.. _preprocessing:
==================
Preprocessing data
==================
.. currentmodule:: sklearn.preprocessing
The ``sklearn.preprocessing`` package provides several common
utility functions and transformer classes to change raw feature vectors
into a representation that is more suitable for the downstream estimators.
.. _preprocessing_scaler:
Standardization, or mean removal and variance scaling
=====================================================
**Standardization** of datasets is a **common requirement for many
machine learning estimators** implemented in the scikit; they might behave
badly if the individual features do not more or less look like standard
normally distributed data: Gaussian with **zero mean and unit variance**.
In practice we often ignore the shape of the distribution and just
transform the data to center it by removing the mean value of each
feature, then scale it by dividing non-constant features by their
standard deviation.
For instance, many elements used in the objective function of
a learning algorithm (such as the RBF kernel of Support Vector
Machines or the l1 and l2 regularizers of linear models) assume that
all features are centered around zero and have variance in the same
order. If a feature has a variance that is orders of magnitude larger
that others, it might dominate the objective function and make the
estimator unable to learn from other features correctly as expected.
The function :func:`scale` provides a quick and easy way to perform this
operation on a single array-like dataset::
>>> from sklearn import preprocessing
>>> import numpy as np
>>> X = np.array([[ 1., -1., 2.],
... [ 2., 0., 0.],
... [ 0., 1., -1.]])
>>> X_scaled = preprocessing.scale(X)
>>> X_scaled # doctest: +ELLIPSIS
array([[ 0. ..., -1.22..., 1.33...],
[ 1.22..., 0. ..., -0.26...],
[-1.22..., 1.22..., -1.06...]])
..
>>> import numpy as np
>>> print_options = np.get_printoptions()
>>> np.set_printoptions(suppress=True)
Scaled data has zero mean and unit variance::
>>> X_scaled.mean(axis=0)
array([ 0., 0., 0.])
>>> X_scaled.std(axis=0)
array([ 1., 1., 1.])
.. >>> print_options = np.set_printoptions(print_options)
The ``preprocessing`` module further provides a utility class
:class:`StandardScaler` that implements the ``Transformer`` API to compute
the mean and standard deviation on a training set so as to be
able to later reapply the same transformation on the testing set.
This class is hence suitable for use in the early steps of a
:class:`sklearn.pipeline.Pipeline`::
>>> scaler = preprocessing.StandardScaler().fit(X)
>>> scaler
StandardScaler(copy=True, with_mean=True, with_std=True)
>>> scaler.mean_ # doctest: +ELLIPSIS
array([ 1. ..., 0. ..., 0.33...])
>>> scaler.std_ # doctest: +ELLIPSIS
array([ 0.81..., 0.81..., 1.24...])
>>> scaler.transform(X) # doctest: +ELLIPSIS
array([[ 0. ..., -1.22..., 1.33...],
[ 1.22..., 0. ..., -0.26...],
[-1.22..., 1.22..., -1.06...]])
The scaler instance can then be used on new data to transform it the
same way it did on the training set::
>>> scaler.transform([[-1., 1., 0.]]) # doctest: +ELLIPSIS
array([[-2.44..., 1.22..., -0.26...]])
It is possible to disable either centering or scaling by either
passing ``with_mean=False`` or ``with_std=False`` to the constructor
of :class:`StandardScaler`.
Scaling features to a range
---------------------------
An alternative standardization is scaling features to
lie between a given minimum and maximum value, often between zero and one,
or so that the maximum absolute value of each feature is scaled to unit size.
This can be achieved using :class:`MinMaxScaler` or :class:`MaxAbsScaler`,
respectively.
The motivation to use this scaling include robustness to very small
standard deviations of features and preserving zero entries in sparse data.
Here is an example to scale a toy data matrix to the ``[0, 1]`` range::
>>> X_train = np.array([[ 1., -1., 2.],
... [ 2., 0., 0.],
... [ 0., 1., -1.]])
...
>>> min_max_scaler = preprocessing.MinMaxScaler()
>>> X_train_minmax = min_max_scaler.fit_transform(X_train)
>>> X_train_minmax
array([[ 0.5 , 0. , 1. ],
[ 1. , 0.5 , 0.33333333],
[ 0. , 1. , 0. ]])
The same instance of the transformer can then be applied to some new test data
unseen during the fit call: the same scaling and shifting operations will be
applied to be consistent with the transformation performed on the train data::
>>> X_test = np.array([[ -3., -1., 4.]])
>>> X_test_minmax = min_max_scaler.transform(X_test)
>>> X_test_minmax
array([[-1.5 , 0. , 1.66666667]])
It is possible to introspect the scaler attributes to find about the exact
nature of the transformation learned on the training data::
>>> min_max_scaler.scale_ # doctest: +ELLIPSIS
array([ 0.5 , 0.5 , 0.33...])
>>> min_max_scaler.min_ # doctest: +ELLIPSIS
array([ 0. , 0.5 , 0.33...])
If :class:`MinMaxScaler` is given an explicit ``feature_range=(min, max)`` the
full formula is::
X_std = (X - X.min(axis=0)) / (X.max(axis=0) - X.min(axis=0))
X_scaled = X_std / (max - min) + min
:class:`MaxAbsScaler` works in a very similar fashion, but scales in a way
that the training data lies within the range ``[-1, 1]`` by dividing through
the largest maximum value in each feature. It is meant for data
that is already centered at zero or sparse data.
Here is how to use the toy data from the previous example with this scaler::
>>> X_train = np.array([[ 1., -1., 2.],
... [ 2., 0., 0.],
... [ 0., 1., -1.]])
...
>>> max_abs_scaler = preprocessing.MaxAbsScaler()
>>> X_train_maxabs = max_abs_scaler.fit_transform(X_train)
>>> X_train_maxabs # doctest +NORMALIZE_WHITESPACE^
array([[ 0.5, -1. , 1. ],
[ 1. , 0. , 0. ],
[ 0. , 1. , -0.5]])
>>> X_test = np.array([[ -3., -1., 4.]])
>>> X_test_maxabs = max_abs_scaler.transform(X_test)
>>> X_test_maxabs # doctest: +NORMALIZE_WHITESPACE
array([[-1.5, -1. , 2. ]])
>>> max_abs_scaler.scale_ # doctest: +ELLIPSIS +NORMALIZE_WHITESPACE
array([ 2., 1., 2.])
As with :func:`scale`, the module further provides a
convenience function :func:`maxabs_scale` if you don't want to
create an object.
Scaling sparse data
-------------------
Centering sparse data would destroy the sparseness structure in the data, and
thus rarely is a sensible thing to do. However, it can make sense to scale
sparse inputs, especially if features are on different scales.
:class:`MaxAbsScaler` and :func:`maxabs_scale` were specifically designed
for scaling sparse data, and are the recommended way to go about this.
However, :func:`scale` and :class:`StandardScaler` can accept ``scipy.sparse``
matrices as input, as long as ``with_centering=False`` is explicitly passed
to the constructor. Otherwise a ``ValueError`` will be raised as
silently centering would break the sparsity and would often crash the
execution by allocating excessive amounts of memory unintentionally.
:class:`RobustScaler` cannot be fited to sparse inputs, but you can use
the ``transform`` method on sparse inputs.
Note that the scalers accept both Compressed Sparse Rows and Compressed
Sparse Columns format (see ``scipy.sparse.csr_matrix`` and
``scipy.sparse.csc_matrix``). Any other sparse input will be **converted to
the Compressed Sparse Rows representation**. To avoid unnecessary memory
copies, it is recommended to choose the CSR or CSC representation upstream.
Finally, if the centered data is expected to be small enough, explicitly
converting the input to an array using the ``toarray`` method of sparse matrices
is another option.
Scaling data with outliers
--------------------------
If your data contains many outliers, scaling using the mean and variance
of the data is likely to not work very well. In these cases, you can use
:func:`robust_scale` and :class:`RobustScaler` as drop-in replacements
instead. They use more robust estimates for the center and range of your
data.
.. topic:: References:
Further discussion on the importance of centering and scaling data is
available on this FAQ: `Should I normalize/standardize/rescale the data?
<http://www.faqs.org/faqs/ai-faq/neural-nets/part2/section-16.html>`_
.. topic:: Scaling vs Whitening
It is sometimes not enough to center and scale the features
independently, since a downstream model can further make some assumption
on the linear independence of the features.
To address this issue you can use :class:`sklearn.decomposition.PCA`
or :class:`sklearn.decomposition.RandomizedPCA` with ``whiten=True``
to further remove the linear correlation across features.
.. topic:: Scaling target variables in regression
:func:`scale` and :class:`StandardScaler` work out-of-the-box with 1d arrays.
This is very useful for scaling the target / response variables used
for regression.
.. _kernel_centering:
Centering kernel matrices
-------------------------
If you have a kernel matrix of a kernel :math:`K` that computes a dot product
in a feature space defined by function :math:`phi`,
a :class:`KernelCenterer` can transform the kernel matrix
so that it contains inner products in the feature space
defined by :math:`phi` followed by removal of the mean in that space.
.. _preprocessing_normalization:
Normalization
=============
**Normalization** is the process of **scaling individual samples to have
unit norm**. This process can be useful if you plan to use a quadratic form
such as the dot-product or any other kernel to quantify the similarity
of any pair of samples.
This assumption is the base of the `Vector Space Model
<http://en.wikipedia.org/wiki/Vector_Space_Model>`_ often used in text
classification and clustering contexts.
The function :func:`normalize` provides a quick and easy way to perform this
operation on a single array-like dataset, either using the ``l1`` or ``l2``
norms::
>>> X = [[ 1., -1., 2.],
... [ 2., 0., 0.],
... [ 0., 1., -1.]]
>>> X_normalized = preprocessing.normalize(X, norm='l2')
>>> X_normalized # doctest: +ELLIPSIS
array([[ 0.40..., -0.40..., 0.81...],
[ 1. ..., 0. ..., 0. ...],
[ 0. ..., 0.70..., -0.70...]])
The ``preprocessing`` module further provides a utility class
:class:`Normalizer` that implements the same operation using the
``Transformer`` API (even though the ``fit`` method is useless in this case:
the class is stateless as this operation treats samples independently).
This class is hence suitable for use in the early steps of a
:class:`sklearn.pipeline.Pipeline`::
>>> normalizer = preprocessing.Normalizer().fit(X) # fit does nothing
>>> normalizer
Normalizer(copy=True, norm='l2')
The normalizer instance can then be used on sample vectors as any transformer::
>>> normalizer.transform(X) # doctest: +ELLIPSIS
array([[ 0.40..., -0.40..., 0.81...],
[ 1. ..., 0. ..., 0. ...],
[ 0. ..., 0.70..., -0.70...]])
>>> normalizer.transform([[-1., 1., 0.]]) # doctest: +ELLIPSIS
array([[-0.70..., 0.70..., 0. ...]])
.. topic:: Sparse input
:func:`normalize` and :class:`Normalizer` accept **both dense array-like
and sparse matrices from scipy.sparse as input**.
For sparse input the data is **converted to the Compressed Sparse Rows
representation** (see ``scipy.sparse.csr_matrix``) before being fed to
efficient Cython routines. To avoid unnecessary memory copies, it is
recommended to choose the CSR representation upstream.
.. _preprocessing_binarization:
Binarization
============
Feature binarization
--------------------
**Feature binarization** is the process of **thresholding numerical
features to get boolean values**. This can be useful for downstream
probabilistic estimators that make assumption that the input data
is distributed according to a multi-variate `Bernoulli distribution
<http://en.wikipedia.org/wiki/Bernoulli_distribution>`_. For instance,
this is the case for the :class:`sklearn.neural_network.BernoulliRBM`.
It is also common among the text processing community to use binary
feature values (probably to simplify the probabilistic reasoning) even
if normalized counts (a.k.a. term frequencies) or TF-IDF valued features
often perform slightly better in practice.
As for the :class:`Normalizer`, the utility class
:class:`Binarizer` is meant to be used in the early stages of
:class:`sklearn.pipeline.Pipeline`. The ``fit`` method does nothing
as each sample is treated independently of others::
>>> X = [[ 1., -1., 2.],
... [ 2., 0., 0.],
... [ 0., 1., -1.]]
>>> binarizer = preprocessing.Binarizer().fit(X) # fit does nothing
>>> binarizer
Binarizer(copy=True, threshold=0.0)
>>> binarizer.transform(X)
array([[ 1., 0., 1.],
[ 1., 0., 0.],
[ 0., 1., 0.]])
It is possible to adjust the threshold of the binarizer::
>>> binarizer = preprocessing.Binarizer(threshold=1.1)
>>> binarizer.transform(X)
array([[ 0., 0., 1.],
[ 1., 0., 0.],
[ 0., 0., 0.]])
As for the :class:`StandardScaler` and :class:`Normalizer` classes, the
preprocessing module provides a companion function :func:`binarize`
to be used when the transformer API is not necessary.
.. topic:: Sparse input
:func:`binarize` and :class:`Binarizer` accept **both dense array-like
and sparse matrices from scipy.sparse as input**.
For sparse input the data is **converted to the Compressed Sparse Rows
representation** (see ``scipy.sparse.csr_matrix``).
To avoid unnecessary memory copies, it is recommended to choose the CSR
representation upstream.
.. _preprocessing_categorical_features:
Encoding categorical features
=============================
Often features are not given as continuous values but categorical.
For example a person could have features ``["male", "female"]``,
``["from Europe", "from US", "from Asia"]``,
``["uses Firefox", "uses Chrome", "uses Safari", "uses Internet Explorer"]``.
Such features can be efficiently coded as integers, for instance
``["male", "from US", "uses Internet Explorer"]`` could be expressed as
``[0, 1, 3]`` while ``["female", "from Asia", "uses Chrome"]`` would be
``[1, 2, 1]``.
Such integer representation can not be used directly with scikit-learn estimators, as these
expect continuous input, and would interpret the categories as being ordered, which is often
not desired (i.e. the set of browsers was ordered arbitrarily).
One possibility to convert categorical features to features that can be used
with scikit-learn estimators is to use a one-of-K or one-hot encoding, which is
implemented in :class:`OneHotEncoder`. This estimator transforms each
categorical feature with ``m`` possible values into ``m`` binary features, with
only one active.
Continuing the example above::
>>> enc = preprocessing.OneHotEncoder()
>>> enc.fit([[0, 0, 3], [1, 1, 0], [0, 2, 1], [1, 0, 2]]) # doctest: +ELLIPSIS
OneHotEncoder(categorical_features='all', dtype=<... 'float'>,
handle_unknown='error', n_values='auto', sparse=True)
>>> enc.transform([[0, 1, 3]]).toarray()
array([[ 1., 0., 0., 1., 0., 0., 0., 0., 1.]])
By default, how many values each feature can take is inferred automatically from the dataset.
It is possible to specify this explicitly using the parameter ``n_values``.
There are two genders, three possible continents and four web browsers in our
dataset.
Then we fit the estimator, and transform a data point.
In the result, the first two numbers encode the gender, the next set of three
numbers the continent and the last four the web browser.
See :ref:`dict_feature_extraction` for categorical features that are represented
as a dict, not as integers.
.. _imputation:
Imputation of missing values
============================
For various reasons, many real world datasets contain missing values, often
encoded as blanks, NaNs or other placeholders. Such datasets however are
incompatible with scikit-learn estimators which assume that all values in an
array are numerical, and that all have and hold meaning. A basic strategy to use
incomplete datasets is to discard entire rows and/or columns containing missing
values. However, this comes at the price of losing data which may be valuable
(even though incomplete). A better strategy is to impute the missing values,
i.e., to infer them from the known part of the data.
The :class:`Imputer` class provides basic strategies for imputing missing
values, either using the mean, the median or the most frequent value of
the row or column in which the missing values are located. This class
also allows for different missing values encodings.
The following snippet demonstrates how to replace missing values,
encoded as ``np.nan``, using the mean value of the columns (axis 0)
that contain the missing values::
>>> import numpy as np
>>> from sklearn.preprocessing import Imputer
>>> imp = Imputer(missing_values='NaN', strategy='mean', axis=0)
>>> imp.fit([[1, 2], [np.nan, 3], [7, 6]])
Imputer(axis=0, copy=True, missing_values='NaN', strategy='mean', verbose=0)
>>> X = [[np.nan, 2], [6, np.nan], [7, 6]]
>>> print(imp.transform(X)) # doctest: +ELLIPSIS
[[ 4. 2. ]
[ 6. 3.666...]
[ 7. 6. ]]
The :class:`Imputer` class also supports sparse matrices::
>>> import scipy.sparse as sp
>>> X = sp.csc_matrix([[1, 2], [0, 3], [7, 6]])
>>> imp = Imputer(missing_values=0, strategy='mean', axis=0)
>>> imp.fit(X)
Imputer(axis=0, copy=True, missing_values=0, strategy='mean', verbose=0)
>>> X_test = sp.csc_matrix([[0, 2], [6, 0], [7, 6]])
>>> print(imp.transform(X_test)) # doctest: +ELLIPSIS
[[ 4. 2. ]
[ 6. 3.666...]
[ 7. 6. ]]
Note that, here, missing values are encoded by 0 and are thus implicitly stored
in the matrix. This format is thus suitable when there are many more missing
values than observed values.
:class:`Imputer` can be used in a Pipeline as a way to build a composite
estimator that supports imputation. See :ref:`example_missing_values.py`
.. _polynomial_features:
Generating polynomial features
==============================
Often it's useful to add complexity to the model by considering nonlinear features of the input data. A simple and common method to use is polynomial features, which can get features' high-order and interaction terms. It is implemented in :class:`PolynomialFeatures`::
>>> import numpy as np
>>> from sklearn.preprocessing import PolynomialFeatures
>>> X = np.arange(6).reshape(3, 2)
>>> X # doctest: +ELLIPSIS
array([[0, 1],
[2, 3],
[4, 5]])
>>> poly = PolynomialFeatures(2)
>>> poly.fit_transform(X) # doctest: +ELLIPSIS
array([[ 1, 0, 1, 0, 0, 1],
[ 1, 2, 3, 4, 6, 9],
[ 1, 4, 5, 16, 20, 25]])
The features of X have been transformed from :math:`(X_1, X_2)` to :math:`(1, X_1, X_2, X_1^2, X_1X_2, X_2^2)`.
In some cases, only interaction terms among features are required, and it can be gotten with the setting ``interaction_only=True``::
>>> X = np.arange(9).reshape(3, 3)
>>> X # doctest: +ELLIPSIS
array([[0, 1, 2],
[3, 4, 5],
[6, 7, 8]])
>>> poly = PolynomialFeatures(degree=3, interaction_only=True)
>>> poly.fit_transform(X) # doctest: +ELLIPSIS
array([[ 1, 0, 1, 2, 0, 0, 2, 0],
[ 1, 3, 4, 5, 12, 15, 20, 60],
[ 1, 6, 7, 8, 42, 48, 56, 336]])
The features of X have been transformed from :math:`(X_1, X_2, X_3)` to :math:`(1, X_1, X_2, X_3, X_1X_2, X_1X_3, X_2X_3, X_1X_2X_3)`.
Note that polynomial features are used implicitily in `kernel methods <http://en.wikipedia.org/wiki/Kernel_method>`_ (e.g., :class:`sklearn.svm.SVC`, :class:`sklearn.decomposition.KernelPCA`) when using polynomial :ref:`svm_kernels`.
See :ref:`example_linear_model_plot_polynomial_interpolation.py` for Ridge regression using created polynomial features.