Bootstrap distribution — PyStatlas: A Python Gallery of Statistical Graphics for the Social Sciences

Warin, Thierry

自助法分布

重抽样得到的相关系数，附百分位数法和 BCa 95% 置信区间。

自助法（Bootstrap）在从数据中有放回抽取的数千个样本上重新估计统计量；这些估计值的离散程度即为其抽样不确定性。直方图是相关系数的自助法分布，橙色线是在完整样本中观测到的值。上方绘有两个 95% 区间：简单的百分位数区间，以及 BCa（偏差校正加速）区间——后者会发生偏移，以校正此处可见的偏差和偏度，因为该分布远非对称。

数据为 statsmodels statecrime 数据集中的美国 50 个州 + 哥伦比亚特区：单亲家庭占比与暴力犯罪率。scipy.stats.bootstrap（配对，9,999 次重抽样）给出两种区间。

代码及图中文字均为英文。

bootstrap-distribution.py · 48 行

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import matplotlib.pyplot as plt
import numpy as np
import statsmodels.api as sm
from scipy import stats

# Data: violent crime vs single-parent households, 51 states
df = sm.datasets.statecrime.load_pandas().data
x, y = df["single"].to_numpy(), df["violent"].to_numpy()
r_obs = stats.pearsonr(x, y)[0]

# Paired bootstrap of Pearson's r: percentile and BCa intervals from the same resamples
r = lambda a, b, axis=-1: stats.pearsonr(a, b, axis=axis)[0]
boot = {m: stats.bootstrap((x, y), r, paired=True, vectorized=True, n_resamples=9999,
                           method=m, random_state=np.random.default_rng(7))
        for m in ["percentile", "BCa"]}
dist = boot["BCa"].bootstrap_distribution
fisher = np.tanh(np.arctanh(r_obs) + np.array([-1, 1]) * 1.96 / np.sqrt(len(x) - 3))

fig, ax = plt.subplots(figsize=(8, 5))
lo, hi = boot["BCa"].confidence_interval
counts, edges, bars = ax.hist(dist, bins=60, color="#C7C7CC", edgecolor="white", lw=0.4)
for b, left in zip(bars, edges[:-1]):
    if lo <= left + (edges[1] - edges[0]) / 2 <= hi:
        b.set_facecolor("#007AFF"); b.set_alpha(0.75)

# Intervals stacked above the histogram
top = counts.max()
for i, (m, c, name) in enumerate([("percentile", "#8E8E93", "Percentile"),
                                  ("BCa", "#007AFF", "BCa")]):
    a, b = boot[m].confidence_interval
    yy = top * (1.12 + 0.12 * i)
    ax.plot([a, b], [yy, yy], color=c, lw=3, solid_capstyle="butt")
    ax.vlines([a, b], yy - top * 0.03, yy + top * 0.03, color=c, lw=2)
    ax.text(a - 0.01, yy, f"{name} 95% CI  [{a:.2f}, {b:.2f}]", ha="right", va="center",
            fontsize=9.5, color=c, fontweight="semibold")

ax.vlines(r_obs, 0, top * 1.34, color="#FF9500", lw=2, zorder=3)
ax.text(r_obs, top * 1.36, f"observed r = {r_obs:.2f}", ha="center", va="bottom",
        fontsize=9.5, color="#FF9500", fontweight="semibold")
ax.set_xlim(0.15, 1)
ax.set_ylim(0, top * 1.46)
ax.set_xlabel("Bootstrap estimate of Pearson's r")
ax.set_ylabel("Resamples")
ax.set_title("Crime and single parenthood: a strong but skewed correlation", pad=26)
ax.text(0, 1.02, f"Paired bootstrap, B = 9,999 · n = {len(x)} states · bias = "
        f"{dist.mean() - r_obs:+.3f} · Fisher-z 95% CI [{fisher[0]:.2f}, {fisher[1]:.2f}] for comparison",
        transform=ax.transAxes, 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

US state crime and social indicators, 2009. U.S. Census Bureau. Statistical Abstract of the United States, state tables for 2009 (crime rates from the FBI Uniform Crime Reporting program). https://www.statsmodels.org/stable/datasets/generated/statecrime.html
Compiled for statsmodels (sm.datasets.statecrime); public domain. See the statsmodels page for each variable's table and footnotes.

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). Bootstrap distribution. In PyStatlas: A Python Gallery of Statistical Graphics for the Social Sciences. https://warint.github.io/pystatlas/inference-uncertainty/bootstrap-distribution/

@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}
}

自助法分布

数据来源

如何引用

更多推断与不确定性图表

95% 置信区间的含义

统计功效曲线

Bland–Altman 图