如何 运行 在 python 中进行非线性回归
How to run non-linear regression in python
我在 python
中有以下信息（数据框）
product baskets scaling_factor
12345 475 95.5
12345 108 57.7
12345 2 1.4
12345 38 21.9
12345 320 88.8
我想运行下面的非线性回归和估计参数。
a,b 和 c
我想要拟合的方程：
scaling_factor = a - (b*np.exp(c*baskets))
在sas中我们通常运行以下模型：（使用高斯牛顿法）
proc nlin data=scaling_factors;
parms a=100 b=100 c=-0.09;
model scaling_factor = a - (b * (exp(c*baskets)));
output out=scaling_equation_parms
parms=a b c;
是否有类似的方法使用非线性回归来估计 Python 中的参数，我怎样才能看到 python 中的图。
对于此类问题，我总是使用 scipy.optimize.minimize 和我自己的最小二乘函数。优化算法不能很好地处理各种输入之间的巨大差异，因此最好缩放函数中的参数，以便暴露给 scipy 的参数都在 1 的数量级上，就像我已经下面完成。
import numpy as np
baskets = np.array([475, 108, 2, 38, 320])
scaling_factor = np.array([95.5, 57.7, 1.4, 21.9, 88.8])
def lsq(arg):
a = arg[0]*100
b = arg[1]*100
c = arg[2]*0.1
now = a - (b*np.exp(c * baskets)) - scaling_factor
return np.sum(now**2)
guesses = [1, 1, -0.9]
res = scipy.optimize.minimize(lsq, guesses)
print(res.message)
# 'Optimization terminated successfully.'
print(res.x)
# [ 0.97336709 0.98685365 -0.07998282]
print([lsq(guesses), lsq(res.x)])
# [7761.0093358076601, 13.055053196410928]
当然，与所有最小化问题一样，使用良好的初始猜测很重要，因为所有算法都可能陷入局部最小值。使用method关键字可以改变优化方式；一些可能性是
- ‘内尔德米德’
- ‘鲍威尔’
- ‘CG’
- 'BFGS'
- ‘牛顿-CG’
根据the documentation默认为BFGS。
同意 Chris Mueller 的观点，我也会使用 scipy 但 scipy.optimize.curve_fit。
代码如下所示：
###the top two lines are required on my linux machine
import matplotlib
matplotlib.use('Qt4Agg')
import matplotlib.pyplot as plt
from matplotlib.pyplot import cm
import numpy as np
from scipy.optimize import curve_fit #we could import more, but this is what we need
###defining your fitfunction
def func(x, a, b, c):
return a - b* np.exp(c * x)
###OP's data
baskets = np.array([475, 108, 2, 38, 320])
scaling_factor = np.array([95.5, 57.7, 1.4, 21.9, 88.8])
###let us guess some start values
initialGuess=[100, 100,-.01]
guessedFactors=[func(x,*initialGuess ) for x in baskets]
###making the actual fit
popt,pcov = curve_fit(func, baskets, scaling_factor,initialGuess)
#one may want to
print popt
print pcov
###preparing data for showing the fit
basketCont=np.linspace(min(baskets),max(baskets),50)
fittedData=[func(x, *popt) for x in basketCont]
###preparing the figure
fig1 = plt.figure(1)
ax=fig1.add_subplot(1,1,1)
###the three sets of data to plot
ax.plot(baskets,scaling_factor,linestyle='',marker='o', color='r',label="data")
ax.plot(baskets,guessedFactors,linestyle='',marker='^', color='b',label="initial guess")
ax.plot(basketCont,fittedData,linestyle='-', color='#900000',label="fit with ({0:0.2g},{1:0.2g},{2:0.2g})".format(*popt))
###beautification
ax.legend(loc=0, title="graphs", fontsize=12)
ax.set_ylabel("factor")
ax.set_xlabel("baskets")
ax.grid()
ax.set_title("$\mathrm{curve}_\mathrm{fit}$")
###putting the covariance matrix nicely
tab= [['{:.2g}'.format(j) for j in i] for i in pcov]
the_table = plt.table(cellText=tab,
colWidths = [0.2]*3,
loc='upper right', bbox=[0.483, 0.35, 0.5, 0.25] )
plt.text(250,65,'covariance:',size=12)
###putting the plot
plt.show()
###done
最终，给你：
我在 python
中有以下信息（数据框）product baskets scaling_factor
12345 475 95.5
12345 108 57.7
12345 2 1.4
12345 38 21.9
12345 320 88.8
我想运行下面的非线性回归和估计参数。
a,b 和 c
我想要拟合的方程：
scaling_factor = a - (b*np.exp(c*baskets))
在sas中我们通常运行以下模型：（使用高斯牛顿法）
proc nlin data=scaling_factors;
parms a=100 b=100 c=-0.09;
model scaling_factor = a - (b * (exp(c*baskets)));
output out=scaling_equation_parms
parms=a b c;
是否有类似的方法使用非线性回归来估计 Python 中的参数，我怎样才能看到 python 中的图。
对于此类问题，我总是使用 scipy.optimize.minimize 和我自己的最小二乘函数。优化算法不能很好地处理各种输入之间的巨大差异，因此最好缩放函数中的参数，以便暴露给 scipy 的参数都在 1 的数量级上，就像我已经下面完成。
import numpy as np
baskets = np.array([475, 108, 2, 38, 320])
scaling_factor = np.array([95.5, 57.7, 1.4, 21.9, 88.8])
def lsq(arg):
a = arg[0]*100
b = arg[1]*100
c = arg[2]*0.1
now = a - (b*np.exp(c * baskets)) - scaling_factor
return np.sum(now**2)
guesses = [1, 1, -0.9]
res = scipy.optimize.minimize(lsq, guesses)
print(res.message)
# 'Optimization terminated successfully.'
print(res.x)
# [ 0.97336709 0.98685365 -0.07998282]
print([lsq(guesses), lsq(res.x)])
# [7761.0093358076601, 13.055053196410928]
当然，与所有最小化问题一样，使用良好的初始猜测很重要，因为所有算法都可能陷入局部最小值。使用method关键字可以改变优化方式；一些可能性是
- ‘内尔德米德’
- ‘鲍威尔’
- ‘CG’
- 'BFGS'
- ‘牛顿-CG’
根据the documentation默认为BFGS。
同意 Chris Mueller 的观点，我也会使用 scipy 但 scipy.optimize.curve_fit。
代码如下所示：
###the top two lines are required on my linux machine
import matplotlib
matplotlib.use('Qt4Agg')
import matplotlib.pyplot as plt
from matplotlib.pyplot import cm
import numpy as np
from scipy.optimize import curve_fit #we could import more, but this is what we need
###defining your fitfunction
def func(x, a, b, c):
return a - b* np.exp(c * x)
###OP's data
baskets = np.array([475, 108, 2, 38, 320])
scaling_factor = np.array([95.5, 57.7, 1.4, 21.9, 88.8])
###let us guess some start values
initialGuess=[100, 100,-.01]
guessedFactors=[func(x,*initialGuess ) for x in baskets]
###making the actual fit
popt,pcov = curve_fit(func, baskets, scaling_factor,initialGuess)
#one may want to
print popt
print pcov
###preparing data for showing the fit
basketCont=np.linspace(min(baskets),max(baskets),50)
fittedData=[func(x, *popt) for x in basketCont]
###preparing the figure
fig1 = plt.figure(1)
ax=fig1.add_subplot(1,1,1)
###the three sets of data to plot
ax.plot(baskets,scaling_factor,linestyle='',marker='o', color='r',label="data")
ax.plot(baskets,guessedFactors,linestyle='',marker='^', color='b',label="initial guess")
ax.plot(basketCont,fittedData,linestyle='-', color='#900000',label="fit with ({0:0.2g},{1:0.2g},{2:0.2g})".format(*popt))
###beautification
ax.legend(loc=0, title="graphs", fontsize=12)
ax.set_ylabel("factor")
ax.set_xlabel("baskets")
ax.grid()
ax.set_title("$\mathrm{curve}_\mathrm{fit}$")
###putting the covariance matrix nicely
tab= [['{:.2g}'.format(j) for j in i] for i in pcov]
the_table = plt.table(cellText=tab,
colWidths = [0.2]*3,
loc='upper right', bbox=[0.483, 0.35, 0.5, 0.25] )
plt.text(250,65,'covariance:',size=12)
###putting the plot
plt.show()
###done
最终，给你：