带 ICE 曲线的部分依赖图
预测的数学成绩如何随贫困程度和生均支出变化——平均值（PDP）叠加在每个学区一条的个体曲线（ICE）之上。
部分依赖图（PDP）显示：在其他特征保持观测值不变的情况下，当某一个特征变化时，模型的预测平均如何变化。平均值背后是 个体条件期望（ICE）曲线——每个观测一条。ICE 曲线相互平行，说明效应对每个人都相同；曲线发散或交叉，则揭示了被平均值掩盖的 交互作用。ICE 曲线在左端对齐（中心化），以便比较形状而非水平；底部的刻度标出数据实际所在之处，在其他地方模型是在外推。
数据为 statsmodels 自带的 1998 年 STAR 考试中加利福尼亚州 303 个学区（sm.datasets.star98）；结果变量为数学成绩高于全国中位数的学生占比，由学区的十项特征预测。两个面板形成对比：随低收入学生占比上升，预测急剧下降；随生均支出变化，预测几乎不动。模型为梯度提升回归，曲线来自 sklearn.inspection.partial_dependence(kind="both")。PDP 描述的是模型，而不是世界：只有模型本身具有因果性时，它才具有因果含义。
代码及图中文字均为英文。
import matplotlib.pyplot as plt
import numpy as np
import statsmodels.api as sm
from sklearn.ensemble import GradientBoostingRegressor
from sklearn.inspection import partial_dependence
# Data: 303 California school districts; outcome = % of students above the national median in maths
d = sm.datasets.star98.load_pandas().data
y = 100 * d["NABOVE"] / (d["NABOVE"] + d["NBELOW"])
names = {"LOWINC": "% low-income students", "PERASIAN": "% Asian students", "PERBLACK": "% Black students",
"PERHISP": "% Hispanic students", "AVYRSEXP": "Teacher experience (years)",
"AVSALK": "Teacher salary ($000)", "PERSPENK": "Spending per pupil ($000)",
"PTRATIO": "Pupils per teacher", "PCTAF": "% in college-prep courses",
"PCTYRRND": "% year-round schools"}
X = d[list(names)]
model = GradientBoostingRegressor(learning_rate=0.05, n_estimators=300, max_depth=2,
subsample=0.8, random_state=0).fit(X, y)
features = {"LOWINC": "Low-income students (% of pupils)", "PERSPENK": "Spending per pupil ($ thousands)"}
change = {}
plt.rcParams["figure.constrained_layout.use"] = False # manual margins leave room for the subtitle
fig, axes = plt.subplots(1, 2, figsize=(10, 5), sharey=True)
fig.subplots_adjust(left=0.10, right=0.97, top=0.80, bottom=0.12, wspace=0.08)
for ax, (f, label) in zip(axes, features.items()):
pd_res = partial_dependence(model, X, [f], kind="both", grid_resolution=40)
grid = pd_res["grid_values"][0]
ice = pd_res["individual"][0]
ice_c = ice - ice[:, [0]] # centred ICE
pdp_c = pd_res["average"][0] - pd_res["average"][0][0]
change[f] = np.ptp(pdp_c) # range of the average effect
ax.plot(grid, ice_c.T, color="#007AFF", alpha=0.07, lw=0.8)
ax.plot(grid, pdp_c, color="#FF9500", lw=3, label="Average (PDP)")
lo, hi = np.percentile(ice_c, [10, 90], axis=0)
ax.plot(grid, lo, color="#007AFF", lw=1, ls="--")
ax.plot(grid, hi, color="#007AFF", lw=1, ls="--", label="10th / 90th percentile of ICE")
ax.plot(X[f], np.full(len(X), -46), "|", color="#3A3A3C", ms=8, alpha=0.4) # rug
ax.set_xlabel(label)
ax.set_title(f"Effect of {label.split(' (')[0].lower()}")
axes[0].set_ylim(-50, 12)
axes[0].set_ylabel("Change in predicted % above median\n(points, relative to the lowest value)")
axes[1].legend(loc="center", fontsize=9)
fig.suptitle(f"Poverty lowers predicted maths results by {change['LOWINC']:.0f} points; "
f"spending per pupil moves them by {change['PERSPENK']:.0f}", x=0.02, y=0.97, ha="left")
fig.text(0.02, 0.885, f"Gradient boosting · n = {len(X)} school districts · thin lines = one district (centred ICE) · "
"ticks = observed values", fontsize=9.5, color="#6E6E73")
plt.show()
图表使用了可选的 PyStatlas 样式：添加 plt.style.use("pystatlas.mplstyle") 即可完全还原其外观。
数据来源
- California STAR test results, 1998 (303 school districts). Gill, J. (2001). Generalized Linear Models: A Unified Approach (Quantitative Applications in the Social Sciences, No. 134). Sage. https://www.statsmodels.org/stable/datasets/generated/star98.html
Original data from the California Department of Education's Standardized Testing and Reporting (STAR) program, 1998. Subset distributed with statsmodels (sm.datasets.star98) with the author's permission.
这些示例用于说明统计方法。图表标题描述的是该图在这些特定数据中呈现的内容——这些数据往往样本较小、年代较早或并不完整——而不是作者对实质问题的立场。关于解释、背景和数据质量，请以此处引用的原始来源为准。
如何引用
PyStatlas 由 Thierry Warin 创建并维护。如果它对你的工作有所帮助，敬请引用。
Warin, T. (2026). Partial dependence with ICE curves. In PyStatlas: A Python Gallery of Statistical Graphics for the Social Sciences. https://warint.github.io/pystatlas/machine-learning/partial-dependence/
@misc{warin2026statlas,
author = {Warin, Thierry},
title = {PyStatlas: A Python Gallery of Statistical Graphics for the Social Sciences},
year = {2026},
howpublished = {\url{https://warint.github.io/pystatlas/}},
note = {Source code: https://github.com/warint/pystatlas}
}
