计算两个函数的重叠面积

Question

我需要计算两个函数重叠的区域。我在这个特殊的简化示例中使用正态分布，但我需要一个更通用的过程来适应其他函数。

请看下图，了解我的意思，红色区域是我想要的：

这是我到目前为止拥有的MWE：

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

# Generate random data uniformly distributed.
a = np.random.normal(1., 0.1, 1000)
b = np.random.normal(1., 0.1, 1000)

# Obtain KDE estimates foe each set of data.
xmin, xmax = -1., 2.
x_pts = np.mgrid[xmin:xmax:1000j]
# Kernels.
ker_a = stats.gaussian_kde(a)
ker_b = stats.gaussian_kde(b)
# KDEs for plotting.
kde_a = np.reshape(ker_a(x_pts).T, x_pts.shape)
kde_b = np.reshape(ker_b(x_pts).T, x_pts.shape)


# Random sample from a KDE distribution.
sample = ker_a.resample(size=1000)

# Compute the points below which to integrate.
iso = ker_b(sample)

# Filter the sample.
insample = ker_a(sample) < iso

# As per Monte Carlo, the integral is equivalent to the
# probability of drawing a point that gets through the
# filter.
integral = insample.sum() / float(insample.shape[0])

print integral

plt.xlim(0.4,1.9)
plt.plot(x_pts, kde_a)
plt.plot(x_pts, kde_b)

plt.show()

这就是我如何应用珍雅的答案

# Calculate overlap between the two KDEs.
def y_pts(pt):
    y_pt = min(ker_a(pt), ker_b(pt))
    return y_pt
# Store overlap value.
overlap = quad(y_pts, -1., 2.) 

Answer 1

图上的红色区域是min(f(x),g(x))的积分，其中f和g是两个函数，绿色和蓝色。要计算积分，可以使用scipy.integrate(quad是默认的）中的任何一个积分器--当然也可以使用MC积分器，但我不太明白这一点。