Python-更复杂函数的曲线拟合

Question

我希望找到下图的最佳拟合曲线的方程：

哪个方程式为：

我已经尝试找到使用numpy here和here进行曲线拟合的示例，但是它们都仅显示了如何仅绘制指数或仅绘制正弦曲线，但是我想绘制结合这两个功能的图形。

我该怎么做？

Answer 1

这是您可能会发现有用的一种方法。这使用了lmfit（http://lmfit.github.io/lmfit-py/），它提供了一种高级的曲线拟合方法：

import numpy as np
import matplotlib.pyplot as plt

from lmfit import Model

def decay_cosine(t, amp, beta, omega, phi):
    """model data as decaying cosine wave"""
    return amp * np.exp(-beta*t)* np.cos(omega*t + phi)

# create fake data to be fitted
t = np.linspace(0, 5, 101)
y = decay_cosine(t, 1.4, 0.9, 7.2, 0.23) + np.random.normal(size=len(t), scale=0.05)

# build model from decay_cosine
mod = Model(decay_cosine)

# create parameters, giving initial values
params = mod.make_params(amp=2.0, beta=0.5, omega=5, phi=0)

# you can place bounds on parameters:
params['phi'].max = np.pi/2
params['phi'].min = -np.pi/2
params['amp'].min = 0

# fit data to model

result = mod.fit(y, params, t=t)

# print out fit results
print(result.fit_report())

# plot data with best fit
plt.plot(t, y, 'bo', label='data')
plt.plot(t, result.best_fit, 'r')
plt.show()

这将打印出这样的报告：

[[Model]]
    Model(decay_cosine)
[[Fit Statistics]]
    # fitting method   = leastsq
    # function evals   = 46
    # data points      = 101
    # variables        = 4
    chi-square         = 0.25540159
    reduced chi-square = 0.00263301
    Akaike info crit   = -595.983903
    Bayesian info crit = -585.523421
[[Variables]]
    amp:    1.38812335 +/- 0.03034640 (2.19%) (init = 2)
    beta:   0.90760648 +/- 0.02820705 (3.11%) (init = 0.5)
    omega:  7.16579292 +/- 0.02891827 (0.40%) (init = 5)
    phi:    0.26249321 +/- 0.02225816 (8.48%) (init = 0)
[[Correlations]] (unreported correlations are < 0.100)
    C(omega, phi)  = -0.713
    C(amp, beta)   =  0.695
    C(amp, phi)    =  0.253
    C(amp, omega)  = -0.183
    C(beta, phi)   =  0.178
    C(beta, omega) = -0.128

并生成如下图：

Answer 2

这是一个非常简单的示例，使用了curve_fit中的leastsq和scipy.optimize。

1。设置参数值，模型和实验数据。

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

np.random.seed(0)  # choosing seed for reproducibility

# ==== theoretical parameter values ====
x0 = 1
beta = .5
omega = 2*np.pi
phi = 0
params = x0, beta, omega, phi

# ==== model ====
def decay_cosine(t, x0, beta, omega, phi):
    x = x0 * np.exp(-beta*t_data) * np.cos(omega*t_data + phi)
    return x

# ==== generating experimental data ====
t_data = np.linspace(0, 5, num=80)
noise = .05 * np.random.randn(t_data.size)
x_data = decay_cosine(t_data, *params) + noise

2。合身。

# ==== fitting using curve_fit ====
params_cf, _ = scipy.optimize.curve_fit(decay_cosine, t_data, x_data)

# ==== fitting using leastsq ====
def residuals(args, t, x):
    return x - decay_cosine(t, *args)

x0 = np.ones(len(params))  # initializing all params at one
params_lsq, _ = scipy.optimize.leastsq(residuals, x0, args=(t_data, x_data))

print(params_cf)
print(params_lsq)

array([ 1.04938794,  0.53877389,  6.30375113, -0.01850761])
array([ 1.04938796,  0.53877389,  6.30375103, -0.01850744])

3。绘图。

plt.plot(t_data, x_data, '.', label='exp data')
plt.plot(t_data, decay_cosine(t_data, *params_cf), label='curve_fit')
plt.plot(t_data, decay_cosine(t_data, *params_lsq), '--', label='leastsq')
plt.legend()
plt.grid(True)
plt.show()