Note
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Compare the effect of different scalers on data with outliers¶
Feature 0 (median income in a block) and feature 5 (number of households) of the California Housing dataset have very different scales and contain some very large outliers. These two characteristics lead to difficulties to visualize the data and, more importantly, they can degrade the predictive performance of many machine learning algorithms. Unscaled data can also slow down or even prevent the convergence of many gradient-based estimators.
Indeed many estimators are designed with the assumption that each feature takes values close to zero or more importantly that all features vary on comparable scales. In particular, metric-based and gradient-based estimators often assume approximately standardized data (centered features with unit variances). A notable exception are decision tree-based estimators that are robust to arbitrary scaling of the data.
This example uses different scalers, transformers, and normalizers to bring the data within a pre-defined range.
Scalers are linear (or more precisely affine) transformers and differ from each other in the way they estimate the parameters used to shift and scale each feature.
QuantileTransformer provides non-linear
transformations in which distances
between marginal outliers and inliers are shrunk.
PowerTransformer provides
non-linear transformations in which data is mapped to a normal distribution to
stabilize variance and minimize skewness.
Unlike the previous transformations, normalization refers to a per sample transformation instead of a per feature transformation.
The following code is a bit verbose, feel free to jump directly to the analysis of the results.
# Author: Raghav RV <rvraghav93@gmail.com>
# Guillaume Lemaitre <g.lemaitre58@gmail.com>
# Thomas Unterthiner
# License: BSD 3 clause
import numpy as np
import matplotlib as mpl
from matplotlib import pyplot as plt
from matplotlib import cm
from sklearn.preprocessing import MinMaxScaler
from sklearn.preprocessing import minmax_scale
from sklearn.preprocessing import MaxAbsScaler
from sklearn.preprocessing import StandardScaler
from sklearn.preprocessing import RobustScaler
from sklearn.preprocessing import Normalizer
from sklearn.preprocessing import QuantileTransformer
from sklearn.preprocessing import PowerTransformer
from sklearn.datasets import fetch_california_housing
print(__doc__)
dataset = fetch_california_housing()
X_full, y_full = dataset.data, dataset.target
# Take only 2 features to make visualization easier
# Feature of 0 has a long tail distribution.
# Feature 5 has a few but very large outliers.
X = X_full[:, [0, 5]]
distributions = [
('Unscaled data', X),
('Data after standard scaling',
StandardScaler().fit_transform(X)),
('Data after min-max scaling',
MinMaxScaler().fit_transform(X)),
('Data after max-abs scaling',
MaxAbsScaler().fit_transform(X)),
('Data after robust scaling',
RobustScaler(quantile_range=(25, 75)).fit_transform(X)),
('Data after power transformation (Yeo-Johnson)',
PowerTransformer(method='yeo-johnson').fit_transform(X)),
('Data after power transformation (Box-Cox)',
PowerTransformer(method='box-cox').fit_transform(X)),
('Data after quantile transformation (uniform pdf)',
QuantileTransformer(output_distribution='uniform')
.fit_transform(X)),
('Data after quantile transformation (gaussian pdf)',
QuantileTransformer(output_distribution='normal')
.fit_transform(X)),
('Data after sample-wise L2 normalizing',
Normalizer().fit_transform(X)),
]
# scale the output between 0 and 1 for the colorbar
y = minmax_scale(y_full)
# plasma does not exist in matplotlib < 1.5
cmap = getattr(cm, 'plasma_r', cm.hot_r)
def create_axes(title, figsize=(16, 6)):
fig = plt.figure(figsize=figsize)
fig.suptitle(title)
# define the axis for the first plot
left, width = 0.1, 0.22
bottom, height = 0.1, 0.7
bottom_h = height + 0.15
left_h = left + width + 0.02
rect_scatter = [left, bottom, width, height]
rect_histx = [left, bottom_h, width, 0.1]
rect_histy = [left_h, bottom, 0.05, height]
ax_scatter = plt.axes(rect_scatter)
ax_histx = plt.axes(rect_histx)
ax_histy = plt.axes(rect_histy)
# define the axis for the zoomed-in plot
left = width + left + 0.2
left_h = left + width + 0.02
rect_scatter = [left, bottom, width, height]
rect_histx = [left, bottom_h, width, 0.1]
rect_histy = [left_h, bottom, 0.05, height]
ax_scatter_zoom = plt.axes(rect_scatter)
ax_histx_zoom = plt.axes(rect_histx)
ax_histy_zoom = plt.axes(rect_histy)
# define the axis for the colorbar
left, width = width + left + 0.13, 0.01
rect_colorbar = [left, bottom, width, height]
ax_colorbar = plt.axes(rect_colorbar)
return ((ax_scatter, ax_histy, ax_histx),
(ax_scatter_zoom, ax_histy_zoom, ax_histx_zoom),
ax_colorbar)
def plot_distribution(axes, X, y, hist_nbins=50, title="",
x0_label="", x1_label=""):
ax, hist_X1, hist_X0 = axes
ax.set_title(title)
ax.set_xlabel(x0_label)
ax.set_ylabel(x1_label)
# The scatter plot
colors = cmap(y)
ax.scatter(X[:, 0], X[:, 1], alpha=0.5, marker='o', s=5, lw=0, c=colors)
# Removing the top and the right spine for aesthetics
# make nice axis layout
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
ax.get_xaxis().tick_bottom()
ax.get_yaxis().tick_left()
ax.spines['left'].set_position(('outward', 10))
ax.spines['bottom'].set_position(('outward', 10))
# Histogram for axis X1 (feature 5)
hist_X1.set_ylim(ax.get_ylim())
hist_X1.hist(X[:, 1], bins=hist_nbins, orientation='horizontal',
color='grey', ec='grey')
hist_X1.axis('off')
# Histogram for axis X0 (feature 0)
hist_X0.set_xlim(ax.get_xlim())
hist_X0.hist(X[:, 0], bins=hist_nbins, orientation='vertical',
color='grey', ec='grey')
hist_X0.axis('off')
Two plots will be shown for each scaler/normalizer/transformer. The left figure will show a scatter plot of the full data set while the right figure will exclude the extreme values considering only 99 % of the data set, excluding marginal outliers. In addition, the marginal distributions for each feature will be shown on the sides of the scatter plot.
def make_plot(item_idx):
title, X = distributions[item_idx]
ax_zoom_out, ax_zoom_in, ax_colorbar = create_axes(title)
axarr = (ax_zoom_out, ax_zoom_in)
plot_distribution(axarr[0], X, y, hist_nbins=200,
x0_label="Median Income",
x1_label="Number of households",
title="Full data")
# zoom-in
zoom_in_percentile_range = (0, 99)
cutoffs_X0 = np.percentile(X[:, 0], zoom_in_percentile_range)
cutoffs_X1 = np.percentile(X[:, 1], zoom_in_percentile_range)
non_outliers_mask = (
np.all(X > [cutoffs_X0[0], cutoffs_X1[0]], axis=1) &
np.all(X < [cutoffs_X0[1], cutoffs_X1[1]], axis=1))
plot_distribution(axarr[1], X[non_outliers_mask], y[non_outliers_mask],
hist_nbins=50,
x0_label="Median Income",
x1_label="Number of households",
title="Zoom-in")
norm = mpl.colors.Normalize(y_full.min(), y_full.max())
mpl.colorbar.ColorbarBase(ax_colorbar, cmap=cmap,
norm=norm, orientation='vertical',
label='Color mapping for values of y')
Original data¶
Each transformation is plotted showing two transformed features, with the left plot showing the entire dataset, and the right zoomed-in to show the dataset without the marginal outliers. A large majority of the samples are compacted to a specific range, [0, 10] for the median income and [0, 6] for the number of households. Note that there are some marginal outliers (some blocks have more than 1200 households). Therefore, a specific pre-processing can be very beneficial depending of the application. In the following, we present some insights and behaviors of those pre-processing methods in the presence of marginal outliers.
make_plot(0)
StandardScaler¶
StandardScaler removes the mean and scales
the data to unit variance. The scaling shrinks the range of the feature
values as shown in the left figure below.
However, the outliers have an influence when computing the empirical mean and
standard deviation. Note in particular that because the outliers on each
feature have different magnitudes, the spread of the transformed data on
each feature is very different: most of the data lie in the [-2, 4] range for
the transformed median income feature while the same data is squeezed in the
smaller [-0.2, 0.2] range for the transformed number of households.
StandardScaler therefore cannot guarantee
balanced feature scales in the
presence of outliers.
make_plot(1)
MinMaxScaler¶
MinMaxScaler rescales the data set such that
all feature values are in
the range [0, 1] as shown in the right panel below. However, this scaling
compresses all inliers into the narrow range [0, 0.005] for the transformed
number of households.
Both StandardScaler and
MinMaxScaler are very sensitive to the
presence of outliers.
make_plot(2)
MaxAbsScaler¶
MaxAbsScaler is similar to
MinMaxScaler except that the
values are mapped in the range [0, 1]. On positive only data, both scalers
behave similarly.
MaxAbsScaler therefore also suffers from
the presence of large outliers.
make_plot(3)
RobustScaler¶
Unlike the previous scalers, the centering and scaling statistics of
RobustScaler
is based on percentiles and are therefore not influenced by a few
number of very large marginal outliers. Consequently, the resulting range of
the transformed feature values is larger than for the previous scalers and,
more importantly, are approximately similar: for both features most of the
transformed values lie in a [-2, 3] range as seen in the zoomed-in figure.
Note that the outliers themselves are still present in the transformed data.
If a separate outlier clipping is desirable, a non-linear transformation is
required (see below).
make_plot(4)
PowerTransformer¶
PowerTransformer applies a power
transformation to each feature to make the data more Gaussian-like in order
to stabilize variance and minimize skewness. Currently the Yeo-Johnson
and Box-Cox transforms are supported and the optimal
scaling factor is determined via maximum likelihood estimation in both
methods. By default, PowerTransformer applies
zero-mean, unit variance normalization. Note that
Box-Cox can only be applied to strictly positive data. Income and number of
households happen to be strictly positive, but if negative values are present
the Yeo-Johnson transformed is preferred.
make_plot(5)
make_plot(6)
QuantileTransformer (uniform output)¶
QuantileTransformer applies a non-linear
transformation such that the
probability density function of each feature will be mapped to a uniform
or Gaussian distribution. In this case, all the data, including outliers,
will be mapped to a uniform distribution with the range [0, 1], making
outliers indistinguishable from inliers.
RobustScaler and
QuantileTransformer are robust to outliers in
the sense that adding or removing outliers in the training set will yield
approximately the same transformation. But contrary to
RobustScaler,
QuantileTransformer will also automatically
collapse any outlier by setting them to the a priori defined range boundaries
(0 and 1). This can result in saturation artifacts for extreme values.
make_plot(7)
QuantileTransformer (Gaussian output)¶
To map to a Gaussian distribution, set the parameter
output_distribution='normal'.
make_plot(8)
Normalizer¶
The Normalizer rescales the vector for each
sample to have unit norm,
independently of the distribution of the samples. It can be seen on both
figures below where all samples are mapped onto the unit circle. In our
example the two selected features have only positive values; therefore the
transformed data only lie in the positive quadrant. This would not be the
case if some original features had a mix of positive and negative values.
make_plot(9)
plt.show()
Total running time of the script: ( 0 minutes 11.973 seconds)
