python - 幂律数据拟合不正确

Question

尝试使用对数 y 轴拟合以下数据时我做错了什么。代码和生成的图表如下。

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
def func(x, a, b, c):
    return a * np.exp(-b * x) + c

fig, ax = plt.subplots()
x = np.array([88.08064516, 264.24193548, 440.40322581, 616.56451613, 792.72580645, 968.88709677, 1145.0483871, 1321.20967742, 1497.37096774, 1673.53225806, 1849.69354839, 2025.85483871, 2202.01612903, 2378.17741935, 2554.33870968, 2730.5, 2906.66129032, 3082.82258065, 3258.98387097, 3435.14516129, 3611.30645161, 3787.46774194, 3963.62903226, 4139.79032258, 4315.9516129, 4492.11290323, 4668.27419355, 4844.43548387, 5020.59677419, 5196.75806452, 5372.91935484, 5549.08064516])
y = np.array([210737, 2175, 514, 158, 90, 46, 27, 22, 10, 11, 3, 7, 3, 2, 0, 1, 1, 1, 0, 0, 1, 0, 0,0, 0, 0, 1, 0, 0, 0, 0,1])
popt, pcov = curve_fit(func, x, y)
ax.plot(x, func(x, *popt), 'g--')
ax.plot(x,  y, 'ro', label='data')

ax.set_yscale('log')  # I need to have the y-axis logarithmic
plt.show()

Answer 1

如果没有初步猜测，您的数据很难拟合。因此，在调用时添加提供猜测（curve_fit如p0文档中所述）：

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

def func(x, a, b, c):
    return a * np.exp(-b * x) + c

x = np.array([88.08064516, 264.24193548, 440.40322581, 616.56451613, 792.72580645, 968.88709677, 1145.0483871, 1321.20967742, 1497.37096774, 1673.53225806, 1849.69354839, 2025.85483871, 2202.01612903, 2378.17741935, 2554.33870968, 2730.5, 2906.66129032,3082.82258065, 3258.98387097, 3435.14516129, 3611.30645161, 3787.46774194, 3963.62903226, 4139.79032258, 4315.9516129, 4492.11290323, 4668.27419355, 4844.43548387, 5020.59677419, 5196.75806452, 5372.91935484, 5549.08064516])
y = np.array([210737, 2175, 514, 158, 90, 46, 27, 22, 10, 11, 3, 7, 3, 2, 0, 1, 1, 1, 0, 0, 1, 0, 0,0, 0, 0, 1, 0, 0, 0, 0,1])

p0 = [20000,0.003,1]
popt, pcov = curve_fit(func, x, y, p0=p0)

fig, ax = plt.subplots()
ax.plot(x, func(x, *popt), 'g--', label = 'fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt))
ax.plot(x,  y, 'ro', label='data')

输出：

请注意，这有助于您解决问题。您仍然必须实现对数轴。对于日志。拟合我建议修剪数据，例如，

y_pruned = np.where(y<1, 1, y)
popt, pcov = curve_fit(func, x, np.log(y_pruned), p0=p0)
ax.plot(x, func(x, *popt), 'g--', label = 'fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt))
ax.plot(x,  np.log(y_pruned), 'ro', label='data')

这产生：

Answer 2

我可以对您的数据做的最好的事情是用对数缩放两组数据值，然后尝试将它们与函数相匹配。我包括了下面的代码和图表。

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
def func(x, a, b, c):
    return a * np.exp(-b * x) + c

fig, ax = plt.subplots()
x = np.array([88.08064516, 264.24193548, 440.40322581, 616.56451613, 792.72580645, 968.88709677, 1145.0483871, 1321.20967742, 1497.37096774, 1673.53225806, 1849.69354839, 2025.85483871, 2202.01612903, 2378.17741935, 2554.33870968, 2730.5, 2906.66129032, 3082.82258065, 3258.98387097, 3435.14516129, 3611.30645161, 3787.46774194, 3963.62903226, 4139.79032258, 4315.9516129, 4492.11290323, 4668.27419355, 4844.43548387, 5020.59677419, 5196.75806452, 5372.91935484, 5549.08064516])
y = np.array([210737, 2175, 514, 158, 90, 46, 27, 22, 10, 11, 3, 7, 3, 2, 0, 1, 1, 1, 0, 0, 1, 0, 0,0, 0, 0, 1, 0, 0, 0, 0,1])

x = np.log(x)
y = np.log(y + 1) # Need to add something to make log work

popt, pcov = curve_fit(func, x, y)
ax.plot(x, func(x, *popt), 'g--')
ax.plot(x,  y, 'ro', label='data')

plt.show()

Answer 3

所有代码实际上都在工作。请参见下图，我运行了您的代码，但在绘图之前没有将缩放比例放在 y 轴上。你可以看到这条线实际上是合适的，除了第一个点它是一个异常值。解决这个问题的方法是在拟合函数之前先缩放 y 值。试试看，如果您需要更多帮助，请告诉我们。