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/ datasets / _samples_generator.py
"""
Generate samples of synthetic data sets.
"""
# Authors: B. Thirion, G. Varoquaux, A. Gramfort, V. Michel, O. Grisel,
# G. Louppe, J. Nothman
# License: BSD 3 clause
import numbers
import array
from collections.abc import Iterable
import numpy as np
from scipy import linalg
import scipy.sparse as sp
from ..preprocessing import MultiLabelBinarizer
from ..utils import check_array, check_random_state
from ..utils import shuffle as util_shuffle
from ..utils.random import sample_without_replacement
def _generate_hypercube(samples, dimensions, rng):
"""Returns distinct binary samples of length dimensions
"""
if dimensions > 30:
return np.hstack([rng.randint(2, size=(samples, dimensions - 30)),
_generate_hypercube(samples, 30, rng)])
out = sample_without_replacement(2 ** dimensions, samples,
random_state=rng).astype(dtype='>u4',
copy=False)
out = np.unpackbits(out.view('>u1')).reshape((-1, 32))[:, -dimensions:]
return out
def make_classification(n_samples=100, n_features=20, n_informative=2,
n_redundant=2, n_repeated=0, n_classes=2,
n_clusters_per_class=2, weights=None, flip_y=0.01,
class_sep=1.0, hypercube=True, shift=0.0, scale=1.0,
shuffle=True, random_state=None):
"""Generate a random n-class classification problem.
This initially creates clusters of points normally distributed (std=1)
about vertices of an ``n_informative``-dimensional hypercube with sides of
length ``2*class_sep`` and assigns an equal number of clusters to each
class. It introduces interdependence between these features and adds
various types of further noise to the data.
Without shuffling, ``X`` horizontally stacks features in the following
order: the primary ``n_informative`` features, followed by ``n_redundant``
linear combinations of the informative features, followed by ``n_repeated``
duplicates, drawn randomly with replacement from the informative and
redundant features. The remaining features are filled with random noise.
Thus, without shuffling, all useful features are contained in the columns
``X[:, :n_informative + n_redundant + n_repeated]``.
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_samples : int, optional (default=100)
The number of samples.
n_features : int, optional (default=20)
The total number of features. These comprise ``n_informative``
informative features, ``n_redundant`` redundant features,
``n_repeated`` duplicated features and
``n_features-n_informative-n_redundant-n_repeated`` useless features
drawn at random.
n_informative : int, optional (default=2)
The number of informative features. Each class is composed of a number
of gaussian clusters each located around the vertices of a hypercube
in a subspace of dimension ``n_informative``. For each cluster,
informative features are drawn independently from N(0, 1) and then
randomly linearly combined within each cluster in order to add
covariance. The clusters are then placed on the vertices of the
hypercube.
n_redundant : int, optional (default=2)
The number of redundant features. These features are generated as
random linear combinations of the informative features.
n_repeated : int, optional (default=0)
The number of duplicated features, drawn randomly from the informative
and the redundant features.
n_classes : int, optional (default=2)
The number of classes (or labels) of the classification problem.
n_clusters_per_class : int, optional (default=2)
The number of clusters per class.
weights : array-like of shape (n_classes,) or (n_classes - 1,),\
(default=None)
The proportions of samples assigned to each class. If None, then
classes are balanced. Note that if ``len(weights) == n_classes - 1``,
then the last class weight is automatically inferred.
More than ``n_samples`` samples may be returned if the sum of
``weights`` exceeds 1.
flip_y : float, optional (default=0.01)
The fraction of samples whose class is assigned randomly. Larger
values introduce noise in the labels and make the classification
task harder.
class_sep : float, optional (default=1.0)
The factor multiplying the hypercube size. Larger values spread
out the clusters/classes and make the classification task easier.
hypercube : boolean, optional (default=True)
If True, the clusters are put on the vertices of a hypercube. If
False, the clusters are put on the vertices of a random polytope.
shift : float, array of shape [n_features] or None, optional (default=0.0)
Shift features by the specified value. If None, then features
are shifted by a random value drawn in [-class_sep, class_sep].
scale : float, array of shape [n_features] or None, optional (default=1.0)
Multiply features by the specified value. If None, then features
are scaled by a random value drawn in [1, 100]. Note that scaling
happens after shifting.
shuffle : boolean, optional (default=True)
Shuffle the samples and the features.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset creation. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape [n_samples, n_features]
The generated samples.
y : array of shape [n_samples]
The integer labels for class membership of each sample.
Notes
-----
The algorithm is adapted from Guyon [1] and was designed to generate
the "Madelon" dataset.
References
----------
.. [1] I. Guyon, "Design of experiments for the NIPS 2003 variable
selection benchmark", 2003.
See also
--------
make_blobs: simplified variant
make_multilabel_classification: unrelated generator for multilabel tasks
"""
generator = check_random_state(random_state)
# Count features, clusters and samples
if n_informative + n_redundant + n_repeated > n_features:
raise ValueError("Number of informative, redundant and repeated "
"features must sum to less than the number of total"
" features")
# Use log2 to avoid overflow errors
if n_informative < np.log2(n_classes * n_clusters_per_class):
msg = "n_classes({}) * n_clusters_per_class({}) must be"
msg += " smaller or equal 2**n_informative({})={}"
raise ValueError(msg.format(n_classes, n_clusters_per_class,
n_informative, 2**n_informative))
if weights is not None:
if len(weights) not in [n_classes, n_classes - 1]:
raise ValueError("Weights specified but incompatible with number "
"of classes.")
if len(weights) == n_classes - 1:
if isinstance(weights, list):
weights = weights + [1.0 - sum(weights)]
else:
weights = np.resize(weights, n_classes)
weights[-1] = 1.0 - sum(weights[:-1])
else:
weights = [1.0 / n_classes] * n_classes
n_useless = n_features - n_informative - n_redundant - n_repeated
n_clusters = n_classes * n_clusters_per_class
# Distribute samples among clusters by weight
n_samples_per_cluster = [
int(n_samples * weights[k % n_classes] / n_clusters_per_class)
for k in range(n_clusters)]
for i in range(n_samples - sum(n_samples_per_cluster)):
n_samples_per_cluster[i % n_clusters] += 1
# Initialize X and y
X = np.zeros((n_samples, n_features))
y = np.zeros(n_samples, dtype=np.int)
# Build the polytope whose vertices become cluster centroids
centroids = _generate_hypercube(n_clusters, n_informative,
generator).astype(float, copy=False)
centroids *= 2 * class_sep
centroids -= class_sep
if not hypercube:
centroids *= generator.rand(n_clusters, 1)
centroids *= generator.rand(1, n_informative)
# Initially draw informative features from the standard normal
X[:, :n_informative] = generator.randn(n_samples, n_informative)
# Create each cluster; a variant of make_blobs
stop = 0
for k, centroid in enumerate(centroids):
start, stop = stop, stop + n_samples_per_cluster[k]
y[start:stop] = k % n_classes # assign labels
X_k = X[start:stop, :n_informative] # slice a view of the cluster
A = 2 * generator.rand(n_informative, n_informative) - 1
X_k[...] = np.dot(X_k, A) # introduce random covariance
X_k += centroid # shift the cluster to a vertex
# Create redundant features
if n_redundant > 0:
B = 2 * generator.rand(n_informative, n_redundant) - 1
X[:, n_informative:n_informative + n_redundant] = \
np.dot(X[:, :n_informative], B)
# Repeat some features
if n_repeated > 0:
n = n_informative + n_redundant
indices = ((n - 1) * generator.rand(n_repeated) + 0.5).astype(np.intp)
X[:, n:n + n_repeated] = X[:, indices]
# Fill useless features
if n_useless > 0:
X[:, -n_useless:] = generator.randn(n_samples, n_useless)
# Randomly replace labels
if flip_y >= 0.0:
flip_mask = generator.rand(n_samples) < flip_y
y[flip_mask] = generator.randint(n_classes, size=flip_mask.sum())
# Randomly shift and scale
if shift is None:
shift = (2 * generator.rand(n_features) - 1) * class_sep
X += shift
if scale is None:
scale = 1 + 100 * generator.rand(n_features)
X *= scale
if shuffle:
# Randomly permute samples
X, y = util_shuffle(X, y, random_state=generator)
# Randomly permute features
indices = np.arange(n_features)
generator.shuffle(indices)
X[:, :] = X[:, indices]
return X, y
def make_multilabel_classification(n_samples=100, n_features=20, n_classes=5,
n_labels=2, length=50, allow_unlabeled=True,
sparse=False, return_indicator='dense',
return_distributions=False,
random_state=None):
"""Generate a random multilabel classification problem.
For each sample, the generative process is:
- pick the number of labels: n ~ Poisson(n_labels)
- n times, choose a class c: c ~ Multinomial(theta)
- pick the document length: k ~ Poisson(length)
- k times, choose a word: w ~ Multinomial(theta_c)
In the above process, rejection sampling is used to make sure that
n is never zero or more than `n_classes`, and that the document length
is never zero. Likewise, we reject classes which have already been chosen.
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_samples : int, optional (default=100)
The number of samples.
n_features : int, optional (default=20)
The total number of features.
n_classes : int, optional (default=5)
The number of classes of the classification problem.
n_labels : int, optional (default=2)
The average number of labels per instance. More precisely, the number
of labels per sample is drawn from a Poisson distribution with
``n_labels`` as its expected value, but samples are bounded (using
rejection sampling) by ``n_classes``, and must be nonzero if
``allow_unlabeled`` is False.
length : int, optional (default=50)
The sum of the features (number of words if documents) is drawn from
a Poisson distribution with this expected value.
allow_unlabeled : bool, optional (default=True)
If ``True``, some instances might not belong to any class.
sparse : bool, optional (default=False)
If ``True``, return a sparse feature matrix
.. versionadded:: 0.17
parameter to allow *sparse* output.
return_indicator : 'dense' (default) | 'sparse' | False
If ``dense`` return ``Y`` in the dense binary indicator format. If
``'sparse'`` return ``Y`` in the sparse binary indicator format.
``False`` returns a list of lists of labels.
return_distributions : bool, optional (default=False)
If ``True``, return the prior class probability and conditional
probabilities of features given classes, from which the data was
drawn.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset creation. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape [n_samples, n_features]
The generated samples.
Y : array or sparse CSR matrix of shape [n_samples, n_classes]
The label sets.
p_c : array, shape [n_classes]
The probability of each class being drawn. Only returned if
``return_distributions=True``.
p_w_c : array, shape [n_features, n_classes]
The probability of each feature being drawn given each class.
Only returned if ``return_distributions=True``.
"""
generator = check_random_state(random_state)
p_c = generator.rand(n_classes)
p_c /= p_c.sum()
cumulative_p_c = np.cumsum(p_c)
p_w_c = generator.rand(n_features, n_classes)
p_w_c /= np.sum(p_w_c, axis=0)
def sample_example():
_, n_classes = p_w_c.shape
# pick a nonzero number of labels per document by rejection sampling
y_size = n_classes + 1
while (not allow_unlabeled and y_size == 0) or y_size > n_classes:
y_size = generator.poisson(n_labels)
# pick n classes
y = set()
while len(y) != y_size:
# pick a class with probability P(c)
c = np.searchsorted(cumulative_p_c,
generator.rand(y_size - len(y)))
y.update(c)
y = list(y)
# pick a non-zero document length by rejection sampling
n_words = 0
while n_words == 0:
n_words = generator.poisson(length)
# generate a document of length n_words
if len(y) == 0:
# if sample does not belong to any class, generate noise word
words = generator.randint(n_features, size=n_words)
return words, y
# sample words with replacement from selected classes
cumulative_p_w_sample = p_w_c.take(y, axis=1).sum(axis=1).cumsum()
cumulative_p_w_sample /= cumulative_p_w_sample[-1]
words = np.searchsorted(cumulative_p_w_sample, generator.rand(n_words))
return words, y
X_indices = array.array('i')
X_indptr = array.array('i', [0])
Y = []
for i in range(n_samples):
words, y = sample_example()
X_indices.extend(words)
X_indptr.append(len(X_indices))
Y.append(y)
X_data = np.ones(len(X_indices), dtype=np.float64)
X = sp.csr_matrix((X_data, X_indices, X_indptr),
shape=(n_samples, n_features))
X.sum_duplicates()
if not sparse:
X = X.toarray()
# return_indicator can be True due to backward compatibility
if return_indicator in (True, 'sparse', 'dense'):
lb = MultiLabelBinarizer(sparse_output=(return_indicator == 'sparse'))
Y = lb.fit([range(n_classes)]).transform(Y)
elif return_indicator is not False:
raise ValueError("return_indicator must be either 'sparse', 'dense' "
'or False.')
if return_distributions:
return X, Y, p_c, p_w_c
return X, Y
def make_hastie_10_2(n_samples=12000, random_state=None):
"""Generates data for binary classification used in
Hastie et al. 2009, Example 10.2.
The ten features are standard independent Gaussian and
the target ``y`` is defined by::
y[i] = 1 if np.sum(X[i] ** 2) > 9.34 else -1
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_samples : int, optional (default=12000)
The number of samples.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset creation. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape [n_samples, 10]
The input samples.
y : array of shape [n_samples]
The output values.
References
----------
.. [1] T. Hastie, R. Tibshirani and J. Friedman, "Elements of Statistical
Learning Ed. 2", Springer, 2009.
See also
--------
make_gaussian_quantiles: a generalization of this dataset approach
"""
rs = check_random_state(random_state)
shape = (n_samples, 10)
X = rs.normal(size=shape).reshape(shape)
y = ((X ** 2.0).sum(axis=1) > 9.34).astype(np.float64, copy=False)
y[y == 0.0] = -1.0
return X, y
def make_regression(n_samples=100, n_features=100, n_informative=10,
n_targets=1, bias=0.0, effective_rank=None,
tail_strength=0.5, noise=0.0, shuffle=True, coef=False,
random_state=None):
"""Generate a random regression problem.
The input set can either be well conditioned (by default) or have a low
rank-fat tail singular profile. See :func:`make_low_rank_matrix` for
more details.
The output is generated by applying a (potentially biased) random linear
regression model with `n_informative` nonzero regressors to the previously
generated input and some gaussian centered noise with some adjustable
scale.
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_samples : int, optional (default=100)
The number of samples.
n_features : int, optional (default=100)
The number of features.
n_informative : int, optional (default=10)
The number of informative features, i.e., the number of features used
to build the linear model used to generate the output.
n_targets : int, optional (default=1)
The number of regression targets, i.e., the dimension of the y output
vector associated with a sample. By default, the output is a scalar.
bias : float, optional (default=0.0)
The bias term in the underlying linear model.
effective_rank : int or None, optional (default=None)
if not None:
The approximate number of singular vectors required to explain most
of the input data by linear combinations. Using this kind of
singular spectrum in the input allows the generator to reproduce
the correlations often observed in practice.
if None:
The input set is well conditioned, centered and gaussian with
unit variance.
tail_strength : float between 0.0 and 1.0, optional (default=0.5)
The relative importance of the fat noisy tail of the singular values
profile if `effective_rank` is not None.
noise : float, optional (default=0.0)
The standard deviation of the gaussian noise applied to the output.
shuffle : boolean, optional (default=True)
Shuffle the samples and the features.
coef : boolean, optional (default=False)
If True, the coefficients of the underlying linear model are returned.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset creation. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape [n_samples, n_features]
The input samples.
y : array of shape [n_samples] or [n_samples, n_targets]
The output values.
coef : array of shape [n_features] or [n_features, n_targets], optional
The coefficient of the underlying linear model. It is returned only if
coef is True.
"""
n_informative = min(n_features, n_informative)
generator = check_random_state(random_state)
if effective_rank is None:
# Randomly generate a well conditioned input set
X = generator.randn(n_samples, n_features)
else:
# Randomly generate a low rank, fat tail input set
X = make_low_rank_matrix(n_samples=n_samples,
n_features=n_features,
effective_rank=effective_rank,
tail_strength=tail_strength,
random_state=generator)
# Generate a ground truth model with only n_informative features being non
# zeros (the other features are not correlated to y and should be ignored
# by a sparsifying regularizers such as L1 or elastic net)
ground_truth = np.zeros((n_features, n_targets))
ground_truth[:n_informative, :] = 100 * generator.rand(n_informative,
n_targets)
y = np.dot(X, ground_truth) + bias
# Add noise
if noise > 0.0:
y += generator.normal(scale=noise, size=y.shape)
# Randomly permute samples and features
if shuffle:
X, y = util_shuffle(X, y, random_state=generator)
indices = np.arange(n_features)
generator.shuffle(indices)
X[:, :] = X[:, indices]
ground_truth = ground_truth[indices]
y = np.squeeze(y)
if coef:
return X, y, np.squeeze(ground_truth)
else:
return X, y
def make_circles(n_samples=100, shuffle=True, noise=None, random_state=None,
factor=.8):
"""Make a large circle containing a smaller circle in 2d.
A simple toy dataset to visualize clustering and classification
algorithms.
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_samples : int, optional (default=100)
The total number of points generated. If odd, the inner circle will
have one point more than the outer circle.
shuffle : bool, optional (default=True)
Whether to shuffle the samples.
noise : double or None (default=None)
Standard deviation of Gaussian noise added to the data.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset shuffling and noise.
Pass an int for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
factor : 0 < double < 1 (default=.8)
Scale factor between inner and outer circle.
Returns
-------
X : array of shape [n_samples, 2]
The generated samples.
y : array of shape [n_samples]
The integer labels (0 or 1) for class membership of each sample.
"""
if factor >= 1 or factor < 0:
raise ValueError("'factor' has to be between 0 and 1.")
n_samples_out = n_samples // 2
n_samples_in = n_samples - n_samples_out
generator = check_random_state(random_state)
# so as not to have the first point = last point, we set endpoint=False
linspace_out = np.linspace(0, 2 * np.pi, n_samples_out, endpoint=False)
linspace_in = np.linspace(0, 2 * np.pi, n_samples_in, endpoint=False)
outer_circ_x = np.cos(linspace_out)
outer_circ_y = np.sin(linspace_out)
inner_circ_x = np.cos(linspace_in) * factor
inner_circ_y = np.sin(linspace_in) * factor
X = np.vstack([np.append(outer_circ_x, inner_circ_x),
np.append(outer_circ_y, inner_circ_y)]).T
y = np.hstack([np.zeros(n_samples_out, dtype=np.intp),
np.ones(n_samples_in, dtype=np.intp)])
if shuffle:
X, y = util_shuffle(X, y, random_state=generator)
if noise is not None:
X += generator.normal(scale=noise, size=X.shape)
return X, y
def make_moons(n_samples=100, shuffle=True, noise=None, random_state=None):
"""Make two interleaving half circles
A simple toy dataset to visualize clustering and classification
algorithms. Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_samples : int, optional (default=100)
The total number of points generated.
shuffle : bool, optional (default=True)
Whether to shuffle the samples.
noise : double or None (default=None)
Standard deviation of Gaussian noise added to the data.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset shuffling and noise.
Pass an int for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape [n_samples, 2]
The generated samples.
y : array of shape [n_samples]
The integer labels (0 or 1) for class membership of each sample.
"""
n_samples_out = n_samples // 2
n_samples_in = n_samples - n_samples_out
generator = check_random_state(random_state)
outer_circ_x = np.cos(np.linspace(0, np.pi, n_samples_out))
outer_circ_y = np.sin(np.linspace(0, np.pi, n_samples_out))
inner_circ_x = 1 - np.cos(np.linspace(0, np.pi, n_samples_in))
inner_circ_y = 1 - np.sin(np.linspace(0, np.pi, n_samples_in)) - .5
X = np.vstack([np.append(outer_circ_x, inner_circ_x),
np.append(outer_circ_y, inner_circ_y)]).T
y = np.hstack([np.zeros(n_samples_out, dtype=np.intp),
np.ones(n_samples_in, dtype=np.intp)])
if shuffle:
X, y = util_shuffle(X, y, random_state=generator)
if noise is not None:
X += generator.normal(scale=noise, size=X.shape)
return X, y
def make_blobs(n_samples=100, n_features=2, centers=None, cluster_std=1.0,
center_box=(-10.0, 10.0), shuffle=True, random_state=None):
"""Generate isotropic Gaussian blobs for clustering.
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_samples : int or array-like, optional (default=100)
If int, it is the total number of points equally divided among
clusters.
If array-like, each element of the sequence indicates
the number of samples per cluster.
n_features : int, optional (default=2)
The number of features for each sample.
centers : int or array of shape [n_centers, n_features], optional
(default=None)
The number of centers to generate, or the fixed center locations.
If n_samples is an int and centers is None, 3 centers are generated.
If n_samples is array-like, centers must be
either None or an array of length equal to the length of n_samples.
cluster_std : float or sequence of floats, optional (default=1.0)
The standard deviation of the clusters.
center_box : pair of floats (min, max), optional (default=(-10.0, 10.0))
The bounding box for each cluster center when centers are
generated at random.
shuffle : boolean, optional (default=True)
Shuffle the samples.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset creation. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape [n_samples, n_features]
The generated samples.
y : array of shape [n_samples]
The integer labels for cluster membership of each sample.
Examples
--------
>>> from sklearn.datasets import make_blobs
>>> X, y = make_blobs(n_samples=10, centers=3, n_features=2,
... random_state=0)
>>> print(X.shape)
(10, 2)
>>> y
array([0, 0, 1, 0, 2, 2, 2, 1, 1, 0])
>>> X, y = make_blobs(n_samples=[3, 3, 4], centers=None, n_features=2,
... random_state=0)
>>> print(X.shape)
(10, 2)
>>> y
array([0, 1, 2, 0, 2, 2, 2, 1, 1, 0])
See also
--------
make_classification: a more intricate variant
"""
generator = check_random_state(random_state)
if isinstance(n_samples, numbers.Integral):
# Set n_centers by looking at centers arg
if centers is None:
centers = 3
if isinstance(centers, numbers.Integral):
n_centers = centers
centers = generator.uniform(center_box[0], center_box[1],
size=(n_centers, n_features))
else:
centers = check_array(centers)
n_features = centers.shape[1]
n_centers = centers.shape[0]
else:
# Set n_centers by looking at [n_samples] arg
n_centers = len(n_samples)
if centers is None:
centers = generator.uniform(center_box[0], center_box[1],
size=(n_centers, n_features))
try:
assert len(centers) == n_centers
except TypeError:
raise ValueError("Parameter `centers` must be array-like. "
"Got {!r} instead".format(centers))
except AssertionError:
raise ValueError("Length of `n_samples` not consistent"
" with number of centers. Got n_samples = {} "
"and centers = {}".format(n_samples, centers))
else:
centers = check_array(centers)
n_features = centers.shape[1]
# stds: if cluster_std is given as list, it must be consistent
# with the n_centers
if (hasattr(cluster_std, "__len__") and len(cluster_std) != n_centers):
raise ValueError("Length of `clusters_std` not consistent with "
"number of centers. Got centers = {} "
"and cluster_std = {}".format(centers, cluster_std))
if isinstance(cluster_std, numbers.Real):
cluster_std = np.full(len(centers), cluster_std)
X = []
y = []
if isinstance(n_samples, Iterable):
n_samples_per_center = n_samples
else:
n_samples_per_center = [int(n_samples // n_centers)] * n_centers
for i in range(n_samples % n_centers):
n_samples_per_center[i] += 1
for i, (n, std) in enumerate(zip(n_samples_per_center, cluster_std)):
X.append(generator.normal(loc=centers[i], scale=std,
size=(n, n_features)))
y += [i] * n
X = np.concatenate(X)
y = np.array(y)
if shuffle:
total_n_samples = np.sum(n_samples)
indices = np.arange(total_n_samples)
generator.shuffle(indices)
X = X[indices]
y = y[indices]
return X, y
def make_friedman1(n_samples=100, n_features=10, noise=0.0, random_state=None):
"""Generate the "Friedman #1" regression problem
This dataset is described in Friedman [1] and Breiman [2].
Inputs `X` are independent features uniformly distributed on the interval
[0, 1]. The output `y` is created according to the formula::
y(X) = 10 * sin(pi * X[:, 0] * X[:, 1]) + 20 * (X[:, 2] - 0.5) ** 2 \
+ 10 * X[:, 3] + 5 * X[:, 4] + noise * N(0, 1).
Out of the `n_features` features, only 5 are actually used to compute
`y`. The remaining features are independent of `y`.
The number of features has to be >= 5.
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_samples : int, optional (default=100)
The number of samples.
n_features : int, optional (default=10)
The number of features. Should be at least 5.
noise : float, optional (default=0.0)
The standard deviation of the gaussian noise applied to the output.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset noise. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape [n_samples, n_features]
The input samples.
y : array of shape [n_samples]
The output values.
References
----------
.. [1] J. Friedman, "Multivariate adaptive regression splines", The Annals
of Statistics 19 (1), pages 1-67, 1991.
.. [2] L. Breiman, "Bagging predictors", Machine Learning 24,
pages 123-140, 1996.
"""
if n_features < 5:
raise ValueError("n_features must be at least five.")
generator = check_random_state(random_state)
X = generator.rand(n_samples, n_features)
y = 10 * np.sin(np.pi * X[:, 0] * X[:, 1]) + 20 * (X[:, 2] - 0.5) ** 2 \
+ 10 * X[:, 3] + 5 * X[:, 4] + noise * generator.randn(n_samples)
return X, y
def make_friedman2(n_samples=100, noise=0.0, random_state=None):
"""Generate the "Friedman #2" regression problem
This dataset is described in Friedman [1] and Breiman [2].
Inputs `X` are 4 independent features uniformly distributed on the
intervals::
0 <= X[:, 0] <= 100,
40 * pi <= X[:, 1] <= 560 * pi,
0 <= X[:, 2] <= 1,
1 <= X[:, 3] <= 11.
The output `y` is created according to the formula::
y(X) = (X[:, 0] ** 2 + (X[:, 1] * X[:, 2] \
- 1 / (X[:, 1] * X[:, 3])) ** 2) ** 0.5 + noise * N(0, 1).
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_samples : int, optional (default=100)
The number of samples.
noise : float, optional (default=0.0)
The standard deviation of the gaussian noise applied to the output.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset noise. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape [n_samples, 4]
The input samples.
y : array of shape [n_samples]
The output values.
References
----------
.. [1] J. Friedman, "Multivariate adaptive regression splines", The Annals
of Statistics 19 (1), pages 1-67, 1991.
.. [2] L. Breiman, "Bagging predictors", Machine Learning 24,
pages 123-140, 1996.
"""
generator = check_random_state(random_state)
X = generator.rand(n_samples, 4)
X[:, 0] *= 100
X[:, 1] *= 520 * np.pi
X[:, 1] += 40 * np.pi
X[:, 3] *= 10
X[:, 3] += 1
y = (X[:, 0] ** 2
+ (X[:, 1] * X[:, 2] - 1 / (X[:, 1] * X[:, 3])) ** 2) ** 0.5 \
+ noise * generator.randn(n_samples)
return X, y
def make_friedman3(n_samples=100, noise=0.0, random_state=None):
"""Generate the "Friedman #3" regression problem
This dataset is described in Friedman [1] and Breiman [2].
Inputs `X` are 4 independent features uniformly distributed on the
intervals::
0 <= X[:, 0] <= 100,
40 * pi <= X[:, 1] <= 560 * pi,
0 <= X[:, 2] <= 1,
1 <= X[:, 3] <= 11.
The output `y` is created according to the formula::
y(X) = arctan((X[:, 1] * X[:, 2] - 1 / (X[:, 1] * X[:, 3])) \
/ X[:, 0]) + noise * N(0, 1).
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_samples : int, optional (default=100)
The number of samples.
noise : float, optional (default=0.0)
The standard deviation of the gaussian noise applied to the output.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset noise. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape [n_samples, 4]
The input samples.
y : array of shape [n_samples]
The output values.
References
----------
.. [1] J. Friedman, "Multivariate adaptive regression splines", The Annals
of Statistics 19 (1), pages 1-67, 1991.
.. [2] L. Breiman, "Bagging predictors", Machine Learning 24,
pages 123-140, 1996.
"""
generator = check_random_state(random_state)
X = generator.rand(n_samples, 4)
X[:, 0] *= 100
X[:, 1] *= 520 * np.pi
X[:, 1] += 40 * np.pi
X[:, 3] *= 10
X[:, 3] += 1
y = np.arctan((X[:, 1] * X[:, 2] - 1 / (X[:, 1] * X[:, 3])) / X[:, 0]) \
+ noise * generator.randn(n_samples)
return X, y
def make_low_rank_matrix(n_samples=100, n_features=100, effective_rank=10,
tail_strength=0.5, random_state=None):
"""Generate a mostly low rank matrix with bell-shaped singular values
Most of the variance can be explained by a bell-shaped curve of width
effective_rank: the low rank part of the singular values profile is::
(1 - tail_strength) * exp(-1.0 * (i / effective_rank) ** 2)
The remaining singular values' tail is fat, decreasing as::
tail_strength * exp(-0.1 * i / effective_rank).
The low rank part of the profile can be considered the structured
signal part of the data while the tail can be considered the noisy
part of the data that cannot be summarized by a low number of linear
components (singular vectors).
This kind of singular profiles is often seen in practice, for instance:
- gray level pictures of faces
- TF-IDF vectors of text documents crawled from the web
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_samples : int, optional (default=100)
The number of samples.
n_features : int, optional (default=100)
The number of features.
effective_rank : int, optional (default=10)
The approximate number of singular vectors required to explain most of
the data by linear combinations.
tail_strength : float between 0.0 and 1.0, optional (default=0.5)
The relative importance of the fat noisy tail of the singular values
profile.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset creation. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape [n_samples, n_features]
The matrix.
"""
generator = check_random_state(random_state)
n = min(n_samples, n_features)
# Random (ortho normal) vectors
u, _ = linalg.qr(generator.randn(n_samples, n), mode='economic')
v, _ = linalg.qr(generator.randn(n_features, n), mode='economic')
# Index of the singular values
singular_ind = np.arange(n, dtype=np.float64)
# Build the singular profile by assembling signal and noise components
low_rank = ((1 - tail_strength) *
np.exp(-1.0 * (singular_ind / effective_rank) ** 2))
tail = tail_strength * np.exp(-0.1 * singular_ind / effective_rank)
s = np.identity(n) * (low_rank + tail)
return np.dot(np.dot(u, s), v.T)
def make_sparse_coded_signal(n_samples, n_components, n_features,
n_nonzero_coefs, random_state=None):
"""Generate a signal as a sparse combination of dictionary elements.
Returns a matrix Y = DX, such as D is (n_features, n_components),
X is (n_components, n_samples) and each column of X has exactly
n_nonzero_coefs non-zero elements.
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_samples : int
number of samples to generate
n_components : int,
number of components in the dictionary
n_features : int
number of features of the dataset to generate
n_nonzero_coefs : int
number of active (non-zero) coefficients in each sample
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset creation. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
data : array of shape [n_features, n_samples]
The encoded signal (Y).
dictionary : array of shape [n_features, n_components]
The dictionary with normalized components (D).
code : array of shape [n_components, n_samples]
The sparse code such that each column of this matrix has exactly
n_nonzero_coefs non-zero items (X).
"""
generator = check_random_state(random_state)
# generate dictionary
D = generator.randn(n_features, n_components)
D /= np.sqrt(np.sum((D ** 2), axis=0))
# generate code
X = np.zeros((n_components, n_samples))
for i in range(n_samples):
idx = np.arange(n_components)
generator.shuffle(idx)
idx = idx[:n_nonzero_coefs]
X[idx, i] = generator.randn(n_nonzero_coefs)
# encode signal
Y = np.dot(D, X)
return map(np.squeeze, (Y, D, X))
def make_sparse_uncorrelated(n_samples=100, n_features=10, random_state=None):
"""Generate a random regression problem with sparse uncorrelated design
This dataset is described in Celeux et al [1]. as::
X ~ N(0, 1)
y(X) = X[:, 0] + 2 * X[:, 1] - 2 * X[:, 2] - 1.5 * X[:, 3]
Only the first 4 features are informative. The remaining features are
useless.
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_samples : int, optional (default=100)
The number of samples.
n_features : int, optional (default=10)
The number of features.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset creation. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape [n_samples, n_features]
The input samples.
y : array of shape [n_samples]
The output values.
References
----------
.. [1] G. Celeux, M. El Anbari, J.-M. Marin, C. P. Robert,
"Regularization in regression: comparing Bayesian and frequentist
methods in a poorly informative situation", 2009.
"""
generator = check_random_state(random_state)
X = generator.normal(loc=0, scale=1, size=(n_samples, n_features))
y = generator.normal(loc=(X[:, 0] +
2 * X[:, 1] -
2 * X[:, 2] -
1.5 * X[:, 3]), scale=np.ones(n_samples))
return X, y
def make_spd_matrix(n_dim, random_state=None):
"""Generate a random symmetric, positive-definite matrix.
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_dim : int
The matrix dimension.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset creation. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape [n_dim, n_dim]
The random symmetric, positive-definite matrix.
See also
--------
make_sparse_spd_matrix
"""
generator = check_random_state(random_state)
A = generator.rand(n_dim, n_dim)
U, s, V = linalg.svd(np.dot(A.T, A))
X = np.dot(np.dot(U, 1.0 + np.diag(generator.rand(n_dim))), V)
return X
def make_sparse_spd_matrix(dim=1, alpha=0.95, norm_diag=False,
smallest_coef=.1, largest_coef=.9,
random_state=None):
"""Generate a sparse symmetric definite positive matrix.
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
dim : integer, optional (default=1)
The size of the random matrix to generate.
alpha : float between 0 and 1, optional (default=0.95)
The probability that a coefficient is zero (see notes). Larger values
enforce more sparsity.
norm_diag : boolean, optional (default=False)
Whether to normalize the output matrix to make the leading diagonal
elements all 1
smallest_coef : float between 0 and 1, optional (default=0.1)
The value of the smallest coefficient.
largest_coef : float between 0 and 1, optional (default=0.9)
The value of the largest coefficient.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset creation. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
prec : sparse matrix of shape (dim, dim)
The generated matrix.
Notes
-----
The sparsity is actually imposed on the cholesky factor of the matrix.
Thus alpha does not translate directly into the filling fraction of
the matrix itself.
See also
--------
make_spd_matrix
"""
random_state = check_random_state(random_state)
chol = -np.eye(dim)
aux = random_state.rand(dim, dim)
aux[aux < alpha] = 0
aux[aux > alpha] = (smallest_coef
+ (largest_coef - smallest_coef)
* random_state.rand(np.sum(aux > alpha)))
aux = np.tril(aux, k=-1)
# Permute the lines: we don't want to have asymmetries in the final
# SPD matrix
permutation = random_state.permutation(dim)
aux = aux[permutation].T[permutation]
chol += aux
prec = np.dot(chol.T, chol)
if norm_diag:
# Form the diagonal vector into a row matrix
d = np.diag(prec).reshape(1, prec.shape[0])
d = 1. / np.sqrt(d)
prec *= d
prec *= d.T
return prec
def make_swiss_roll(n_samples=100, noise=0.0, random_state=None):
"""Generate a swiss roll dataset.
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_samples : int, optional (default=100)
The number of sample points on the S curve.
noise : float, optional (default=0.0)
The standard deviation of the gaussian noise.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset creation. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape [n_samples, 3]
The points.
t : array of shape [n_samples]
The univariate position of the sample according to the main dimension
of the points in the manifold.
Notes
-----
The algorithm is from Marsland [1].
References
----------
.. [1] S. Marsland, "Machine Learning: An Algorithmic Perspective",
Chapter 10, 2009.
http://seat.massey.ac.nz/personal/s.r.marsland/Code/10/lle.py
"""
generator = check_random_state(random_state)
t = 1.5 * np.pi * (1 + 2 * generator.rand(1, n_samples))
x = t * np.cos(t)
y = 21 * generator.rand(1, n_samples)
z = t * np.sin(t)
X = np.concatenate((x, y, z))
X += noise * generator.randn(3, n_samples)
X = X.T
t = np.squeeze(t)
return X, t
def make_s_curve(n_samples=100, noise=0.0, random_state=None):
"""Generate an S curve dataset.
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
n_samples : int, optional (default=100)
The number of sample points on the S curve.
noise : float, optional (default=0.0)
The standard deviation of the gaussian noise.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset creation. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape [n_samples, 3]
The points.
t : array of shape [n_samples]
The univariate position of the sample according to the main dimension
of the points in the manifold.
"""
generator = check_random_state(random_state)
t = 3 * np.pi * (generator.rand(1, n_samples) - 0.5)
x = np.sin(t)
y = 2.0 * generator.rand(1, n_samples)
z = np.sign(t) * (np.cos(t) - 1)
X = np.concatenate((x, y, z))
X += noise * generator.randn(3, n_samples)
X = X.T
t = np.squeeze(t)
return X, t
def make_gaussian_quantiles(mean=None, cov=1., n_samples=100,
n_features=2, n_classes=3,
shuffle=True, random_state=None):
r"""Generate isotropic Gaussian and label samples by quantile
This classification dataset is constructed by taking a multi-dimensional
standard normal distribution and defining classes separated by nested
concentric multi-dimensional spheres such that roughly equal numbers of
samples are in each class (quantiles of the :math:`\chi^2` distribution).
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
mean : array of shape [n_features], optional (default=None)
The mean of the multi-dimensional normal distribution.
If None then use the origin (0, 0, ...).
cov : float, optional (default=1.)
The covariance matrix will be this value times the unit matrix. This
dataset only produces symmetric normal distributions.
n_samples : int, optional (default=100)
The total number of points equally divided among classes.
n_features : int, optional (default=2)
The number of features for each sample.
n_classes : int, optional (default=3)
The number of classes
shuffle : boolean, optional (default=True)
Shuffle the samples.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset creation. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape [n_samples, n_features]
The generated samples.
y : array of shape [n_samples]
The integer labels for quantile membership of each sample.
Notes
-----
The dataset is from Zhu et al [1].
References
----------
.. [1] J. Zhu, H. Zou, S. Rosset, T. Hastie, "Multi-class AdaBoost", 2009.
"""
if n_samples < n_classes:
raise ValueError("n_samples must be at least n_classes")
generator = check_random_state(random_state)
if mean is None:
mean = np.zeros(n_features)
else:
mean = np.array(mean)
# Build multivariate normal distribution
X = generator.multivariate_normal(mean, cov * np.identity(n_features),
(n_samples,))
# Sort by distance from origin
idx = np.argsort(np.sum((X - mean[np.newaxis, :]) ** 2, axis=1))
X = X[idx, :]
# Label by quantile
step = n_samples // n_classes
y = np.hstack([np.repeat(np.arange(n_classes), step),
np.repeat(n_classes - 1, n_samples - step * n_classes)])
if shuffle:
X, y = util_shuffle(X, y, random_state=generator)
return X, y
def _shuffle(data, random_state=None):
generator = check_random_state(random_state)
n_rows, n_cols = data.shape
row_idx = generator.permutation(n_rows)
col_idx = generator.permutation(n_cols)
result = data[row_idx][:, col_idx]
return result, row_idx, col_idx
def make_biclusters(shape, n_clusters, noise=0.0, minval=10,
maxval=100, shuffle=True, random_state=None):
"""Generate an array with constant block diagonal structure for
biclustering.
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
shape : iterable (n_rows, n_cols)
The shape of the result.
n_clusters : integer
The number of biclusters.
noise : float, optional (default=0.0)
The standard deviation of the gaussian noise.
minval : int, optional (default=10)
Minimum value of a bicluster.
maxval : int, optional (default=100)
Maximum value of a bicluster.
shuffle : boolean, optional (default=True)
Shuffle the samples.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset creation. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape `shape`
The generated array.
rows : array of shape (n_clusters, X.shape[0],)
The indicators for cluster membership of each row.
cols : array of shape (n_clusters, X.shape[1],)
The indicators for cluster membership of each column.
References
----------
.. [1] Dhillon, I. S. (2001, August). Co-clustering documents and
words using bipartite spectral graph partitioning. In Proceedings
of the seventh ACM SIGKDD international conference on Knowledge
discovery and data mining (pp. 269-274). ACM.
See also
--------
make_checkerboard
"""
generator = check_random_state(random_state)
n_rows, n_cols = shape
consts = generator.uniform(minval, maxval, n_clusters)
# row and column clusters of approximately equal sizes
row_sizes = generator.multinomial(n_rows,
np.repeat(1.0 / n_clusters,
n_clusters))
col_sizes = generator.multinomial(n_cols,
np.repeat(1.0 / n_clusters,
n_clusters))
row_labels = np.hstack(list(np.repeat(val, rep) for val, rep in
zip(range(n_clusters), row_sizes)))
col_labels = np.hstack(list(np.repeat(val, rep) for val, rep in
zip(range(n_clusters), col_sizes)))
result = np.zeros(shape, dtype=np.float64)
for i in range(n_clusters):
selector = np.outer(row_labels == i, col_labels == i)
result[selector] += consts[i]
if noise > 0:
result += generator.normal(scale=noise, size=result.shape)
if shuffle:
result, row_idx, col_idx = _shuffle(result, random_state)
row_labels = row_labels[row_idx]
col_labels = col_labels[col_idx]
rows = np.vstack([row_labels == c for c in range(n_clusters)])
cols = np.vstack([col_labels == c for c in range(n_clusters)])
return result, rows, cols
def make_checkerboard(shape, n_clusters, noise=0.0, minval=10,
maxval=100, shuffle=True, random_state=None):
"""Generate an array with block checkerboard structure for
biclustering.
Read more in the :ref:`User Guide <sample_generators>`.
Parameters
----------
shape : iterable (n_rows, n_cols)
The shape of the result.
n_clusters : integer or iterable (n_row_clusters, n_column_clusters)
The number of row and column clusters.
noise : float, optional (default=0.0)
The standard deviation of the gaussian noise.
minval : int, optional (default=10)
Minimum value of a bicluster.
maxval : int, optional (default=100)
Maximum value of a bicluster.
shuffle : boolean, optional (default=True)
Shuffle the samples.
random_state : int, RandomState instance or None (default)
Determines random number generation for dataset creation. Pass an int
for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Returns
-------
X : array of shape `shape`
The generated array.
rows : array of shape (n_clusters, X.shape[0],)
The indicators for cluster membership of each row.
cols : array of shape (n_clusters, X.shape[1],)
The indicators for cluster membership of each column.
References
----------
.. [1] Kluger, Y., Basri, R., Chang, J. T., & Gerstein, M. (2003).
Spectral biclustering of microarray data: coclustering genes
and conditions. Genome research, 13(4), 703-716.
See also
--------
make_biclusters
"""
generator = check_random_state(random_state)
if hasattr(n_clusters, "__len__"):
n_row_clusters, n_col_clusters = n_clusters
else:
n_row_clusters = n_col_clusters = n_clusters
# row and column clusters of approximately equal sizes
n_rows, n_cols = shape
row_sizes = generator.multinomial(n_rows,
np.repeat(1.0 / n_row_clusters,
n_row_clusters))
col_sizes = generator.multinomial(n_cols,
np.repeat(1.0 / n_col_clusters,
n_col_clusters))
row_labels = np.hstack(list(np.repeat(val, rep) for val, rep in
zip(range(n_row_clusters), row_sizes)))
col_labels = np.hstack(list(np.repeat(val, rep) for val, rep in
zip(range(n_col_clusters), col_sizes)))
result = np.zeros(shape, dtype=np.float64)
for i in range(n_row_clusters):
for j in range(n_col_clusters):
selector = np.outer(row_labels == i, col_labels == j)
result[selector] += generator.uniform(minval, maxval)
if noise > 0:
result += generator.normal(scale=noise, size=result.shape)
if shuffle:
result, row_idx, col_idx = _shuffle(result, random_state)
row_labels = row_labels[row_idx]
col_labels = col_labels[col_idx]
rows = np.vstack([row_labels == label
for label in range(n_row_clusters)
for _ in range(n_col_clusters)])
cols = np.vstack([col_labels == label
for _ in range(n_row_clusters)
for label in range(n_col_clusters)])
return result, rows, cols