如何用Python自动化实现方程组中的累积分布函数近似计算？

我们需要根据观测数据，计算从t=0到数据最大值的所有整数t对应的CDF近似值，核心公式为：
F(t) = (观测值≤t的数量) / 总观测数

实现方式 #

1. 纯Python实现（无依赖库）

适合小型数据集，无需额外安装库：

def compute_cdf(observations):
    total = len(observations)
    if not total:
        return {}
    max_val = max(observations)
    cdf_results = {}
    for t in range(0, max_val + 1):
        # 统计观测值≤t的数量
        count = sum(1 for obs in observations if obs <= t)
        cdf_results[f"F({t})"] = count / total
    return cdf_results

# 示例调用
list_goals = [1, 2, 2, 1, 2]
cdf = compute_cdf(list_goals)
# 输出结果
for key, value in cdf.items():
    print(f"{key} = {value}")

运行后输出：

F(0) = 0.0
F(1) = 0.4
F(2) = 1.0

2. NumPy实现（高效处理大数据）

利用NumPy的向量化操作提升计算效率，适合大规模数据集：

import numpy as np

def compute_cdf_numpy(observations):
    obs_array = np.array(observations)
    total = len(obs_array)
    if total == 0:
        return {}
    max_val = obs_array.max()
    t_values = np.arange(0, max_val + 1)
    # 批量计算每个t对应的观测值计数
    counts = np.array([np.sum(obs_array <= t) for t in t_values])
    cdf_results = {f"F({t})": count / total for t, count in zip(t_values, counts)}
    return cdf_results

# 示例调用
list_goals = [1, 2, 2, 1, 2]
cdf_np = compute_cdf_numpy(list_goals)
for key, value in cdf_np.items():
    print(f"{key} = {value}")

3. Pandas实现（便捷的数据分析流程）

适合需要整合到数据分析流水线的场景，利用Pandas的统计函数简化操作：

import pandas as pd

def compute_cdf_pandas(observations):
    df = pd.DataFrame({"values": observations})
    total = len(df)
    if total == 0:
        return {}
    max_val = df["values"].max()
    # 生成所有需要计算的t值
    all_t = pd.Series(range(0, max_val + 1), name="t")
    # 统计每个值的出现次数，缺失值补0
    count_series = df["values"].value_counts().reindex(all_t, fill_value=0)
    # 计算累积和并得到CDF值
    cumsum = count_series.cumsum()
    cdf_results = {f"F({t})": cumsum[t] / total for t in all_t}
    return cdf_results

# 示例调用
list_goals = [1, 2, 2, 1, 2]
cdf_pd = compute_cdf_pandas(list_goals)
for key, value in cdf_pd.items():
    print(f"{key} = {value}")

内容的提问来源于stack exchange，提问作者Tappetinoorange