame in the code_book/
Parameters
----------
obs : ndarray
Expect a rank 2 array. Each row is one observation.
code_book : ndarray
Code book to use. Same format than obs. Should have same number of
features (eg columns) than obs.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
code : ndarray
code[i] gives the label of the ith obversation, that its code is
code_book[code[i]].
mind_dist : ndarray
min_dist[i] gives the distance between the ith observation and its
corresponding code.
Notes
-----
This could be faster when number of codebooks is small, but it
becomes a real memory hog when codebook is large. It requires
N by M by O storage where N=number of obs, M = number of
features, and O = number of codes.
"""
obs = _asarray_validated(obs, check_finite=check_finite)
code_book = _asarray_validated(code_book, check_finite=check_finite)
d = shape(obs)[1]
# code books and observations should have same number of features
if not d == code_book.shape[1]:
raise ValueError(
"""
code book(%d) and obs(%d) should have the same
number of features (eg columns)"""
% (code_book.shape[1], d)
)
diff = obs[newaxis, :, :] - code_book[:, newaxis, :]
dist = sqrt(np.sum(diff * diff, -1))
code = argmin(dist, 0)
min_dist = minimum.reduce(dist, 0)
# The next line I think is equivalent and should be faster than the one
# above, but in practice didn't seem to make much difference:
# min_dist = choose(code,dist)
return code, min_dist
def py_vq2(obs, code_book, check_finite=True):
"""2nd Python version of vq algorithm.
The algorithm simply computes the euclidian distance between each
observation and every frame in the code_book/
Parameters
----------
obs : ndarray
Expect a rank 2 array. Each row is one observation.
code_book : ndarray
Code book to use. Same format than obs. Should have same number of
features (eg columns) than obs.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
code : ndarray
code[i] gives the label of the ith obversation, that its code is
code_book[code[i]].
mind_dist : ndarray
min_dist[i] gives the distance between the ith observation and its
corresponding code.
Notes
-----
This could be faster when number of codebooks is small, but it
becomes a real memory hog when codebook is large. It requires
N by M by O storage where N=number of obs, M = number of
features, and O = number of codes.
"""
obs = _asarray_validated(obs, check_finite=check_finite)
code_book = _asarray_validated(code_book, check_finite=check_finite)
d = shape(obs)[1]
# code books and observations should have same number of features
if not d == code_book.shape[1]:
raise ValueError("""
code book(%d) and obs(%d) should have the same
number of features (eg columns)""" % (code_book.shape[1], d))
diff = obs[newaxis, :, :] - code_book[:, newaxis, :]
dist = sqrt(np.sum(diff * diff, -1))
code = argmin(dist, 0)
min_dist = minimum.reduce(dist, 0)
# The next line I think is equivalent and should be faster than the one
# above, but in practice didn't seem to make much difference:
# min_dist = choose(code,dist)
return code, min_dist
def py_vq(obs, code_book, check_finite=True):
""" Python version of vq algorithm.
The algorithm computes the Euclidean distance between each
observation and every frame in the code_book.
Parameters
----------
obs : ndarray
Expects a rank 2 array. Each row is one observation.
code_book : ndarray
Code book to use. Same format than obs. Should have same number of
features (e.g., columns) than obs.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
code : ndarray
code[i] gives the label of the ith obversation; its code is
code_book[code[i]].
mind_dist : ndarray
min_dist[i] gives the distance between the ith observation and its
corresponding code.
Notes
-----
This function is slower than the C version but works for
all input types. If the inputs have the wrong types for the
C versions of the function, this one is called as a last resort.
It is about 20 times slower than the C version.
"""
obs = _asarray_validated(obs, check_finite=check_finite)
code_book = _asarray_validated(code_book, check_finite=check_finite)
if obs.ndim != code_book.ndim:
raise ValueError("Observation and code_book should have the same rank")
if obs.ndim == 1:
obs = obs[:, np.newaxis]
code_book = code_book[:, np.newaxis]
dist = cdist(obs, code_book)
code = dist.argmin(axis=1)
min_dist = dist[np.arange(len(code)), code]
return code, min_dist
def py_vq(obs, code_book, check_finite=True):
""" Python version of vq algorithm.
The algorithm computes the euclidian distance between each
observation and every frame in the code_book.
Parameters
----------
obs : ndarray
Expects a rank 2 array. Each row is one observation.
code_book : ndarray
Code book to use. Same format than obs. Should have same number of
features (eg columns) than obs.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
code : ndarray
code[i] gives the label of the ith obversation, that its code is
code_book[code[i]].
mind_dist : ndarray
min_dist[i] gives the distance between the ith observation and its
corresponding code.
Notes
-----
This function is slower than the C version but works for
all input types. If the inputs have the wrong types for the
C versions of the function, this one is called as a last resort.
It is about 20 times slower than the C version.
"""
obs = _asarray_validated(obs, check_finite=check_finite)
code_book = _asarray_validated(code_book, check_finite=check_finite)
if obs.ndim != code_book.ndim:
raise ValueError("Observation and code_book should have the same rank")
if obs.ndim == 1:
obs = obs[:, np.newaxis]
code_book = code_book[:, np.newaxis]
dist = cdist(obs, code_book)
code = dist.argmin(axis=1)
min_dist = dist[np.arange(len(code)), code]
return code, min_dist
def __init__(self, x, y, axis=0, extrapolate=None):
x = _asarray_validated(x, check_finite=False, as_inexact=True)
y = _asarray_validated(y, check_finite=False, as_inexact=True)
axis = axis % y.ndim
xp = x.reshape((x.shape[0],) + (1,) * (y.ndim - 1))
yp = np.rollaxis(y, axis)
dk = self._find_derivatives(xp, yp)
data = np.hstack((yp[:, None, ...], dk[:, None, ...]))
_b = BPoly.from_derivatives(x, data, orders=None)
super(PchipInterpolator_new, self).__init__(_b.c, _b.x, extrapolate=extrapolate)
self.axis = axis
def auxFixedPoint(function, x0, maxit, tol, args=()):
"""
Encontrar el punto fijo de un vector columna nx1.
Dada una función de una o más varaiables y un punto de inicio, encontrar
el punto fijo de la función. Dondé ``func(x0) == x0``.
Parametros
----------
func : function
Función a evaluar.
x0 : vector
Punto fijo de la función-
args : tuple
Argumentos para la función.
tol : float
Tolerancia de convergencia, recomendado: 1e-08.
maxit : int
Maximo de iteraciones.
References
----------
.. [1] Burden, Faires, "Numerical Analysis", 5th edition, pg. 80
--------
"""
x0 = _asarray_validated(x0, as_inexact=True) #Verifica si el input es valido
return fixedPoint(function, x0, args, tol, maxit)
def fixed_point(func, x0, args=(), xtol=1e-8, maxiter=500, method='iteration'):
"""
Find a fixed point of the function.
Given a function of one or more variables and a starting point, find a
fixed-point of the function: i.e. where ``func(x0) == x0``.
Parameters
----------
func : function
Function to evaluate.
x0 : array_like
Fixed point of function.
args : tuple, optional
Extra arguments to `func`.
xtol : float, optional
Convergence tolerance, defaults to 1e-08.
maxiter : int, optional
Maximum number of iterations, defaults to 500.
method : {"del2", "iteration"}, optional
Method of finding the fixed-point, defaults to "del2"
which uses Steffensen's Method with Aitken's ``Del^2``
convergence acceleration [1]_. The "iteration" method simply iterates
the function until convergence is detected, without attempting to
accelerate the convergence.
References
----------
.. [1] Burden, Faires, "Numerical Analysis", 5th edition, pg. 80
Examples
--------
"""
use_accel = {'del2': True, 'iteration': True}[method]
x0 = _asarray_validated(x0, as_inexact=True)
return _fixed_point_helper(func, x0, args, xtol, maxiter, use_accel)
def __init__(self, x, y, axis=0, extrapolate=None):
x = _asarray_validated(x, check_finite=False, as_inexact=True)
y = _asarray_validated(y, check_finite=False, as_inexact=True)
axis = axis % y.ndim
xp = x.reshape((x.shape[0],) + (1,)*(y.ndim-1))
yp = np.rollaxis(y, axis)
dk = self._find_derivatives(xp, yp)
data = np.hstack((yp[:, None, ...], dk[:, None, ...]))
_b = BPoly.from_derivatives(x, data, orders=None)
super(PchipInterpolator, self).__init__(_b.c, _b.x,
extrapolate=extrapolate)
self.axis = axis
def whiten(obs, check_finite=True):
"""
Normalize a group of observations on a per feature basis.
Before running k-means, it is beneficial to rescale each feature
dimension of the observation set by its standard deviation (i.e. "whiten"
it - as in "white noise" where each frequency has equal power).
Each feature is divided by its standard deviation across all observations
to give it unit variance.
Parameters
----------
obs : ndarray
Each row of the array is an observation. The
columns are the features seen during each observation.
>>> # f0 f1 f2
>>> obs = [[ 1., 1., 1.], #o0
... [ 2., 2., 2.], #o1
... [ 3., 3., 3.], #o2
... [ 4., 4., 4.]] #o3
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
result : ndarray
Contains the values in `obs` scaled by the standard deviation
of each column.
Examples
--------
>>> import numpy as np
>>> from scipy.cluster.vq import whiten
>>> features = np.array([[1.9, 2.3, 1.7],
... [1.5, 2.5, 2.2],
... [0.8, 0.6, 1.7,]])
>>> whiten(features)
array([[ 4.17944278, 2.69811351, 7.21248917],
[ 3.29956009, 2.93273208, 9.33380951],
[ 1.75976538, 0.7038557 , 7.21248917]])
"""
obs = _asarray_validated(obs, check_finite=check_finite)
std_dev = obs.std(axis=0)
zero_std_mask = std_dev == 0
if zero_std_mask.any():
std_dev[zero_std_mask] = 1.0
warnings.warn(
"Some columns have standard deviation zero. "
"The values of these columns will not change.", RuntimeWarning)
return obs / std_dev
def whiten(obs, check_finite=True):
"""
Normalize a group of observations on a per feature basis.
Before running k-means, it is beneficial to rescale each feature
dimension of the observation set with whitening. Each feature is
divided by its standard deviation across all observations to give
it unit variance.
Parameters
----------
obs : ndarray
Each row of the array is an observation. The
columns are the features seen during each observation.
>>> # f0 f1 f2
>>> obs = [[ 1., 1., 1.], #o0
... [ 2., 2., 2.], #o1
... [ 3., 3., 3.], #o2
... [ 4., 4., 4.]] #o3
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
result : ndarray
Contains the values in `obs` scaled by the standard deviation
of each column.
Examples
--------
>>> from scipy.cluster.vq import whiten
>>> features = np.array([[1.9, 2.3, 1.7],
... [1.5, 2.5, 2.2],
... [0.8, 0.6, 1.7,]])
>>> whiten(features)
array([[ 4.17944278, 2.69811351, 7.21248917],
[ 3.29956009, 2.93273208, 9.33380951],
[ 1.75976538, 0.7038557 , 7.21248917]])
"""
obs = _asarray_validated(obs, check_finite=check_finite)
std_dev = std(obs, axis=0)
zero_std_mask = std_dev == 0
if zero_std_mask.any():
std_dev[zero_std_mask] = 1.0
warnings.warn("Some columns have standard deviation zero. "
"The values of these columns will not change.",
RuntimeWarning)
return obs / std_dev
def whiten(obs, check_finite=True, std_dev=None):
obs = _asarray_validated(obs, check_finite=check_finite)
if std_dev is None:
std_dev = np.std(obs, axis=0)
zero_std_mask = std_dev == 0
if zero_std_mask.any():
std_dev[zero_std_mask] = 1.0
warnings.warn(
"Some columns have standard deviation zero. "
"The values of these columns will not change.", RuntimeWarning)
return (std_dev, obs / std_dev)
def whiten(obs, check_finite=False):
"""Adapted from c:/python27/lib/site-
packages/skimage/filters/thresholding.py to return array and std_dev."""
obs = _asarray_validated(obs, check_finite=check_finite)
std_dev = np.std(obs, axis=0)
zero_std_mask = std_dev == 0
if zero_std_mask.any():
std_dev[zero_std_mask] = 1.0
raise RuntimeWarning(
'Some columns have standard deviation zero. The values of these columns will not change.'
)
return obs / std_dev, std_dev
def fixed_point(func, x0, args=(), xtol=1e-8, maxiter=500, method='del2'):
"""
Find a fixed point of the function.
Given a function of one or more variables and a starting point, find a
fixed-point of the function: i.e. where ``func(x0) == x0``.
Parameters
----------
func : function
Function to evaluate.
x0 : array_like
Fixed point of function.
args : tuple, optional
Extra arguments to `func`.
xtol : float, optional
Convergence tolerance, defaults to 1e-08.
maxiter : int, optional
Maximum number of iterations, defaults to 500.
method : {"del2", "iteration"}, optional
Method of finding the fixed-point, defaults to "del2"
which uses Steffensen's Method with Aitken's ``Del^2``
convergence acceleration [1]_. The "iteration" method simply iterates
the function until convergence is detected, without attempting to
accelerate the convergence.
References
----------
.. [1] Burden, Faires, "Numerical Analysis", 5th edition, pg. 80
Examples
--------
>>> from scipy import optimize
>>> def func(x, c1, c2):
... return np.sqrt(c1/(x+c2))
>>> c1 = np.array([10,12.])
>>> c2 = np.array([3, 5.])
>>> optimize.fixed_point(func, [1.2, 1.3], args=(c1,c2))
array([ 1.4920333 , 1.37228132])
"""
use_accel = {'del2': True, 'iteration': False}[method]
x0 = _asarray_validated(x0, as_inexact=True)
return _fixed_point_helper(func, x0, args, xtol, maxiter, use_accel)
def eigh(a, overwrite_a=False, check_finite=True):
"""Efficient wrapper for eigh.
Parameters
----------
a : ndarray, shape (n_components, n_components)
The symmetric array operate on.
overwrite_a : bool
If True, the contents of a can be overwritten for efficiency.
check_finite : bool
If True, check that all elements are finite.
Returns
-------
w : ndarray, shape (n_components,)
The N eigenvalues, in ascending order, each repeated according to
its multiplicity.
v : ndarray, shape (n_components, n_components)
The normalized eigenvector corresponding to the eigenvalue ``w[i]``
is the column ``v[:, i]``.
"""
# We use SYEVD, see https://github.com/scipy/scipy/issues/9212
if check_finite:
a = _asarray_validated(a, check_finite=check_finite)
if a.dtype == np.float64:
evr, driver = dsyevd, 'syevd'
else:
assert a.dtype == np.complex128
evr, driver = zheevd, 'heevd'
w, v, info = evr(a, lower=1, overwrite_a=overwrite_a)
if info == 0:
return w, v
if info < 0:
raise ValueError('illegal value in argument %d of internal %s' %
(-info, driver))
else:
raise LinAlgError("internal fortran routine failed to converge: "
"%i off-diagonal elements of an "
"intermediate tridiagonal form did not converge"
" to zero." % info)
def logsumexp(a, axis=None, b=None, keepdims=False, return_sign=False):
a = _asarray_validated(a, check_finite=False)
if b is not None:
a, b = np.broadcast_arrays(a, b)
if np.any(b == 0):
a = a + 0. # promote to at least float
a[b == 0] = -np.inf
a_max = np.amax(a, axis=axis, keepdims=True)
if a_max.ndim > 0:
a_max[~np.isfinite(a_max)] = 0
elif not np.isfinite(a_max):
a_max = 0
if b is not None:
b = np.asarray(b)
tmp = b * np.exp(a - a_max)
else:
tmp = np.exp(a - a_max)
# suppress warnings about log of zero
with np.errstate(divide='ignore'):
s = np.sum(tmp, axis=axis, keepdims=keepdims)
if return_sign:
sgn = np.sign(s)
s *= sgn # /= makes more sense but we need zero -> zero
out = np.log(s)
if not keepdims:
a_max = np.squeeze(a_max, axis=axis)
out += a_max
if return_sign:
return out, sgn
else:
return out
def inv(a, overwrite_a=False):
"""
Inverts a matrix
Parameters
----------
a : (N, N) array_like
the matrix to be inverted.
overwrite_a : bool, optional
whether we are allowed to overwrite the matrix `a`
Returns
-------
x : (N, N) ndarray
The inverted matrix
"""
a1 = atleast_2d(_asarray_validated(a, check_finite=False))
overwrite_a = overwrite_a or _datacopied(a1, a)
if a1.shape[0] != a1.shape[1]:
raise ValueError('Input a needs to be a square matrix.')
getrf, getri, getri_lwork = get_lapack_funcs(
('getrf', 'getri', 'getri_lwork'), (a1, ))
lu, piv, info = getrf(a1, overwrite_a=overwrite_a)
if info == 0:
lwork = _compute_lwork(getri_lwork, a1.shape[0])
lwork = int(1.01 * lwork)
x, info = getri(lu, piv, lwork=lwork, overwrite_lu=True)
if info > 0:
raise LinAlgError("Singular matrix")
if info < 0:
raise ValueError('illegal value in %d-th argument of internal '
'getrf|getri' % -info)
return x
def py_vq(obs, code_book, check_finite=True):
""" Python version of vq algorithm.
The algorithm computes the euclidian distance between each
observation and every frame in the code_book.
Parameters
----------
obs : ndarray
Expects a rank 2 array. Each row is one observation.
code_book : ndarray
Code book to use. Same format than obs. Should have same number of
features (eg columns) than obs.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
code : ndarray
code[i] gives the label of the ith obversation, that its code is
code_book[code[i]].
mind_dist : ndarray
min_dist[i] gives the distance between the ith observation and its
corresponding code.
Notes
-----
This function is slower than the C version but works for
all input types. If the inputs have the wrong types for the
C versions of the function, this one is called as a last resort.
It is about 20 times slower than the C version.
"""
obs = _asarray_validated(obs, check_finite=check_finite)
code_book = _asarray_validated(code_book, check_finite=check_finite)
# n = number of observations
# d = number of features
if np.ndim(obs) == 1:
if not np.ndim(obs) == np.ndim(code_book):
raise ValueError(
"Observation and code_book should have the same rank")
else:
return _py_vq_1d(obs, code_book)
else:
(n, d) = shape(obs)
# code books and observations should have same number of features and same
# shape
if not np.ndim(obs) == np.ndim(code_book):
raise ValueError("Observation and code_book should have the same rank")
elif not d == code_book.shape[1]:
raise ValueError("Code book(%d) and obs(%d) should have the same "
"number of features (eg columns)""" %
(code_book.shape[1], d))
code = zeros(n, dtype=int)
min_dist = zeros(n)
for i in range(n):
dist = np.sum((obs[i] - code_book) ** 2, 1)
code[i] = argmin(dist)
min_dist[i] = dist[code[i]]
return code, sqrt(min_dist)
def sqrtm(A, disp=True, blocksize=64):
"""
Matrix square root.
Parameters
----------
A : (N, N) array_like
Matrix whose square root to evaluate
disp : bool, optional
Print warning if error in the result is estimated large
instead of returning estimated error. (Default: True)
blocksize : integer, optional
If the blocksize is not degenerate with respect to the
size of the input array, then use a blocked algorithm. (Default: 64)
Returns
-------
sqrtm : (N, N) ndarray
Value of the sqrt function at `A`
errest : float
(if disp == False)
Frobenius norm of the estimated error, ||err||_F / ||A||_F
References
----------
.. [1] Edvin Deadman, Nicholas J. Higham, Rui Ralha (2013)
"Blocked Schur Algorithms for Computing the Matrix Square Root,
Lecture Notes in Computer Science, 7782. pp. 171-182.
Examples
--------
>>> from scipy.linalg import sqrtm
>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
>>> r = sqrtm(a)
>>> r
array([[ 0.75592895, 1.13389342],
[ 0.37796447, 1.88982237]])
>>> r.dot(r)
array([[ 1., 3.],
[ 1., 4.]])
"""
A = _asarray_validated(A, check_finite=True, as_inexact=True)
if len(A.shape) != 2:
raise ValueError("Non-matrix input to matrix function.")
if blocksize < 1:
raise ValueError("The blocksize should be at least 1.")
keep_it_real = np.isrealobj(A)
if keep_it_real:
T, Z = schur(A)
if not np.array_equal(T, np.triu(T)):
T, Z = rsf2csf(T, Z)
else:
T, Z = schur(A, output='complex')
failflag = False
try:
R = _sqrtm_triu(T, blocksize=blocksize)
ZH = np.conjugate(Z).T
X = Z.dot(R).dot(ZH)
except SqrtmError:
failflag = True
X = np.empty_like(A)
X.fill(np.nan)
if disp:
nzeig = np.any(np.diag(T) == 0)
if nzeig:
print("Matrix is singular and may not have a square root.")
elif failflag:
print("Failed to find a square root.")
return X
else:
try:
arg2 = norm(X.dot(X) - A, 'fro')**2 / norm(A, 'fro')
except ValueError:
# NaNs in matrix
arg2 = np.inf
return X, arg2
def vq(obs, code_book, check_finite=True):
"""
Assign codes from a code book to observations.
Assigns a code from a code book to each observation. Each
observation vector in the 'M' by 'N' `obs` array is compared with the
centroids in the code book and assigned the code of the closest
centroid.
The features in `obs` should have unit variance, which can be
achieved by passing them through the whiten function. The code
book can be created with the k-means algorithm or a different
encoding algorithm.
Parameters
----------
obs : ndarray
Each row of the 'M' x 'N' array is an observation. The columns are
the "features" seen during each observation. The features must be
whitened first using the whiten function or something equivalent.
code_book : ndarray
The code book is usually generated using the k-means algorithm.
Each row of the array holds a different code, and the columns are
the features of the code.
>>> # f0 f1 f2 f3
>>> code_book = [
... [ 1., 2., 3., 4.], #c0
... [ 1., 2., 3., 4.], #c1
... [ 1., 2., 3., 4.]] #c2
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
code : ndarray
A length M array holding the code book index for each observation.
dist : ndarray
The distortion (distance) between the observation and its nearest
code.
Examples
--------
>>> from numpy import array
>>> from scipy.cluster.vq import vq
>>> code_book = array([[1.,1.,1.],
... [2.,2.,2.]])
>>> features = array([[ 1.9,2.3,1.7],
... [ 1.5,2.5,2.2],
... [ 0.8,0.6,1.7]])
>>> vq(features,code_book)
(array([1, 1, 0],'i'), array([ 0.43588989, 0.73484692, 0.83066239]))
"""
obs = _asarray_validated(obs, check_finite=check_finite)
code_book = _asarray_validated(code_book, check_finite=check_finite)
ct = common_type(obs, code_book)
c_obs = obs.astype(ct, copy=False)
if code_book.dtype != ct:
c_code_book = code_book.astype(ct)
else:
c_code_book = code_book
if ct in (single, double):
results = _vq.vq(c_obs, c_code_book)
else:
results = py_vq(obs, code_book)
return results
def kmeans2(data,
k,
iter=10,
thresh=1e-5,
minit='random',
missing='warn',
check_finite=True):
"""
Classify a set of observations into k clusters using the k-means algorithm.
The algorithm attempts to minimize the Euclidean distance between
observations and centroids. Several initialization methods are
included.
Parameters
----------
data : ndarray
A 'M' by 'N' array of 'M' observations in 'N' dimensions or a length
'M' array of 'M' 1-D observations.
k : int or ndarray
The number of clusters to form as well as the number of
centroids to generate. If `minit` initialization string is
'matrix', or if a ndarray is given instead, it is
interpreted as initial cluster to use instead.
iter : int, optional
Number of iterations of the k-means algorithm to run. Note
that this differs in meaning from the iters parameter to
the kmeans function.
thresh : float, optional
(not used yet)
minit : str, optional
Method for initialization. Available methods are 'random',
'points', '++' and 'matrix':
'random': generate k centroids from a Gaussian with mean and
variance estimated from the data.
'points': choose k observations (rows) at random from data for
the initial centroids.
'++': choose k observations accordingly to the kmeans++ method
(careful seeding)
'matrix': interpret the k parameter as a k by M (or length k
array for 1-D data) array of initial centroids.
missing : str, optional
Method to deal with empty clusters. Available methods are
'warn' and 'raise':
'warn': give a warning and continue.
'raise': raise an ClusterError and terminate the algorithm.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
centroid : ndarray
A 'k' by 'N' array of centroids found at the last iteration of
k-means.
label : ndarray
label[i] is the code or index of the centroid the
ith observation is closest to.
See Also
--------
kmeans
References
----------
.. [1] D. Arthur and S. Vassilvitskii, "k-means++: the advantages of
careful seeding", Proceedings of the Eighteenth Annual ACM-SIAM Symposium
on Discrete Algorithms, 2007.
Examples
--------
>>> from scipy.cluster.vq import kmeans2
>>> import matplotlib.pyplot as plt
Create z, an array with shape (100, 2) containing a mixture of samples
from three multivariate normal distributions.
>>> np.random.seed(12345678)
>>> a = np.random.multivariate_normal([0, 6], [[2, 1], [1, 1.5]], size=45)
>>> b = np.random.multivariate_normal([2, 0], [[1, -1], [-1, 3]], size=30)
>>> c = np.random.multivariate_normal([6, 4], [[5, 0], [0, 1.2]], size=25)
>>> z = np.concatenate((a, b, c))
>>> np.random.shuffle(z)
Compute three clusters.
>>> centroid, label = kmeans2(z, 3, minit='points')
>>> centroid
array([[-0.35770296, 5.31342524],
[ 2.32210289, -0.50551972],
[ 6.17653859, 4.16719247]])
How many points are in each cluster?
>>> counts = np.bincount(label)
>>> counts
array([52, 27, 21])
Plot the clusters.
>>> w0 = z[label == 0]
>>> w1 = z[label == 1]
>>> w2 = z[label == 2]
>>> plt.plot(w0[:, 0], w0[:, 1], 'o', alpha=0.5, label='cluster 0')
>>> plt.plot(w1[:, 0], w1[:, 1], 'd', alpha=0.5, label='cluster 1')
>>> plt.plot(w2[:, 0], w2[:, 1], 's', alpha=0.5, label='cluster 2')
>>> plt.plot(centroid[:, 0], centroid[:, 1], 'k*', label='centroids')
>>> plt.axis('equal')
>>> plt.legend(shadow=True)
>>> plt.show()
"""
if int(iter) < 1:
raise ValueError("Invalid iter (%s), "
"must be a positive integer." % iter)
try:
miss_meth = _valid_miss_meth[missing]
except KeyError:
raise ValueError("Unknown missing method %r" % (missing, ))
data = _asarray_validated(data, check_finite=check_finite)
if data.ndim == 1:
d = 1
elif data.ndim == 2:
d = data.shape[1]
else:
raise ValueError("Input of rank > 2 is not supported.")
if data.size < 1:
raise ValueError("Empty input is not supported.")
# If k is not a single value, it should be compatible with data's shape
if minit == 'matrix' or not np.isscalar(k):
code_book = np.array(k, copy=True)
if data.ndim != code_book.ndim:
raise ValueError("k array doesn't match data rank")
nc = len(code_book)
if data.ndim > 1 and code_book.shape[1] != d:
raise ValueError("k array doesn't match data dimension")
else:
nc = int(k)
if nc < 1:
raise ValueError("Cannot ask kmeans2 for %d clusters"
" (k was %s)" % (nc, k))
elif nc != k:
warnings.warn("k was not an integer, was converted.")
try:
init_meth = _valid_init_meth[minit]
except KeyError:
raise ValueError("Unknown init method %r" % (minit, ))
else:
code_book = init_meth(data, k)
for i in range(iter):
# Compute the nearest neighbor for each obs using the current code book
label = vq(data, code_book)[0]
# Update the code book by computing centroids
new_code_book, has_members = _vq.update_cluster_means(data, label, nc)
if not has_members.all():
miss_meth()
# Set the empty clusters to their previous positions
new_code_book[~has_members] = code_book[~has_members]
code_book = new_code_book
return code_book, label
def eigh(
a,
b=None,
lower=True,
eigvals_only=False,
overwrite_a=False,
overwrite_b=False,
turbo=True,
eigvals=None,
type=1,
check_finite=True,
):
"""
Solve an ordinary or generalized eigenvalue problem for a complex
Hermitian or real symmetric matrix.
Find eigenvalues w and optionally eigenvectors v of matrix `a`, where
`b` is positive definite::
a v[:,i] = w[i] b v[:,i]
v[i,:].conj() a v[:,i] = w[i]
v[i,:].conj() b v[:,i] = 1
Parameters
----------
a : (M, M) array_like
A complex Hermitian or real symmetric matrix whose eigenvalues and
eigenvectors will be computed.
b : (M, M) array_like, optional
A complex Hermitian or real symmetric definite positive matrix in.
If omitted, identity matrix is assumed.
lower : bool, optional
Whether the pertinent array data is taken from the lower or upper
triangle of `a`. (Default: lower)
eigvals_only : bool, optional
Whether to calculate only eigenvalues and no eigenvectors.
(Default: both are calculated)
turbo : bool, optional
Use divide and conquer algorithm (faster but expensive in memory,
only for generalized eigenvalue problem and if eigvals=None)
eigvals : tuple (lo, hi), optional
Indexes of the smallest and largest (in ascending order) eigenvalues
and corresponding eigenvectors to be returned: 0 <= lo <= hi <= M-1.
If omitted, all eigenvalues and eigenvectors are returned.
type : int, optional
Specifies the problem type to be solved:
type = 1: a v[:,i] = w[i] b v[:,i]
type = 2: a b v[:,i] = w[i] v[:,i]
type = 3: b a v[:,i] = w[i] v[:,i]
overwrite_a : bool, optional
Whether to overwrite data in `a` (may improve performance)
overwrite_b : bool, optional
Whether to overwrite data in `b` (may improve performance)
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
w : (N,) float ndarray
The N (1<=N<=M) selected eigenvalues, in ascending order, each
repeated according to its multiplicity.
v : (M, N) complex ndarray
(if eigvals_only == False)
The normalized selected eigenvector corresponding to the
eigenvalue w[i] is the column v[:,i].
Normalization:
type 1 and 3: v.conj() a v = w
type 2: inv(v).conj() a inv(v) = w
type = 1 or 2: v.conj() b v = I
type = 3: v.conj() inv(b) v = I
Raises
------
LinAlgError
If eigenvalue computation does not converge,
an error occurred, or b matrix is not definite positive. Note that
if input matrices are not symmetric or hermitian, no error is reported
but results will be wrong.
See Also
--------
eig : eigenvalues and right eigenvectors for non-symmetric arrays
"""
a1 = _asarray_validated(a, check_finite=check_finite)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError("expected square matrix")
overwrite_a = overwrite_a or (_datacopied(a1, a))
if iscomplexobj(a1):
cplx = True
else:
cplx = False
if b is not None:
b1 = _asarray_validated(b, check_finite=check_finite)
overwrite_b = overwrite_b or _datacopied(b1, b)
if len(b1.shape) != 2 or b1.shape[0] != b1.shape[1]:
raise ValueError("expected square matrix")
if b1.shape != a1.shape:
raise ValueError("wrong b dimensions %s, should " "be %s" % (str(b1.shape), str(a1.shape)))
if iscomplexobj(b1):
cplx = True
else:
cplx = cplx or False
else:
b1 = None
# Set job for fortran routines
_job = (eigvals_only and "N") or "V"
# port eigenvalue range from python to fortran convention
if eigvals is not None:
lo, hi = eigvals
if lo < 0 or hi >= a1.shape[0]:
raise ValueError(
"The eigenvalue range specified is not valid.\n" "Valid range is [%s,%s]" % (0, a1.shape[0] - 1)
)
lo += 1
hi += 1
eigvals = (lo, hi)
# set lower
if lower:
uplo = "L"
else:
uplo = "U"
# fix prefix for lapack routines
if cplx:
pfx = "he"
else:
pfx = "sy"
# Standard Eigenvalue Problem
# Use '*evr' routines
# FIXME: implement calculation of optimal lwork
# for all lapack routines
if b1 is None:
(evr,) = get_lapack_funcs((pfx + "evr",), (a1,))
if eigvals is None:
w, v, info = evr(a1, uplo=uplo, jobz=_job, range="A", il=1, iu=a1.shape[0], overwrite_a=overwrite_a)
else:
(lo, hi) = eigvals
w_tot, v, info = evr(a1, uplo=uplo, jobz=_job, range="I", il=lo, iu=hi, overwrite_a=overwrite_a)
w = w_tot[0 : hi - lo + 1]
# Generalized Eigenvalue Problem
else:
# Use '*gvx' routines if range is specified
if eigvals is not None:
(gvx,) = get_lapack_funcs((pfx + "gvx",), (a1, b1))
(lo, hi) = eigvals
w_tot, v, ifail, info = gvx(
a1, b1, uplo=uplo, iu=hi, itype=type, jobz=_job, il=lo, overwrite_a=overwrite_a, overwrite_b=overwrite_b
)
w = w_tot[0 : hi - lo + 1]
# Use '*gvd' routine if turbo is on and no eigvals are specified
elif turbo:
(gvd,) = get_lapack_funcs((pfx + "gvd",), (a1, b1))
v, w, info = gvd(a1, b1, uplo=uplo, itype=type, jobz=_job, overwrite_a=overwrite_a, overwrite_b=overwrite_b)
# Use '*gv' routine if turbo is off and no eigvals are specified
else:
(gv,) = get_lapack_funcs((pfx + "gv",), (a1, b1))
v, w, info = gv(a1, b1, uplo=uplo, itype=type, jobz=_job, overwrite_a=overwrite_a, overwrite_b=overwrite_b)
# Check if we had a successful exit
if info == 0:
if eigvals_only:
return w
else:
return w, v
elif info < 0:
raise LinAlgError("illegal value in %i-th argument of internal" " fortran routine." % (-info))
elif info > 0 and b1 is None:
raise LinAlgError("unrecoverable internal error.")
# The algorithm failed to converge.
elif 0 < info <= b1.shape[0]:
if eigvals is not None:
raise LinAlgError("the eigenvectors %s failed to" " converge." % nonzero(ifail) - 1)
else:
raise LinAlgError(
"internal fortran routine failed to converge: "
"%i off-diagonal elements of an "
"intermediate tridiagonal form did not converge"
" to zero." % info
)
# This occurs when b is not positive definite
else:
raise LinAlgError(
"the leading minor of order %i"
" of 'b' is not positive definite. The"
" factorization of 'b' could not be completed"
" and no eigenvalues or eigenvectors were"
" computed." % (info - b1.shape[0])
)
def _mul_triangular(c,
b,
trans=False,
lower=True,
overwrite_b=False,
check_finite=True):
"""wrapper for BLAS function trmv to perform triangular matrix
multiplications
Parameters
----------
a : (M, M) array_like
A triangular matrix
b : (M,) or (M, N) array_like
vector/matrix being multiplied
lower : bool, optional
Use only data contained in the lower triangle of `a`.
Default is to use upper triangle.
trans : bool, optional
type of multiplication,
======== =========
trans system
======== =========
False a b
True a^T b
overwrite_b : bool, optional
Allow overwriting data in `b` (may enhance performance)
not fully tested
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
"""
a1 = _asarray_validated(c, check_finite=check_finite)
b1 = _asarray_validated(b, check_finite=check_finite)
n = c.shape[1]
if c.shape[0] != n:
raise ValueError("Triangular matrix passed must be square")
if b.shape[0] != n:
raise ValueError(f"shapes {c.shape} and {b.shape} not aligned: "
f"{n} (dim 1) != {b.shape[0]} (dim 0)")
trmv, = get_blas_funcs(('trmv', ), (a1, b1))
if a1.flags.f_contiguous:
def _trmv(a1, b1, overwrite_x):
return trmv(a1,
b1,
lower=lower,
trans=trans,
overwrite_x=overwrite_x)
else:
# transposed system is solved since trmv expects Fortran ordering
def _trmv(a1, b1, overwrite_x=overwrite_b):
return trmv(a1.T,
b1,
lower=not lower,
trans=not trans,
overwrite_x=overwrite_x)
if b1.ndim == 1:
return _trmv(a1, b1, overwrite_b)
elif b1.ndim == 2:
# trmv only works for vector multiplications
# set Fortran order so memory contiguous
b2 = np.array(b1, order='F')
for i in range(b2.shape[1]):
# overwrite results
_trmv(a1, b2[:, i], True)
if overwrite_b:
b1[:] = b2
return b1
else:
return b2
else:
raise ValueError("b must have 1 or 2 dimensions, has {b.ndim}")
def eig_banded(a_band,
lower=False,
eigvals_only=False,
overwrite_a_band=False,
select='a',
select_range=None,
max_ev=0,
check_finite=True):
"""
Solve real symmetric or complex hermitian band matrix eigenvalue problem.
Find eigenvalues w and optionally right eigenvectors v of a::
a v[:,i] = w[i] v[:,i]
v.H v = identity
The matrix a is stored in a_band either in lower diagonal or upper
diagonal ordered form:
a_band[u + i - j, j] == a[i,j] (if upper form; i <= j)
a_band[ i - j, j] == a[i,j] (if lower form; i >= j)
where u is the number of bands above the diagonal.
Example of a_band (shape of a is (6,6), u=2)::
upper form:
* * a02 a13 a24 a35
* a01 a12 a23 a34 a45
a00 a11 a22 a33 a44 a55
lower form:
a00 a11 a22 a33 a44 a55
a10 a21 a32 a43 a54 *
a20 a31 a42 a53 * *
Cells marked with * are not used.
Parameters
----------
a_band : (u+1, M) array_like
The bands of the M by M matrix a.
lower : bool, optional
Is the matrix in the lower form. (Default is upper form)
eigvals_only : bool, optional
Compute only the eigenvalues and no eigenvectors.
(Default: calculate also eigenvectors)
overwrite_a_band : bool, optional
Discard data in a_band (may enhance performance)
select : {'a', 'v', 'i'}, optional
Which eigenvalues to calculate
====== ========================================
select calculated
====== ========================================
'a' All eigenvalues
'v' Eigenvalues in the interval (min, max]
'i' Eigenvalues with indices min <= i <= max
====== ========================================
select_range : (min, max), optional
Range of selected eigenvalues
max_ev : int, optional
For select=='v', maximum number of eigenvalues expected.
For other values of select, has no meaning.
In doubt, leave this parameter untouched.
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
w : (M,) ndarray
The eigenvalues, in ascending order, each repeated according to its
multiplicity.
v : (M, M) float or complex ndarray
The normalized eigenvector corresponding to the eigenvalue w[i] is
the column v[:,i].
Raises
------
LinAlgError
If eigenvalue computation does not converge.
See Also
--------
eigvals_banded : eigenvalues for symmetric/Hermitian band matrices
eig : eigenvalues and right eigenvectors of general arrays.
eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
eigh_tridiagonal : eigenvalues and right eiegenvectors for
symmetric/Hermitian tridiagonal matrices
Examples
--------
>>> from scipy.linalg import eig_banded
>>> A = np.array([[1, 5, 2, 0], [5, 2, 5, 2], [2, 5, 3, 5], [0, 2, 5, 4]])
>>> Ab = np.array([[1, 2, 3, 4], [5, 5, 5, 0], [2, 2, 0, 0]])
>>> w, v = eig_banded(Ab, lower=True)
>>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
True
>>> w = eig_banded(Ab, lower=True, eigvals_only=True)
>>> w
array([-4.26200532, -2.22987175, 3.95222349, 12.53965359])
Request only the eigenvalues between ``[-3, 4]``
>>> w, v = eig_banded(Ab, lower=True, select='v', select_range=[-3, 4])
>>> w
array([-2.22987175, 3.95222349])
"""
if eigvals_only or overwrite_a_band:
a1 = _asarray_validated(a_band, check_finite=check_finite)
overwrite_a_band = overwrite_a_band or (_datacopied(a1, a_band))
else:
a1 = array(a_band)
if issubclass(a1.dtype.type, inexact) and not isfinite(a1).all():
raise ValueError("array must not contain infs or NaNs")
overwrite_a_band = 1
if len(a1.shape) != 2:
raise ValueError('expected two-dimensional array')
select, vl, vu, il, iu, max_ev = _check_select(select, select_range,
max_ev, a1.shape[1])
del select_range
if select == 0:
if a1.dtype.char in 'GFD':
# FIXME: implement this somewhen, for now go with builtin values
# FIXME: calc optimal lwork by calling ?hbevd(lwork=-1)
# or by using calc_lwork.f ???
# lwork = calc_lwork.hbevd(bevd.typecode, a1.shape[0], lower)
internal_name = 'hbevd'
else: # a1.dtype.char in 'fd':
# FIXME: implement this somewhen, for now go with builtin values
# see above
# lwork = calc_lwork.sbevd(bevd.typecode, a1.shape[0], lower)
internal_name = 'sbevd'
bevd, = get_lapack_funcs((internal_name, ), (a1, ))
w, v, info = bevd(a1,
compute_v=not eigvals_only,
lower=lower,
overwrite_ab=overwrite_a_band)
else: # select in [1, 2]
if eigvals_only:
max_ev = 1
# calculate optimal abstol for dsbevx (see manpage)
if a1.dtype.char in 'fF': # single precision
lamch, = get_lapack_funcs(('lamch', ), (array(0, dtype='f'), ))
else:
lamch, = get_lapack_funcs(('lamch', ), (array(0, dtype='d'), ))
abstol = 2 * lamch('s')
if a1.dtype.char in 'GFD':
internal_name = 'hbevx'
else: # a1.dtype.char in 'gfd'
internal_name = 'sbevx'
bevx, = get_lapack_funcs((internal_name, ), (a1, ))
w, v, m, ifail, info = bevx(a1,
vl,
vu,
il,
iu,
compute_v=not eigvals_only,
mmax=max_ev,
range=select,
lower=lower,
overwrite_ab=overwrite_a_band,
abstol=abstol)
# crop off w and v
w = w[:m]
if not eigvals_only:
v = v[:, :m]
_check_info(info, internal_name)
if eigvals_only:
return w
return w, v
def py_vq(obs, code_book, check_finite=True):
""" Python version of vq algorithm.
The algorithm computes the euclidian distance between each
observation and every frame in the code_book.
Parameters
----------
obs : ndarray
Expects a rank 2 array. Each row is one observation.
code_book : ndarray
Code book to use. Same format than obs. Should have same number of
features (eg columns) than obs.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
code : ndarray
code[i] gives the label of the ith obversation, that its code is
code_book[code[i]].
mind_dist : ndarray
min_dist[i] gives the distance between the ith observation and its
corresponding code.
Notes
-----
This function is slower than the C version but works for
all input types. If the inputs have the wrong types for the
C versions of the function, this one is called as a last resort.
It is about 20 times slower than the C version.
"""
obs = _asarray_validated(obs, check_finite=check_finite)
code_book = _asarray_validated(code_book, check_finite=check_finite)
# n = number of observations
# d = number of features
if np.ndim(obs) == 1:
if not np.ndim(obs) == np.ndim(code_book):
raise ValueError(
"Observation and code_book should have the same rank")
else:
return _py_vq_1d(obs, code_book)
else:
(n, d) = shape(obs)
# code books and observations should have same number of features and same
# shape
if not np.ndim(obs) == np.ndim(code_book):
raise ValueError("Observation and code_book should have the same rank")
elif not d == code_book.shape[1]:
raise ValueError("Code book(%d) and obs(%d) should have the same "
"number of features (eg columns)"
"" % (code_book.shape[1], d))
code = zeros(n, dtype=int)
min_dist = zeros(n)
for i in range(n):
dist = np.sum((obs[i] - code_book)**2, 1)
code[i] = argmin(dist)
min_dist[i] = dist[code[i]]
return code, sqrt(min_dist)
def eig(a, b=None, left=False, right=True, overwrite_a=False,
overwrite_b=False, check_finite=True, homogeneous_eigvals=False):
"""
Solve an ordinary or generalized eigenvalue problem of a square matrix.
Find eigenvalues w and right or left eigenvectors of a general matrix::
a vr[:,i] = w[i] b vr[:,i]
a.H vl[:,i] = w[i].conj() b.H vl[:,i]
where ``.H`` is the Hermitian conjugation.
Parameters
----------
a : (M, M) array_like
A complex or real matrix whose eigenvalues and eigenvectors
will be computed.
b : (M, M) array_like, optional
Right-hand side matrix in a generalized eigenvalue problem.
Default is None, identity matrix is assumed.
left : bool, optional
Whether to calculate and return left eigenvectors. Default is False.
right : bool, optional
Whether to calculate and return right eigenvectors. Default is True.
overwrite_a : bool, optional
Whether to overwrite `a`; may improve performance. Default is False.
overwrite_b : bool, optional
Whether to overwrite `b`; may improve performance. Default is False.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
homogeneous_eigvals : bool, optional
If True, return the eigenvalues in homogeneous coordinates.
In this case ``w`` is a (2, M) array so that::
w[1,i] a vr[:,i] = w[0,i] b vr[:,i]
Default is False.
Returns
-------
w : (M,) or (2, M) double or complex ndarray
The eigenvalues, each repeated according to its
multiplicity. The shape is (M,) unless
``homogeneous_eigvals=True``.
vl : (M, M) double or complex ndarray
The normalized left eigenvector corresponding to the eigenvalue
``w[i]`` is the column vl[:,i]. Only returned if ``left=True``.
vr : (M, M) double or complex ndarray
The normalized right eigenvector corresponding to the eigenvalue
``w[i]`` is the column ``vr[:,i]``. Only returned if ``right=True``.
Raises
------
LinAlgError
If eigenvalue computation does not converge.
See Also
--------
eigvals : eigenvalues of general arrays
eigh : Eigenvalues and right eigenvectors for symmetric/Hermitian arrays.
eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian
band matrices
eigh_tridiagonal : eigenvalues and right eiegenvectors for
symmetric/Hermitian tridiagonal matrices
Examples
--------
>>> from scipy import linalg
>>> a = np.array([[0., -1.], [1., 0.]])
>>> linalg.eigvals(a)
array([0.+1.j, 0.-1.j])
>>> b = np.array([[0., 1.], [1., 1.]])
>>> linalg.eigvals(a, b)
array([ 1.+0.j, -1.+0.j])
>>> a = np.array([[3., 0., 0.], [0., 8., 0.], [0., 0., 7.]])
>>> linalg.eigvals(a, homogeneous_eigvals=True)
array([[3.+0.j, 8.+0.j, 7.+0.j],
[1.+0.j, 1.+0.j, 1.+0.j]])
"""
a1 = _asarray_validated(a, check_finite=check_finite)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or (_datacopied(a1, a))
if b is not None:
b1 = _asarray_validated(b, check_finite=check_finite)
overwrite_b = overwrite_b or _datacopied(b1, b)
if len(b1.shape) != 2 or b1.shape[0] != b1.shape[1]:
raise ValueError('expected square matrix')
if b1.shape != a1.shape:
raise ValueError('a and b must have the same shape')
return _geneig(a1, b1, left, right, overwrite_a, overwrite_b,
homogeneous_eigvals)
geev, geev_lwork = get_lapack_funcs(('geev', 'geev_lwork'), (a1,))
compute_vl, compute_vr = left, right
lwork = _compute_lwork(geev_lwork, a1.shape[0],
compute_vl=compute_vl,
compute_vr=compute_vr)
if geev.typecode in 'cz':
w, vl, vr, info = geev(a1, lwork=lwork,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
w = _make_eigvals(w, None, homogeneous_eigvals)
else:
wr, wi, vl, vr, info = geev(a1, lwork=lwork,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
t = {'f': 'F', 'd': 'D'}[wr.dtype.char]
w = wr + _I * wi
w = _make_eigvals(w, None, homogeneous_eigvals)
_check_info(info, 'eig algorithm (geev)',
positive='did not converge (only eigenvalues '
'with order >= %d have converged)')
only_real = numpy.all(w.imag == 0.0)
if not (geev.typecode in 'cz' or only_real):
t = w.dtype.char
if left:
vl = _make_complex_eigvecs(w, vl, t)
if right:
vr = _make_complex_eigvecs(w, vr, t)
if not (left or right):
return w
if left:
if right:
return w, vl, vr
return w, vl
return w, vr
def kmeans2(data,
k,
iter=10,
thresh=1e-5,
minit='random',
missing='warn',
check_finite=True):
"""
Classify a set of observations into k clusters using the k-means algorithm.
The algorithm attempts to minimize the Euclidian distance between
observations and centroids. Several initialization methods are
included.
Parameters
----------
data : ndarray
A 'M' by 'N' array of 'M' observations in 'N' dimensions or a length
'M' array of 'M' one-dimensional observations.
k : int or ndarray
The number of clusters to form as well as the number of
centroids to generate. If `minit` initialization string is
'matrix', or if a ndarray is given instead, it is
interpreted as initial cluster to use instead.
iter : int, optional
Number of iterations of the k-means algorithm to run. Note
that this differs in meaning from the iters parameter to
the kmeans function.
thresh : float, optional
(not used yet)
minit : str, optional
Method for initialization. Available methods are 'random',
'points', and 'matrix':
'random': generate k centroids from a Gaussian with mean and
variance estimated from the data.
'points': choose k observations (rows) at random from data for
the initial centroids.
'matrix': interpret the k parameter as a k by M (or length k
array for one-dimensional data) array of initial centroids.
missing : str, optional
Method to deal with empty clusters. Available methods are
'warn' and 'raise':
'warn': give a warning and continue.
'raise': raise an ClusterError and terminate the algorithm.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
centroid : ndarray
A 'k' by 'N' array of centroids found at the last iteration of
k-means.
label : ndarray
label[i] is the code or index of the centroid the
i'th observation is closest to.
"""
if int(iter) < 1:
raise ValueError("Invalid iter (%s), "
"must be a positive integer." % iter)
try:
miss_meth = _valid_miss_meth[missing]
except KeyError:
raise ValueError("Unknown missing method %r" % (missing, ))
data = _asarray_validated(data, check_finite=check_finite)
if data.ndim == 1:
d = 1
elif data.ndim == 2:
d = data.shape[1]
else:
raise ValueError("Input of rank > 2 is not supported.")
if data.size < 1:
raise ValueError("Empty input is not supported.")
# If k is not a single value it should be compatible with data's shape
if minit == 'matrix' or not np.isscalar(k):
code_book = np.array(k, copy=True)
if data.ndim != code_book.ndim:
raise ValueError("k array doesn't match data rank")
nc = len(code_book)
if data.ndim > 1 and code_book.shape[1] != d:
raise ValueError("k array doesn't match data dimension")
else:
nc = int(k)
if nc < 1:
raise ValueError("Cannot ask kmeans2 for %d clusters"
" (k was %s)" % (nc, k))
elif nc != k:
warnings.warn("k was not an integer, was converted.")
try:
init_meth = _valid_init_meth[minit]
except KeyError:
raise ValueError("Unknown init method %r" % (minit, ))
else:
code_book = init_meth(data, k)
for i in xrange(iter):
# Compute the nearest neighbor for each obs using the current code book
label = vq(data, code_book)[0]
# Update the code book by computing centroids
new_code_book, has_members = _vq.update_cluster_means(data, label, nc)
if not has_members.all():
miss_meth()
# Set the empty clusters to their previous positions
new_code_book[~has_members] = code_book[~has_members]
code_book = new_code_book
return code_book, label
def softmax(x, axis=None):
r"""Compute the softmax function.
The softmax function transforms each element of a collection by
computing the exponential of each element divided by the sum of the
exponentials of all the elements. That is, if `x` is a one-dimensional
numpy array::
softmax(x) = np.exp(x)/sum(np.exp(x))
Parameters
----------
x : array_like
Input array.
axis : int or tuple of ints, optional
Axis to compute values along. Default is None and softmax will be
computed over the entire array `x`.
Returns
-------
s : ndarray
An array the same shape as `x`. The result will sum to 1 along the
specified axis.
Notes
-----
The formula for the softmax function :math:`\sigma(x)` for a vector
:math:`x = \{x_0, x_1, ..., x_{n-1}\}` is
.. math:: \sigma(x)_j = \frac{e^{x_j}}{\sum_k e^{x_k}}
The `softmax` function is the gradient of `logsumexp`.
The implementation uses shifting to avoid overflow. See [1]_ for more
details.
.. versionadded:: 1.2.0
References
----------
.. [1] P. Blanchard, D.J. Higham, N.J. Higham, "Accurately computing the
log-sum-exp and softmax functions", IMA Journal of Numerical Analysis,
Vol.41(4), :doi:`10.1093/imanum/draa038`.
Examples
--------
>>> from scipy.special import softmax
>>> np.set_printoptions(precision=5)
>>> x = np.array([[1, 0.5, 0.2, 3],
... [1, -1, 7, 3],
... [2, 12, 13, 3]])
...
Compute the softmax transformation over the entire array.
>>> m = softmax(x)
>>> m
array([[ 4.48309e-06, 2.71913e-06, 2.01438e-06, 3.31258e-05],
[ 4.48309e-06, 6.06720e-07, 1.80861e-03, 3.31258e-05],
[ 1.21863e-05, 2.68421e-01, 7.29644e-01, 3.31258e-05]])
>>> m.sum()
1.0
Compute the softmax transformation along the first axis (i.e., the
columns).
>>> m = softmax(x, axis=0)
>>> m
array([[ 2.11942e-01, 1.01300e-05, 2.75394e-06, 3.33333e-01],
[ 2.11942e-01, 2.26030e-06, 2.47262e-03, 3.33333e-01],
[ 5.76117e-01, 9.99988e-01, 9.97525e-01, 3.33333e-01]])
>>> m.sum(axis=0)
array([ 1., 1., 1., 1.])
Compute the softmax transformation along the second axis (i.e., the rows).
>>> m = softmax(x, axis=1)
>>> m
array([[ 1.05877e-01, 6.42177e-02, 4.75736e-02, 7.82332e-01],
[ 2.42746e-03, 3.28521e-04, 9.79307e-01, 1.79366e-02],
[ 1.22094e-05, 2.68929e-01, 7.31025e-01, 3.31885e-05]])
>>> m.sum(axis=1)
array([ 1., 1., 1.])
"""
x = _asarray_validated(x, check_finite=False)
x_max = np.amax(x, axis=axis, keepdims=True)
exp_x_shifted = np.exp(x - x_max)
return exp_x_shifted / np.sum(exp_x_shifted, axis=axis, keepdims=True)
def eig(a, b=None, left=False, right=True, overwrite_a=False, overwrite_b=False, check_finite=True):
"""
Solve an ordinary or generalized eigenvalue problem of a square matrix.
Find eigenvalues w and right or left eigenvectors of a general matrix::
a vr[:,i] = w[i] b vr[:,i]
a.H vl[:,i] = w[i].conj() b.H vl[:,i]
where ``.H`` is the Hermitian conjugation.
Parameters
----------
a : (M, M) array_like
A complex or real matrix whose eigenvalues and eigenvectors
will be computed.
b : (M, M) array_like, optional
Right-hand side matrix in a generalized eigenvalue problem.
Default is None, identity matrix is assumed.
left : bool, optional
Whether to calculate and return left eigenvectors. Default is False.
right : bool, optional
Whether to calculate and return right eigenvectors. Default is True.
overwrite_a : bool, optional
Whether to overwrite `a`; may improve performance. Default is False.
overwrite_b : bool, optional
Whether to overwrite `b`; may improve performance. Default is False.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
w : (M,) double or complex ndarray
The eigenvalues, each repeated according to its multiplicity.
vl : (M, M) double or complex ndarray
The normalized left eigenvector corresponding to the eigenvalue
``w[i]`` is the column vl[:,i]. Only returned if ``left=True``.
vr : (M, M) double or complex ndarray
The normalized right eigenvector corresponding to the eigenvalue
``w[i]`` is the column ``vr[:,i]``. Only returned if ``right=True``.
Raises
------
LinAlgError
If eigenvalue computation does not converge.
See Also
--------
eigh : Eigenvalues and right eigenvectors for symmetric/Hermitian arrays.
"""
a1 = _asarray_validated(a, check_finite=check_finite)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError("expected square matrix")
overwrite_a = overwrite_a or (_datacopied(a1, a))
if b is not None:
b1 = _asarray_validated(b, check_finite=check_finite)
overwrite_b = overwrite_b or _datacopied(b1, b)
if len(b1.shape) != 2 or b1.shape[0] != b1.shape[1]:
raise ValueError("expected square matrix")
if b1.shape != a1.shape:
raise ValueError("a and b must have the same shape")
return _geneig(a1, b1, left, right, overwrite_a, overwrite_b)
geev, geev_lwork = get_lapack_funcs(("geev", "geev_lwork"), (a1,))
compute_vl, compute_vr = left, right
lwork = _compute_lwork(geev_lwork, a1.shape[0], compute_vl=compute_vl, compute_vr=compute_vr)
if geev.typecode in "cz":
w, vl, vr, info = geev(a1, lwork=lwork, compute_vl=compute_vl, compute_vr=compute_vr, overwrite_a=overwrite_a)
else:
wr, wi, vl, vr, info = geev(
a1, lwork=lwork, compute_vl=compute_vl, compute_vr=compute_vr, overwrite_a=overwrite_a
)
t = {"f": "F", "d": "D"}[wr.dtype.char]
w = wr + _I * wi
if info < 0:
raise ValueError("illegal value in %d-th argument of internal geev" % -info)
if info > 0:
raise LinAlgError("eig algorithm did not converge (only eigenvalues " "with order >= %d have converged)" % info)
only_real = numpy.logical_and.reduce(numpy.equal(w.imag, 0.0))
if not (geev.typecode in "cz" or only_real):
t = w.dtype.char
if left:
vl = _make_complex_eigvecs(w, vl, t)
if right:
vr = _make_complex_eigvecs(w, vr, t)
if not (left or right):
return w
if left:
if right:
return w, vl, vr
return w, vl
return w, vr
def hessenberg(a, calc_q=False, overwrite_a=False, check_finite=True):
"""
Compute Hessenberg form of a matrix.
The Hessenberg decomposition is::
A = Q H Q^H
where `Q` is unitary/orthogonal and `H` has only zero elements below
the first sub-diagonal.
Parameters
----------
a : (M, M) array_like
Matrix to bring into Hessenberg form.
calc_q : bool, optional
Whether to compute the transformation matrix. Default is False.
overwrite_a : bool, optional
Whether to overwrite `a`; may improve performance.
Default is False.
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
H : (M, M) ndarray
Hessenberg form of `a`.
Q : (M, M) ndarray
Unitary/orthogonal similarity transformation matrix ``A = Q H Q^H``.
Only returned if ``calc_q=True``.
"""
a1 = _asarray_validated(a, check_finite=check_finite)
if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]):
raise ValueError("expected square matrix")
overwrite_a = overwrite_a or (_datacopied(a1, a))
# if 2x2 or smaller: already in Hessenberg
if a1.shape[0] <= 2:
if calc_q:
return a1, numpy.eye(a1.shape[0])
return a1
gehrd, gebal, gehrd_lwork = get_lapack_funcs(("gehrd", "gebal", "gehrd_lwork"), (a1,))
ba, lo, hi, pivscale, info = gebal(a1, permute=0, overwrite_a=overwrite_a)
if info < 0:
raise ValueError("illegal value in %d-th argument of internal gebal " "(hessenberg)" % -info)
n = len(a1)
lwork = _compute_lwork(gehrd_lwork, ba.shape[0], lo=lo, hi=hi)
hq, tau, info = gehrd(ba, lo=lo, hi=hi, lwork=lwork, overwrite_a=1)
if info < 0:
raise ValueError("illegal value in %d-th argument of internal gehrd " "(hessenberg)" % -info)
h = numpy.triu(hq, -1)
if not calc_q:
return h
# use orghr/unghr to compute q
orghr, orghr_lwork = get_lapack_funcs(("orghr", "orghr_lwork"), (a1,))
lwork = _compute_lwork(orghr_lwork, n, lo=lo, hi=hi)
q, info = orghr(a=hq, tau=tau, lo=lo, hi=hi, lwork=lwork, overwrite_a=1)
if info < 0:
raise ValueError("illegal value in %d-th argument of internal orghr " "(hessenberg)" % -info)
return h, q
def eig_banded(
a_band,
lower=False,
eigvals_only=False,
overwrite_a_band=False,
select="a",
select_range=None,
max_ev=0,
check_finite=True,
):
"""
Solve real symmetric or complex hermitian band matrix eigenvalue problem.
Find eigenvalues w and optionally right eigenvectors v of a::
a v[:,i] = w[i] v[:,i]
v.H v = identity
The matrix a is stored in a_band either in lower diagonal or upper
diagonal ordered form:
a_band[u + i - j, j] == a[i,j] (if upper form; i <= j)
a_band[ i - j, j] == a[i,j] (if lower form; i >= j)
where u is the number of bands above the diagonal.
Example of a_band (shape of a is (6,6), u=2)::
upper form:
* * a02 a13 a24 a35
* a01 a12 a23 a34 a45
a00 a11 a22 a33 a44 a55
lower form:
a00 a11 a22 a33 a44 a55
a10 a21 a32 a43 a54 *
a20 a31 a42 a53 * *
Cells marked with * are not used.
Parameters
----------
a_band : (u+1, M) array_like
The bands of the M by M matrix a.
lower : bool, optional
Is the matrix in the lower form. (Default is upper form)
eigvals_only : bool, optional
Compute only the eigenvalues and no eigenvectors.
(Default: calculate also eigenvectors)
overwrite_a_band : bool, optional
Discard data in a_band (may enhance performance)
select : {'a', 'v', 'i'}, optional
Which eigenvalues to calculate
====== ========================================
select calculated
====== ========================================
'a' All eigenvalues
'v' Eigenvalues in the interval (min, max]
'i' Eigenvalues with indices min <= i <= max
====== ========================================
select_range : (min, max), optional
Range of selected eigenvalues
max_ev : int, optional
For select=='v', maximum number of eigenvalues expected.
For other values of select, has no meaning.
In doubt, leave this parameter untouched.
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
w : (M,) ndarray
The eigenvalues, in ascending order, each repeated according to its
multiplicity.
v : (M, M) float or complex ndarray
The normalized eigenvector corresponding to the eigenvalue w[i] is
the column v[:,i].
Raises LinAlgError if eigenvalue computation does not converge
"""
if eigvals_only or overwrite_a_band:
a1 = _asarray_validated(a_band, check_finite=check_finite)
overwrite_a_band = overwrite_a_band or (_datacopied(a1, a_band))
else:
a1 = array(a_band)
if issubclass(a1.dtype.type, inexact) and not isfinite(a1).all():
raise ValueError("array must not contain infs or NaNs")
overwrite_a_band = 1
if len(a1.shape) != 2:
raise ValueError("expected two-dimensional array")
if select.lower() not in [0, 1, 2, "a", "v", "i", "all", "value", "index"]:
raise ValueError("invalid argument for select")
if select.lower() in [0, "a", "all"]:
if a1.dtype.char in "GFD":
bevd, = get_lapack_funcs(("hbevd",), (a1,))
# FIXME: implement this somewhen, for now go with builtin values
# FIXME: calc optimal lwork by calling ?hbevd(lwork=-1)
# or by using calc_lwork.f ???
# lwork = calc_lwork.hbevd(bevd.typecode, a1.shape[0], lower)
internal_name = "hbevd"
else: # a1.dtype.char in 'fd':
bevd, = get_lapack_funcs(("sbevd",), (a1,))
# FIXME: implement this somewhen, for now go with builtin values
# see above
# lwork = calc_lwork.sbevd(bevd.typecode, a1.shape[0], lower)
internal_name = "sbevd"
w, v, info = bevd(a1, compute_v=not eigvals_only, lower=lower, overwrite_ab=overwrite_a_band)
if select.lower() in [1, 2, "i", "v", "index", "value"]:
# calculate certain range only
if select.lower() in [2, "i", "index"]:
select = 2
vl, vu, il, iu = 0.0, 0.0, min(select_range), max(select_range)
if min(il, iu) < 0 or max(il, iu) >= a1.shape[1]:
raise ValueError("select_range out of bounds")
max_ev = iu - il + 1
else: # 1, 'v', 'value'
select = 1
vl, vu, il, iu = min(select_range), max(select_range), 0, 0
if max_ev == 0:
max_ev = a_band.shape[1]
if eigvals_only:
max_ev = 1
# calculate optimal abstol for dsbevx (see manpage)
if a1.dtype.char in "fF": # single precision
lamch, = get_lapack_funcs(("lamch",), (array(0, dtype="f"),))
else:
lamch, = get_lapack_funcs(("lamch",), (array(0, dtype="d"),))
abstol = 2 * lamch("s")
if a1.dtype.char in "GFD":
bevx, = get_lapack_funcs(("hbevx",), (a1,))
internal_name = "hbevx"
else: # a1.dtype.char in 'gfd'
bevx, = get_lapack_funcs(("sbevx",), (a1,))
internal_name = "sbevx"
# il+1, iu+1: translate python indexing (0 ... N-1) into Fortran
# indexing (1 ... N)
w, v, m, ifail, info = bevx(
a1,
vl,
vu,
il + 1,
iu + 1,
compute_v=not eigvals_only,
mmax=max_ev,
range=select,
lower=lower,
overwrite_ab=overwrite_a_band,
abstol=abstol,
)
# crop off w and v
w = w[:m]
if not eigvals_only:
v = v[:, :m]
if info < 0:
raise ValueError("illegal value in %d-th argument of internal %s" % (-info, internal_name))
if info > 0:
raise LinAlgError("eig algorithm did not converge")
if eigvals_only:
return w
return w, v
def kmeans(obs, k_or_guess, iter=20, thresh=1e-5, check_finite=True):
"""
Performs k-means on a set of observation vectors forming k clusters.
The k-means algorithm adjusts the centroids until sufficient
progress cannot be made, i.e. the change in distortion since
the last iteration is less than some threshold. This yields
a code book mapping centroids to codes and vice versa.
Distortion is defined as the sum of the squared differences
between the observations and the corresponding centroid.
Parameters
----------
obs : ndarray
Each row of the M by N array is an observation vector. The
columns are the features seen during each observation.
The features must be whitened first with the `whiten` function.
k_or_guess : int or ndarray
The number of centroids to generate. A code is assigned to
each centroid, which is also the row index of the centroid
in the code_book matrix generated.
The initial k centroids are chosen by randomly selecting
observations from the observation matrix. Alternatively,
passing a k by N array specifies the initial k centroids.
iter : int, optional
The number of times to run k-means, returning the codebook
with the lowest distortion. This argument is ignored if
initial centroids are specified with an array for the
``k_or_guess`` parameter. This parameter does not represent the
number of iterations of the k-means algorithm.
thresh : float, optional
Terminates the k-means algorithm if the change in
distortion since the last k-means iteration is less than
or equal to thresh.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
codebook : ndarray
A k by N array of k centroids. The i'th centroid
codebook[i] is represented with the code i. The centroids
and codes generated represent the lowest distortion seen,
not necessarily the globally minimal distortion.
distortion : float
The distortion between the observations passed and the
centroids generated.
See Also
--------
kmeans2 : a different implementation of k-means clustering
with more methods for generating initial centroids but without
using a distortion change threshold as a stopping criterion.
whiten : must be called prior to passing an observation matrix
to kmeans.
Examples
--------
>>> from numpy import array
>>> from scipy.cluster.vq import vq, kmeans, whiten
>>> features = array([[ 1.9,2.3],
... [ 1.5,2.5],
... [ 0.8,0.6],
... [ 0.4,1.8],
... [ 0.1,0.1],
... [ 0.2,1.8],
... [ 2.0,0.5],
... [ 0.3,1.5],
... [ 1.0,1.0]])
>>> whitened = whiten(features)
>>> book = array((whitened[0],whitened[2]))
>>> kmeans(whitened,book)
(array([[ 2.3110306 , 2.86287398], # random
[ 0.93218041, 1.24398691]]), 0.85684700941625547)
>>> from numpy import random
>>> random.seed((1000,2000))
>>> codes = 3
>>> kmeans(whitened,codes)
(array([[ 2.3110306 , 2.86287398], # random
[ 1.32544402, 0.65607529],
[ 0.40782893, 2.02786907]]), 0.5196582527686241)
"""
obs = _asarray_validated(obs, check_finite=check_finite)
if int(iter) < 1:
raise ValueError('iter must be at least 1.')
# Determine whether a count (scalar) or an initial guess (array) was passed.
k = None
guess = None
try:
k = int(k_or_guess)
except TypeError:
guess = _asarray_validated(k_or_guess, check_finite=check_finite)
if guess is not None:
if guess.size < 1:
raise ValueError("Asked for 0 cluster ? initial book was %s" %
guess)
result = _kmeans(obs, guess, thresh=thresh)
else:
if k != k_or_guess:
raise ValueError('if k_or_guess is a scalar, it must be an integer')
# initialize best distance value to a large value
best_dist = np.inf
No = obs.shape[0]
k = k_or_guess
if k < 1:
raise ValueError("Asked for 0 cluster ? ")
for i in range(iter):
# the initial code book is randomly selected from observations
guess = take(obs, randint(0, No, k), 0)
book, dist = _kmeans(obs, guess, thresh=thresh)
if dist < best_dist:
best_book = book
best_dist = dist
result = best_book, best_dist
return result
def kmeans2(data, k, iter=10, thresh=1e-5, minit='random',
missing='warn', check_finite=True):
"""
Classify a set of observations into k clusters using the k-means algorithm.
The algorithm attempts to minimize the Euclidian distance between
observations and centroids. Several initialization methods are
included.
Parameters
----------
data : ndarray
A 'M' by 'N' array of 'M' observations in 'N' dimensions or a length
'M' array of 'M' one-dimensional observations.
k : int or ndarray
The number of clusters to form as well as the number of
centroids to generate. If `minit` initialization string is
'matrix', or if a ndarray is given instead, it is
interpreted as initial cluster to use instead.
iter : int, optional
Number of iterations of the k-means algorithm to run. Note
that this differs in meaning from the iters parameter to
the kmeans function.
thresh : float, optional
(not used yet)
minit : str, optional
Method for initialization. Available methods are 'random',
'points', and 'matrix':
'random': generate k centroids from a Gaussian with mean and
variance estimated from the data.
'points': choose k observations (rows) at random from data for
the initial centroids.
'matrix': interpret the k parameter as a k by M (or length k
array for one-dimensional data) array of initial centroids.
missing : str, optional
Method to deal with empty clusters. Available methods are
'warn' and 'raise':
'warn': give a warning and continue.
'raise': raise an ClusterError and terminate the algorithm.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
centroid : ndarray
A 'k' by 'N' array of centroids found at the last iteration of
k-means.
label : ndarray
label[i] is the code or index of the centroid the
i'th observation is closest to.
"""
if int(iter) < 1:
raise ValueError("Invalid iter (%s), "
"must be a positive integer." % iter)
try:
miss_meth = _valid_miss_meth[missing]
except KeyError:
raise ValueError("Unknown missing method %r" % (missing,))
data = _asarray_validated(data, check_finite=check_finite)
if data.ndim == 1:
d = 1
elif data.ndim == 2:
d = data.shape[1]
else:
raise ValueError("Input of rank > 2 is not supported.")
if data.size < 1:
raise ValueError("Empty input is not supported.")
# If k is not a single value it should be compatible with data's shape
if minit == 'matrix' or not np.isscalar(k):
code_book = np.array(k, copy=True)
if data.ndim != code_book.ndim:
raise ValueError("k array doesn't match data rank")
nc = len(code_book)
if data.ndim > 1 and code_book.shape[1] != d:
raise ValueError("k array doesn't match data dimension")
else:
nc = int(k)
if nc < 1:
raise ValueError("Cannot ask kmeans2 for %d clusters"
" (k was %s)" % (nc, k))
elif nc != k:
warnings.warn("k was not an integer, was converted.")
try:
init_meth = _valid_init_meth[minit]
except KeyError:
raise ValueError("Unknown init method %r" % (minit,))
else:
code_book = init_meth(data, k)
for i in xrange(iter):
# Compute the nearest neighbor for each obs using the current code book
label = vq(data, code_book)[0]
# Update the code book by computing centroids
new_code_book, has_members = _vq.update_cluster_means(data, label, nc)
if not has_members.all():
miss_meth()
# Set the empty clusters to their previous positions
new_code_book[~has_members] = code_book[~has_members]
code_book = new_code_book
return code_book, label
def kmeans2(data, k, iter=10, thresh=1e-5, minit='random',
missing='warn', check_finite=True):
"""
Classify a set of observations into k clusters using the k-means algorithm.
The algorithm attempts to minimize the Euclidian distance between
observations and centroids. Several initialization methods are
included.
Parameters
----------
data : ndarray
A 'M' by 'N' array of 'M' observations in 'N' dimensions or a length
'M' array of 'M' one-dimensional observations.
k : int or ndarray
The number of clusters to form as well as the number of
centroids to generate. If `minit` initialization string is
'matrix', or if a ndarray is given instead, it is
interpreted as initial cluster to use instead.
iter : int, optional
Number of iterations of the k-means algrithm to run. Note
that this differs in meaning from the iters parameter to
the kmeans function.
thresh : float, optional
(not used yet)
minit : str, optional
Method for initialization. Available methods are 'random',
'points', 'uniform', and 'matrix':
'random': generate k centroids from a Gaussian with mean and
variance estimated from the data.
'points': choose k observations (rows) at random from data for
the initial centroids.
'uniform': generate k observations from the data from a uniform
distribution defined by the data set (unsupported).
'matrix': interpret the k parameter as a k by M (or length k
array for one-dimensional data) array of initial centroids.
missing : str, optional
Method to deal with empty clusters. Available methods are
'warn' and 'raise':
'warn': give a warning and continue.
'raise': raise an ClusterError and terminate the algorithm.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
centroid : ndarray
A 'k' by 'N' array of centroids found at the last iteration of
k-means.
label : ndarray
label[i] is the code or index of the centroid the
i'th observation is closest to.
"""
data = _asarray_validated(data, check_finite=check_finite)
if missing not in _valid_miss_meth:
raise ValueError("Unkown missing method: %s" % str(missing))
# If data is rank 1, then we have 1 dimension problem.
nd = np.ndim(data)
if nd == 1:
d = 1
# raise ValueError("Input of rank 1 not supported yet")
elif nd == 2:
d = data.shape[1]
else:
raise ValueError("Input of rank > 2 not supported")
if np.size(data) < 1:
raise ValueError("Input has 0 items.")
# If k is not a single value, then it should be compatible with data's
# shape
if np.size(k) > 1 or minit == 'matrix':
if not nd == np.ndim(k):
raise ValueError("k is not an int and has not same rank than data")
if d == 1:
nc = len(k)
else:
(nc, dc) = k.shape
if not dc == d:
raise ValueError("k is not an int and has not same rank than\
data")
clusters = k.copy()
else:
try:
nc = int(k)
except TypeError:
raise ValueError("k (%s) could not be converted to an integer " % str(k))
if nc < 1:
raise ValueError("kmeans2 for 0 clusters ? (k was %s)" % str(k))
if not nc == k:
warnings.warn("k was not an integer, was converted.")
try:
init = _valid_init_meth[minit]
except KeyError:
raise ValueError("unknown init method %s" % str(minit))
clusters = init(data, k)
if int(iter) < 1:
raise ValueError("iter = %s is not valid. iter must be a positive integer." % iter)
return _kmeans2(data, clusters, iter, nc, _valid_miss_meth[missing])
def eig(a,
b=None,
left=False,
right=True,
overwrite_a=False,
overwrite_b=False,
check_finite=True,
homogeneous_eigvals=False):
"""
Solve an ordinary or generalized eigenvalue problem of a square matrix.
Find eigenvalues w and right or left eigenvectors of a general matrix::
a vr[:,i] = w[i] b vr[:,i]
a.H vl[:,i] = w[i].conj() b.H vl[:,i]
where ``.H`` is the Hermitian conjugation.
Parameters
----------
a : (M, M) array_like
A complex or real matrix whose eigenvalues and eigenvectors
will be computed.
b : (M, M) array_like, optional
Right-hand side matrix in a generalized eigenvalue problem.
Default is None, identity matrix is assumed.
left : bool, optional
Whether to calculate and return left eigenvectors. Default is False.
right : bool, optional
Whether to calculate and return right eigenvectors. Default is True.
overwrite_a : bool, optional
Whether to overwrite `a`; may improve performance. Default is False.
overwrite_b : bool, optional
Whether to overwrite `b`; may improve performance. Default is False.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
homogeneous_eigvals : bool, optional
If True, return the eigenvalues in homogeneous coordinates.
In this case ``w`` is a (2, M) array so that::
w[1,i] a vr[:,i] = w[0,i] b vr[:,i]
Default is False.
Returns
-------
w : (M,) or (2, M) double or complex ndarray
The eigenvalues, each repeated according to its
multiplicity. The shape is (M,) unless
``homogeneous_eigvals=True``.
vl : (M, M) double or complex ndarray
The normalized left eigenvector corresponding to the eigenvalue
``w[i]`` is the column vl[:,i]. Only returned if ``left=True``.
vr : (M, M) double or complex ndarray
The normalized right eigenvector corresponding to the eigenvalue
``w[i]`` is the column ``vr[:,i]``. Only returned if ``right=True``.
Raises
------
LinAlgError
If eigenvalue computation does not converge.
See Also
--------
eigh : Eigenvalues and right eigenvectors for symmetric/Hermitian arrays.
"""
a1 = _asarray_validated(a, check_finite=check_finite)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or (_datacopied(a1, a))
if b is not None:
b1 = _asarray_validated(b, check_finite=check_finite)
overwrite_b = overwrite_b or _datacopied(b1, b)
if len(b1.shape) != 2 or b1.shape[0] != b1.shape[1]:
raise ValueError('expected square matrix')
if b1.shape != a1.shape:
raise ValueError('a and b must have the same shape')
return _geneig(a1, b1, left, right, overwrite_a, overwrite_b,
homogeneous_eigvals)
geev, geev_lwork = get_lapack_funcs(('geev', 'geev_lwork'), (a1, ))
compute_vl, compute_vr = left, right
lwork = _compute_lwork(geev_lwork,
a1.shape[0],
compute_vl=compute_vl,
compute_vr=compute_vr)
if geev.typecode in 'cz':
w, vl, vr, info = geev(a1,
lwork=lwork,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
w = _make_eigvals(w, None, homogeneous_eigvals)
else:
wr, wi, vl, vr, info = geev(a1,
lwork=lwork,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
t = {'f': 'F', 'd': 'D'}[wr.dtype.char]
w = wr + _I * wi
w = _make_eigvals(w, None, homogeneous_eigvals)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal geev' %
-info)
if info > 0:
raise LinAlgError("eig algorithm did not converge (only eigenvalues "
"with order >= %d have converged)" % info)
only_real = numpy.all(w.imag == 0.0)
if not (geev.typecode in 'cz' or only_real):
t = w.dtype.char
if left:
vl = _make_complex_eigvecs(w, vl, t)
if right:
vr = _make_complex_eigvecs(w, vr, t)
if not (left or right):
return w
if left:
if right:
return w, vl, vr
return w, vl
return w, vr
def vq(obs, code_book, check_finite=True):
"""
Assign codes from a code book to observations.
Assigns a code from a code book to each observation. Each
observation vector in the 'M' by 'N' `obs` array is compared with the
centroids in the code book and assigned the code of the closest
centroid.
The features in `obs` should have unit variance, which can be
achieved by passing them through the whiten function. The code
book can be created with the k-means algorithm or a different
encoding algorithm.
Parameters
----------
obs : ndarray
Each row of the 'M' x 'N' array is an observation. The columns are
the "features" seen during each observation. The features must be
whitened first using the whiten function or something equivalent.
code_book : ndarray
The code book is usually generated using the k-means algorithm.
Each row of the array holds a different code, and the columns are
the features of the code.
>>> # f0 f1 f2 f3
>>> code_book = [
... [ 1., 2., 3., 4.], #c0
... [ 1., 2., 3., 4.], #c1
... [ 1., 2., 3., 4.]] #c2
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
code : ndarray
A length M array holding the code book index for each observation.
dist : ndarray
The distortion (distance) between the observation and its nearest
code.
Examples
--------
>>> from numpy import array
>>> from scipy.cluster.vq import vq
>>> code_book = array([[1.,1.,1.],
... [2.,2.,2.]])
>>> features = array([[ 1.9,2.3,1.7],
... [ 1.5,2.5,2.2],
... [ 0.8,0.6,1.7]])
>>> vq(features,code_book)
(array([1, 1, 0],'i'), array([ 0.43588989, 0.73484692, 0.83066239]))
"""
obs = _asarray_validated(obs, check_finite=check_finite)
code_book = _asarray_validated(code_book, check_finite=check_finite)
ct = np.common_type(obs, code_book)
c_obs = obs.astype(ct, copy=False)
c_code_book = code_book.astype(ct, copy=False)
if np.issubdtype(ct, np.float64) or np.issubdtype(ct, np.float32):
return _vq.vq(c_obs, c_code_book)
return py_vq(obs, code_book, check_finite=False)
def eig_banded(a_band,
lower=False,
eigvals_only=False,
overwrite_a_band=False,
select='a',
select_range=None,
max_ev=0,
check_finite=True):
"""
Solve real symmetric or complex hermitian band matrix eigenvalue problem.
Find eigenvalues w and optionally right eigenvectors v of a::
a v[:,i] = w[i] v[:,i]
v.H v = identity
The matrix a is stored in a_band either in lower diagonal or upper
diagonal ordered form:
a_band[u + i - j, j] == a[i,j] (if upper form; i <= j)
a_band[ i - j, j] == a[i,j] (if lower form; i >= j)
where u is the number of bands above the diagonal.
Example of a_band (shape of a is (6,6), u=2)::
upper form:
* * a02 a13 a24 a35
* a01 a12 a23 a34 a45
a00 a11 a22 a33 a44 a55
lower form:
a00 a11 a22 a33 a44 a55
a10 a21 a32 a43 a54 *
a20 a31 a42 a53 * *
Cells marked with * are not used.
Parameters
----------
a_band : (u+1, M) array_like
The bands of the M by M matrix a.
lower : bool, optional
Is the matrix in the lower form. (Default is upper form)
eigvals_only : bool, optional
Compute only the eigenvalues and no eigenvectors.
(Default: calculate also eigenvectors)
overwrite_a_band : bool, optional
Discard data in a_band (may enhance performance)
select : {'a', 'v', 'i'}, optional
Which eigenvalues to calculate
====== ========================================
select calculated
====== ========================================
'a' All eigenvalues
'v' Eigenvalues in the interval (min, max]
'i' Eigenvalues with indices min <= i <= max
====== ========================================
select_range : (min, max), optional
Range of selected eigenvalues
max_ev : int, optional
For select=='v', maximum number of eigenvalues expected.
For other values of select, has no meaning.
In doubt, leave this parameter untouched.
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
w : (M,) ndarray
The eigenvalues, in ascending order, each repeated according to its
multiplicity.
v : (M, M) float or complex ndarray
The normalized eigenvector corresponding to the eigenvalue w[i] is
the column v[:,i].
Raises LinAlgError if eigenvalue computation does not converge
"""
if eigvals_only or overwrite_a_band:
a1 = _asarray_validated(a_band, check_finite=check_finite)
overwrite_a_band = overwrite_a_band or (_datacopied(a1, a_band))
else:
a1 = array(a_band)
if issubclass(a1.dtype.type, inexact) and not isfinite(a1).all():
raise ValueError("array must not contain infs or NaNs")
overwrite_a_band = 1
if len(a1.shape) != 2:
raise ValueError('expected two-dimensional array')
if select.lower() not in [0, 1, 2, 'a', 'v', 'i', 'all', 'value', 'index']:
raise ValueError('invalid argument for select')
if select.lower() in [0, 'a', 'all']:
if a1.dtype.char in 'GFD':
bevd, = get_lapack_funcs(('hbevd', ), (a1, ))
# FIXME: implement this somewhen, for now go with builtin values
# FIXME: calc optimal lwork by calling ?hbevd(lwork=-1)
# or by using calc_lwork.f ???
# lwork = calc_lwork.hbevd(bevd.typecode, a1.shape[0], lower)
internal_name = 'hbevd'
else: # a1.dtype.char in 'fd':
bevd, = get_lapack_funcs(('sbevd', ), (a1, ))
# FIXME: implement this somewhen, for now go with builtin values
# see above
# lwork = calc_lwork.sbevd(bevd.typecode, a1.shape[0], lower)
internal_name = 'sbevd'
w, v, info = bevd(a1,
compute_v=not eigvals_only,
lower=lower,
overwrite_ab=overwrite_a_band)
if select.lower() in [1, 2, 'i', 'v', 'index', 'value']:
# calculate certain range only
if select.lower() in [2, 'i', 'index']:
select = 2
vl, vu, il, iu = 0.0, 0.0, min(select_range), max(select_range)
if min(il, iu) < 0 or max(il, iu) >= a1.shape[1]:
raise ValueError('select_range out of bounds')
max_ev = iu - il + 1
else: # 1, 'v', 'value'
select = 1
vl, vu, il, iu = min(select_range), max(select_range), 0, 0
if max_ev == 0:
max_ev = a_band.shape[1]
if eigvals_only:
max_ev = 1
# calculate optimal abstol for dsbevx (see manpage)
if a1.dtype.char in 'fF': # single precision
lamch, = get_lapack_funcs(('lamch', ), (array(0, dtype='f'), ))
else:
lamch, = get_lapack_funcs(('lamch', ), (array(0, dtype='d'), ))
abstol = 2 * lamch('s')
if a1.dtype.char in 'GFD':
bevx, = get_lapack_funcs(('hbevx', ), (a1, ))
internal_name = 'hbevx'
else: # a1.dtype.char in 'gfd'
bevx, = get_lapack_funcs(('sbevx', ), (a1, ))
internal_name = 'sbevx'
# il+1, iu+1: translate python indexing (0 ... N-1) into Fortran
# indexing (1 ... N)
w, v, m, ifail, info = bevx(a1,
vl,
vu,
il + 1,
iu + 1,
compute_v=not eigvals_only,
mmax=max_ev,
range=select,
lower=lower,
overwrite_ab=overwrite_a_band,
abstol=abstol)
# crop off w and v
w = w[:m]
if not eigvals_only:
v = v[:, :m]
if info < 0:
raise ValueError('illegal value in %d-th argument of internal %s' %
(-info, internal_name))
if info > 0:
raise LinAlgError("eig algorithm did not converge")
if eigvals_only:
return w
return w, v
def kmeans(obs, k_or_guess, iter=20, thresh=1e-5, check_finite=True):
"""
Performs k-means on a set of observation vectors forming k clusters.
The k-means algorithm adjusts the classification of the observations
into clusters and updates the cluster centroids until the position of
the centroids is stable over successive iterations. In this
implementation of the algorithm, the stability of the centroids is
determined by comparing the absolute value of the change in the average
Euclidean distance between the observations and their corresponding
centroids against a threshold. This yields
a code book mapping centroids to codes and vice versa.
Parameters
----------
obs : ndarray
Each row of the M by N array is an observation vector. The
columns are the features seen during each observation.
The features must be whitened first with the `whiten` function.
k_or_guess : int or ndarray
The number of centroids to generate. A code is assigned to
each centroid, which is also the row index of the centroid
in the code_book matrix generated.
The initial k centroids are chosen by randomly selecting
observations from the observation matrix. Alternatively,
passing a k by N array specifies the initial k centroids.
iter : int, optional
The number of times to run k-means, returning the codebook
with the lowest distortion. This argument is ignored if
initial centroids are specified with an array for the
``k_or_guess`` parameter. This parameter does not represent the
number of iterations of the k-means algorithm.
thresh : float, optional
Terminates the k-means algorithm if the change in
distortion since the last k-means iteration is less than
or equal to threshold.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
codebook : ndarray
A k by N array of k centroids. The ith centroid
codebook[i] is represented with the code i. The centroids
and codes generated represent the lowest distortion seen,
not necessarily the globally minimal distortion.
distortion : float
The mean (non-squared) Euclidean distance between the observations
passed and the centroids generated. Note the difference to the standard
definition of distortion in the context of the k-means algorithm, which
is the sum of the squared distances.
See Also
--------
kmeans2 : a different implementation of k-means clustering
with more methods for generating initial centroids but without
using a distortion change threshold as a stopping criterion.
whiten : must be called prior to passing an observation matrix
to kmeans.
Examples
--------
>>> from numpy import array
>>> from scipy.cluster.vq import vq, kmeans, whiten
>>> import matplotlib.pyplot as plt
>>> features = array([[ 1.9,2.3],
... [ 1.5,2.5],
... [ 0.8,0.6],
... [ 0.4,1.8],
... [ 0.1,0.1],
... [ 0.2,1.8],
... [ 2.0,0.5],
... [ 0.3,1.5],
... [ 1.0,1.0]])
>>> whitened = whiten(features)
>>> book = np.array((whitened[0],whitened[2]))
>>> kmeans(whitened,book)
(array([[ 2.3110306 , 2.86287398], # random
[ 0.93218041, 1.24398691]]), 0.85684700941625547)
>>> from numpy import random
>>> random.seed((1000,2000))
>>> codes = 3
>>> kmeans(whitened,codes)
(array([[ 2.3110306 , 2.86287398], # random
[ 1.32544402, 0.65607529],
[ 0.40782893, 2.02786907]]), 0.5196582527686241)
>>> # Create 50 datapoints in two clusters a and b
>>> pts = 50
>>> a = np.random.multivariate_normal([0, 0], [[4, 1], [1, 4]], size=pts)
>>> b = np.random.multivariate_normal([30, 10],
... [[10, 2], [2, 1]],
... size=pts)
>>> features = np.concatenate((a, b))
>>> # Whiten data
>>> whitened = whiten(features)
>>> # Find 2 clusters in the data
>>> codebook, distortion = kmeans(whitened, 2)
>>> # Plot whitened data and cluster centers in red
>>> plt.scatter(whitened[:, 0], whitened[:, 1])
>>> plt.scatter(codebook[:, 0], codebook[:, 1], c='r')
>>> plt.show()
"""
obs = _asarray_validated(obs, check_finite=check_finite)
if iter < 1:
raise ValueError("iter must be at least 1, got %s" % iter)
# Determine whether a count (scalar) or an initial guess (array) was passed.
if not np.isscalar(k_or_guess):
guess = _asarray_validated(k_or_guess, check_finite=check_finite)
if guess.size < 1:
raise ValueError("Asked for 0 clusters. Initial book was %s" %
guess)
return _kmeans(obs, guess, thresh=thresh)
# k_or_guess is a scalar, now verify that it's an integer
k = int(k_or_guess)
if k != k_or_guess:
raise ValueError("If k_or_guess is a scalar, it must be an integer.")
if k < 1:
raise ValueError("Asked for %d clusters." % k)
# initialize best distance value to a large value
best_dist = np.inf
for i in range(iter):
# the initial code book is randomly selected from observations
guess = _kpoints(obs, k)
book, dist = _kmeans(obs, guess, thresh=thresh)
if dist < best_dist:
best_book = book
best_dist = dist
return best_book, best_dist
def eig_banded(a_band, lower=False, eigvals_only=False, overwrite_a_band=False,
select='a', select_range=None, max_ev=0, check_finite=True):
"""
Solve real symmetric or complex hermitian band matrix eigenvalue problem.
Find eigenvalues w and optionally right eigenvectors v of a::
a v[:,i] = w[i] v[:,i]
v.H v = identity
The matrix a is stored in a_band either in lower diagonal or upper
diagonal ordered form:
a_band[u + i - j, j] == a[i,j] (if upper form; i <= j)
a_band[ i - j, j] == a[i,j] (if lower form; i >= j)
where u is the number of bands above the diagonal.
Example of a_band (shape of a is (6,6), u=2)::
upper form:
* * a02 a13 a24 a35
* a01 a12 a23 a34 a45
a00 a11 a22 a33 a44 a55
lower form:
a00 a11 a22 a33 a44 a55
a10 a21 a32 a43 a54 *
a20 a31 a42 a53 * *
Cells marked with * are not used.
Parameters
----------
a_band : (u+1, M) array_like
The bands of the M by M matrix a.
lower : bool, optional
Is the matrix in the lower form. (Default is upper form)
eigvals_only : bool, optional
Compute only the eigenvalues and no eigenvectors.
(Default: calculate also eigenvectors)
overwrite_a_band : bool, optional
Discard data in a_band (may enhance performance)
select : {'a', 'v', 'i'}, optional
Which eigenvalues to calculate
====== ========================================
select calculated
====== ========================================
'a' All eigenvalues
'v' Eigenvalues in the interval (min, max]
'i' Eigenvalues with indices min <= i <= max
====== ========================================
select_range : (min, max), optional
Range of selected eigenvalues
max_ev : int, optional
For select=='v', maximum number of eigenvalues expected.
For other values of select, has no meaning.
In doubt, leave this parameter untouched.
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
w : (M,) ndarray
The eigenvalues, in ascending order, each repeated according to its
multiplicity.
v : (M, M) float or complex ndarray
The normalized eigenvector corresponding to the eigenvalue w[i] is
the column v[:,i].
Raises
------
LinAlgError
If eigenvalue computation does not converge.
See Also
--------
eigvals_banded : eigenvalues for symmetric/Hermitian band matrices
eig : eigenvalues and right eigenvectors of general arrays.
eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
eigh_tridiagonal : eigenvalues and right eiegenvectors for
symmetric/Hermitian tridiagonal matrices
Examples
--------
>>> from scipy.linalg import eig_banded
>>> A = np.array([[1, 5, 2, 0], [5, 2, 5, 2], [2, 5, 3, 5], [0, 2, 5, 4]])
>>> Ab = np.array([[1, 2, 3, 4], [5, 5, 5, 0], [2, 2, 0, 0]])
>>> w, v = eig_banded(Ab, lower=True)
>>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
True
>>> w = eig_banded(Ab, lower=True, eigvals_only=True)
>>> w
array([-4.26200532, -2.22987175, 3.95222349, 12.53965359])
Request only the eigenvalues between ``[-3, 4]``
>>> w, v = eig_banded(Ab, lower=True, select='v', select_range=[-3, 4])
>>> w
array([-2.22987175, 3.95222349])
"""
if eigvals_only or overwrite_a_band:
a1 = _asarray_validated(a_band, check_finite=check_finite)
overwrite_a_band = overwrite_a_band or (_datacopied(a1, a_band))
else:
a1 = array(a_band)
if issubclass(a1.dtype.type, inexact) and not isfinite(a1).all():
raise ValueError("array must not contain infs or NaNs")
overwrite_a_band = 1
if len(a1.shape) != 2:
raise ValueError('expected two-dimensional array')
select, vl, vu, il, iu, max_ev = _check_select(
select, select_range, max_ev, a1.shape[1])
del select_range
if select == 0:
if a1.dtype.char in 'GFD':
# FIXME: implement this somewhen, for now go with builtin values
# FIXME: calc optimal lwork by calling ?hbevd(lwork=-1)
# or by using calc_lwork.f ???
# lwork = calc_lwork.hbevd(bevd.typecode, a1.shape[0], lower)
internal_name = 'hbevd'
else: # a1.dtype.char in 'fd':
# FIXME: implement this somewhen, for now go with builtin values
# see above
# lwork = calc_lwork.sbevd(bevd.typecode, a1.shape[0], lower)
internal_name = 'sbevd'
bevd, = get_lapack_funcs((internal_name,), (a1,))
w, v, info = bevd(a1, compute_v=not eigvals_only,
lower=lower, overwrite_ab=overwrite_a_band)
else: # select in [1, 2]
if eigvals_only:
max_ev = 1
# calculate optimal abstol for dsbevx (see manpage)
if a1.dtype.char in 'fF': # single precision
lamch, = get_lapack_funcs(('lamch',), (array(0, dtype='f'),))
else:
lamch, = get_lapack_funcs(('lamch',), (array(0, dtype='d'),))
abstol = 2 * lamch('s')
if a1.dtype.char in 'GFD':
internal_name = 'hbevx'
else: # a1.dtype.char in 'gfd'
internal_name = 'sbevx'
bevx, = get_lapack_funcs((internal_name,), (a1,))
w, v, m, ifail, info = bevx(
a1, vl, vu, il, iu, compute_v=not eigvals_only, mmax=max_ev,
range=select, lower=lower, overwrite_ab=overwrite_a_band,
abstol=abstol)
# crop off w and v
w = w[:m]
if not eigvals_only:
v = v[:, :m]
_check_info(info, internal_name)
if eigvals_only:
return w
return w, v
def logsumexp(a, axis=None, b=None, keepdims=False, return_sign=False):
"""Compute the log of the sum of exponentials of input elements.
Parameters
----------
a : array_like
Input array.
axis : None or int or tuple of ints, optional
Axis or axes over which the sum is taken. By default `axis` is None,
and all elements are summed.
.. versionadded:: 0.11.0
keepdims : bool, optional
If this is set to True, the axes which are reduced are left in the
result as dimensions with size one. With this option, the result
will broadcast correctly against the original array.
.. versionadded:: 0.15.0
b : array-like, optional
Scaling factor for exp(`a`) must be of the same shape as `a` or
broadcastable to `a`. These values may be negative in order to
implement subtraction.
.. versionadded:: 0.12.0
return_sign : bool, optional
If this is set to True, the result will be a pair containing sign
information; if False, results that are negative will be returned
as NaN. Default is False (no sign information).
.. versionadded:: 0.16.0
Returns
-------
res : ndarray
The result, ``np.log(np.sum(np.exp(a)))`` calculated in a numerically
more stable way. If `b` is given then ``np.log(np.sum(b*np.exp(a)))``
is returned.
sgn : ndarray
If return_sign is True, this will be an array of floating-point
numbers matching res and +1, 0, or -1 depending on the sign
of the result. If False, only one result is returned.
See Also
--------
numpy.logaddexp, numpy.logaddexp2
Notes
-----
Numpy has a logaddexp function which is very similar to `logsumexp`, but
only handles two arguments. `logaddexp.reduce` is similar to this
function, but may be less stable.
Examples
--------
>>> from scipy.special import logsumexp
>>> a = np.arange(10)
>>> np.log(np.sum(np.exp(a)))
9.4586297444267107
>>> logsumexp(a)
9.4586297444267107
With weights
>>> a = np.arange(10)
>>> b = np.arange(10, 0, -1)
>>> logsumexp(a, b=b)
9.9170178533034665
>>> np.log(np.sum(b*np.exp(a)))
9.9170178533034647
Returning a sign flag
>>> logsumexp([1,2],b=[1,-1],return_sign=True)
(1.5413248546129181, -1.0)
Notice that `logsumexp` does not directly support masked arrays. To use it
on a masked array, convert the mask into zero weights:
>>> a = np.ma.array([np.log(2), 2, np.log(3)],
... mask=[False, True, False])
>>> b = (~a.mask).astype(int)
>>> logsumexp(a.data, b=b), np.log(5)
1.6094379124341005, 1.6094379124341005
"""
a = _asarray_validated(a, check_finite=False)
if b is not None:
a, b = np.broadcast_arrays(a, b)
if np.any(b == 0):
a = a + 0. # promote to at least float
a[b == 0] = -np.inf
a_max = np.amax(a, axis=axis, keepdims=True)
if a_max.ndim > 0:
a_max[~np.isfinite(a_max)] = 0
elif not np.isfinite(a_max):
a_max = 0
if b is not None:
b = np.asarray(b)
tmp = b * np.exp(a - a_max)
else:
tmp = np.exp(a - a_max)
# suppress warnings about log of zero
with np.errstate(divide='ignore'):
s = np.sum(tmp, axis=axis, keepdims=keepdims)
if return_sign:
sgn = np.sign(s)
s *= sgn # /= makes more sense but we need zero -> zero
out = np.log(s)
if not keepdims:
a_max = np.squeeze(a_max, axis=axis)
out += a_max
if return_sign:
return out, sgn
else:
return out
def log_softmax(x, axis=None):
r"""Compute the logarithm of the softmax function.
In principle::
log_softmax(x) = log(softmax(x))
but using a more accurate implementation.
Parameters
----------
x : array_like
Input array.
axis : int or tuple of ints, optional
Axis to compute values along. Default is None and softmax will be
computed over the entire array `x`.
Returns
-------
s : ndarray or scalar
An array with the same shape as `x`. Exponential of the result will
sum to 1 along the specified axis. If `x` is a scalar, a scalar is
returned.
Notes
-----
`log_softmax` is more accurate than ``np.log(softmax(x))`` with inputs that
make `softmax` saturate (see examples below).
.. versionadded:: 1.5.0
Examples
--------
>>> from scipy.special import log_softmax
>>> from scipy.special import softmax
>>> np.set_printoptions(precision=5)
>>> x = np.array([1000.0, 1.0])
>>> y = log_softmax(x)
>>> y
array([ 0., -999.])
>>> with np.errstate(divide='ignore'):
... y = np.log(softmax(x))
...
>>> y
array([ 0., -inf])
"""
x = _asarray_validated(x, check_finite=False)
x_max = np.amax(x, axis=axis, keepdims=True)
if x_max.ndim > 0:
x_max[~np.isfinite(x_max)] = 0
elif not np.isfinite(x_max):
x_max = 0
tmp = x - x_max
exp_tmp = np.exp(tmp)
# suppress warnings about log of zero
with np.errstate(divide='ignore'):
s = np.sum(exp_tmp, axis=axis, keepdims=True)
out = np.log(s)
out = tmp - out
return out
def sqrtm(A, disp=True, blocksize=64):
"""
Matrix square root.
Parameters
----------
A : (N, N) array_like
Matrix whose square root to evaluate
disp : bool, optional
Print warning if error in the result is estimated large
instead of returning estimated error. (Default: True)
blocksize : integer, optional
If the blocksize is not degenerate with respect to the
size of the input array, then use a blocked algorithm. (Default: 64)
Returns
-------
sqrtm : (N, N) ndarray
Value of the sqrt function at `A`
errest : float
(if disp == False)
Frobenius norm of the estimated error, ||err||_F / ||A||_F
References
----------
.. [1] Edvin Deadman, Nicholas J. Higham, Rui Ralha (2013)
"Blocked Schur Algorithms for Computing the Matrix Square Root,
Lecture Notes in Computer Science, 7782. pp. 171-182.
Examples
--------
>>> from scipy.linalg import sqrtm
>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
>>> r = sqrtm(a)
>>> r
array([[ 0.75592895, 1.13389342],
[ 0.37796447, 1.88982237]])
>>> r.dot(r)
array([[ 1., 3.],
[ 1., 4.]])
"""
A = _asarray_validated(A, check_finite=True, as_inexact=True)
if len(A.shape) != 2:
raise ValueError("Non-matrix input to matrix function.")
if blocksize < 1:
raise ValueError("The blocksize should be at least 1.")
keep_it_real = np.isrealobj(A)
if keep_it_real:
T, Z = schur(A)
if not np.array_equal(T, np.triu(T)):
T, Z = rsf2csf(T, Z)
else:
T, Z = schur(A, output='complex')
failflag = False
try:
R = _sqrtm_triu(T, blocksize=blocksize)
ZH = np.conjugate(Z).T
X = Z.dot(R).dot(ZH)
except SqrtmError:
failflag = True
X = np.empty_like(A)
X.fill(np.nan)
if disp:
if failflag:
print("Failed to find a square root.")
return X
else:
try:
arg2 = norm(X.dot(X) - A, 'fro')**2 / norm(A, 'fro')
except ValueError:
# NaNs in matrix
arg2 = np.inf
return X, arg2
def kmeans(obs, k_or_guess, iter=20, thresh=1e-5, check_finite=True):
"""
Performs k-means on a set of observation vectors forming k clusters.
The k-means algorithm adjusts the classification of the observations
into clusters and updates the cluster centroids until the position of
the centroids is stable over successive iterations. In this
implementation of the algorithm, the stability of the centroids is
determined by comparing the absolute value of the change in the average
Euclidean distance between the observations and their corresponding
centroids against a threshold. This yields
a code book mapping centroids to codes and vice versa.
Parameters
----------
obs : ndarray
Each row of the M by N array is an observation vector. The
columns are the features seen during each observation.
The features must be whitened first with the `whiten` function.
k_or_guess : int or ndarray
The number of centroids to generate. A code is assigned to
each centroid, which is also the row index of the centroid
in the code_book matrix generated.
The initial k centroids are chosen by randomly selecting
observations from the observation matrix. Alternatively,
passing a k by N array specifies the initial k centroids.
iter : int, optional
The number of times to run k-means, returning the codebook
with the lowest distortion. This argument is ignored if
initial centroids are specified with an array for the
``k_or_guess`` parameter. This parameter does not represent the
number of iterations of the k-means algorithm.
thresh : float, optional
Terminates the k-means algorithm if the change in
distortion since the last k-means iteration is less than
or equal to thresh.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
codebook : ndarray
A k by N array of k centroids. The i'th centroid
codebook[i] is represented with the code i. The centroids
and codes generated represent the lowest distortion seen,
not necessarily the globally minimal distortion.
distortion : float
The mean (non-squared) Euclidean distance between the observations
passed and the centroids generated. Note the difference to the standard
definition of distortion in the context of the K-means algorithm, which
is the sum of the squared distances.
See Also
--------
kmeans2 : a different implementation of k-means clustering
with more methods for generating initial centroids but without
using a distortion change threshold as a stopping criterion.
whiten : must be called prior to passing an observation matrix
to kmeans.
Examples
--------
>>> from numpy import array
>>> from scipy.cluster.vq import vq, kmeans, whiten
>>> import matplotlib.pyplot as plt
>>> features = array([[ 1.9,2.3],
... [ 1.5,2.5],
... [ 0.8,0.6],
... [ 0.4,1.8],
... [ 0.1,0.1],
... [ 0.2,1.8],
... [ 2.0,0.5],
... [ 0.3,1.5],
... [ 1.0,1.0]])
>>> whitened = whiten(features)
>>> book = np.array((whitened[0],whitened[2]))
>>> kmeans(whitened,book)
(array([[ 2.3110306 , 2.86287398], # random
[ 0.93218041, 1.24398691]]), 0.85684700941625547)
>>> from numpy import random
>>> random.seed((1000,2000))
>>> codes = 3
>>> kmeans(whitened,codes)
(array([[ 2.3110306 , 2.86287398], # random
[ 1.32544402, 0.65607529],
[ 0.40782893, 2.02786907]]), 0.5196582527686241)
>>> # Create 50 datapoints in two clusters a and b
>>> pts = 50
>>> a = np.random.multivariate_normal([0, 0], [[4, 1], [1, 4]], size=pts)
>>> b = np.random.multivariate_normal([30, 10],
... [[10, 2], [2, 1]],
... size=pts)
>>> features = np.concatenate((a, b))
>>> # Whiten data
>>> whitened = whiten(features)
>>> # Find 2 clusters in the data
>>> codebook, distortion = kmeans(whitened, 2)
>>> # Plot whitened data and cluster centers in red
>>> plt.scatter(whitened[:, 0], whitened[:, 1])
>>> plt.scatter(codebook[:, 0], codebook[:, 1], c='r')
>>> plt.show()
"""
obs = _asarray_validated(obs, check_finite=check_finite)
if iter < 1:
raise ValueError("iter must be at least 1, got %s" % iter)
# Determine whether a count (scalar) or an initial guess (array) was passed.
if not np.isscalar(k_or_guess):
guess = _asarray_validated(k_or_guess, check_finite=check_finite)
if guess.size < 1:
raise ValueError("Asked for 0 clusters. Initial book was %s" %
guess)
return _kmeans(obs, guess, thresh=thresh)
# k_or_guess is a scalar, now verify that it's an integer
k = int(k_or_guess)
if k != k_or_guess:
raise ValueError("If k_or_guess is a scalar, it must be an integer.")
if k < 1:
raise ValueError("Asked for %d clusters." % k)
# initialize best distance value to a large value
best_dist = np.inf
for i in xrange(iter):
# the initial code book is randomly selected from observations
guess = _kpoints(obs, k)
book, dist = _kmeans(obs, guess, thresh=thresh)
if dist < best_dist:
best_book = book
best_dist = dist
return best_book, best_dist
def kmeans(obs, k_or_guess, iter=20, thresh=1e-5, check_finite=True):
"""
Performs k-means on a set of observation vectors forming k clusters.
The k-means algorithm adjusts the centroids until sufficient
progress cannot be made, i.e. the change in distortion since
the last iteration is less than some threshold. This yields
a code book mapping centroids to codes and vice versa.
Distortion is defined as the sum of the squared differences
between the observations and the corresponding centroid.
Parameters
----------
obs : ndarray
Each row of the M by N array is an observation vector. The
columns are the features seen during each observation.
The features must be whitened first with the `whiten` function.
k_or_guess : int or ndarray
The number of centroids to generate. A code is assigned to
each centroid, which is also the row index of the centroid
in the code_book matrix generated.
The initial k centroids are chosen by randomly selecting
observations from the observation matrix. Alternatively,
passing a k by N array specifies the initial k centroids.
iter : int, optional
The number of times to run k-means, returning the codebook
with the lowest distortion. This argument is ignored if
initial centroids are specified with an array for the
``k_or_guess`` parameter. This parameter does not represent the
number of iterations of the k-means algorithm.
thresh : float, optional
Terminates the k-means algorithm if the change in
distortion since the last k-means iteration is less than
or equal to thresh.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
codebook : ndarray
A k by N array of k centroids. The i'th centroid
codebook[i] is represented with the code i. The centroids
and codes generated represent the lowest distortion seen,
not necessarily the globally minimal distortion.
distortion : float
The distortion between the observations passed and the
centroids generated.
See Also
--------
kmeans2 : a different implementation of k-means clustering
with more methods for generating initial centroids but without
using a distortion change threshold as a stopping criterion.
whiten : must be called prior to passing an observation matrix
to kmeans.
Examples
--------
>>> from numpy import array
>>> from scipy.cluster.vq import vq, kmeans, whiten
>>> features = array([[ 1.9,2.3],
... [ 1.5,2.5],
... [ 0.8,0.6],
... [ 0.4,1.8],
... [ 0.1,0.1],
... [ 0.2,1.8],
... [ 2.0,0.5],
... [ 0.3,1.5],
... [ 1.0,1.0]])
>>> whitened = whiten(features)
>>> book = array((whitened[0],whitened[2]))
>>> kmeans(whitened,book)
(array([[ 2.3110306 , 2.86287398], # random
[ 0.93218041, 1.24398691]]), 0.85684700941625547)
>>> from numpy import random
>>> random.seed((1000,2000))
>>> codes = 3
>>> kmeans(whitened,codes)
(array([[ 2.3110306 , 2.86287398], # random
[ 1.32544402, 0.65607529],
[ 0.40782893, 2.02786907]]), 0.5196582527686241)
"""
obs = _asarray_validated(obs, check_finite=check_finite)
if int(iter) < 1:
raise ValueError('iter must be at least 1.')
# Determine whether a count (scalar) or an initial guess (array) was passed.
k = None
guess = None
try:
k = int(k_or_guess)
except TypeError:
guess = _asarray_validated(k_or_guess, check_finite=check_finite)
if guess is not None:
if guess.size < 1:
raise ValueError("Asked for 0 cluster ? initial book was %s" %
guess)
result = _kmeans(obs, guess, thresh=thresh)
else:
if k != k_or_guess:
raise ValueError(
'if k_or_guess is a scalar, it must be an integer')
# initialize best distance value to a large value
best_dist = np.inf
No = obs.shape[0]
k = k_or_guess
if k < 1:
raise ValueError("Asked for 0 cluster ? ")
for i in range(iter):
# the initial code book is randomly selected from observations
guess = take(obs, randint(0, No, k), 0)
book, dist = _kmeans(obs, guess, thresh=thresh)
if dist < best_dist:
best_book = book
best_dist = dist
result = best_book, best_dist
return result
def kmeans2(data,
k,
iter=10,
thresh=1e-5,
minit='random',
missing='warn',
check_finite=True):
"""
Classify a set of observations into k clusters using the k-means algorithm.
The algorithm attempts to minimize the Euclidian distance between
observations and centroids. Several initialization methods are
included.
Parameters
----------
data : ndarray
A 'M' by 'N' array of 'M' observations in 'N' dimensions or a length
'M' array of 'M' one-dimensional observations.
k : int or ndarray
The number of clusters to form as well as the number of
centroids to generate. If `minit` initialization string is
'matrix', or if a ndarray is given instead, it is
interpreted as initial cluster to use instead.
iter : int, optional
Number of iterations of the k-means algrithm to run. Note
that this differs in meaning from the iters parameter to
the kmeans function.
thresh : float, optional
(not used yet)
minit : str, optional
Method for initialization. Available methods are 'random',
'points', and 'matrix':
'random': generate k centroids from a Gaussian with mean and
variance estimated from the data.
'points': choose k observations (rows) at random from data for
the initial centroids.
'matrix': interpret the k parameter as a k by M (or length k
array for one-dimensional data) array of initial centroids.
missing : str, optional
Method to deal with empty clusters. Available methods are
'warn' and 'raise':
'warn': give a warning and continue.
'raise': raise an ClusterError and terminate the algorithm.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default: True
Returns
-------
centroid : ndarray
A 'k' by 'N' array of centroids found at the last iteration of
k-means.
label : ndarray
label[i] is the code or index of the centroid the
i'th observation is closest to.
"""
data = _asarray_validated(data, check_finite=check_finite)
if missing not in _valid_miss_meth:
raise ValueError("Unkown missing method: %s" % str(missing))
# If data is rank 1, then we have 1 dimension problem.
nd = np.ndim(data)
if nd == 1:
d = 1
# raise ValueError("Input of rank 1 not supported yet")
elif nd == 2:
d = data.shape[1]
else:
raise ValueError("Input of rank > 2 not supported")
if np.size(data) < 1:
raise ValueError("Input has 0 items.")
# If k is not a single value, then it should be compatible with data's
# shape
if np.size(k) > 1 or minit == 'matrix':
if not nd == np.ndim(k):
raise ValueError("k is not an int and has not same rank than data")
if d == 1:
nc = len(k)
else:
(nc, dc) = k.shape
if not dc == d:
raise ValueError("k is not an int and has not same rank than\
data")
clusters = k.copy()
else:
try:
nc = int(k)
except TypeError:
raise ValueError("k (%s) could not be converted to an integer " %
str(k))
if nc < 1:
raise ValueError("kmeans2 for 0 clusters ? (k was %s)" % str(k))
if not nc == k:
warnings.warn("k was not an integer, was converted.")
try:
init = _valid_init_meth[minit]
except KeyError:
raise ValueError("unknown init method %s" % str(minit))
clusters = init(data, k)
if int(iter) < 1:
raise ValueError(
"iter = %s is not valid. iter must be a positive integer." % iter)
return _kmeans2(data, clusters, iter, nc, _valid_miss_meth[missing])
def _prepare_x(self, x):
"""Reshape input x array to 1-D"""
x = _asarray_validated(x, check_finite=False, as_inexact=True)
x_shape = x.shape
return x.ravel(), x_shape
def eigh_tridiagonal(d, e, eigvals_only=False, select='a', select_range=None,
check_finite=True, tol=0., lapack_driver='auto'):
"""
Solve eigenvalue problem for a real symmetric tridiagonal matrix.
Find eigenvalues `w` and optionally right eigenvectors `v` of ``a``::
a v[:,i] = w[i] v[:,i]
v.H v = identity
For a real symmetric matrix ``a`` with diagonal elements `d` and
off-diagonal elements `e`.
Parameters
----------
d : ndarray, shape (ndim,)
The diagonal elements of the array.
e : ndarray, shape (ndim-1,)
The off-diagonal elements of the array.
select : {'a', 'v', 'i'}, optional
Which eigenvalues to calculate
====== ========================================
select calculated
====== ========================================
'a' All eigenvalues
'v' Eigenvalues in the interval (min, max]
'i' Eigenvalues with indices min <= i <= max
====== ========================================
select_range : (min, max), optional
Range of selected eigenvalues
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
tol : float
The absolute tolerance to which each eigenvalue is required
(only used when 'stebz' is the `lapack_driver`).
An eigenvalue (or cluster) is considered to have converged if it
lies in an interval of this width. If <= 0. (default),
the value ``eps*|a|`` is used where eps is the machine precision,
and ``|a|`` is the 1-norm of the matrix ``a``.
lapack_driver : str
LAPACK function to use, can be 'auto', 'stemr', 'stebz', 'sterf',
or 'stev'. When 'auto' (default), it will use 'stemr' if ``select='a'``
and 'stebz' otherwise. When 'stebz' is used to find the eigenvalues and
``eigvals_only=False``, then a second LAPACK call (to ``?STEIN``) is
used to find the corresponding eigenvectors. 'sterf' can only be
used when ``eigvals_only=True`` and ``select='a'``. 'stev' can only
be used when ``select='a'``.
Returns
-------
w : (M,) ndarray
The eigenvalues, in ascending order, each repeated according to its
multiplicity.
v : (M, M) ndarray
The normalized eigenvector corresponding to the eigenvalue ``w[i]`` is
the column ``v[:,i]``.
Raises
------
LinAlgError
If eigenvalue computation does not converge.
See Also
--------
eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal
matrices
eig : eigenvalues and right eigenvectors for non-symmetric arrays
eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian
band matrices
Notes
-----
This function makes use of LAPACK ``S/DSTEMR`` routines.
Examples
--------
>>> from scipy.linalg import eigh_tridiagonal
>>> d = 3*np.ones(4)
>>> e = -1*np.ones(3)
>>> w, v = eigh_tridiagonal(d, e)
>>> A = np.diag(d) + np.diag(e, k=1) + np.diag(e, k=-1)
>>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
True
"""
d = _asarray_validated(d, check_finite=check_finite)
e = _asarray_validated(e, check_finite=check_finite)
for check in (d, e):
if check.ndim != 1:
raise ValueError('expected one-dimensional array')
if check.dtype.char in 'GFD': # complex
raise TypeError('Only real arrays currently supported')
if d.size != e.size + 1:
raise ValueError('d (%s) must have one more element than e (%s)'
% (d.size, e.size))
select, vl, vu, il, iu, _ = _check_select(
select, select_range, 0, d.size)
if not isinstance(lapack_driver, string_types):
raise TypeError('lapack_driver must be str')
drivers = ('auto', 'stemr', 'sterf', 'stebz', 'stev')
if lapack_driver not in drivers:
raise ValueError('lapack_driver must be one of %s, got %s'
% (drivers, lapack_driver))
if lapack_driver == 'auto':
lapack_driver = 'stemr' if select == 0 else 'stebz'
func, = get_lapack_funcs((lapack_driver,), (d, e))
compute_v = not eigvals_only
if lapack_driver == 'sterf':
if select != 0:
raise ValueError('sterf can only be used when select == "a"')
if not eigvals_only:
raise ValueError('sterf can only be used when eigvals_only is '
'True')
w, info = func(d, e)
m = len(w)
elif lapack_driver == 'stev':
if select != 0:
raise ValueError('stev can only be used when select == "a"')
w, v, info = func(d, e, compute_v=compute_v)
m = len(w)
elif lapack_driver == 'stebz':
tol = float(tol)
internal_name = 'stebz'
stebz, = get_lapack_funcs((internal_name,), (d, e))
# If getting eigenvectors, needs to be block-ordered (B) instead of
# matirx-ordered (E), and we will reorder later
order = 'E' if eigvals_only else 'B'
m, w, iblock, isplit, info = stebz(d, e, select, vl, vu, il, iu, tol,
order)
else: # 'stemr'
# ?STEMR annoyingly requires size N instead of N-1
e_ = empty(e.size+1, e.dtype)
e_[:-1] = e
stemr_lwork, = get_lapack_funcs(('stemr_lwork',), (d, e))
lwork, liwork, info = stemr_lwork(d, e_, select, vl, vu, il, iu,
compute_v=compute_v)
_check_info(info, 'stemr_lwork')
m, w, v, info = func(d, e_, select, vl, vu, il, iu,
compute_v=compute_v, lwork=lwork, liwork=liwork)
_check_info(info, lapack_driver + ' (eigh_tridiagonal)')
w = w[:m]
if eigvals_only:
return w
else:
# Do we still need to compute the eigenvalues?
if lapack_driver == 'stebz':
func, = get_lapack_funcs(('stein',), (d, e))
v, info = func(d, e, w, iblock, isplit)
_check_info(info, 'stein (eigh_tridiagonal)',
positive='%d eigenvectors failed to converge')
# Convert block-order to matrix-order
order = argsort(w)
w, v = w[order], v[:, order]
else:
v = v[:, :m]
return w, v
def eigh(a,
b=None,
lower=True,
eigvals_only=False,
overwrite_a=False,
overwrite_b=False,
turbo=True,
eigvals=None,
type=1,
check_finite=True):
"""
Solve an ordinary or generalized eigenvalue problem for a complex
Hermitian or real symmetric matrix.
Find eigenvalues w and optionally eigenvectors v of matrix `a`, where
`b` is positive definite::
a v[:,i] = w[i] b v[:,i]
v[i,:].conj() a v[:,i] = w[i]
v[i,:].conj() b v[:,i] = 1
Parameters
----------
a : (M, M) array_like
A complex Hermitian or real symmetric matrix whose eigenvalues and
eigenvectors will be computed.
b : (M, M) array_like, optional
A complex Hermitian or real symmetric definite positive matrix in.
If omitted, identity matrix is assumed.
lower : bool, optional
Whether the pertinent array data is taken from the lower or upper
triangle of `a`. (Default: lower)
eigvals_only : bool, optional
Whether to calculate only eigenvalues and no eigenvectors.
(Default: both are calculated)
turbo : bool, optional
Use divide and conquer algorithm (faster but expensive in memory,
only for generalized eigenvalue problem and if eigvals=None)
eigvals : tuple (lo, hi), optional
Indexes of the smallest and largest (in ascending order) eigenvalues
and corresponding eigenvectors to be returned: 0 <= lo <= hi <= M-1.
If omitted, all eigenvalues and eigenvectors are returned.
type : int, optional
Specifies the problem type to be solved:
type = 1: a v[:,i] = w[i] b v[:,i]
type = 2: a b v[:,i] = w[i] v[:,i]
type = 3: b a v[:,i] = w[i] v[:,i]
overwrite_a : bool, optional
Whether to overwrite data in `a` (may improve performance)
overwrite_b : bool, optional
Whether to overwrite data in `b` (may improve performance)
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
w : (N,) float ndarray
The N (1<=N<=M) selected eigenvalues, in ascending order, each
repeated according to its multiplicity.
v : (M, N) complex ndarray
(if eigvals_only == False)
The normalized selected eigenvector corresponding to the
eigenvalue w[i] is the column v[:,i].
Normalization:
type 1 and 3: v.conj() a v = w
type 2: inv(v).conj() a inv(v) = w
type = 1 or 2: v.conj() b v = I
type = 3: v.conj() inv(b) v = I
Raises
------
LinAlgError
If eigenvalue computation does not converge,
an error occurred, or b matrix is not definite positive. Note that
if input matrices are not symmetric or hermitian, no error is reported
but results will be wrong.
See Also
--------
eig : eigenvalues and right eigenvectors for non-symmetric arrays
"""
a1 = _asarray_validated(a, check_finite=check_finite)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or (_datacopied(a1, a))
if iscomplexobj(a1):
cplx = True
else:
cplx = False
if b is not None:
b1 = _asarray_validated(b, check_finite=check_finite)
overwrite_b = overwrite_b or _datacopied(b1, b)
if len(b1.shape) != 2 or b1.shape[0] != b1.shape[1]:
raise ValueError('expected square matrix')
if b1.shape != a1.shape:
raise ValueError("wrong b dimensions %s, should "
"be %s" % (str(b1.shape), str(a1.shape)))
if iscomplexobj(b1):
cplx = True
else:
cplx = cplx or False
else:
b1 = None
# Set job for fortran routines
_job = (eigvals_only and 'N') or 'V'
# port eigenvalue range from python to fortran convention
if eigvals is not None:
lo, hi = eigvals
if lo < 0 or hi >= a1.shape[0]:
raise ValueError('The eigenvalue range specified is not valid.\n'
'Valid range is [%s,%s]' % (0, a1.shape[0] - 1))
lo += 1
hi += 1
eigvals = (lo, hi)
# set lower
if lower:
uplo = 'L'
else:
uplo = 'U'
# fix prefix for lapack routines
if cplx:
pfx = 'he'
else:
pfx = 'sy'
# Standard Eigenvalue Problem
# Use '*evr' routines
# FIXME: implement calculation of optimal lwork
# for all lapack routines
if b1 is None:
(evr, ) = get_lapack_funcs((pfx + 'evr', ), (a1, ))
if eigvals is None:
w, v, info = evr(a1,
uplo=uplo,
jobz=_job,
range="A",
il=1,
iu=a1.shape[0],
overwrite_a=overwrite_a)
else:
(lo, hi) = eigvals
w_tot, v, info = evr(a1,
uplo=uplo,
jobz=_job,
range="I",
il=lo,
iu=hi,
overwrite_a=overwrite_a)
w = w_tot[0:hi - lo + 1]
# Generalized Eigenvalue Problem
else:
# Use '*gvx' routines if range is specified
if eigvals is not None:
(gvx, ) = get_lapack_funcs((pfx + 'gvx', ), (a1, b1))
(lo, hi) = eigvals
w_tot, v, ifail, info = gvx(a1,
b1,
uplo=uplo,
iu=hi,
itype=type,
jobz=_job,
il=lo,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
w = w_tot[0:hi - lo + 1]
# Use '*gvd' routine if turbo is on and no eigvals are specified
elif turbo:
(gvd, ) = get_lapack_funcs((pfx + 'gvd', ), (a1, b1))
v, w, info = gvd(a1,
b1,
uplo=uplo,
itype=type,
jobz=_job,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
# Use '*gv' routine if turbo is off and no eigvals are specified
else:
(gv, ) = get_lapack_funcs((pfx + 'gv', ), (a1, b1))
v, w, info = gv(a1,
b1,
uplo=uplo,
itype=type,
jobz=_job,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
# Check if we had a successful exit
if info == 0:
if eigvals_only:
return w
else:
return w, v
elif info < 0:
raise LinAlgError("illegal value in %i-th argument of internal"
" fortran routine." % (-info))
elif info > 0 and b1 is None:
raise LinAlgError("unrecoverable internal error.")
# The algorithm failed to converge.
elif 0 < info <= b1.shape[0]:
if eigvals is not None:
raise LinAlgError("the eigenvectors %s failed to"
" converge." % nonzero(ifail) - 1)
else:
raise LinAlgError("internal fortran routine failed to converge: "
"%i off-diagonal elements of an "
"intermediate tridiagonal form did not converge"
" to zero." % info)
# This occurs when b is not positive definite
else:
raise LinAlgError("the leading minor of order %i"
" of 'b' is not positive definite. The"
" factorization of 'b' could not be completed"
" and no eigenvalues or eigenvectors were"
" computed." % (info - b1.shape[0]))
def _prepare_x(self, x):
"""Reshape input x array to 1-D"""
x = _asarray_validated(x, check_finite=False, as_inexact=True)
x_shape = x.shape
return x.ravel(), x_shape
def hessenberg(a, calc_q=False, overwrite_a=False, check_finite=True):
"""
Compute Hessenberg form of a matrix.
The Hessenberg decomposition is::
A = Q H Q^H
where `Q` is unitary/orthogonal and `H` has only zero elements below
the first sub-diagonal.
Parameters
----------
a : (M, M) array_like
Matrix to bring into Hessenberg form.
calc_q : bool, optional
Whether to compute the transformation matrix. Default is False.
overwrite_a : bool, optional
Whether to overwrite `a`; may improve performance.
Default is False.
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
H : (M, M) ndarray
Hessenberg form of `a`.
Q : (M, M) ndarray
Unitary/orthogonal similarity transformation matrix ``A = Q H Q^H``.
Only returned if ``calc_q=True``.
"""
a1 = _asarray_validated(a, check_finite=check_finite)
if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]):
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or (_datacopied(a1, a))
# if 2x2 or smaller: already in Hessenberg
if a1.shape[0] <= 2:
if calc_q:
return a1, numpy.eye(a1.shape[0])
return a1
gehrd, gebal, gehrd_lwork = get_lapack_funcs(
('gehrd', 'gebal', 'gehrd_lwork'), (a1, ))
ba, lo, hi, pivscale, info = gebal(a1, permute=0, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal gebal '
'(hessenberg)' % -info)
n = len(a1)
lwork = _compute_lwork(gehrd_lwork, ba.shape[0], lo=lo, hi=hi)
hq, tau, info = gehrd(ba, lo=lo, hi=hi, lwork=lwork, overwrite_a=1)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal gehrd '
'(hessenberg)' % -info)
h = numpy.triu(hq, -1)
if not calc_q:
return h
# use orghr/unghr to compute q
orghr, orghr_lwork = get_lapack_funcs(('orghr', 'orghr_lwork'), (a1, ))
lwork = _compute_lwork(orghr_lwork, n, lo=lo, hi=hi)
q, info = orghr(a=hq, tau=tau, lo=lo, hi=hi, lwork=lwork, overwrite_a=1)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal orghr '
'(hessenberg)' % -info)
return h, q
def hessenberg(a, calc_q=False, overwrite_a=False, check_finite=True):
"""
Compute Hessenberg form of a matrix.
The Hessenberg decomposition is::
A = Q H Q^H
where `Q` is unitary/orthogonal and `H` has only zero elements below
the first sub-diagonal.
Parameters
----------
a : (M, M) array_like
Matrix to bring into Hessenberg form.
calc_q : bool, optional
Whether to compute the transformation matrix. Default is False.
overwrite_a : bool, optional
Whether to overwrite `a`; may improve performance.
Default is False.
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
H : (M, M) ndarray
Hessenberg form of `a`.
Q : (M, M) ndarray
Unitary/orthogonal similarity transformation matrix ``A = Q H Q^H``.
Only returned if ``calc_q=True``.
Examples
--------
>>> from scipy.linalg import hessenberg
>>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
>>> H, Q = hessenberg(A, calc_q=True)
>>> H
array([[ 2. , -11.65843866, 1.42005301, 0.25349066],
[ -9.94987437, 14.53535354, -5.31022304, 2.43081618],
[ 0. , -1.83299243, 0.38969961, -0.51527034],
[ 0. , 0. , -3.83189513, 1.07494686]])
>>> np.allclose(Q @ H @ Q.conj().T - A, np.zeros((4, 4)))
True
"""
a1 = _asarray_validated(a, check_finite=check_finite)
if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]):
raise ValueError('expected square matrix')
overwrite_a = overwrite_a or (_datacopied(a1, a))
# if 2x2 or smaller: already in Hessenberg
if a1.shape[0] <= 2:
if calc_q:
return a1, numpy.eye(a1.shape[0])
return a1
gehrd, gebal, gehrd_lwork = get_lapack_funcs(('gehrd', 'gebal',
'gehrd_lwork'), (a1,))
ba, lo, hi, pivscale, info = gebal(a1, permute=0, overwrite_a=overwrite_a)
_check_info(info, 'gebal (hessenberg)', positive=False)
n = len(a1)
lwork = _compute_lwork(gehrd_lwork, ba.shape[0], lo=lo, hi=hi)
hq, tau, info = gehrd(ba, lo=lo, hi=hi, lwork=lwork, overwrite_a=1)
_check_info(info, 'gehrd (hessenberg)', positive=False)
h = numpy.triu(hq, -1)
if not calc_q:
return h
# use orghr/unghr to compute q
orghr, orghr_lwork = get_lapack_funcs(('orghr', 'orghr_lwork'), (a1,))
lwork = _compute_lwork(orghr_lwork, n, lo=lo, hi=hi)
q, info = orghr(a=hq, tau=tau, lo=lo, hi=hi, lwork=lwork, overwrite_a=1)
_check_info(info, 'orghr (hessenberg)', positive=False)
return h, q