回归模型中转换目标的影响

在本示例中，我们将概述 TransformedTargetRegressor。我们将使用两个示例来说明在学习线性回归模型之前转换目标的好处。第一个示例使用合成数据，而第二个示例基于 Ames 房屋数据集。

# Author: Guillaume Lemaitre <[email protected]>
# License: BSD 3 clause

print(__doc__)

合成示例

生成一个合成的随机回归数据集。目标 y 通过以下方式修改：

1. 将所有目标平移，使所有条目都为非负数（通过添加最低 y 的绝对值），以及

2. 应用指数函数以获得无法使用简单线性模型拟合的非线性目标。

因此，将使用对数函数 (np.log1p) 和指数函数 (np.expm1) 在训练线性回归模型之前转换目标，并将其用于预测。

import numpy as np

from sklearn.datasets import make_regression

X, y = make_regression(n_samples=10_000, noise=100, random_state=0)
y = np.expm1((y + abs(y.min())) / 200)
y_trans = np.log1p(y)

下面我们将绘制应用对数函数前后目标的概率密度函数。

import matplotlib.pyplot as plt

from sklearn.model_selection import train_test_split

f, (ax0, ax1) = plt.subplots(1, 2)

ax0.hist(y, bins=100, density=True)
ax0.set_xlim([0, 2000])
ax0.set_ylabel("Probability")
ax0.set_xlabel("Target")
ax0.set_title("Target distribution")

ax1.hist(y_trans, bins=100, density=True)
ax1.set_ylabel("Probability")
ax1.set_xlabel("Target")
ax1.set_title("Transformed target distribution")

f.suptitle("Synthetic data", y=1.05)
plt.tight_layout()

X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)

首先，将线性模型应用于原始目标。由于非线性，训练的模型在预测期间将不精确。随后，使用对数函数来线性化目标，即使使用类似的线性模型，也能获得更好的预测结果，如中位数绝对误差 (MedAE) 所示。

from sklearn.metrics import median_absolute_error, r2_score


def compute_score(y_true, y_pred):
    return {
        "R2": f"{r2_score(y_true, y_pred):.3f}",
        "MedAE": f"{median_absolute_error(y_true, y_pred):.3f}",
    }

from sklearn.compose import TransformedTargetRegressor
from sklearn.linear_model import RidgeCV
from sklearn.metrics import PredictionErrorDisplay

f, (ax0, ax1) = plt.subplots(1, 2, sharey=True)

ridge_cv = RidgeCV().fit(X_train, y_train)
y_pred_ridge = ridge_cv.predict(X_test)

ridge_cv_with_trans_target = TransformedTargetRegressor(
    regressor=RidgeCV(), func=np.log1p, inverse_func=np.expm1
).fit(X_train, y_train)
y_pred_ridge_with_trans_target = ridge_cv_with_trans_target.predict(X_test)

PredictionErrorDisplay.from_predictions(
    y_test,
    y_pred_ridge,
    kind="actual_vs_predicted",
    ax=ax0,
    scatter_kwargs={"alpha": 0.5},
)
PredictionErrorDisplay.from_predictions(
    y_test,
    y_pred_ridge_with_trans_target,
    kind="actual_vs_predicted",
    ax=ax1,
    scatter_kwargs={"alpha": 0.5},
)

# Add the score in the legend of each axis
for ax, y_pred in zip([ax0, ax1], [y_pred_ridge, y_pred_ridge_with_trans_target]):
    for name, score in compute_score(y_test, y_pred).items():
        ax.plot([], [], " ", label=f"{name}={score}")
    ax.legend(loc="upper left")

ax0.set_title("Ridge regression \n without target transformation")
ax1.set_title("Ridge regression \n with target transformation")
f.suptitle("Synthetic data", y=1.05)
plt.tight_layout()

真实世界数据集

以类似的方式，Ames 房屋数据集用于展示在学习模型之前转换目标的影响。在本示例中，要预测的目标是每栋房子的售价。

from sklearn.datasets import fetch_openml
from sklearn.preprocessing import quantile_transform

ames = fetch_openml(name="house_prices", as_frame=True)
# Keep only numeric columns
X = ames.data.select_dtypes(np.number)
# Remove columns with NaN or Inf values
X = X.drop(columns=["LotFrontage", "GarageYrBlt", "MasVnrArea"])
# Let the price be in k$
y = ames.target / 1000
y_trans = quantile_transform(
    y.to_frame(), n_quantiles=900, output_distribution="normal", copy=True
).squeeze()

一个 QuantileTransformer 用于在应用 RidgeCV 模型之前规范化目标分布。

f, (ax0, ax1) = plt.subplots(1, 2)

ax0.hist(y, bins=100, density=True)
ax0.set_ylabel("Probability")
ax0.set_xlabel("Target")
ax0.set_title("Target distribution")

ax1.hist(y_trans, bins=100, density=True)
ax1.set_ylabel("Probability")
ax1.set_xlabel("Target")
ax1.set_title("Transformed target distribution")

f.suptitle("Ames housing data: selling price", y=1.05)
plt.tight_layout()

X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=1)

转换器的影响比合成数据弱。但是，转换导致 \(R^2\) 增加，MedAE 大幅下降。没有目标转换的残差图（预测目标 - 真实目标与预测目标）呈现出弯曲的“反微笑”形状，因为残差值会根据预测目标的值而变化。使用目标转换后，形状更加线性，表明模型拟合更好。

from sklearn.preprocessing import QuantileTransformer

f, (ax0, ax1) = plt.subplots(2, 2, sharey="row", figsize=(6.5, 8))

ridge_cv = RidgeCV().fit(X_train, y_train)
y_pred_ridge = ridge_cv.predict(X_test)

ridge_cv_with_trans_target = TransformedTargetRegressor(
    regressor=RidgeCV(),
    transformer=QuantileTransformer(n_quantiles=900, output_distribution="normal"),
).fit(X_train, y_train)
y_pred_ridge_with_trans_target = ridge_cv_with_trans_target.predict(X_test)

# plot the actual vs predicted values
PredictionErrorDisplay.from_predictions(
    y_test,
    y_pred_ridge,
    kind="actual_vs_predicted",
    ax=ax0[0],
    scatter_kwargs={"alpha": 0.5},
)
PredictionErrorDisplay.from_predictions(
    y_test,
    y_pred_ridge_with_trans_target,
    kind="actual_vs_predicted",
    ax=ax0[1],
    scatter_kwargs={"alpha": 0.5},
)

# Add the score in the legend of each axis
for ax, y_pred in zip([ax0[0], ax0[1]], [y_pred_ridge, y_pred_ridge_with_trans_target]):
    for name, score in compute_score(y_test, y_pred).items():
        ax.plot([], [], " ", label=f"{name}={score}")
    ax.legend(loc="upper left")

ax0[0].set_title("Ridge regression \n without target transformation")
ax0[1].set_title("Ridge regression \n with target transformation")

# plot the residuals vs the predicted values
PredictionErrorDisplay.from_predictions(
    y_test,
    y_pred_ridge,
    kind="residual_vs_predicted",
    ax=ax1[0],
    scatter_kwargs={"alpha": 0.5},
)
PredictionErrorDisplay.from_predictions(
    y_test,
    y_pred_ridge_with_trans_target,
    kind="residual_vs_predicted",
    ax=ax1[1],
    scatter_kwargs={"alpha": 0.5},
)
ax1[0].set_title("Ridge regression \n without target transformation")
ax1[1].set_title("Ridge regression \n with target transformation")

f.suptitle("Ames housing data: selling price", y=1.05)
plt.tight_layout()
plt.show()

相关示例

等距回归

等距回归

管道：将 PCA 和逻辑回归链接起来

管道：将 PCA 和逻辑回归链接起来

使用堆叠组合预测器

使用堆叠组合预测器

绘制交叉验证预测

绘制交叉验证预测