1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297
|
"""Incremental Principal Components Analysis."""
# Author: Kyle Kastner <kastnerkyle@gmail.com>
# Giorgio Patrini
# License: BSD 3 clause
from __future__ import division
import numpy as np
from scipy import linalg
from .base import _BasePCA
from ..utils import check_array, gen_batches
from ..utils.extmath import svd_flip, _incremental_mean_and_var
class IncrementalPCA(_BasePCA):
"""Incremental principal components analysis (IPCA).
Linear dimensionality reduction using Singular Value Decomposition of
centered data, keeping only the most significant singular vectors to
project the data to a lower dimensional space.
Depending on the size of the input data, this algorithm can be much more
memory efficient than a PCA.
This algorithm has constant memory complexity, on the order
of ``batch_size``, enabling use of np.memmap files without loading the
entire file into memory.
The computational overhead of each SVD is
``O(batch_size * n_features ** 2)``, but only 2 * batch_size samples
remain in memory at a time. There will be ``n_samples / batch_size`` SVD
computations to get the principal components, versus 1 large SVD of
complexity ``O(n_samples * n_features ** 2)`` for PCA.
Read more in the :ref:`User Guide <IncrementalPCA>`.
Parameters
----------
n_components : int or None, (default=None)
Number of components to keep. If ``n_components `` is ``None``,
then ``n_components`` is set to ``min(n_samples, n_features)``.
whiten : bool, optional
When True (False by default) the ``components_`` vectors are divided
by ``n_samples`` times ``components_`` to ensure uncorrelated outputs
with unit component-wise variances.
Whitening will remove some information from the transformed signal
(the relative variance scales of the components) but can sometimes
improve the predictive accuracy of the downstream estimators by
making data respect some hard-wired assumptions.
copy : bool, (default=True)
If False, X will be overwritten. ``copy=False`` can be used to
save memory but is unsafe for general use.
batch_size : int or None, (default=None)
The number of samples to use for each batch. Only used when calling
``fit``. If ``batch_size`` is ``None``, then ``batch_size``
is inferred from the data and set to ``5 * n_features``, to provide a
balance between approximation accuracy and memory consumption.
Attributes
----------
components_ : array, shape (n_components, n_features)
Components with maximum variance.
explained_variance_ : array, shape (n_components,)
Variance explained by each of the selected components.
explained_variance_ratio_ : array, shape (n_components,)
Percentage of variance explained by each of the selected components.
If all components are stored, the sum of explained variances is equal
to 1.0.
singular_values_ : array, shape (n_components,)
The singular values corresponding to each of the selected components.
The singular values are equal to the 2-norms of the ``n_components``
variables in the lower-dimensional space.
mean_ : array, shape (n_features,)
Per-feature empirical mean, aggregate over calls to ``partial_fit``.
var_ : array, shape (n_features,)
Per-feature empirical variance, aggregate over calls to
``partial_fit``.
noise_variance_ : float
The estimated noise covariance following the Probabilistic PCA model
from Tipping and Bishop 1999. See "Pattern Recognition and
Machine Learning" by C. Bishop, 12.2.1 p. 574 or
http://www.miketipping.com/papers/met-mppca.pdf.
n_components_ : int
The estimated number of components. Relevant when
``n_components=None``.
n_samples_seen_ : int
The number of samples processed by the estimator. Will be reset on
new calls to fit, but increments across ``partial_fit`` calls.
Examples
--------
>>> from sklearn.datasets import load_digits
>>> from sklearn.decomposition import IncrementalPCA
>>> X, _ = load_digits(return_X_y=True)
>>> transformer = IncrementalPCA(n_components=7, batch_size=200)
>>> # either partially fit on smaller batches of data
>>> transformer.partial_fit(X[:100, :])
IncrementalPCA(batch_size=200, copy=True, n_components=7, whiten=False)
>>> # or let the fit function itself divide the data into batches
>>> X_transformed = transformer.fit_transform(X)
>>> X_transformed.shape
(1797, 7)
Notes
-----
Implements the incremental PCA model from:
`D. Ross, J. Lim, R. Lin, M. Yang, Incremental Learning for Robust Visual
Tracking, International Journal of Computer Vision, Volume 77, Issue 1-3,
pp. 125-141, May 2008.`
See http://www.cs.toronto.edu/~dross/ivt/RossLimLinYang_ijcv.pdf
This model is an extension of the Sequential Karhunen-Loeve Transform from:
`A. Levy and M. Lindenbaum, Sequential Karhunen-Loeve Basis Extraction and
its Application to Images, IEEE Transactions on Image Processing, Volume 9,
Number 8, pp. 1371-1374, August 2000.`
See http://www.cs.technion.ac.il/~mic/doc/skl-ip.pdf
We have specifically abstained from an optimization used by authors of both
papers, a QR decomposition used in specific situations to reduce the
algorithmic complexity of the SVD. The source for this technique is
`Matrix Computations, Third Edition, G. Holub and C. Van Loan, Chapter 5,
section 5.4.4, pp 252-253.`. This technique has been omitted because it is
advantageous only when decomposing a matrix with ``n_samples`` (rows)
>= 5/3 * ``n_features`` (columns), and hurts the readability of the
implemented algorithm. This would be a good opportunity for future
optimization, if it is deemed necessary.
References
----------
D. Ross, J. Lim, R. Lin, M. Yang. Incremental Learning for Robust Visual
Tracking, International Journal of Computer Vision, Volume 77,
Issue 1-3, pp. 125-141, May 2008.
G. Golub and C. Van Loan. Matrix Computations, Third Edition, Chapter 5,
Section 5.4.4, pp. 252-253.
See also
--------
PCA
KernelPCA
SparsePCA
TruncatedSVD
"""
def __init__(self, n_components=None, whiten=False, copy=True,
batch_size=None):
self.n_components = n_components
self.whiten = whiten
self.copy = copy
self.batch_size = batch_size
def fit(self, X, y=None):
"""Fit the model with X, using minibatches of size batch_size.
Parameters
----------
X : array-like, shape (n_samples, n_features)
Training data, where n_samples is the number of samples and
n_features is the number of features.
y : Ignored
Returns
-------
self : object
Returns the instance itself.
"""
self.components_ = None
self.n_samples_seen_ = 0
self.mean_ = .0
self.var_ = .0
self.singular_values_ = None
self.explained_variance_ = None
self.explained_variance_ratio_ = None
self.singular_values_ = None
self.noise_variance_ = None
X = check_array(X, copy=self.copy, dtype=[np.float64, np.float32])
n_samples, n_features = X.shape
if self.batch_size is None:
self.batch_size_ = 5 * n_features
else:
self.batch_size_ = self.batch_size
for batch in gen_batches(n_samples, self.batch_size_,
min_batch_size=self.n_components or 0):
self.partial_fit(X[batch], check_input=False)
return self
def partial_fit(self, X, y=None, check_input=True):
"""Incremental fit with X. All of X is processed as a single batch.
Parameters
----------
X : array-like, shape (n_samples, n_features)
Training data, where n_samples is the number of samples and
n_features is the number of features.
check_input : bool
Run check_array on X.
y : Ignored
Returns
-------
self : object
Returns the instance itself.
"""
if check_input:
X = check_array(X, copy=self.copy, dtype=[np.float64, np.float32])
n_samples, n_features = X.shape
if not hasattr(self, 'components_'):
self.components_ = None
if self.n_components is None:
if self.components_ is None:
self.n_components_ = min(n_samples, n_features)
else:
self.n_components_ = self.components_.shape[0]
elif not 1 <= self.n_components <= n_features:
raise ValueError("n_components=%r invalid for n_features=%d, need "
"more rows than columns for IncrementalPCA "
"processing" % (self.n_components, n_features))
elif not self.n_components <= n_samples:
raise ValueError("n_components=%r must be less or equal to "
"the batch number of samples "
"%d." % (self.n_components, n_samples))
else:
self.n_components_ = self.n_components
if (self.components_ is not None) and (self.components_.shape[0] !=
self.n_components_):
raise ValueError("Number of input features has changed from %i "
"to %i between calls to partial_fit! Try "
"setting n_components to a fixed value." %
(self.components_.shape[0], self.n_components_))
# This is the first partial_fit
if not hasattr(self, 'n_samples_seen_'):
self.n_samples_seen_ = 0
self.mean_ = .0
self.var_ = .0
# Update stats - they are 0 if this is the fisrt step
col_mean, col_var, n_total_samples = \
_incremental_mean_and_var(
X, last_mean=self.mean_, last_variance=self.var_,
last_sample_count=np.repeat(self.n_samples_seen_, X.shape[1]))
n_total_samples = n_total_samples[0]
# Whitening
if self.n_samples_seen_ == 0:
# If it is the first step, simply whiten X
X -= col_mean
else:
col_batch_mean = np.mean(X, axis=0)
X -= col_batch_mean
# Build matrix of combined previous basis and new data
mean_correction = \
np.sqrt((self.n_samples_seen_ * n_samples) /
n_total_samples) * (self.mean_ - col_batch_mean)
X = np.vstack((self.singular_values_.reshape((-1, 1)) *
self.components_, X, mean_correction))
U, S, V = linalg.svd(X, full_matrices=False)
U, V = svd_flip(U, V, u_based_decision=False)
explained_variance = S ** 2 / (n_total_samples - 1)
explained_variance_ratio = S ** 2 / np.sum(col_var * n_total_samples)
self.n_samples_seen_ = n_total_samples
self.components_ = V[:self.n_components_]
self.singular_values_ = S[:self.n_components_]
self.mean_ = col_mean
self.var_ = col_var
self.explained_variance_ = explained_variance[:self.n_components_]
self.explained_variance_ratio_ = \
explained_variance_ratio[:self.n_components_]
if self.n_components_ < n_features:
self.noise_variance_ = \
explained_variance[self.n_components_:].mean()
else:
self.noise_variance_ = 0.
return self
|