Scipy优化曲线拟合极限

Question

有什么方法可以为Scipy的优化曲线拟合提供限制吗？

我的例子：

    def optimized_formula(x, m_1, m_2, y_1, y_2, ratio_2):
        return (log(x[0]) * m_1 + m_2)*((1 - x[1]/max_age)*(1-ratio_2)) + ((log(x[1]) * y_1 + y_2)*(x[1]/max_age)*ratio_2)

    popt, pcov = optimize.curve_fit(optimized_formula, usage_and_age, prices)

x[0]是年龄，max_age是常数。考虑到这一点，当x[0]接近最大值时，x[1]/max_age接近1。

是否可以提供x[1]/max_age>；0.3和x[1]/max_age<；0.7等约束/限制，以及m_1<；0、m_2>；0等其他约束。

Answer 1

注：SciPy 0.17版中的新版本

假设您希望将模型与如下所示的数据相匹配：

y=a*t**alpha+b

在阿尔法的约束下

0<alpha<2

其他参数a和b保持自由。然后我们应该使用optimize.curve_fit的bounds选项：

import numpy as np
from scipy.optimize import curve_fit
def func(t, a,alpha,b):
     return a*t**alpha+b
param_bounds=([-np.inf,0,-np.inf],[np.inf,2,np.inf])
popt, pcov = optimize.curve_fit(func, xdata,ydata,bounds=param_bounds)

来源是here

Answer 2

尝试lmfit模块（http://lmfit.github.io/lmfit-py/）。它为scipy.optimize中的许多优化例程（包括最小二乘法）添加了一种修复或设置参数边界的方法，并提供了许多使拟合更容易的工具。

Answer 3

正如另一个答案中所建议的，您可以使用lmfit来解决此类问题。因此，我添加了一个例子来说明如何使用它，以防有人也对这个主题感兴趣。

假设您有一个数据集，如下所示：

xdata = np.array([177.,180.,183.,187.,189.,190.,196.,197.,201.,202.,203.,204.,206.,218.,225.,231.,234.,
          252.,262.,266.,267.,268.,277.,286.,303.])

ydata = np.array([0.81,0.74,0.78,0.75,0.77,0.81,0.73,0.76,0.71,0.74,0.81,0.71,0.74,0.71,
      0.72,0.69,0.75,0.59,0.61,0.63,0.64,0.63,0.35,0.27,0.26])

您希望将模型与如下所示的数据相匹配：

model = n1 + (n2 * x + n3) * 1./ (1. + np.exp(n4 * (n5 - x)))

限制条件是

0.2 < n1 < 0.8
-0.3 < n2 < 0

使用lmfit（0.8.3版）可以获得以下输出：

n1:   0.26564921 +/- 0.024765 (9.32%) (init= 0.2)
n2:  -0.00195398 +/- 0.000311 (15.93%) (init=-0.005)
n3:   0.87261892 +/- 0.068601 (7.86%) (init= 1.0766)
n4:  -1.43507072 +/- 1.223086 (85.23%) (init=-0.36379)
n5:   277.684530 +/- 3.768676 (1.36%) (init= 274)

如您所见，fit很好地再现了数据，并且参数在请求的范围内。

下面是复制情节的完整代码，并附带一些注释：

from lmfit import minimize, Parameters, Parameter, report_fit
import numpy as np

xdata = np.array([177.,180.,183.,187.,189.,190.,196.,197.,201.,202.,203.,204.,206.,218.,225.,231.,234.,
      252.,262.,266.,267.,268.,277.,286.,303.])

ydata = np.array([0.81,0.74,0.78,0.75,0.77,0.81,0.73,0.76,0.71,0.74,0.81,0.71,0.74,0.71,
      0.72,0.69,0.75,0.59,0.61,0.63,0.64,0.63,0.35,0.27,0.26])

def fit_fc(params, x, data):

    n1 = params['n1'].value
    n2 = params['n2'].value
    n3 = params['n3'].value
    n4 = params['n4'].value
    n5 = params['n5'].value

    model = n1 + (n2 * x + n3) * 1./ (1. + np.exp(n4 * (n5 - x)))

    return model - data #that's what you want to minimize

# create a set of Parameters
# 'value' is the initial condition
# 'min' and 'max' define your boundaries
params = Parameters()
params.add('n1', value= 0.2, min=0.2, max=0.8)
params.add('n2', value= -0.005, min=-0.3, max=10**(-10))
params.add('n3', value= 1.0766, min=-1000., max=1000.)
params.add('n4', value= -0.36379, min=-1000., max=1000.)
params.add('n5', value= 274.0, min=0., max=1000.)

# do fit, here with leastsq model
result = minimize(fit_fc, params, args=(xdata, ydata))

# write error report
report_fit(params)

xplot = np.linspace(min(xdata), max(xdata), 1000)
yplot = result.values['n1'] + (result.values['n2'] * xplot + result.values['n3']) * \
                              1./ (1. + np.exp(result.values['n4'] * (result.values['n5'] - xplot)))
#plot results
try:
    import pylab
    pylab.plot(xdata, ydata, 'k+')
    pylab.plot(xplot, yplot, 'r')
    pylab.show()
except:
    pass

编辑：

如果使用0.9.x版本，则需要相应地调整代码；请检查here哪些更改已从0.8.3更改为0.9.x