Standardization, or mean removal and variance scaling

　　4.2.1. 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 feature 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 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

　　array([[ 0. ..., -1.22..., 1.33...],

　　[ 1.22..., 0. ..., -0.26...],

　　[-1.22..., 1.22..., -1.06...]])

　　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.])

　　The preprocessing module further provides a utility 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 sklearn.pipeline.Pipeline:

　　>>> scaler = preprocessing.StandardScaler().fit(X)

　　>>> scaler

　　StandardScaler(copy=True, with_mean=True, with_std=True)

　　>>> scaler.mean_

　　array([ 1. ..., 0. ..., 0.33...])

　　>>> scaler.std_

　　array([ 0.81..., 0.81..., 1.24...])

　　>>> scaler.transform(X)

　　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.]])

　　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 StandardScaler.

　　4.2.1.1. 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. This can be achieved using MinMaxScaler.

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　　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: