如何在 python 曲线拟合阻尼余弦波中有效地使用 "LMFIT"
How to efficiently use "LMFIT" in python curve fitting to a damped cosine wave
我有数据集 (x & y)，我想将其拟合为阻尼余弦，例如 (1-A+Acos(Kx))exp(- B*x) 通过使用 LMFIT（“非线性最小二乘法”）作为 link (https://lmfit.github.io/lmfit-py/intro.html)。为此，我尝试使用以下代码，但无法正确安装。我的代码出了什么问题？任何帮助或建议将不胜感激。
x = [0, 1.3, 1.7, 1.72, 1.84, 1.98, 2.02, 2.16, 2.2, 2.2, 2.3, 2.38, 2.5, 2.55, 2.75, 2.8, 2.82, 2.84, 2.9, 2.92, 3.1, 3.13, 3.18, 3.19, 3.22, 3.3, 3.38, 3.44, 3.49, 3.62, 3.64, 3.72, 3.72, 3.75, 3.8, 3.82, 3.86, 3.92, 4.0, 4.07, 4.1, 4.1, 4.13, 4.14, 4.14, 4.17, 4.21, 4.24, 4.24, 4.24, 4.28, 4.3, 4.38, 4.49, 4.62, 4.62, 4.67, 4.72, 4.73, 4.74, 4.76, 4.76, 4.81, 4.81, 4.88, 4.89, 4.9, 4.9, 4.94, 4.96, 5.03, 5.05, 5.06, 5.1, 5.1, 5.15, 5.16, 5.16, 5.19, 5.22, 5.22, 5.3, 5.37, 5.41, 5.46, 5.56, 5.63, 5.65, 5.65, 5.73, 5.76, 5.81, 5.86, 5.91, 5.98, 6.03, 6.05, 6.05, 6.06, 6.11, 6.14, 6.22, 6.25, 6.27, 6.27, 6.3, 6.3, 6.31, 6.36, 6.42, 6.42, 6.47, 6.48, 6.5, 6.51, 6.58, 6.59, 6.62, 6.65, 6.66, 6.67, 6.69, 6.72, 6.77, 6.8, 6.84, 6.87, 6.91, 6.94, 6.94, 6.94, 7.05, 7.14, 7.17, 7.22, 7.23, 7.24, 7.32, 7.32, 7.35, 7.38, 7.4, 7.41, 7.42, 7.44, 7.45, 7.49, 7.5, 7.52, 7.54, 7.6, 7.72, 7.75, 7.81, 7.9, 7.92, 7.95, 7.97, 7.98, 7.99, 8.02, 8.03, 8.03, 8.05, 8.06, 8.07, 8.1, 8.12, 8.14, 8.19, 8.2, 8.21, 8.24, 8.25, 8.28, 8.28, 8.29, 8.32, 8.38, 8.38, 8.43, 8.49, 8.52, 8.54, 8.54, 8.57, 8.7, 8.75, 8.75, 8.78, 8.79, 8.88, 8.88, 8.93, 8.95, 9.0, 9.01, 9.02, 9.03, 9.06, 9.07, 9.11, 9.14, 9.16, 9.17, 9.18, 9.19, 9.2, 9.3, 9.33, 9.44, 9.46, 9.59, 9.62, 9.62, 9.64, 9.66, 9.71, 9.73, 9.73, 9.75, 9.76, 9.76, 9.79, 9.88, 9.9, 9.93, 9.93, 9.95, 9.99, 10.01, 10.03, 10.04, 10.05, 10.07, 10.11, 10.13, 10.18, 10.22, 10.22, 10.31, 10.37, 10.38, 10.41, 10.42, 10.44, 10.5, 10.52, 10.55, 10.56, 10.56, 10.58, 10.6, 10.66, 10.68, 10.68, 10.69, 10.7, 10.73, 10.75, 10.81, 10.93, 10.96, 10.98, 10.98, 11.02, 11.04, 11.1, 11.14, 11.15, 11.15, 11.17, 11.19, 11.21, 11.23, 11.24, 11.28, 11.3, 11.31, 11.32, 11.33, 11.4, 11.42, 11.48, 11.5, 11.51, 11.6, 11.62, 11.62, 11.63, 11.65, 11.72, 11.74, 11.74, 11.94, 11.95, 11.98, 12.02, 12.02, 12.03, 12.04, 12.09, 12.11, 12.17, 12.2, 12.23, 12.26, 12.3, 12.31, 12.33, 12.33, 12.37, 12.38, 12.61, 12.63, 12.69, 12.7, 12.74, 12.79, 12.8, 12.84, 12.87, 12.9, 12.91, 12.92, 12.94, 13.0, 13.19, 13.2, 13.26, 13.29, 13.3, 13.31, 13.31, 13.34, 13.35, 13.36, 13.44, 13.48, 13.52, 13.59, 13.78, 13.83, 13.88, 13.98, 14.02, 14.05, 14.07, 14.1, 14.14, 14.19, 14.25, 14.33, 14.36, 14.38, 14.41, 14.46, 14.47, 14.53, 14.54, 14.57, 14.69, 14.72, 14.77, 14.78, 14.78, 14.8, 14.82, 14.82, 14.91, 14.92, 14.96, 14.96, 15.05, 15.09, 15.17, 15.2, 15.21, 15.25, 15.26, 15.31, 15.32, 15.36, 15.36, 15.4, 15.4, 15.4, 15.41, 15.47, 15.52, 15.6, 15.61, 15.61, 15.63, 15.65, 15.71, 15.77, 15.8, 15.84, 15.86, 15.88, 15.94, 15.94, 15.97, 15.98, 16.02, 16.03, 16.27, 16.43, 16.56, 16.64, 16.64, 16.64, 16.68, 16.88, 16.91, 16.92, 16.93, 16.97, 16.99, 17.0, 17.01, 17.02, 17.05, 17.13, 17.21, 17.32, 17.45, 17.59, 17.79, 17.8, 17.81, 17.87, 17.9, 17.92, 17.93, 17.93, 17.97, 17.98, 18.02, 18.05, 18.08, 18.11, 18.2, 18.24, 18.4, 18.48, 18.5, 18.51, 18.59, 18.65, 18.76, 18.76, 18.86, 18.86, 18.86, 18.87, 18.9, 18.92, 18.93, 19.05, 19.06, 19.17, 19.26, 19.27, 19.41, 19.47, 19.48, 19.54, 19.6, 19.66, 19.67, 19.68, 19.8, 19.9, 20.01, 20.04, 20.1, 20.49, 20.49, 20.5, 20.56, 20.65, 20.65, 20.7, 20.71, 20.78, 20.91, 21.11, 21.19, 21.2, 21.28, 21.29, 21.58, 21.62, 21.7, 21.7, 21.76, 21.76, 21.84, 21.85, 21.87, 21.9, 21.94, 22.0, 22.02, 22.09, 22.16, 22.3, 22.3, 22.41, 22.51, 22.53, 22.71, 22.77, 22.94, 23.17, 23.25, 23.33, 23.72, 23.87, 24.12, 24.14, 24.19, 24.34, 24.4, 24.6, 24.62, 24.62, 24.8, 25.01, 25.13, 25.4, 25.42, 25.81, 25.85, 25.89, 26.03, 26.17, 26.22, 26.41, 26.98, 27.01, 27.02, 27.06, 27.17, 27.49, 27.73, 28.14, 28.23, 28.37, 28.56, 28.83, 28.84, 30.32, 30.57, 31.95, 33.23, 33.46, 33.81, 33.85, 34.44]
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import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import least_squares
from lmfit import minimize, Parameters, Parameter, report_fit
x = np.asarray(x); y = np.asarray(y)
def fit_fc(params, t, data):
A = params['A'].value
K = params['K'].value
B = params['B'].value
model = (1-A+A*np.cos(K * t))*np.exp(-1 * B * t)
return model - data
params = Parameters()
params.add('A', value=.9, min =0, max =1)
params.add('K', value=0.42, min=-0.2, max=0.8)
params.add('B', value=0.1, min=.01, max=.1)
result = minimize(fit_fc, params, args=(x, y), method='leastsq')
report_fit(params) # write error report
y_lsq = (1-result.params['A'] + (result.params['A'] *np.cos(result.params['K']*x))*np.exp(result.params['B'])*x*(-1))
#plot results
plt.plot(x, y, 'o', label='Data')
plt.plot(x, y_lsq, label='Least_Square_Method')
plt.xlabel('$x$')
plt.ylabel('$y$')
plt.legend()
你试过画图吗？与您尝试的拟合相比，您的数据确实“嘈杂”。
我将您的脚本更改为使用 lmfit.Model，这样更容易进行曲线拟合。看起来像（为简单起见删除数据）：
import numpy as np
import matplotlib.pyplot as plt
from lmfit import Model, Parameters
x = np.asarray(x)
y = np.asarray(y)
def damped_cosine(t, a, k, b):
return (1-a+a*np.cos(k*t))*np.exp(-b*t)
params = Parameters()
params.add('a', value=0.9, min=0)
params.add('k', value=0.42)
params.add('b', value=0.1)
dmodel = Model(damped_cosine)
result = dmodel.fit(y, params, t=x)
print(result.fit_report())
result.plot_fit(show_init=True)
plt.show()
我已经删除了大部分参数边界——不要使用任意和严格的边界，除非你知道你需要这样做，如果你甚至没有绘制你的数据，那么你肯定不知道.
为此配合打印的报告是
[[Model]]
Model(damped_cosine)
[[Fit Statistics]]
# fitting method = leastsq
# function evals = 42
# data points = 562
# variables = 3
chi-square = 42.2473566
reduced chi-square = 0.07557667
Akaike info crit = -1448.43356
Bayesian info crit = -1435.43905
[[Variables]]
a: 1.1104e-10 +/- 0.00205742 (1852835584.87%) (init = 0.9)
k: 0.45910709 +/- 170843.547 (37212134.66%) (init = 0.42)
b: 0.05936966 +/- 0.00309385 (5.21%) (init = 0.1)
[[Correlations]] (unreported correlations are < 0.100)
C(a, b) = -0.707
数据图和拟合（以及初始猜测）是
我认为最明显的结论是阻尼振荡函数不能非常令人信服地表示您的数据。没有任何明显的完整振荡。
我有数据集 (x & y)，我想将其拟合为阻尼余弦，例如 (1-A+Acos(Kx))exp(- B*x) 通过使用 LMFIT（“非线性最小二乘法”）作为 link (https://lmfit.github.io/lmfit-py/intro.html)。为此，我尝试使用以下代码，但无法正确安装。我的代码出了什么问题？任何帮助或建议将不胜感激。
x = [0, 1.3, 1.7, 1.72, 1.84, 1.98, 2.02, 2.16, 2.2, 2.2, 2.3, 2.38, 2.5, 2.55, 2.75, 2.8, 2.82, 2.84, 2.9, 2.92, 3.1, 3.13, 3.18, 3.19, 3.22, 3.3, 3.38, 3.44, 3.49, 3.62, 3.64, 3.72, 3.72, 3.75, 3.8, 3.82, 3.86, 3.92, 4.0, 4.07, 4.1, 4.1, 4.13, 4.14, 4.14, 4.17, 4.21, 4.24, 4.24, 4.24, 4.28, 4.3, 4.38, 4.49, 4.62, 4.62, 4.67, 4.72, 4.73, 4.74, 4.76, 4.76, 4.81, 4.81, 4.88, 4.89, 4.9, 4.9, 4.94, 4.96, 5.03, 5.05, 5.06, 5.1, 5.1, 5.15, 5.16, 5.16, 5.19, 5.22, 5.22, 5.3, 5.37, 5.41, 5.46, 5.56, 5.63, 5.65, 5.65, 5.73, 5.76, 5.81, 5.86, 5.91, 5.98, 6.03, 6.05, 6.05, 6.06, 6.11, 6.14, 6.22, 6.25, 6.27, 6.27, 6.3, 6.3, 6.31, 6.36, 6.42, 6.42, 6.47, 6.48, 6.5, 6.51, 6.58, 6.59, 6.62, 6.65, 6.66, 6.67, 6.69, 6.72, 6.77, 6.8, 6.84, 6.87, 6.91, 6.94, 6.94, 6.94, 7.05, 7.14, 7.17, 7.22, 7.23, 7.24, 7.32, 7.32, 7.35, 7.38, 7.4, 7.41, 7.42, 7.44, 7.45, 7.49, 7.5, 7.52, 7.54, 7.6, 7.72, 7.75, 7.81, 7.9, 7.92, 7.95, 7.97, 7.98, 7.99, 8.02, 8.03, 8.03, 8.05, 8.06, 8.07, 8.1, 8.12, 8.14, 8.19, 8.2, 8.21, 8.24, 8.25, 8.28, 8.28, 8.29, 8.32, 8.38, 8.38, 8.43, 8.49, 8.52, 8.54, 8.54, 8.57, 8.7, 8.75, 8.75, 8.78, 8.79, 8.88, 8.88, 8.93, 8.95, 9.0, 9.01, 9.02, 9.03, 9.06, 9.07, 9.11, 9.14, 9.16, 9.17, 9.18, 9.19, 9.2, 9.3, 9.33, 9.44, 9.46, 9.59, 9.62, 9.62, 9.64, 9.66, 9.71, 9.73, 9.73, 9.75, 9.76, 9.76, 9.79, 9.88, 9.9, 9.93, 9.93, 9.95, 9.99, 10.01, 10.03, 10.04, 10.05, 10.07, 10.11, 10.13, 10.18, 10.22, 10.22, 10.31, 10.37, 10.38, 10.41, 10.42, 10.44, 10.5, 10.52, 10.55, 10.56, 10.56, 10.58, 10.6, 10.66, 10.68, 10.68, 10.69, 10.7, 10.73, 10.75, 10.81, 10.93, 10.96, 10.98, 10.98, 11.02, 11.04, 11.1, 11.14, 11.15, 11.15, 11.17, 11.19, 11.21, 11.23, 11.24, 11.28, 11.3, 11.31, 11.32, 11.33, 11.4, 11.42, 11.48, 11.5, 11.51, 11.6, 11.62, 11.62, 11.63, 11.65, 11.72, 11.74, 11.74, 11.94, 11.95, 11.98, 12.02, 12.02, 12.03, 12.04, 12.09, 12.11, 12.17, 12.2, 12.23, 12.26, 12.3, 12.31, 12.33, 12.33, 12.37, 12.38, 12.61, 12.63, 12.69, 12.7, 12.74, 12.79, 12.8, 12.84, 12.87, 12.9, 12.91, 12.92, 12.94, 13.0, 13.19, 13.2, 13.26, 13.29, 13.3, 13.31, 13.31, 13.34, 13.35, 13.36, 13.44, 13.48, 13.52, 13.59, 13.78, 13.83, 13.88, 13.98, 14.02, 14.05, 14.07, 14.1, 14.14, 14.19, 14.25, 14.33, 14.36, 14.38, 14.41, 14.46, 14.47, 14.53, 14.54, 14.57, 14.69, 14.72, 14.77, 14.78, 14.78, 14.8, 14.82, 14.82, 14.91, 14.92, 14.96, 14.96, 15.05, 15.09, 15.17, 15.2, 15.21, 15.25, 15.26, 15.31, 15.32, 15.36, 15.36, 15.4, 15.4, 15.4, 15.41, 15.47, 15.52, 15.6, 15.61, 15.61, 15.63, 15.65, 15.71, 15.77, 15.8, 15.84, 15.86, 15.88, 15.94, 15.94, 15.97, 15.98, 16.02, 16.03, 16.27, 16.43, 16.56, 16.64, 16.64, 16.64, 16.68, 16.88, 16.91, 16.92, 16.93, 16.97, 16.99, 17.0, 17.01, 17.02, 17.05, 17.13, 17.21, 17.32, 17.45, 17.59, 17.79, 17.8, 17.81, 17.87, 17.9, 17.92, 17.93, 17.93, 17.97, 17.98, 18.02, 18.05, 18.08, 18.11, 18.2, 18.24, 18.4, 18.48, 18.5, 18.51, 18.59, 18.65, 18.76, 18.76, 18.86, 18.86, 18.86, 18.87, 18.9, 18.92, 18.93, 19.05, 19.06, 19.17, 19.26, 19.27, 19.41, 19.47, 19.48, 19.54, 19.6, 19.66, 19.67, 19.68, 19.8, 19.9, 20.01, 20.04, 20.1, 20.49, 20.49, 20.5, 20.56, 20.65, 20.65, 20.7, 20.71, 20.78, 20.91, 21.11, 21.19, 21.2, 21.28, 21.29, 21.58, 21.62, 21.7, 21.7, 21.76, 21.76, 21.84, 21.85, 21.87, 21.9, 21.94, 22.0, 22.02, 22.09, 22.16, 22.3, 22.3, 22.41, 22.51, 22.53, 22.71, 22.77, 22.94, 23.17, 23.25, 23.33, 23.72, 23.87, 24.12, 24.14, 24.19, 24.34, 24.4, 24.6, 24.62, 24.62, 24.8, 25.01, 25.13, 25.4, 25.42, 25.81, 25.85, 25.89, 26.03, 26.17, 26.22, 26.41, 26.98, 27.01, 27.02, 27.06, 27.17, 27.49, 27.73, 28.14, 28.23, 28.37, 28.56, 28.83, 28.84, 30.32, 30.57, 31.95, 33.23, 33.46, 33.81, 33.85, 34.44]
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import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import least_squares
from lmfit import minimize, Parameters, Parameter, report_fit
x = np.asarray(x); y = np.asarray(y)
def fit_fc(params, t, data):
A = params['A'].value
K = params['K'].value
B = params['B'].value
model = (1-A+A*np.cos(K * t))*np.exp(-1 * B * t)
return model - data
params = Parameters()
params.add('A', value=.9, min =0, max =1)
params.add('K', value=0.42, min=-0.2, max=0.8)
params.add('B', value=0.1, min=.01, max=.1)
result = minimize(fit_fc, params, args=(x, y), method='leastsq')
report_fit(params) # write error report
y_lsq = (1-result.params['A'] + (result.params['A'] *np.cos(result.params['K']*x))*np.exp(result.params['B'])*x*(-1))
#plot results
plt.plot(x, y, 'o', label='Data')
plt.plot(x, y_lsq, label='Least_Square_Method')
plt.xlabel('$x$')
plt.ylabel('$y$')
plt.legend()
你试过画图吗？与您尝试的拟合相比，您的数据确实“嘈杂”。
我将您的脚本更改为使用 lmfit.Model，这样更容易进行曲线拟合。看起来像（为简单起见删除数据）：
import numpy as np
import matplotlib.pyplot as plt
from lmfit import Model, Parameters
x = np.asarray(x)
y = np.asarray(y)
def damped_cosine(t, a, k, b):
return (1-a+a*np.cos(k*t))*np.exp(-b*t)
params = Parameters()
params.add('a', value=0.9, min=0)
params.add('k', value=0.42)
params.add('b', value=0.1)
dmodel = Model(damped_cosine)
result = dmodel.fit(y, params, t=x)
print(result.fit_report())
result.plot_fit(show_init=True)
plt.show()
我已经删除了大部分参数边界——不要使用任意和严格的边界，除非你知道你需要这样做，如果你甚至没有绘制你的数据，那么你肯定不知道.
为此配合打印的报告是
[[Model]]
Model(damped_cosine)
[[Fit Statistics]]
# fitting method = leastsq
# function evals = 42
# data points = 562
# variables = 3
chi-square = 42.2473566
reduced chi-square = 0.07557667
Akaike info crit = -1448.43356
Bayesian info crit = -1435.43905
[[Variables]]
a: 1.1104e-10 +/- 0.00205742 (1852835584.87%) (init = 0.9)
k: 0.45910709 +/- 170843.547 (37212134.66%) (init = 0.42)
b: 0.05936966 +/- 0.00309385 (5.21%) (init = 0.1)
[[Correlations]] (unreported correlations are < 0.100)
C(a, b) = -0.707
数据图和拟合（以及初始猜测）是
我认为最明显的结论是阻尼振荡函数不能非常令人信服地表示您的数据。没有任何明显的完整振荡。