Compare the effect of different scalers on data with outliers

Feature 0 (median income in a block) and feature 5 (average house occupancy) 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.

# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause

import matplotlib as mpl
import numpy as np
from matplotlib import cm
from matplotlib import pyplot as plt

from sklearn.datasets import fetch_california_housing
from sklearn.preprocessing import (
    MaxAbsScaler,
    MinMaxScaler,
    Normalizer,
    PowerTransformer,
    QuantileTransformer,
    RobustScaler,
    StandardScaler,
    minmax_scale,
)

dataset = fetch_california_housing()
X_full, y_full = dataset.data, dataset.target
feature_names = dataset.feature_names

feature_mapping = {
    "MedInc": "Median income in block",
    "HouseAge": "Median house age in block",
    "AveRooms": "Average number of rooms",
    "AveBedrms": "Average number of bedrooms",
    "Population": "Block population",
    "AveOccup": "Average house occupancy",
    "Latitude": "House block latitude",
    "Longitude": "House block longitude",
}

# Take only 2 features to make visualization easier
# Feature MedInc has a long tail distribution.
# Feature AveOccup has a few but very large outliers.
features = ["MedInc", "AveOccup"]
features_idx = [feature_names.index(feature) for feature in features]
X = X_full[:, features_idx]
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", random_state=42
        ).fit_transform(X),
    ),
    (
        "Data after quantile transformation (gaussian pdf)",
        QuantileTransformer(
            output_distribution="normal", random_state=42
        ).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=feature_mapping[features[0]],
        x1_label=feature_mapping[features[1]],
        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=feature_mapping[features[0]],
        x1_label=feature_mapping[features[1]],
        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 average house occupancy. Note that there are some marginal outliers (some blocks have average occupancy of more than 1200). 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)