python - curve_fit、python和excel之间的幂律回归问题

Question

抱歉，我是 python 和堆栈流方面的新手。所以我无法发布图像。

我想用Python中的curve_fit函数进行幂律回归。但结果对我来说很奇怪。我用excel进一步检查了一下。看起来这两者差别很大。黑线是curve_fit的结果，红线的参数来自excel。有人可以让我知道其中的区别吗？谢谢你!

x=[164000,400,13000,700,57000,108,12000]
y=[0.011970,0.000098,0.066100,0.004300,0.042600,0.000061,0.002858 ]

def f(x,a,b):
    return a*x**b

popt,pocv=curve_fit(f,x,y)

ax.set_xscale("log")
ax.set_yscale("log")
ax.set_ylim(0.00001,0.1)
ax.set_xlim(10,1000000)

ax.scatter(x,y)

px=np.linspace(10,1000000,1000)

#parameter form curve_fit
py=a*px**b
[enter image description here][1]
#parameter from excel
pyy=3E-6*px**0.8305

ax.loglog(px,pyy,color="red")
ax.loglog(px,py,color="k")

Answer 1

您在双对数空间中绘制数据这一事实应该会给您一个很好的提示:适合对数空间。也就是说，将 np.log(a*x**b) 拟合到 np.log(y) 中。对实际运行并获得良好适合的脚本的修改将是:

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

x=[164000,400,13000,700,57000,108,12000]
y=[0.011970,0.000098,0.066100,0.004300,0.042600,0.000061,0.002858 ]

def f(x, a, b):
    return np.log(a*x**b)

popt,pcov=curve_fit(f, x, np.log(y), [1.e-6, 0.9])

ax = plt.gca()

ax.set_xscale("log")
ax.set_yscale("log")
ax.set_ylim(0.00001,0.1)
ax.set_xlim(10,1000000)

ax.scatter(x,y)

px = np.linspace(10,1000000,1000)
a, b = popt
print("Parameters: a=%g,  b=%g" % (a, b))

#parameter form curve_fit
py=a*px**b

#parameter from excel
pyy=3e-6*px**0.8305

ax.loglog(px,pyy, color="red")
ax.loglog(px,py,  color="k")
plt.show()

始终确保提供参数的初始值，并确保打印出结果。例如，运行此命令将打印出 Parameters: a=2.78612e-06, b=0.829763 并显示两条预测线几乎彼此重叠。

为了获得更好的曲线拟合体验，您可能会发现 lmfit ( https://lmfit.github.io/lmfit-py/ ) 很有用(是的，我是主要作者并且有偏见)。使用 lmfit，您的合身度可能是:

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

x=[164000,400,13000,700,57000,108,12000]
y=[0.011970,0.000098,0.066100,0.004300,0.042600,0.000061,0.002858 ]

def f(x, a, b):
    return np.log(a*x**b)

model = Model(f)
params = model.make_params(a=1.e-6, b=0.9)
result = model.fit(np.log(y), params, x=x)

print(result.fit_report())

px = np.linspace(10,1000000,1000)
plt.scatter(x,y)
plt.loglog(px, np.exp(result.eval(x=px)),  color="k")
plt.show()

请注意，对于 lmfit，参数是使用 f() 模型函数中的名称​​命名的。这将打印出一份拟合报告，其中包括估计的不确定性:

[[Model]]
    Model(f)
[[Fit Statistics]]
    # fitting method   = leastsq
    # function evals   = 16
    # data points      = 7
    # variables        = 2
    chi-square         = 14.7591170
    reduced chi-square = 2.95182340
    Akaike info crit   = 9.22165592
    Bayesian info crit = 9.11347621
[[Variables]]
    a:  2.7861e-06 +/- 6.3053e-06 (226.31%) (init = 1e-06)
    b:  0.82976271 +/- 0.25700150 (30.97%) (init = 0.9)
[[Correlations]] (unreported correlations are < 0.100)
    C(a, b) = -0.958

并生成一个图