Map data to a normal distribution

This example demonstrates the use of the Box-Cox and Yeo-Johnson transforms through PowerTransformer to map data from various distributions to a normal distribution.

The power transform is useful as a transformation in modeling problems where homoscedasticity and normality are desired. Below are examples of Box-Cox and Yeo-Johnwon applied to six different probability distributions: Lognormal, Chi-squared, Weibull, Gaussian, Uniform, and Bimodal.

Note that the transformations successfully map the data to a normal distribution when applied to certain datasets, but are ineffective with others. This highlights the importance of visualizing the data before and after transformation.

Also note that even though Box-Cox seems to perform better than Yeo-Johnson for lognormal and chi-squared distributions, keep in mind that Box-Cox does not support inputs with negative values.

For comparison, we also add the output from QuantileTransformer. It can force any arbitrary distribution into a gaussian, provided that there are enough training samples (thousands). Because it is a non-parametric method, it is harder to interpret than the parametric ones (Box-Cox and Yeo-Johnson).

On “small” datasets (less than a few hundred points), the quantile transformer is prone to overfitting. The use of the power transform is then recommended.

Lognormal, Chi-squared, Weibull, After Box-Cox $\lambda$ = -0.0, After Box-Cox $\lambda$ = 0.27, After Box-Cox $\lambda$ = 12.59, After Yeo-Johnson $\lambda$ = -0.83, After Yeo-Johnson $\lambda$ = -0.12, After Yeo-Johnson $\lambda$ = 24.53, After Quantile transform, After Quantile transform, After Quantile transform, Gaussian, Uniform, Bimodal, After Box-Cox $\lambda$ = 0.54, After Box-Cox $\lambda$ = 0.63, After Box-Cox $\lambda$ = 1.69, After Yeo-Johnson $\lambda$ = 0.54, After Yeo-Johnson $\lambda$ = 0.42, After Yeo-Johnson $\lambda$ = 1.7, After Quantile transform, After Quantile transform, After Quantile transform
# Author: Eric Chang <ericchang2017@u.northwestern.edu>
#         Nicolas Hug <contact@nicolas-hug.com>
# License: BSD 3 clause

import matplotlib.pyplot as plt
import numpy as np

from sklearn.model_selection import train_test_split
from sklearn.preprocessing import PowerTransformer, QuantileTransformer

N_SAMPLES = 1000
FONT_SIZE = 6
BINS = 30


rng = np.random.RandomState(304)
bc = PowerTransformer(method="box-cox")
yj = PowerTransformer(method="yeo-johnson")
# n_quantiles is set to the training set size rather than the default value
# to avoid a warning being raised by this example
qt = QuantileTransformer(
    n_quantiles=500, output_distribution="normal", random_state=rng
)
size = (N_SAMPLES, 1)


# lognormal distribution
X_lognormal = rng.lognormal(size=size)

# chi-squared distribution
df = 3
X_chisq = rng.chisquare(df=df, size=size)

# weibull distribution
a = 50
X_weibull = rng.weibull(a=a, size=size)

# gaussian distribution
loc = 100
X_gaussian = rng.normal(loc=loc, size=size)

# uniform distribution
X_uniform = rng.uniform(low=0, high=1, size=size)

# bimodal distribution
loc_a, loc_b = 100, 105
X_a, X_b = rng.normal(loc=loc_a, size=size), rng.normal(loc=loc_b, size=size)
X_bimodal = np.concatenate([X_a, X_b], axis=0)


# create plots
distributions = [
    ("Lognormal", X_lognormal),
    ("Chi-squared", X_chisq),
    ("Weibull", X_weibull),
    ("Gaussian", X_gaussian),
    ("Uniform", X_uniform),
    ("Bimodal", X_bimodal),
]

colors = ["#D81B60", "#0188FF", "#FFC107", "#B7A2FF", "#000000", "#2EC5AC"]

fig, axes = plt.subplots(nrows=8, ncols=3, figsize=plt.figaspect(2))
axes = axes.flatten()
axes_idxs = [
    (0, 3, 6, 9),
    (1, 4, 7, 10),
    (2, 5, 8, 11),
    (12, 15, 18, 21),
    (13, 16, 19, 22),
    (14, 17, 20, 23),
]
axes_list = [(axes[i], axes[j], axes[k], axes[l]) for (i, j, k, l) in axes_idxs]


for distribution, color, axes in zip(distributions, colors, axes_list):
    name, X = distribution
    X_train, X_test = train_test_split(X, test_size=0.5)

    # perform power transforms and quantile transform
    X_trans_bc = bc.fit(X_train).transform(X_test)
    lmbda_bc = round(bc.lambdas_[0], 2)
    X_trans_yj = yj.fit(X_train).transform(X_test)
    lmbda_yj = round(yj.lambdas_[0], 2)
    X_trans_qt = qt.fit(X_train).transform(X_test)

    ax_original, ax_bc, ax_yj, ax_qt = axes

    ax_original.hist(X_train, color=color, bins=BINS)
    ax_original.set_title(name, fontsize=FONT_SIZE)
    ax_original.tick_params(axis="both", which="major", labelsize=FONT_SIZE)

    for ax, X_trans, meth_name, lmbda in zip(
        (ax_bc, ax_yj, ax_qt),
        (X_trans_bc, X_trans_yj, X_trans_qt),
        ("Box-Cox", "Yeo-Johnson", "Quantile transform"),
        (lmbda_bc, lmbda_yj, None),
    ):
        ax.hist(X_trans, color=color, bins=BINS)
        title = "After {}".format(meth_name)
        if lmbda is not None:
            title += "\n$\\lambda$ = {}".format(lmbda)
        ax.set_title(title, fontsize=FONT_SIZE)
        ax.tick_params(axis="both", which="major", labelsize=FONT_SIZE)
        ax.set_xlim([-3.5, 3.5])


plt.tight_layout()
plt.show()

Total running time of the script: (0 minutes 1.975 seconds)

Related examples

Compare the effect of different scalers on data with outliers

Compare the effect of different scalers on data with outliers

Curve Fitting with Bayesian Ridge Regression

Curve Fitting with Bayesian Ridge Regression

Early stopping in Gradient Boosting

Early stopping in Gradient Boosting

Test with permutations the significance of a classification score

Test with permutations the significance of a classification score

Plot classification probability

Plot classification probability

Gallery generated by Sphinx-Gallery