Note

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# 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.

```
# Author: Eric Chang <ericchang2017@u.northwestern.edu>
# Nicolas Hug <contact@nicolas-hug.com>
# License: BSD 3 clause
import numpy as np
import matplotlib.pyplot as plt
from sklearn.preprocessing import PowerTransformer
from sklearn.preprocessing import QuantileTransformer
from sklearn.model_selection import train_test_split
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 1.684 seconds)