scipy curve_fit 提高 "OptimizeWarning: Covariance of the parameters could not be estimated"

Question

我正在尝试使这个函数适合一些数据：

但是当我使用我的代码时

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

def f(x, start, end):
    res = np.empty_like(x)
    res[x < start] =-1
    res[x > end] = 1
    linear = np.all([[start <= x], [x <= end]], axis=0)[0]
    res[linear] = np.linspace(-1., 1., num=np.sum(linear))
    return res

if __name__ == '__main__':

    xdata = np.linspace(0., 1000., 1000)
    ydata = -np.ones(1000)
    ydata[500:1000] = 1.
    ydata = ydata + np.random.normal(0., 0.25, len(ydata))

    popt, pcov = curve_fit(f, xdata, ydata, p0=[495., 505.])
    print(popt, pcov)
    plt.figure()
    plt.plot(xdata, f(xdata, *popt), 'r-', label='fit')
    plt.plot(xdata, ydata, 'b-', label='data')
    plt.show()

我收到错误

OptimizeWarning: Covariance of the parameters could not be estimated

输出：

在此示例中，开始和结束应该接近 500，但它们与我最初的猜测完全没有变化。

Answer 1

的警告（非错误）

OptimizeWarning: Covariance of the parameters could not be estimated

表示拟合无法确定拟合参数的不确定性（方差）。

主要问题是您的模型函数 f 将参数 start 和 end 视为离散值——它们被用作函数形式变化的整数位置。 scipy 的 curve_fit（以及 scipy.optimize 中的所有其他优化例程）假设参数是 连续的 变量，而不是离散的。

拟合过程将尝试在参数中采取小步长（通常围绕机器精度）以获得残差相对于变量（雅可比行列式）的数值导数。将值用作离散变量时，这些导数将为零，并且拟合过程将不知道如何更改值以改善拟合。

您似乎在尝试将阶跃函数拟合到某些数据。请允许我推荐尝试 lmfit (https://lmfit.github.io/lmfit-py)，它提供了更高级别的曲线拟合接口，并具有许多内置模型。例如，它包含一个 StepModel 应该能够为您的数据建模。

为了对您的数据进行轻微修改（使其具有有限步长），以下带有 lmfit 的脚本可以适合此类数据：

#!/usr/bin/python
import numpy as np
from lmfit.models import StepModel, LinearModel
import matplotlib.pyplot as plt

np.random.seed(0)
xdata = np.linspace(0., 1000., 1000)
ydata = -np.ones(1000)
ydata[500:1000] = 1.
# note that a linear step is added here:
ydata[490:510] = -1 + np.arange(20)/10.0
ydata = ydata + np.random.normal(size=len(xdata), scale=0.1)

# model data as Step + Line
step_mod = StepModel(form='linear', prefix='step_')
line_mod = LinearModel(prefix='line_')

model = step_mod + line_mod

# make named parameters, giving initial values:
pars = model.make_params(line_intercept=ydata.min(),
                         line_slope=0,
                         step_center=xdata.mean(),
                         step_amplitude=ydata.std(),
                         step_sigma=2.0)

# fit data to this model with these parameters
out = model.fit(ydata, pars, x=xdata)

# print results
print(out.fit_report())

# plot data and best-fit
plt.plot(xdata, ydata, 'b')
plt.plot(xdata, out.best_fit, 'r-')
plt.show()

打印出

的报告

[[Model]]
    (Model(step, prefix='step_', form='linear') + Model(linear, prefix='line_'))
[[Fit Statistics]]
    # fitting method   = leastsq
    # function evals   = 49
    # data points      = 1000
    # variables        = 5
    chi-square         = 9.72660131
    reduced chi-square = 0.00977548
    Akaike info crit   = -4622.89074
    Bayesian info crit = -4598.35197
[[Variables]]
    step_sigma:      20.6227793 +/- 0.77214167 (3.74%) (init = 2)
    step_center:     490.167878 +/- 0.44804412 (0.09%) (init = 500)
    step_amplitude:  1.98946656 +/- 0.01304854 (0.66%) (init = 0.996283)
    line_intercept: -1.00628058 +/- 0.00706005 (0.70%) (init = -1.277259)
    line_slope:      1.3947e-05 +/- 2.2340e-05 (160.18%) (init = 0)
[[Correlations]] (unreported correlations are < 0.100)
    C(step_amplitude, line_slope)     = -0.875
    C(step_sigma, step_center)        = -0.863
    C(line_intercept, line_slope)     = -0.774
    C(step_amplitude, line_intercept) =  0.461
    C(step_sigma, step_amplitude)     =  0.170
    C(step_sigma, line_slope)         = -0.147
    C(step_center, step_amplitude)    = -0.146
    C(step_center, line_slope)        =  0.127

并生成一个图

Lmfit 有很多额外的功能。例如，如果你想对某些参数值设置边界或修复一些参数值的变化，你可以执行以下操作：

# make named parameters, giving initial values:
pars = model.make_params(line_intercept=ydata.min(),
                         line_slope=0,
                         step_center=xdata.mean(),
                         step_amplitude=ydata.std(),
                         step_sigma=2.0)

# now set max and min values for step amplitude"
pars['step_amplitude'].min = 0
pars['step_amplitude'].max = 100

# fix the offset of the line to be -1.0
pars['line_offset'].value = -1.0
pars['line_offset'].vary = False

# then run fit with these parameters
out = model.fit(ydata, pars, x=xdata)

如果您知道模型应该是 Step+Constant 并且常数应该是固定的，您也可以将模型修改为

from lmfit.models import ConstantModel
# model data as Step + Constant
step_mod = StepModel(form='linear', prefix='step_')
const_mod = ConstantModel(prefix='const_')

model = step_mod + const_mod

pars = model.make_params(const_c=-1,
                         step_center=xdata.mean(),
                         step_amplitude=ydata.std(),
                         step_sigma=2.0)
pars['const_c'].vary = False

scipy curve_fit 提高 "OptimizeWarning: Covariance of the parameters could not be estimated"

scipy curve_fit raises "OptimizeWarning: Covariance of the parameters could not be estimated"

python

scipy

curve-fitting

lmfit