寻找更好的 scipy curve_fit！ - 指数函数？

Question

我正在尝试使用 scipy curve_fit 拟合数据。我相信负指数可能是最好的，因为这对我的其他一些（类似生成的）数据很有效——但我得到的结果并不理想。

我已经对数据集进行了标准化以避免提供初始值，并正在应用如下指数函数：

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
data = np.array([[0.32,0.38],[0.61,0.32],[0.28,0.50],[0.60,0.32],[0.26,0.45],[0.19,0.57],[0.61,0.32],[0.59,0.29],[0.39,0.42],[0.61,0.32],[0.20,0.46],[0.24,0.45],[0.59,0.29],[0.39,0.42],[0.56,0.39],[0.32,0.43],[0.38,0.44],[0.54,0.34],[0.61,0.32],[0.20,0.46],[0.28,0.51],[0.54,0.34],[0.60,0.32],[0.30,0.42],[0.28,0.43],[0.14,0.57],[0.24,0.54],[0.39,0.42],[0.20,0.56],[0.56,0.39],[0.24,0.54],[0.33,0.37],[0.33,0.51],[0.20,0.46],[0.32,0.39],[0.20,0.56],[0.19,0.57],[0.32,0.39],[0.30,0.42],[0.33,0.50],[0.54,0.34],[0.28,0.50],[0.32,0.39],[0.28,0.43],[0.27,0.42],[0.56,0.39],[0.19,0.57],[0.19,0.57],[0.60,0.32],[0.44,0.41],[0.27,0.42],[0.19,0.57],[0.24,0.38],[0.24,0.54],[0.61,0.32],[0.39,0.40],[0.30,0.41],[0.19,0.57],[0.14,0.57],[0.32,0.43],[0.14,0.57],[0.59,0.29],[0.44,0.41],[0.30,0.41],[0.32,0.38],[0.61,0.32],[0.20,0.46],[0.20,0.56],[0.30,0.41],[0.33,0.36],[0.14,0.57],[0.19,0.57],[0.46,0.38],[0.36,0.44],[0.61,0.32],[0.31,0.48],[0.60,0.32],[0.39,0.40],[0.14,0.57],[0.44,0.41],[0.24,0.49],[0.41,0.40],[0.19,0.57],[0.19,0.57],[0.31,0.49],[0.31,0.43],[0.35,0.35],[0.20,0.46],[0.54,0.34],[0.20,0.56],[0.39,0.44],[0.33,0.36],[0.20,0.56],[0.30,0.41],[0.56,0.39],[0.31,0.48],[0.28,0.51],[0.14,0.57],[0.61,0.32],[0.30,0.50],[0.20,0.56],[0.19,0.57],[0.59,0.31],[0.20,0.56],[0.27,0.42],[0.29,0.48],[0.56,0.39],[0.32,0.39],[0.20,0.56],[0.59,0.29],[0.24,0.49],[0.56,0.39],[0.60,0.32],[0.35,0.35],[0.28,0.50],[0.46,0.38],[0.14,0.57],[0.54,0.34],[0.32,0.38],[0.26,0.45],[0.26,0.45],[0.39,0.42],[0.19,0.57],[0.28,0.51],[0.27,0.42],[0.33,0.50],[0.54,0.34],[0.39,0.40],[0.19,0.57],[0.33,0.36],[0.22,0.44],[0.33,0.51],[0.61,0.32],[0.28,0.51],[0.25,0.50],[0.39,0.40],[0.34,0.35],[0.59,0.31],[0.31,0.49],[0.20,0.46],[0.39,0.46],[0.20,0.50],[0.32,0.39],[0.30,0.41],[0.23,0.44],[0.29,0.53],[0.28,0.50],[0.31,0.48],[0.61,0.32],[0.54,0.34],[0.28,0.53],[0.56,0.39],[0.19,0.57],[0.14,0.57],[0.59,0.29],[0.29,0.48],[0.44,0.41],[0.27,0.51],[0.50,0.29],[0.14,0.57],[0.60,0.32],[0.32,0.39],[0.19,0.57],[0.24,0.38],[0.56,0.39],[0.14,0.57],[0.54,0.34],[0.61,0.38],[0.27,0.53],[0.20,0.46],[0.61,0.32],[0.27,0.42],[0.27,0.42],[0.20,0.56],[0.30,0.41],[0.31,0.51],[0.32,0.39],[0.31,0.51],[0.29,0.48],[0.20,0.46],[0.33,0.51],[0.31,0.43],[0.30,0.41],[0.27,0.44],[0.31,0.51],[0.29,0.48],[0.35,0.35],[0.46,0.38],[0.28,0.51],[0.61,0.38],[0.31,0.49],[0.33,0.51],[0.59,0.29],[0.14,0.57],[0.31,0.51],[0.39,0.40],[0.32,0.39],[0.20,0.56],[0.55,0.31],[0.56,0.39],[0.24,0.49],[0.56,0.39],[0.27,0.50],[0.60,0.32],[0.54,0.34],[0.19,0.57],[0.28,0.51],[0.54,0.34],[0.56,0.39],[0.19,0.57],[0.59,0.31],[0.37,0.45],[0.19,0.57],[0.44,0.41],[0.32,0.43],[0.35,0.48],[0.24,0.49],[0.26,0.45],[0.14,0.57],[0.59,0.30],[0.26,0.45],[0.26,0.45],[0.14,0.57],[0.20,0.50],[0.31,0.45],[0.27,0.51],[0.30,0.41],[0.19,0.57],[0.30,0.41],[0.27,0.50],[0.34,0.35],[0.30,0.42],[0.27,0.42],[0.27,0.42],[0.34,0.35],[0.35,0.35],[0.14,0.57],[0.45,0.36],[0.26,0.45],[0.56,0.39],[0.34,0.35],[0.19,0.57],[0.30,0.41],[0.19,0.57],[0.26,0.45],[0.26,0.45],[0.59,0.29],[0.19,0.57],[0.26,0.45],[0.32,0.39],[0.30,0.50],[0.28,0.50],[0.32,0.39],[0.59,0.29],[0.32,0.51],[0.56,0.39],[0.59,0.29],[0.61,0.38],[0.33,0.51],[0.22,0.44],[0.33,0.36],[0.27,0.42],[0.20,0.56],[0.28,0.51],[0.31,0.48],[0.20,0.56],[0.61,0.32],[0.24,0.54],[0.59,0.29],[0.32,0.43],[0.61,0.32],[0.19,0.57],[0.61,0.38],[0.55,0.31],[0.19,0.57],[0.31,0.46],[0.32,0.52],[0.30,0.41],[0.28,0.51],[0.28,0.50],[0.60,0.32],[0.61,0.32],[0.27,0.50],[0.59,0.29],[0.41,0.47],[0.39,0.42],[0.20,0.46],[0.19,0.57],[0.14,0.57],[0.23,0.47],[0.54,0.34],[0.28,0.51],[0.19,0.57],[0.33,0.37],[0.46,0.38],[0.27,0.42],[0.20,0.56],[0.39,0.42],[0.30,0.47],[0.26,0.45],[0.61,0.32],[0.61,0.38],[0.35,0.35],[0.14,0.57],[0.35,0.35],[0.28,0.51],[0.61,0.32],[0.24,0.54],[0.54,0.34],[0.28,0.43],[0.24,0.54],[0.30,0.41],[0.56,0.39],[0.23,0.52],[0.14,0.57],[0.26,0.45],[0.30,0.42],[0.32,0.43],[0.19,0.57],[0.45,0.36],[0.27,0.42],[0.29,0.48],[0.28,0.43],[0.27,0.51],[0.39,0.44],[0.32,0.49],[0.24,0.49],[0.56,0.39],[0.20,0.56],[0.30,0.42],[0.24,0.38],[0.46,0.38],[0.28,0.50],[0.26,0.45],[0.27,0.50],[0.23,0.47],[0.39,0.42],[0.28,0.51],[0.24,0.49],[0.27,0.42],[0.26,0.45],[0.60,0.32],[0.32,0.43],[0.39,0.42],[0.28,0.50],[0.28,0.52],[0.61,0.32],[0.32,0.39],[0.24,0.50],[0.39,0.40],[0.33,0.36],[0.24,0.38],[0.54,0.33],[0.19,0.57],[0.61,0.32],[0.33,0.36],[0.19,0.57],[0.30,0.41],[0.19,0.57],[0.34,0.35],[0.24,0.42],[0.27,0.42],[0.54,0.34],[0.54,0.34],[0.24,0.49],[0.27,0.42],[0.56,0.39],[0.19,0.57],[0.20,0.50],[0.14,0.57],[0.30,0.41],[0.30,0.41],[0.33,0.36],[0.26,0.45],[0.26,0.45],[0.23,0.47],[0.32,0.39],[0.27,0.53],[0.30,0.41],[0.20,0.46],[0.34,0.35],[0.34,0.35],[0.14,0.57],[0.46,0.38],[0.27,0.42],[0.36,0.44],[0.17,0.51],[0.60,0.32],[0.27,0.42],[0.20,0.56],[0.24,0.49],[0.41,0.40],[0.61,0.38],[0.19,0.57],[0.28,0.50],[0.23,0.52],[0.61,0.32],[0.39,0.46],[0.33,0.51],[0.19,0.57],[0.39,0.44],[0.56,0.39],[0.35,0.35],[0.28,0.43],[0.54,0.34],[0.36,0.44],[0.14,0.57],[0.61,0.38],[0.46,0.38],[0.61,0.32],[0.19,0.57],[0.54,0.34],[0.27,0.53],[0.33,0.51],[0.31,0.51],[0.59,0.29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x = data[:,0]
y = data[:,1]

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

popt, pcov = curve_fit(func, x, y)
a, b, c = popt

x_line = np.arange(min(x), max(x), 0.01)
x_line =np.reshape(x_line,(-1,1))

y_line = func(x_line, a, b, c)
y_line = np.reshape(y_line,(-1,1))

plt.scatter(x,y)
plt.plot(x_line,y_line)

plt.show()

如您所见，拟合偏向高 x 值。 example plot 我知道有很多类似的问题，我也读过很多 - 但我的数学技能并不出色，所以我正在努力为我的特定问题想出更好的解决方案。

我不依赖于指数函数 - 任何人都可以提出更好的建议吗？ 我需要为数百个数据集半自动地执行此操作，因此理想情况下我想要尽可能灵活的东西。

非常感谢任何帮助！

附言我很抱歉发布这么大的样本数据集 - 但我认为这类问题需要实际数据，我不想发布指向可疑文件的链接.. =)

Answer 1

这不是最佳解决方案，但这应该适用于数据中的任何密度分布。这个想法是通过计算沿 x 轴的局部平均值来对数据进行给定次数的重新采样，以得到均匀分布的点。

#!/usr/bin/python3.6
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

def my_floor(a, precision=0):
    return np.round(a - 0.5 * 10**(-precision), precision)

data = np.array([[0.32,0.38],[0.61,0.32],[0.28,0.50],[0.60,0.32],[0.26,0.45],[0.19,0.57],[0.61,0.32],[0.59,0.29],[0.39,0.42],[0.61,0.32],[0.20,0.46],[0.24,0.45],[0.59,0.29],[0.39,0.42],[0.56,0.39],[0.32,0.43],[0.38,0.44],[0.54,0.34],[0.61,0.32],[0.20,0.46],[0.28,0.51],[0.54,0.34],[0.60,0.32],[0.30,0.42],[0.28,0.43],[0.14,0.57],[0.24,0.54],[0.39,0.42],[0.20,0.56],[0.56,0.39],[0.24,0.54],[0.33,0.37],[0.33,0.51],[0.20,0.46],[0.32,0.39],[0.20,0.56],[0.19,0.57],[0.32,0.39],[0.30,0.42],[0.33,0.50],[0.54,0.34],[0.28,0.50],[0.32,0.39],[0.28,0.43],[0.27,0.42],[0.56,0.39],[0.19,0.57],[0.19,0.57],[0.60,0.32],[0.44,0.41],[0.27,0.42],[0.19,0.57],[0.24,0.38],[0.24,0.54],[0.61,0.32],[0.39,0.40],[0.30,0.41],[0.19,0.57],[0.14,0.57],[0.32,0.43],[0.14,0.57],[0.59,0.29],[0.44,0.41],[0.30,0.41],[0.32,0.38],[0.61,0.32],[0.20,0.46],[0.20,0.56],[0.30,0.41],[0.33,0.36],[0.14,0.57],[0.19,0.57],[0.46,0.38],[0.36,0.44],[0.61,0.32],[0.31,0.48],[0.60,0.32],[0.39,0.40],[0.14,0.57],[0.44,0.41],[0.24,0.49],[0.41,0.40],[0.19,0.57],[0.19,0.57],[0.31,0.49],[0.31,0.43],[0.35,0.35],[0.20,0.46],[0.54,0.34],[0.20,0.56],[0.39,0.44],[0.33,0.36],[0.20,0.56],[0.30,0.41],[0.56,0.39],[0.31,0.48],[0.28,0.51],[0.14,0.57],[0.61,0.32],[0.30,0.50],[0.20,0.56],[0.19,0.57],[0.59,0.31],[0.20,0.56],[0.27,0.42],[0.29,0.48],[0.56,0.39],[0.32,0.39],[0.20,0.56],[0.59,0.29],[0.24,0.49],[0.56,0.39],[0.60,0.32],[0.35,0.35],[0.28,0.50],[0.46,0.38],[0.14,0.57],[0.54,0.34],[0.32,0.38],[0.26,0.45],[0.26,0.45],[0.39,0.42],[0.19,0.57],[0.28,0.51],[0.27,0.42],[0.33,0.50],[0.54,0.34],[0.39,0.40],[0.19,0.57],[0.33,0.36],[0.22,0.44],[0.33,0.51],[0.61,0.32],[0.28,0.51],[0.25,0.50],[0.39,0.40],[0.34,0.35],[0.59,0.31],[0.31,0.49],[0.20,0.46],[0.39,0.46],[0.20,0.50],[0.32,0.39],[0.30,0.41],[0.23,0.44],[0.29,0.53],[0.28,0.50],[0.31,0.48],[0.61,0.32],[0.54,0.34],[0.28,0.53],[0.56,0.39],[0.19,0.57],[0.14,0.57],[0.59,0.29],[0.29,0.48],[0.44,0.41],[0.27,0.51],[0.50,0.29],[0.14,0.57],[0.60,0.32],[0.32,0.39],[0.19,0.57],[0.24,0.38],[0.56,0.39],[0.14,0.57],[0.54,0.34],[0.61,0.38],[0.27,0.53],[0.20,0.46],[0.61,0.32],[0.27,0.42],[0.27,0.42],[0.20,0.56],[0.30,0.41],[0.31,0.51],[0.32,0.39],[0.31,0.51],[0.29,0.48],[0.20,0.46],[0.33,0.51],[0.31,0.43],[0.30,0.41],[0.27,0.44],[0.31,0.51],[0.29,0.48],[0.35,0.35],[0.46,0.38],[0.28,0.51],[0.61,0.38],[0.31,0.49],[0.33,0.51],[0.59,0.29],[0.14,0.57],[0.31,0.51],[0.39,0.40],[0.32,0.39],[0.20,0.56],[0.55,0.31],[0.56,0.39],[0.24,0.49],[0.56,0.39],[0.27,0.50],[0.60,0.32],[0.54,0.34],[0.19,0.57],[0.28,0.51],[0.54,0.34],[0.56,0.39],[0.19,0.57],[0.59,0.31],[0.37,0.45],[0.19,0.57],[0.44,0.41],[0.32,0.43],[0.35,0.48],[0.24,0.49],[0.26,0.45],[0.14,0.57],[0.59,0.30],[0.26,0.45],[0.26,0.45],[0.14,0.57],[0.20,0.50],[0.31,0.45],[0.27,0.51],[0.30,0.41],[0.19,0.57],[0.30,0.41],[0.27,0.50],[0.34,0.35],[0.30,0.42],[0.27,0.42],[0.27,0.42],[0.34,0.35],[0.35,0.35],[0.14,0.57],[0.45,0.36],[0.26,0.45],[0.56,0.39],[0.34,0.35],[0.19,0.57],[0.30,0.41],[0.19,0.57],[0.26,0.45],[0.26,0.45],[0.59,0.29],[0.19,0.57],[0.26,0.45],[0.32,0.39],[0.30,0.50],[0.28,0.50],[0.32,0.39],[0.59,0.29],[0.32,0.51],[0.56,0.39],[0.59,0.29],[0.61,0.38],[0.33,0.51],[0.22,0.44],[0.33,0.36],[0.27,0.42],[0.20,0.56],[0.28,0.51],[0.31,0.48],[0.20,0.56],[0.61,0.32],[0.24,0.54],[0.59,0.29],[0.32,0.43],[0.61,0.32],[0.19,0.57],[0.61,0.38],[0.55,0.31],[0.19,0.57],[0.31,0.46],[0.32,0.52],[0.30,0.41],[0.28,0.51],[0.28,0.50],[0.60,0.32],[0.61,0.32],[0.27,0.50],[0.59,0.29],[0.41,0.47],[0.39,0.42],[0.20,0.46],[0.19,0.57],[0.14,0.57],[0.23,0.47],[0.54,0.34],[0.28,0.51],[0.19,0.57],[0.33,0.37],[0.46,0.38],[0.27,0.42],[0.20,0.56],[0.39,0.42],[0.30,0.47],[0.26,0.45],[0.61,0.32],[0.61,0.38],[0.35,0.35],[0.14,0.57],[0.35,0.35],[0.28,0.51],[0.61,0.32],[0.24,0.54],[0.54,0.34],[0.28,0.43],[0.24,0.54],[0.30,0.41],[0.56,0.39],[0.23,0.52],[0.14,0.57],[0.26,0.45],[0.30,0.42],[0.32,0.43],[0.19,0.57],[0.45,0.36],[0.27,0.42],[0.29,0.48],[0.28,0.43],[0.27,0.51],[0.39,0.44],[0.32,0.49],[0.24,0.49],[0.56,0.39],[0.20,0.56],[0.30,0.42],[0.24,0.38],[0.46,0.38],[0.28,0.50],[0.26,0.45],[0.27,0.50],[0.23,0.47],[0.39,0.42],[0.28,0.51],[0.24,0.49],[0.27,0.42],[0.26,0.45],[0.60,0.32],[0.32,0.43],[0.39,0.42],[0.28,0.50],[0.28,0.52],[0.61,0.32],[0.32,0.39],[0.24,0.50],[0.39,0.40],[0.33,0.36],[0.24,0.38],[0.54,0.33],[0.19,0.57],[0.61,0.32],[0.33,0.36],[0.19,0.57],[0.30,0.41],[0.19,0.57],[0.34,0.35],[0.24,0.42],[0.27,0.42],[0.54,0.34],[0.54,0.34],[0.24,0.49],[0.27,0.42],[0.56,0.39],[0.19,0.57],[0.20,0.50],[0.14,0.57],[0.30,0.41],[0.30,0.41],[0.33,0.36],[0.26,0.45],[0.26,0.45],[0.23,0.47],[0.32,0.39],[0.27,0.53],[0.30,0.41],[0.20,0.46],[0.34,0.35],[0.34,0.35],[0.14,0.57],[0.46,0.38],[0.27,0.42],[0.36,0.44],[0.17,0.51],[0.60,0.32],[0.27,0.42],[0.20,0.56],[0.24,0.49],[0.41,0.40],[0.61,0.38],[0.19,0.57],[0.28,0.50],[0.23,0.52],[0.61,0.32],[0.39,0.46],[0.33,0.51],[0.19,0.57],[0.39,0.44],[0.56,0.39],[0.35,0.35],[0.28,0.43],[0.54,0.34],[0.36,0.44],[0.14,0.57],[0.61,0.38],[0.46,0.38],[0.61,0.32],[0.19,0.57],[0.54,0.34],[0.27,0.53],[0.33,0.51],[0.31,0.51],[0.59,0.29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x = data[:,0]
y = data[:,1]

#---------------------------ADD THIS---------------------------

# Define how to resample the data 
n_bins = 20 # choose how many samples to use 
bin_size = (max(x) - min(x))/n_bins

# Prepare empty arrays for resampled x and y 
x_res, y_res = [],[] 

# Resample the data with consistent density 
for i in range(n_bins-1): 
    lower = x >= min(x)+i*bin_size
    higher = x < min(x)+(i+1)*bin_size
    x_res.append(np.mean(x[np.where(lower & higher)]))
    y_res.append(np.mean(y[np.where(lower & higher)]))

#------------------------------------------------------

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

popt, pcov = curve_fit(func, x_res, y_res)
a, b, c = popt

x_line = np.arange(min(x), max(x), 0.01)
x_line = np.reshape(x_line,(-1,1))

y_line = func(x_line, a, b, c)
y_line = np.reshape(y_line,(-1,1))

plt.scatter(x,y,alpha=0.5)
plt.scatter(x_res, y_res)
plt.plot(x_line,y_line, c='red')

plt.show()

这给出了输出：

Answer 2

curve_fit 不能让您对拟合进行大量控制，您可能希望查看更通用但使用起来更复杂的least_squares： https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least_squares.html

你可以控制很多事情。 curve_fit 确实为您提供了一个 sigma 参数，该参数可让您对分数中的“不确定性”进行加权。这里的诀窍是为x=1 周围拟合不佳的点分配较低的不确定性。通过降低不确定性，拟合者会更加努力地拟合它们。

经过一些试验，如果将您的 ...curve_fit... 行替换为

uncertainty = np.exp(-5*x*x)
popt, pcov = curve_fit(func, x, y, sigma = uncertainty)

我得到了以下配合

您可以尝试通过使用上面的uncertainty 向量来改进这一点