Python - 查找正态曲线的最小值、最大值和拐点

Question

我试图在植被曲线中找到最小值、季节开始、生长旺季、最大生长、衰老、季节结束、最小值（即拐点）的位置（即 x 值）。我在这里以正态曲线为例。我确实遇到了一些代码来找到斜率和一阶/二阶导数的变化，但无法为我的案例实现它们。如果有任何相关示例，请指导我，并感谢您的帮助。谢谢！

## Version 2 code
import matplotlib.pyplot as plt
import numpy as np
from  scipy.stats import norm

x_min = 0.0
x_max = 16.0

mean = 8
std = 2

x = np.linspace(x_min, x_max, 100)
y = norm.pdf(x, mean, std)

# Slice the group in 3
def group_in_threes(slicable):
    for i in range(len(slicable)-2):
        yield slicable[i:i+3]

# Locate the change in slope
def turns(L):
    for index, three in enumerate(group_in_threes(L)):
        if (three[0] > three[1] < three[2]) or (three[0] < three[1] > three[2]):
            yield index + 1

# 1st inflection point estimation
dy = np.diff(y, n=1) # first derivative
idx_max_dy = np.argmax(dy)
ix = list(turns(dy))
print(ix)

# All inflection point estimation
dy2 = np.diff(dy, n=2) # Second derivative?
idx_max_dy2 = np.argmax(dy2)
ix2 = list(turns(dy2))
print(ix2)

# Graph
plt.plot(x, y)
#plt.plot(x[ix], y[ix], 'or', label='estimated inflection point')
plt.plot(x[ix2], y[ix2], 'or', label='estimated inflection point - 2')
plt.xlabel('x'); plt.ylabel('y'); plt.legend(loc='best');

Answer 1

这是一种非常简单且不可靠的方法来找到非噪声曲线的拐点：

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

x_min = 0.0
x_max = 16.0

mean = 8
std = 2

x = np.linspace(x_min, x_max, 100)
y = norm.pdf(x, mean, std)

# 1st inflection point estimation
dy = np.diff(y) # first derivative
idx_max_dy = np.argmax(dy)

# Graph
plt.plot(x, y)
plt.plot(x[idx_max_dy], y[idx_max_dy], 'or', label='estimated inflection point')
plt.xlabel('x'); plt.ylabel('y'); plt.legend();

对于高斯曲线，拐点的实际位置是x1 = mean - std。

为了处理真实数据，必须在寻找最大值之前对其进行平滑处理，例如应用简单的移动平均线、gaussian filter 或Savitzky-Golay filter，它们可以直接输出二阶导数...选择合适的过滤器取决于数据