Feature 0 (median income in a block) and feature 5 (number of households) ofthe California housing dataset have verydifferent scales and contain some very large outliers. These twocharacteristics lead to difficulties to visualize the data and, moreimportantly, they can degrade the predictive performance of many machinelearning algorithms. Unscaled data can also slow down or even prevent theconvergence of many gradient-based estimators.
Indeed many estimators are designed with the assumption that each feature takesvalues close to zero or more importantly that all features vary on comparablescales. In particular, metric-based and gradient-based estimators often assumeapproximately standardized data (centered features with unit variances). Anotable exception are decision tree-based estimators that are robust toarbitrary scaling of the data.
This example uses different scalers, transformers, and normalizers to bring thedata within a pre-defined range.
Scalers are linear (or more precisely affine) transformers and differ from eachother in the way to estimate the parameters used to shift and scale eachfeature.
QuantileTransformer provides non-linear transformations in which distancesbetween marginal outliers and inliers are shrunk. PowerTransformer providesnon-linear transformations in which data is mapped to a normal distribution tostabilize variance and minimize skewness.
Unlike the previous transformations, normalization refers to a per sampletransformation instead of a per feature transformation.
The following code is a bit verbose, feel free to jump directly to the analysisof the results.
# Author: Raghav RV <rvraghav93@gmail.com># Guillaume Lemaitre <g.lemaitre58@gmail.com># Thomas Unterthiner# License: BSD 3 clauseimport numpy as npimport matplotlib as mplfrom matplotlib import pyplot as pltfrom matplotlib import cmfrom sklearn.preprocessing import MinMaxScalerfrom sklearn.preprocessing import minmax_scalefrom sklearn.preprocessing import MaxAbsScalerfrom sklearn.preprocessing import StandardScalerfrom sklearn.preprocessing import RobustScalerfrom sklearn.preprocessing import Normalizerfrom sklearn.preprocessing import QuantileTransformerfrom sklearn.preprocessing import PowerTransformerfrom sklearn.datasets import fetch_california_housingprint(__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 (gaussian pdf)', QuantileTransformer(output_distribution='normal') .fit_transform(X)), ('Data after quantile transformation (uniform pdf)', QuantileTransformer(output_distribution='uniform') .fit_transform(X)), ('Data after sample-wise L2 normalizing', Normalizer().fit_transform(X)),]# scale the output between 0 and 1 for the colorbary = minmax_scale(y_full)# plasma does not exist in matplotlib < 1.5cmap = 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 leftfigure will show a scatter plot of the full data set while the right figurewill exclude the extreme values considering only 99 % of the data set,excluding marginal outliers. In addition, the marginal distributions for eachfeature will be shown on the side 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')
Each transformation is plotted showing two transformed features, with theleft plot showing the entire dataset, and the right zoomed-in to show thedataset without the marginal outliers. A large majority of the samples arecompacted to a specific range, [0, 10] for the median income and [0, 6] forthe number of households. Note that there are some marginal outliers (someblocks have more than 1200 households). Therefore, a specific pre-processingcan be very beneficial depending of the application. In the following, wepresent some insights and behaviors of those pre-processing methods in thepresence of marginal outliers.
make_plot(0)
StandardScaler removes the mean and scales the data to unit variance.However, the outliers have an influence when computing the empirical mean andstandard deviation which shrink the range of the feature values as shown inthe left figure below. Note in particular that because the outliers on eachfeature have different magnitudes, the spread of the transformed data oneach feature is very different: most of the data lie in the [-2, 4] range forthe transformed median income feature while the same data is squeezed in thesmaller [-0.2, 0.2] range for the transformed number of households.
StandardScaler therefore cannot guarantee balanced feature scales in thepresence of outliers.
make_plot(1)
MinMaxScaler rescales the data set such that all feature values are inthe range [0, 1] as shown in the right panel below. However, this scalingcompress all inliers in the narrow range [0, 0.005] for the transformednumber of households.
As StandardScaler, MinMaxScaler is very sensitive to the presence ofoutliers.
make_plot(2)
MaxAbsScaler differs from the previous scaler such that the absolutevalues are mapped in the range [0, 1]. On positive only data, this scalerbehaves similarly to MinMaxScaler and therefore also suffers from thepresence of large outliers.
make_plot(3)
Unlike the previous scalers, the centering and scaling statistics of thisscaler are based on percentiles and are therefore not influenced by a fewnumber of very large marginal outliers. Consequently, the resulting range ofthe transformed feature values is larger than for the previous scalers and,more importantly, are approximately similar: for both features most of thetransformed 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 isrequired (see below).
make_plot(4)
PowerTransformer applies a power transformation to each feature to makethe data more Gaussian-like. Currently, PowerTransformer implements theYeo-Johnson and Box-Cox transforms. The power transform finds the optimalscaling factor to stabilize variance and mimimize skewness through maximumlikelihood estimation. By default, PowerTransformer also applieszero-mean, unit variance normalization to the transformed output. Note thatBox-Cox can only be applied to strictly positive data. Income and number ofhouseholds happen to be strictly positive, but if negative values are presentthe Yeo-Johnson transformed is to be preferred.
make_plot(5)make_plot(6)
QuantileTransformer has an additional output_distribution parameterallowing to match a Gaussian distribution instead of a uniform distribution.Note that this non-parametetric transformer introduces saturation artifactsfor extreme values.
make_plot(7)
QuantileTransformer applies a non-linear transformation such that theprobability density function of each feature will be mapped to a uniformdistribution. In this case, all the data will be mapped in the range [0, 1],even the outliers which cannot be distinguished anymore from the inliers.
As RobustScaler, QuantileTransformer is robust to outliers in thesense that adding or removing outliers in the training set will yieldapproximately the same transformation on held out data. But contrary toRobustScaler, QuantileTransformer will also automatically collapseany outlier by setting them to the a priori defined range boundaries (0 and1).
make_plot(8)
The Normalizer rescales the vector for each sample to have unit norm,independently of the distribution of the samples. It can be seen on bothfigures below where all samples are mapped onto the unit circle. In ourexample the two selected features have only positive values; therefore thetransformed data only lie in the positive quadrant. This would not be thecase 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 4.218 seconds)
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