使用lmfit模型最小化在Python中拟合两个洛伦兹函数

Question

也许有人可以帮我。我花了好几天的时间，但无法解决这个问题。谢谢您提前的帮助。

我想要将两个洛仑兹函数拟合到我的实验数据中。我将我的方程简化为两个洛仑兹函数lorentz1和lorentz2。然后，我定义了另外两个函数L1和L2，只需将常数cnst乘以它们。我有所有需要拟合的四个参数: cnst1、cnst2、tau1、tau2。

我使用lmfit：Model and minimize（可能都使用同一种方法）。

初始拟合参数设置得更接近一个良好的拟合结果，但使用lmfit进行最小化时会失控（如下面第一个图所示）：

---

使用这些参数：

params.add('cnst1', value=1e3 , min=1e2, max=1e5)
params.add('cnst2', value=3e5, min=1e2, max=1e6)
params.add('tau1', value=2e0, min=0, max=1e2)
params.add('tau2', value=5e-3, min=0, max=10)

但错误率很低:

cnst1:   117.459806 +/- 14.67188 (12.49%) (init= 1000)
cnst2:   413.959032 +/- 44.21042 (10.68%) (init= 300000)
tau1:    11.0343531 +/- 1.065570 (9.66%) (init= 2)
tau2:    1.55259664 +/- 0.125853 (8.11%) (init= 0.005)

---

另一方面，将参数限制在非常接近初始值的范围内（强制为初始值）：

使用以下参数：

#params.add('cnst1', value=1e3 , min=0.1e3, max=1e3)
#params.add('cnst2', value=3e5, min=1e3, max=1e6)
#params.add('tau1', value=2e0, min=0, max=2)
#params.add('tau2', value=5e-3, min=0, max=10)

虽然 fit 看起来更好，但误差值非常大：

[[Variables]]
cnst1:   752.988629 +/- 221.3098 (29.39%) (init= 1000)
cnst2:   3.0159e+05 +/- 3.05e+07 (10113.40%) (init= 300000)
tau1:    1.99684317 +/- 0.600748 (30.08%) (init= 2)
tau2:    0.00497806 +/- 0.289651 (5818.56%) (init= 0.005)

这是完整的代码：

import numpy as np
from lmfit import Model, minimize, Parameters, report_fit
import matplotlib.pyplot as plt

x = np.array([0.02988, 0.07057,0.19365,0.4137,0.91078,1.85075,3.44353,6.39428,\
        11.99302,24.37024,52.58804,121.71927,221.53799,358.27392,464.70405])

y = 1.0 / np.array([4.60362E-4,5.63559E-4,8.44538E-4,0.00138,0.00287,0.00657,0.01506,\
            0.03119,0.0584,0.09153,0.12538,0.19389,0.34391,0.68869,1.0])

def lorentz1(x, tau):
    L =  tau  / ( 1 + (x*tau)**2 )  
    return(L)

def lorentz2(x, tau):
    L =  tau**2  / ( 1 + (x*tau)**2 )  
    return(L)

def L1(x,cnst1,tau1):
    L1 =  cnst1 * lorentz1(x,tau1)
    return (L1)

def L2(x, cnst2, tau2):
    L2 =  cnst2 * lorentz2(x,tau2)
    return (L2)    

def L_min(params, x, y):
    cnst1 = params['cnst1'].value
    cnst2 = params['cnst2'].value
    tau1 = params['tau1'].value
    tau2 = params['tau2'].value

    L_total = L1(x, cnst1, tau1) + L2(x, cnst2, tau2)
    resids = L_total - y
    return resids

#params  = mod.make_params( cnst1=10e2, cnst2=3e5, tau1=2e0, tau2=0.5e-2)
params = Parameters()
#params.add('cnst1', value=1e3 , min=0.1e3, max=1e3)
#params.add('cnst2', value=3e5, min=1e3, max=1e6)
#params.add('tau1', value=2e0, min=0, max=2)
#params.add('tau2', value=5e-3, min=0, max=10)

params.add('cnst1', value=1e3 , min=1e2, max=1e5)
params.add('cnst2', value=3e5, min=1e2, max=1e6)
params.add('tau1', value=2e0, min=0, max=1e2)
params.add('tau2', value=5e-3, min=0, max=10)


#1-----Model--------------------
mod = Model(L1) + Model(L2)
result_mod = mod.fit(y, params, x=x)
print('---results from lmfit.Model----')
print(result_mod.fit_report())

#2---minimize-----------
result_min = minimize(L_min, params, args=(x,y))
final_min = y + result_min.residual
print('---results from lmfit.minimize----')
report_fit(params)

#-------Plot------
plt.close('all')
plt.loglog(x, y,'bo' , label='experimental data')
plt.loglog(x, result_mod.init_fit, 'k--', label='initial')
plt.loglog(x, result_mod.best_fit, 'r-', label='final')
plt.legend()
plt.show()

- Amin

---

你的y轴是对数尺度，而你正在拟合最小二乘线性残差。虽然它可能在视觉上不太美观，但最佳匹配会匹配最大的数字并忽略较小的数字 - 你会发现平方残差更小。 - mdurant

Answer 1

我在搜索时，谷歌找到了我以前问过的问题。现在我知道答案了，我会在这里分享出来，希望能帮助到别人。:)

我将考虑使用lmfit.minimize函数。所做的更改是绘制lmfit.minimize的结果。为了解决对数y轴比例（这也是@mdurant提到的主要问题）的问题，我只是将残差除以其y值（某种方式规范化所有数据以便在取残差时进行比较）。我称之为加权残差。

def L_min(params, x, y):
    ...
    ..
    .
    resids = L_total - y
    weighted_resids = resids/y
    return weighted_resids

因此，结果如蓝线所示：

完整代码：

import numpy as np
from lmfit import Model, minimize, Parameters, report_fit
import matplotlib.pyplot as plt

x = np.array([0.02988, 0.07057,0.19365,0.4137,0.91078,1.85075,3.44353,6.39428,\
        11.99302,24.37024,52.58804,121.71927,221.53799,358.27392,464.70405])

y = 1.0 / np.array([4.60362E-4,5.63559E-4,8.44538E-4,0.00138,0.00287,0.00657,0.01506,\
            0.03119,0.0584,0.09153,0.12538,0.19389,0.34391,0.68869,1.0])

def lorentz1(x, tau):
    L =  tau  / ( 1 + (x*tau)**2 )  
    return(L)

def lorentz2(x, tau):
    L =  tau**2  / ( 1 + (x*tau)**2 )  
    return(L)

def L1(x,cnst1,tau1):
    L1 =  cnst1 * lorentz1(x,tau1)
    return (L1)

def L2(x, cnst2, tau2):
    L2 =  cnst2 * lorentz2(x,tau2)
    return (L2)    

def L_min(params, x, y):
    cnst1 = params['cnst1'].value
    cnst2 = params['cnst2'].value
    tau1 = params['tau1'].value
    tau2 = params['tau2'].value

    L_total = L1(x, cnst1, tau1) + L2(x, cnst2, tau2)
    resids = L_total - y
    weighted_resids = resids/y
    return weighted_resids
#    return resids

#params  = mod.make_params( cnst1=10e2, cnst2=3e5, tau1=2e0, tau2=0.5e-2)
params = Parameters()
#params.add('cnst1', value=1e3 , min=0.1e3, max=1e3)
#params.add('cnst2', value=3e5, min=1e3, max=1e6)
#params.add('tau1', value=2e0, min=0, max=2)
#params.add('tau2', value=5e-3, min=0, max=10)

params.add('cnst1', value=1e3 , min=1e2, max=1e5)
params.add('cnst2', value=3e5, min=1e2, max=1e6)
params.add('tau1', value=2e0, min=0, max=1e2)
params.add('tau2', value=5e-3, min=0, max=10)


#1-----Model--------------------
mod = Model(L1) + Model(L2)
result_mod = mod.fit(y, params, x=x)
print('---results from lmfit.Model----')
print(result_mod.fit_report())

#2---minimize-----------
result_min = minimize(L_min, params, args=(x,y))
final_min = y + result_min.residual
print('---results from lmfit.minimize----')
report_fit(params)

#-------Plot------
plt.close('all')
plt.loglog(x, y,'bo' , label='experimental data')
plt.loglog(x, result_mod.init_fit, 'k--', label='initial')
plt.loglog(x, result_mod.best_fit, 'r-', label='lmfit.Model')

min_result = L1(x, params['cnst1'].value, params['tau1'].value) + \
            L2(x, params['cnst2'].value, params['tau2'].value)
plt.loglog(x, min_result, 'b-', label='lmfit.Minimize')
plt.legend()
plt.show()

拟合误差很小：

    cnst1:   832.592441 +/- 77.32939 (9.29%) (init= 1000)
    cnst2:   2.0836e+05 +/- 3.55e+04 (17.04%) (init= 300000)
    tau1:    1.64355457 +/- 0.221466 (13.47%) (init= 2)
    tau2:    0.00700899 +/- 0.000935 (13.34%) (init= 0.005)
[[Correlations]] (unreported correlations are <  0.100)
    C(tau1, tau2)                =  0.151 

- Amin