SciPy 中的指数曲线拟合 [英] Exponential curve fitting in SciPy
问题描述
我有两个 NumPy 数组 x
和 y
.当我尝试使用指数函数和 curve_fit
(SciPy) 与这个简单的代码
I have two NumPy arrays x
and y
. When I try to fit my data using exponential function and curve_fit
(SciPy) with this simple code
#!/usr/bin/env python
from pylab import *
from scipy.optimize import curve_fit
x = np.array([399.75, 989.25, 1578.75, 2168.25, 2757.75, 3347.25, 3936.75, 4526.25, 5115.75, 5705.25])
y = np.array([109,62,39,13,10,4,2,0,1,2])
def func(x, a, b, c, d):
return a*np.exp(b-c*x)+d
popt, pcov = curve_fit(func, x, y)
我得到了错误的系数 popt
[a,b,c,d] = [1., 1., 1., 24.19999988]
有什么问题?
推荐答案
第一条评论:因为 a*exp(b - c*x) = (a*exp(b))*exp(-c*x) = A*exp(-c*x)
、a
或 b
是多余的.我将删除 b
并使用:
First comment: since a*exp(b - c*x) = (a*exp(b))*exp(-c*x) = A*exp(-c*x)
, a
or b
is redundant. I'll drop b
and use:
def func(x, a, c, d):
return a*np.exp(-c*x)+d
这不是主要问题.问题很简单,当您使用默认的初始猜测(全为 1)时,curve_fit
无法收敛到此问题的解决方案.检查pcov
;你会看到它是inf
.这并不奇怪,因为如果 c
为 1,exp(-c*x)
的大部分值下溢为 0:
That isn't the main issue. The problem is simply that curve_fit
fails to converge to a solution to this problem when you use the default initial guess (which is all 1s). Check pcov
; you'll see that it is inf
. This is not surprising, because if c
is 1, most of the values of exp(-c*x)
underflow to 0:
In [32]: np.exp(-x)
Out[32]:
array([ 2.45912644e-174, 0.00000000e+000, 0.00000000e+000,
0.00000000e+000, 0.00000000e+000, 0.00000000e+000,
0.00000000e+000, 0.00000000e+000, 0.00000000e+000,
0.00000000e+000])
这表明 c
应该很小.例如,更好的初始猜测是 p0 = (1, 1e-6, 1)
.然后我得到:
This suggests that c
should be small. A better initial guess is, say, p0 = (1, 1e-6, 1)
. Then I get:
In [36]: popt, pcov = curve_fit(func, x, y, p0=(1, 1e-6, 1))
In [37]: popt
Out[37]: array([ 1.63561656e+02, 9.71142196e-04, -1.16854450e+00])
这看起来很合理:
In [42]: xx = np.linspace(300, 6000, 1000)
In [43]: yy = func(xx, *popt)
In [44]: plot(x, y, 'ko')
Out[44]: [<matplotlib.lines.Line2D at 0x41c5ad0>]
In [45]: plot(xx, yy)
Out[45]: [<matplotlib.lines.Line2D at 0x41c5c10>]
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