方差缩减——分层抽样

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原创最新推荐文章于 2026-08-22 16:58:19 发布 2.5k 阅读 AI 写同款· GEO 优化 ›

第二话统计计算之蒙特卡洛积分和方差缩减技术（未完待续）统计计算例如：第一章 Python 机器学习入门之pandas的使用提示：写完文章后，目录可以自动生成，如何生成可参考右边的帮助文档蒙特卡洛积分和方差缩减技术统计计算前言1 蒙特卡洛积分2 方差缩减1.引入库2.读入数据3 对偶变量法4 控制变量法5 重要性抽样（Importance Sampling,IS）6 分层抽样（Stratified Sampling,SS）7 分层重要性抽样前言提示：这里可以添加本文要记录的大概内容：例如：随着人工智能的不断发展，机器学习这门技术也越来越重要，阅读详情

方差缩减——分层抽样

分成4个子区间

import numpy as np

n = 500
estimates = np.empty([100, 2])

def g(x):
    if x >= 0 and x <= 1:
        y = np.exp(-x) / (1 + x ** 2)
    else:
        y = 0
    return y

def get_mean(X):
    results = []
    for x in X:
        results.append(g(x))

    return np.mean(results)

for i in range(100):
    estimates[i, 0] = get_mean(np.random.uniform(0, 1, n))
    t2 = []
    t2.append(get_mean(np.random.uniform(0, 0.25, int(n / 4))))
    t2.append(get_mean(np.random.uniform(0.25, 0.5, int(n / 4))))
    t2.append(get_mean(np.random.uniform(0.5, 0.75, int(n / 4))))
    t2.append(get_mean(np.random.uniform(0.75, 1, int(n / 4))))
    estimates[i, 1] = np.mean(t2)

print(np.var(estimates, axis=0))

输出：[1.11658751e-04 8.00280336e-06]

分成100个子区间

import numpy as np

n = 500
estimates = np.empty([100, 2])

def g(x):
    if x >= 0 and x <= 1:
        y = np.exp(-x) / (1 + x ** 2)
    else:
        y = 0
    return y

def get_mean(X):
    results = []
    for x in X:
        results.append(g(x))

    return np.mean(results)

intervals = np.linspace(0, 1, 100)

for i in range(100):
    estimates[i, 0] = get_mean(np.random.uniform(0, 1, n))
    t2 = []
    for j in range(99):
        t2.append(get_mean(np.random.uniform(intervals[j], intervals[j + 1], int(n / 100))))
    estimates[i, 1] = np.mean(t2)

print(np.var(estimates, axis=0))