查找两个分布的非重叠边缘区域

Question

我正在尝试找到一种方法来计算两个分布的重叠（交集）和非重叠区域。根据一些帖子（例如：FINDING AREA），我可以弄清楚如何计算两个图的交集。但是我没有找到不重叠的部分。
鉴于此示例分布：

import pandas as pd
import numpy as np
import seaborn as  sns

data1=np.random.normal(0, 0.1, 100)
data2=np.random.normal(0, 0.3, 100)

x0=pd.Series(data1)
x1=pd.Series(data2)
kwargs = dict(hist_kws={'alpha':.01}, kde_kws={'linewidth':2})
sns.distplot(x0,bins=3, color="dodgerblue", **kwargs)
sns.distplot(x1,bins=3, color="dodgerblue", **kwargs)

我想知道如何计算两个分布在边缘的面积？
我应该提到我也有数据（如果这是我可以用数据本身做的事情）。

Answer 1

从answer you linked 中汲取灵感，从数据中您应该通过scipy.stats.gaussian_kde 计算kde 曲线：

data1 = np.random.normal(0, 0.1, 100)
data2 = np.random.normal(0, 0.3, 100)

kde1 = gaussian_kde(data1, bw_method = 0.5)
kde2 = gaussian_kde(data2, bw_method = 0.5)

xmin = min(data1.min(), data2.min())
xmax = max(data1.max(), data2.max())
dx = 0.2*(xmax - xmin)
xmin -= dx
xmax += dx

x = np.linspace(xmin, xmax, 1000)
kde1_x = kde1(x)
kde2_x = kde2(x)

然后你需要找到kde曲线之间的交点：

idx = np.argwhere(np.diff(np.sign(kde1_x - kde2_x))).flatten()

其中idx 是交点索引列表。
最后，您可以通过numpy.trapz 计算面积，使用先前计算的索引对x、kde1_x 和kde2_x 进行切片：

area1 = np.trapz(kde2_x[:idx[0]] - kde1_x[:idx[0]], x[:idx[0]]) # area under the lower kde, from the first leftmost point to the first intersection point
area2 = np.trapz(kde2_x[idx[1]:] - kde1_x[idx[1]:], x[idx[1]:]) # area under the lower kde, from the second intersection point to the last rightmost point
area3 = np.trapz(np.minimum(kde1_x, kde2_x), x) # intersection area between of the kde curves, between the two intersection points
area4 = np.trapz(kde1_x[idx[0]:idx[1]] - kde2_x[idx[0]:idx[1]], x[idx[0]:idx[1]]) # area under the highest kde, excluding the lower one, between the two intersection points

完整代码

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import gaussian_kde


data1 = np.random.normal(0, 0.1, 100)
data2 = np.random.normal(0, 0.3, 100)

kde1 = gaussian_kde(data1, bw_method = 0.5)
kde2 = gaussian_kde(data2, bw_method = 0.5)

xmin = min(data1.min(), data2.min())
xmax = max(data1.max(), data2.max())
dx = 0.2*(xmax - xmin)
xmin -= dx
xmax += dx

x = np.linspace(xmin, xmax, 1000)
kde1_x = kde1(x)
kde2_x = kde2(x)


idx = np.argwhere(np.diff(np.sign(kde1_x - kde2_x))).flatten()

area1 = np.trapz(kde2_x[:idx[0]] - kde1_x[:idx[0]], x[:idx[0]]) # area under the lower kde, from the first leftmost point to the first intersection point
area2 = np.trapz(kde2_x[idx[1]:] - kde1_x[idx[1]:], x[idx[1]:]) # area under the lower kde, from the second intersection point to the last rightmost point
area3 = np.trapz(np.minimum(kde1_x, kde2_x), x) # intersection area between of the kde curves, between the two intersection points
area4 = np.trapz(kde1_x[idx[0]:idx[1]] - kde2_x[idx[0]:idx[1]], x[idx[0]:idx[1]]) # area under the highest kde, excluding the lower one, between the two intersection points


fig, ax = plt.subplots()

ax.plot(x, kde1_x, color = 'dodgerblue', linewidth = 2)
ax.plot(x, kde2_x, color = 'orangered', linewidth = 2)

ax.fill_between(x[:idx[0]], kde2_x[:idx[0]], kde1_x[:idx[0]], color = 'dodgerblue', alpha = 0.3, label = 'area1')
ax.fill_between(x[idx[1]:], kde2_x[idx[1]:], kde1_x[idx[1]:], color = 'orangered', alpha = 0.3, label = 'area2')
ax.fill_between(x, np.minimum(kde1_x, kde2_x), 0, color = 'lime', alpha = 0.3, label = 'area3')
ax.fill_between(x[idx[0]:idx[1]], kde1_x[idx[0]:idx[1]], kde2_x[idx[0]:idx[1]], color = 'gold', alpha = 0.3, label = 'area4')

ax.plot(x[idx], kde2_x[idx], 'ko')

handles, labels = ax.get_legend_handles_labels()
labels[0] += f': {area1 * 100:.1f}%'
labels[1] += f': {area2 * 100:.1f}%'
labels[2] += f': {area3 * 100:.1f}%'
labels[3] += f': {area4 * 100:.1f}%'
ax.legend(handles, labels)

plt.show()