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将数据映射到正态分布#

本示例演示了通过 PowerTransformer 使用 Box-Cox 和 Yeo-Johnson 变换将各种分布的数据映射到正态分布。

在建模问题中，当需要同方差性和正态性时，幂变换是有用的。以下是 Box-Cox 和 Yeo-Johnson 应用于六种不同概率分布的示例：对数正态分布、卡方分布、韦伯分布、高斯分布、均匀分布和双峰分布。

请注意，当应用于某些数据集时，这些变换成功地将数据映射到正态分布，但对其他数据集则无效。这突显了在变换前后可视化数据的重要性。

还要注意，尽管 Box-Cox 似乎在对数正态分布和卡方分布上表现优于 Yeo-Johnson，但请记住 Box-Cox 不支持负值输入。

为了比较，我们还添加了 QuantileTransformer 的输出。只要有足够的训练样本（成千上万），它可以将任意分布强制转换为高斯分布。由于它是一种非参数方法，因此比参数方法（Box-Cox 和 Yeo-Johnson）更难解释。

在“小”数据集（少于几百个点）上，分位数变换器容易过拟合。因此，建议使用幂变换。

$Lognormal, Chi-squared, Weibull, After Box-Cox $\lambda$ = 0.04, After Box-Cox $\lambda$ = 0.27, After Box-Cox $\lambda$ = 12.83, After Yeo-Johnson $\lambda$ = -0.79, After Yeo-Johnson $\lambda$ = -0.13, After Yeo-Johnson $\lambda$ = 25.0, After Quantile transform, After Quantile transform, After Quantile transform, Gaussian, Uniform, Bimodal, After Box-Cox $\lambda$ = 4.92, After Box-Cox $\lambda$ = 0.63, After Box-Cox $\lambda$ = -1.66, After Yeo-Johnson $\lambda$ = 4.96, After Yeo-Johnson $\lambda$ = 0.37, After Yeo-Johnson $\lambda$ = -1.69, After Quantile transform, After Quantile transform, After Quantile transform$

# 作者：scikit-learn 开发者
# SPDX-License-Identifier: 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 被设置为训练集大小而不是默认值，以避免此示例引发警告
qt = QuantileTransformer(
    n_quantiles=500, output_distribution="normal", random_state=rng
)
size = (N_SAMPLES, 1)


# 对数正态分布
X_lognormal = rng.lognormal(size=size)

# 卡方分布
df = 3
X_chisq = rng.chisquare(df=df, size=size)

# 威布尔分布
a = 50
X_weibull = rng.weibull(a=a, size=size)

# 高斯分布
loc = 100
X_gaussian = rng.normal(loc=loc, size=size)

# 均匀分布
X_uniform = rng.uniform(low=0, high=1, size=size)

# 双峰分布
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)


# 创建图表
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)

    # 执行幂变换和分位数变换
    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()