带有约束和固定点的 scipy curve_fit

如何解决带有约束和固定点的 scipy curve_fit

我正在尝试使用 SciPy 的 optimize.curve_fit 将函数拟合到一些散点数据，但我需要拟合曲线下的面积与基于散点数据计算的面积相同，并且曲线通过数据的起点和终点。为了做到这一点，我在惩罚公式中使用由散点数据定义的面积（积分），如this answer，同时权衡与参数sigma的拟合here .

不幸的是，当包含积分约束时，我无法通过初始点和结束点。如果我忽略积分约束，拟合效果很好并通过该点。不能同时满足积分和点约束吗？我在 Windows 10 上使用 Python 3.7.10。

import scipy
import numpy as np
import matplotlib.pyplot as plt

x = scipy.linspace(0,scipy.pi,100)
y = scipy.sin(x) + (0. + scipy.rand(len(x))*0.4)

def Func(x,a,b,c):
    return a*x**2 + b*x + c

# modified function definition with penalization
def FuncPen(x,c):
    integral = scipy.integrate.quad(Func,x[0],x[-1],args=(a,c))[0]
    penalization = abs(np.trapz(y,x)-integral)*10000
    return a*x**2 + b*x + c + penalization

sigma = np.ones(len(x))
sigma[[0,-1]] = 0.0001 # first and last points

popt1,_ = scipy.optimize.curve_fit(Func,x,y,sigma=sigma)
popt2,_ = scipy.optimize.curve_fit(FuncPen,sigma=sigma)

y_fit1 = Func(x,*popt1)
y_fit2 = Func(x,*popt2)    

fig,ax = plt.subplots(1)
ax.scatter(x,y)
ax.plot(x,y_fit1,color='g',alpha=0.75,label='curve_fit')
ax.plot(x,y_fit2,color='b',label='constrained')
plt.legend()

解决方法

非常感谢 Erwin Kalvelagen 对这个问题的启发性评论。我在这里发布我的解决方案：

import scipy
import numpy as np
import matplotlib.pyplot as plt

x = scipy.linspace(0,scipy.pi,100)
y = scipy.sin(x) + (0. + scipy.rand(len(x))*0.4)

def Func(x,a,b,c):
    return a*x**2 + b*x + c

# modified function definition with penalization
def FuncPen(x,c):
    integral = scipy.integrate.quad(Func,x[0],x[-1],args=(a,c))[0]
    penalization = abs(np.trapz(y,x)-integral)*10000
    return a*x**2 + b*x + c + penalization

# Writing as a general constraint problem
def FuncNew(x,params):
    return params[2]*x**2 + params[1]*x + params[0]

def ConstraintIntegral(params):
    integral = integr.quad(FuncNew,args=(params,))[0]
    return integral- np.trapz(y,x)

def ConstraintBegin(params):
    return y[0] - FuncNew(x[0],params)

def ConstraintEnd(params):
    return y[-1] - FuncNew(x[-1],params)

def Objective(params,x,y):
    y_pred = FuncNew(x,params)
    return np.sum((y_pred - y) ** 2) # least squares

cons = [{'type':'eq','fun': ConstraintIntegral},{'type':'eq','fun': ConstraintBegin},'fun': ConstraintEnd}]

new = scipy.optimize.minimize(Objective,x0=popt1,args=(x,y),constraints=cons)
popt3 = new.x
y_fit3 = FuncNew(x,popt3)
#####

sigma = np.ones(len(x))
sigma[[0,-1]] = 0.0001 # first and last points

popt1,_ = scipy.optimize.curve_fit(Func,y,sigma=sigma)
popt2,_ = scipy.optimize.curve_fit(FuncPen,sigma=sigma)

y_fit1 = Func(x,*popt1)
y_fit2 = Func(x,*popt2)    

fig,ax = plt.subplots(1)
ax.scatter(x,y)
ax.plot(x,y_fit1,color='g',alpha=0.75,label='curve_fit')
ax.plot(x,y_fit2,color='b',label='constrained')
ax.plot(x,y_fit3,color='r',label='generally constrained')
plt.legend()