Learn more »
Push, build, and install
RubyGems
npm packages
Python packages
Maven artifacts
PHP packages
Go Modules
Bower components
Debian packages
RPM packages
NuGet packages
/ utils / extmath.py
"""
Extended math utilities.
"""
# Authors: Gael Varoquaux
# Alexandre Gramfort
# Alexandre T. Passos
# Olivier Grisel
# Lars Buitinck
# Stefan van der Walt
# Kyle Kastner
# Giorgio Patrini
# License: BSD 3 clause
import warnings
import numpy as np
from scipy import linalg, sparse
from . import check_random_state
from ._logistic_sigmoid import _log_logistic_sigmoid
from .sparsefuncs_fast import csr_row_norms
from .validation import check_array
from .deprecation import deprecated
def squared_norm(x):
"""Squared Euclidean or Frobenius norm of x.
Faster than norm(x) ** 2.
Parameters
----------
x : array_like
Returns
-------
float
The Euclidean norm when x is a vector, the Frobenius norm when x
is a matrix (2-d array).
"""
x = np.ravel(x, order='K')
if np.issubdtype(x.dtype, np.integer):
warnings.warn('Array type is integer, np.dot may overflow. '
'Data should be float type to avoid this issue',
UserWarning)
return np.dot(x, x)
def row_norms(X, squared=False):
"""Row-wise (squared) Euclidean norm of X.
Equivalent to np.sqrt((X * X).sum(axis=1)), but also supports sparse
matrices and does not create an X.shape-sized temporary.
Performs no input validation.
Parameters
----------
X : array_like
The input array
squared : bool, optional (default = False)
If True, return squared norms.
Returns
-------
array_like
The row-wise (squared) Euclidean norm of X.
"""
if sparse.issparse(X):
if not isinstance(X, sparse.csr_matrix):
X = sparse.csr_matrix(X)
norms = csr_row_norms(X)
else:
norms = np.einsum('ij,ij->i', X, X)
if not squared:
np.sqrt(norms, norms)
return norms
def fast_logdet(A):
"""Compute log(det(A)) for A symmetric
Equivalent to : np.log(nl.det(A)) but more robust.
It returns -Inf if det(A) is non positive or is not defined.
Parameters
----------
A : array_like
The matrix
"""
sign, ld = np.linalg.slogdet(A)
if not sign > 0:
return -np.inf
return ld
def density(w, **kwargs):
"""Compute density of a sparse vector
Parameters
----------
w : array_like
The sparse vector
Returns
-------
float
The density of w, between 0 and 1
"""
if hasattr(w, "toarray"):
d = float(w.nnz) / (w.shape[0] * w.shape[1])
else:
d = 0 if w is None else float((w != 0).sum()) / w.size
return d
def safe_sparse_dot(a, b, dense_output=False):
"""Dot product that handle the sparse matrix case correctly
Parameters
----------
a : array or sparse matrix
b : array or sparse matrix
dense_output : boolean, (default=False)
When False, ``a`` and ``b`` both being sparse will yield sparse output.
When True, output will always be a dense array.
Returns
-------
dot_product : array or sparse matrix
sparse if ``a`` and ``b`` are sparse and ``dense_output=False``.
"""
if a.ndim > 2 or b.ndim > 2:
if sparse.issparse(a):
# sparse is always 2D. Implies b is 3D+
# [i, j] @ [k, ..., l, m, n] -> [i, k, ..., l, n]
b_ = np.rollaxis(b, -2)
b_2d = b_.reshape((b.shape[-2], -1))
ret = a @ b_2d
ret = ret.reshape(a.shape[0], *b_.shape[1:])
elif sparse.issparse(b):
# sparse is always 2D. Implies a is 3D+
# [k, ..., l, m] @ [i, j] -> [k, ..., l, j]
a_2d = a.reshape(-1, a.shape[-1])
ret = a_2d @ b
ret = ret.reshape(*a.shape[:-1], b.shape[1])
else:
ret = np.dot(a, b)
else:
ret = a @ b
if (sparse.issparse(a) and sparse.issparse(b)
and dense_output and hasattr(ret, "toarray")):
return ret.toarray()
return ret
def randomized_range_finder(A, size, n_iter,
power_iteration_normalizer='auto',
random_state=None):
"""Computes an orthonormal matrix whose range approximates the range of A.
Parameters
----------
A : 2D array
The input data matrix
size : integer
Size of the return array
n_iter : integer
Number of power iterations used to stabilize the result
power_iteration_normalizer : 'auto' (default), 'QR', 'LU', 'none'
Whether the power iterations are normalized with step-by-step
QR factorization (the slowest but most accurate), 'none'
(the fastest but numerically unstable when `n_iter` is large, e.g.
typically 5 or larger), or 'LU' factorization (numerically stable
but can lose slightly in accuracy). The 'auto' mode applies no
normalization if `n_iter` <= 2 and switches to LU otherwise.
.. versionadded:: 0.18
random_state : int, RandomState instance or None, optional (default=None)
The seed of the pseudo random number generator to use when shuffling
the data. If int, random_state is the seed used by the random number
generator; If RandomState instance, random_state is the random number
generator; If None, the random number generator is the RandomState
instance used by `np.random`.
Returns
-------
Q : 2D array
A (size x size) projection matrix, the range of which
approximates well the range of the input matrix A.
Notes
-----
Follows Algorithm 4.3 of
Finding structure with randomness: Stochastic algorithms for constructing
approximate matrix decompositions
Halko, et al., 2009 (arXiv:909) https://arxiv.org/pdf/0909.4061.pdf
An implementation of a randomized algorithm for principal component
analysis
A. Szlam et al. 2014
"""
random_state = check_random_state(random_state)
# Generating normal random vectors with shape: (A.shape[1], size)
Q = random_state.normal(size=(A.shape[1], size))
if A.dtype.kind == 'f':
# Ensure f32 is preserved as f32
Q = Q.astype(A.dtype, copy=False)
# Deal with "auto" mode
if power_iteration_normalizer == 'auto':
if n_iter <= 2:
power_iteration_normalizer = 'none'
else:
power_iteration_normalizer = 'LU'
# Perform power iterations with Q to further 'imprint' the top
# singular vectors of A in Q
for i in range(n_iter):
if power_iteration_normalizer == 'none':
Q = safe_sparse_dot(A, Q)
Q = safe_sparse_dot(A.T, Q)
elif power_iteration_normalizer == 'LU':
Q, _ = linalg.lu(safe_sparse_dot(A, Q), permute_l=True)
Q, _ = linalg.lu(safe_sparse_dot(A.T, Q), permute_l=True)
elif power_iteration_normalizer == 'QR':
Q, _ = linalg.qr(safe_sparse_dot(A, Q), mode='economic')
Q, _ = linalg.qr(safe_sparse_dot(A.T, Q), mode='economic')
# Sample the range of A using by linear projection of Q
# Extract an orthonormal basis
Q, _ = linalg.qr(safe_sparse_dot(A, Q), mode='economic')
return Q
def randomized_svd(M, n_components, n_oversamples=10, n_iter='auto',
power_iteration_normalizer='auto', transpose='auto',
flip_sign=True, random_state=0):
"""Computes a truncated randomized SVD
Parameters
----------
M : ndarray or sparse matrix
Matrix to decompose
n_components : int
Number of singular values and vectors to extract.
n_oversamples : int (default is 10)
Additional number of random vectors to sample the range of M so as
to ensure proper conditioning. The total number of random vectors
used to find the range of M is n_components + n_oversamples. Smaller
number can improve speed but can negatively impact the quality of
approximation of singular vectors and singular values.
n_iter : int or 'auto' (default is 'auto')
Number of power iterations. It can be used to deal with very noisy
problems. When 'auto', it is set to 4, unless `n_components` is small
(< .1 * min(X.shape)) `n_iter` in which case is set to 7.
This improves precision with few components.
.. versionchanged:: 0.18
power_iteration_normalizer : 'auto' (default), 'QR', 'LU', 'none'
Whether the power iterations are normalized with step-by-step
QR factorization (the slowest but most accurate), 'none'
(the fastest but numerically unstable when `n_iter` is large, e.g.
typically 5 or larger), or 'LU' factorization (numerically stable
but can lose slightly in accuracy). The 'auto' mode applies no
normalization if `n_iter` <= 2 and switches to LU otherwise.
.. versionadded:: 0.18
transpose : True, False or 'auto' (default)
Whether the algorithm should be applied to M.T instead of M. The
result should approximately be the same. The 'auto' mode will
trigger the transposition if M.shape[1] > M.shape[0] since this
implementation of randomized SVD tend to be a little faster in that
case.
.. versionchanged:: 0.18
flip_sign : boolean, (True by default)
The output of a singular value decomposition is only unique up to a
permutation of the signs of the singular vectors. If `flip_sign` is
set to `True`, the sign ambiguity is resolved by making the largest
loadings for each component in the left singular vectors positive.
random_state : int, RandomState instance or None, optional (default=None)
The seed of the pseudo random number generator to use when shuffling
the data. If int, random_state is the seed used by the random number
generator; If RandomState instance, random_state is the random number
generator; If None, the random number generator is the RandomState
instance used by `np.random`.
Notes
-----
This algorithm finds a (usually very good) approximate truncated
singular value decomposition using randomization to speed up the
computations. It is particularly fast on large matrices on which
you wish to extract only a small number of components. In order to
obtain further speed up, `n_iter` can be set <=2 (at the cost of
loss of precision).
References
----------
* Finding structure with randomness: Stochastic algorithms for constructing
approximate matrix decompositions
Halko, et al., 2009 https://arxiv.org/abs/0909.4061
* A randomized algorithm for the decomposition of matrices
Per-Gunnar Martinsson, Vladimir Rokhlin and Mark Tygert
* An implementation of a randomized algorithm for principal component
analysis
A. Szlam et al. 2014
"""
if isinstance(M, (sparse.lil_matrix, sparse.dok_matrix)):
warnings.warn("Calculating SVD of a {} is expensive. "
"csr_matrix is more efficient.".format(
type(M).__name__),
sparse.SparseEfficiencyWarning)
random_state = check_random_state(random_state)
n_random = n_components + n_oversamples
n_samples, n_features = M.shape
if n_iter == 'auto':
# Checks if the number of iterations is explicitly specified
# Adjust n_iter. 7 was found a good compromise for PCA. See #5299
n_iter = 7 if n_components < .1 * min(M.shape) else 4
if transpose == 'auto':
transpose = n_samples < n_features
if transpose:
# this implementation is a bit faster with smaller shape[1]
M = M.T
Q = randomized_range_finder(M, n_random, n_iter,
power_iteration_normalizer, random_state)
# project M to the (k + p) dimensional space using the basis vectors
B = safe_sparse_dot(Q.T, M)
# compute the SVD on the thin matrix: (k + p) wide
Uhat, s, V = linalg.svd(B, full_matrices=False)
del B
U = np.dot(Q, Uhat)
if flip_sign:
if not transpose:
U, V = svd_flip(U, V)
else:
# In case of transpose u_based_decision=false
# to actually flip based on u and not v.
U, V = svd_flip(U, V, u_based_decision=False)
if transpose:
# transpose back the results according to the input convention
return V[:n_components, :].T, s[:n_components], U[:, :n_components].T
else:
return U[:, :n_components], s[:n_components], V[:n_components, :]
def weighted_mode(a, w, axis=0):
"""Returns an array of the weighted modal (most common) value in a
If there is more than one such value, only the first is returned.
The bin-count for the modal bins is also returned.
This is an extension of the algorithm in scipy.stats.mode.
Parameters
----------
a : array_like
n-dimensional array of which to find mode(s).
w : array_like
n-dimensional array of weights for each value
axis : int, optional
Axis along which to operate. Default is 0, i.e. the first axis.
Returns
-------
vals : ndarray
Array of modal values.
score : ndarray
Array of weighted counts for each mode.
Examples
--------
>>> from sklearn.utils.extmath import weighted_mode
>>> x = [4, 1, 4, 2, 4, 2]
>>> weights = [1, 1, 1, 1, 1, 1]
>>> weighted_mode(x, weights)
(array([4.]), array([3.]))
The value 4 appears three times: with uniform weights, the result is
simply the mode of the distribution.
>>> weights = [1, 3, 0.5, 1.5, 1, 2] # deweight the 4's
>>> weighted_mode(x, weights)
(array([2.]), array([3.5]))
The value 2 has the highest score: it appears twice with weights of
1.5 and 2: the sum of these is 3.5.
See Also
--------
scipy.stats.mode
"""
if axis is None:
a = np.ravel(a)
w = np.ravel(w)
axis = 0
else:
a = np.asarray(a)
w = np.asarray(w)
if a.shape != w.shape:
w = np.full(a.shape, w, dtype=w.dtype)
scores = np.unique(np.ravel(a)) # get ALL unique values
testshape = list(a.shape)
testshape[axis] = 1
oldmostfreq = np.zeros(testshape)
oldcounts = np.zeros(testshape)
for score in scores:
template = np.zeros(a.shape)
ind = (a == score)
template[ind] = w[ind]
counts = np.expand_dims(np.sum(template, axis), axis)
mostfrequent = np.where(counts > oldcounts, score, oldmostfreq)
oldcounts = np.maximum(counts, oldcounts)
oldmostfreq = mostfrequent
return mostfrequent, oldcounts
def cartesian(arrays, out=None):
"""Generate a cartesian product of input arrays.
Parameters
----------
arrays : list of array-like
1-D arrays to form the cartesian product of.
out : ndarray
Array to place the cartesian product in.
Returns
-------
out : ndarray
2-D array of shape (M, len(arrays)) containing cartesian products
formed of input arrays.
Examples
--------
>>> cartesian(([1, 2, 3], [4, 5], [6, 7]))
array([[1, 4, 6],
[1, 4, 7],
[1, 5, 6],
[1, 5, 7],
[2, 4, 6],
[2, 4, 7],
[2, 5, 6],
[2, 5, 7],
[3, 4, 6],
[3, 4, 7],
[3, 5, 6],
[3, 5, 7]])
"""
arrays = [np.asarray(x) for x in arrays]
shape = (len(x) for x in arrays)
dtype = arrays[0].dtype
ix = np.indices(shape)
ix = ix.reshape(len(arrays), -1).T
if out is None:
out = np.empty_like(ix, dtype=dtype)
for n, arr in enumerate(arrays):
out[:, n] = arrays[n][ix[:, n]]
return out
def svd_flip(u, v, u_based_decision=True):
"""Sign correction to ensure deterministic output from SVD.
Adjusts the columns of u and the rows of v such that the loadings in the
columns in u that are largest in absolute value are always positive.
Parameters
----------
u : ndarray
u and v are the output of `linalg.svd` or
:func:`~sklearn.utils.extmath.randomized_svd`, with matching inner
dimensions so one can compute `np.dot(u * s, v)`.
v : ndarray
u and v are the output of `linalg.svd` or
:func:`~sklearn.utils.extmath.randomized_svd`, with matching inner
dimensions so one can compute `np.dot(u * s, v)`.
u_based_decision : boolean, (default=True)
If True, use the columns of u as the basis for sign flipping.
Otherwise, use the rows of v. The choice of which variable to base the
decision on is generally algorithm dependent.
Returns
-------
u_adjusted, v_adjusted : arrays with the same dimensions as the input.
"""
if u_based_decision:
# columns of u, rows of v
max_abs_cols = np.argmax(np.abs(u), axis=0)
signs = np.sign(u[max_abs_cols, range(u.shape[1])])
u *= signs
v *= signs[:, np.newaxis]
else:
# rows of v, columns of u
max_abs_rows = np.argmax(np.abs(v), axis=1)
signs = np.sign(v[range(v.shape[0]), max_abs_rows])
u *= signs
v *= signs[:, np.newaxis]
return u, v
def log_logistic(X, out=None):
"""Compute the log of the logistic function, ``log(1 / (1 + e ** -x))``.
This implementation is numerically stable because it splits positive and
negative values::
-log(1 + exp(-x_i)) if x_i > 0
x_i - log(1 + exp(x_i)) if x_i <= 0
For the ordinary logistic function, use ``scipy.special.expit``.
Parameters
----------
X : array-like, shape (M, N) or (M, )
Argument to the logistic function
out : array-like, shape: (M, N) or (M, ), optional:
Preallocated output array.
Returns
-------
out : array, shape (M, N) or (M, )
Log of the logistic function evaluated at every point in x
Notes
-----
See the blog post describing this implementation:
http://fa.bianp.net/blog/2013/numerical-optimizers-for-logistic-regression/
"""
is_1d = X.ndim == 1
X = np.atleast_2d(X)
X = check_array(X, dtype=np.float64)
n_samples, n_features = X.shape
if out is None:
out = np.empty_like(X)
_log_logistic_sigmoid(n_samples, n_features, X, out)
if is_1d:
return np.squeeze(out)
return out
def softmax(X, copy=True):
"""
Calculate the softmax function.
The softmax function is calculated by
np.exp(X) / np.sum(np.exp(X), axis=1)
This will cause overflow when large values are exponentiated.
Hence the largest value in each row is subtracted from each data
point to prevent this.
Parameters
----------
X : array-like of floats, shape (M, N)
Argument to the logistic function
copy : bool, optional
Copy X or not.
Returns
-------
out : array, shape (M, N)
Softmax function evaluated at every point in x
"""
if copy:
X = np.copy(X)
max_prob = np.max(X, axis=1).reshape((-1, 1))
X -= max_prob
np.exp(X, X)
sum_prob = np.sum(X, axis=1).reshape((-1, 1))
X /= sum_prob
return X
@deprecated("safe_min is deprecated in version 0.22 and will be removed "
"in version 0.24.")
def safe_min(X):
"""Returns the minimum value of a dense or a CSR/CSC matrix.
Adapated from https://stackoverflow.com/q/13426580
.. deprecated:: 0.22.0
Parameters
----------
X : array_like
The input array or sparse matrix
Returns
-------
Float
The min value of X
"""
if sparse.issparse(X):
if len(X.data) == 0:
return 0
m = X.data.min()
return m if X.getnnz() == X.size else min(m, 0)
else:
return X.min()
def make_nonnegative(X, min_value=0):
"""Ensure `X.min()` >= `min_value`.
Parameters
----------
X : array_like
The matrix to make non-negative
min_value : float
The threshold value
Returns
-------
array_like
The thresholded array
Raises
------
ValueError
When X is sparse
"""
min_ = X.min()
if min_ < min_value:
if sparse.issparse(X):
raise ValueError("Cannot make the data matrix"
" nonnegative because it is sparse."
" Adding a value to every entry would"
" make it no longer sparse.")
X = X + (min_value - min_)
return X
# Use at least float64 for the accumulating functions to avoid precision issue
# see https://github.com/numpy/numpy/issues/9393. The float64 is also retained
# as it is in case the float overflows
def _safe_accumulator_op(op, x, *args, **kwargs):
"""
This function provides numpy accumulator functions with a float64 dtype
when used on a floating point input. This prevents accumulator overflow on
smaller floating point dtypes.
Parameters
----------
op : function
A numpy accumulator function such as np.mean or np.sum
x : numpy array
A numpy array to apply the accumulator function
*args : positional arguments
Positional arguments passed to the accumulator function after the
input x
**kwargs : keyword arguments
Keyword arguments passed to the accumulator function
Returns
-------
result : The output of the accumulator function passed to this function
"""
if np.issubdtype(x.dtype, np.floating) and x.dtype.itemsize < 8:
result = op(x, *args, **kwargs, dtype=np.float64)
else:
result = op(x, *args, **kwargs)
return result
def _incremental_mean_and_var(X, last_mean, last_variance, last_sample_count):
"""Calculate mean update and a Youngs and Cramer variance update.
last_mean and last_variance are statistics computed at the last step by the
function. Both must be initialized to 0.0. In case no scaling is required
last_variance can be None. The mean is always required and returned because
necessary for the calculation of the variance. last_n_samples_seen is the
number of samples encountered until now.
From the paper "Algorithms for computing the sample variance: analysis and
recommendations", by Chan, Golub, and LeVeque.
Parameters
----------
X : array-like, shape (n_samples, n_features)
Data to use for variance update
last_mean : array-like, shape: (n_features,)
last_variance : array-like, shape: (n_features,)
last_sample_count : array-like, shape (n_features,)
Returns
-------
updated_mean : array, shape (n_features,)
updated_variance : array, shape (n_features,)
If None, only mean is computed
updated_sample_count : array, shape (n_features,)
Notes
-----
NaNs are ignored during the algorithm.
References
----------
T. Chan, G. Golub, R. LeVeque. Algorithms for computing the sample
variance: recommendations, The American Statistician, Vol. 37, No. 3,
pp. 242-247
Also, see the sparse implementation of this in
`utils.sparsefuncs.incr_mean_variance_axis` and
`utils.sparsefuncs_fast.incr_mean_variance_axis0`
"""
# old = stats until now
# new = the current increment
# updated = the aggregated stats
last_sum = last_mean * last_sample_count
new_sum = _safe_accumulator_op(np.nansum, X, axis=0)
new_sample_count = np.sum(~np.isnan(X), axis=0)
updated_sample_count = last_sample_count + new_sample_count
updated_mean = (last_sum + new_sum) / updated_sample_count
if last_variance is None:
updated_variance = None
else:
new_unnormalized_variance = (
_safe_accumulator_op(np.nanvar, X, axis=0) * new_sample_count)
last_unnormalized_variance = last_variance * last_sample_count
with np.errstate(divide='ignore', invalid='ignore'):
last_over_new_count = last_sample_count / new_sample_count
updated_unnormalized_variance = (
last_unnormalized_variance + new_unnormalized_variance +
last_over_new_count / updated_sample_count *
(last_sum / last_over_new_count - new_sum) ** 2)
zeros = last_sample_count == 0
updated_unnormalized_variance[zeros] = new_unnormalized_variance[zeros]
updated_variance = updated_unnormalized_variance / updated_sample_count
return updated_mean, updated_variance, updated_sample_count
def _deterministic_vector_sign_flip(u):
"""Modify the sign of vectors for reproducibility
Flips the sign of elements of all the vectors (rows of u) such that
the absolute maximum element of each vector is positive.
Parameters
----------
u : ndarray
Array with vectors as its rows.
Returns
-------
u_flipped : ndarray with same shape as u
Array with the sign flipped vectors as its rows.
"""
max_abs_rows = np.argmax(np.abs(u), axis=1)
signs = np.sign(u[range(u.shape[0]), max_abs_rows])
u *= signs[:, np.newaxis]
return u
def stable_cumsum(arr, axis=None, rtol=1e-05, atol=1e-08):
"""Use high precision for cumsum and check that final value matches sum
Parameters
----------
arr : array-like
To be cumulatively summed as flat
axis : int, optional
Axis along which the cumulative sum is computed.
The default (None) is to compute the cumsum over the flattened array.
rtol : float
Relative tolerance, see ``np.allclose``
atol : float
Absolute tolerance, see ``np.allclose``
"""
out = np.cumsum(arr, axis=axis, dtype=np.float64)
expected = np.sum(arr, axis=axis, dtype=np.float64)
if not np.all(np.isclose(out.take(-1, axis=axis), expected, rtol=rtol,
atol=atol, equal_nan=True)):
warnings.warn('cumsum was found to be unstable: '
'its last element does not correspond to sum',
RuntimeWarning)
return out