将逻辑曲线拟合到数据

Question

我想用 scipy 为一些数据拟合一个对数函数。

不幸的是，我收到以下错误：无法估计参数的协方差

如何防止这种情况发生？

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

x = [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0]
y = [0.073, 2.521, 15.879, 48.365, 72.68, 90.298, 92.111, 93.44, 93.439, 93.389, 93.381, 93.367, 93.94, 93.269, 96.376]

def f(x, a, b, c, d):
    return a / (1. + np.exp(-c * (x - d))) + b

(a_, b_, c_, d_), _ = opt.curve_fit(f, x, y)

y_fit = f(x, a_, b_, c_, d_)
fig, ax = plt.subplots(1, 1, figsize=(6, 4))
ax.plot(x, y, 'o')
ax.plot(x, y_fit, '-')

Answer 1

经过几次尝试，我发现在计算与您的数据的协方差时存在问题。我试图删除 0.0 以防这是原因但不是。

我发现的唯一选择是将计算方法从 lm 更改为 trf ：

x = np.array(x)
y = np.array(y)

popt, pcov = opt.curve_fit(f, x, y, method="trf")
y_fit = f(x, *popt)
fig, ax = plt.subplots(1, 1, figsize=(6, 4))
ax.plot(x, y, 'o')
ax.plot(x, y_fit, '-')
plt.show()

曲线与这些参数正确拟合[96.2823169 -2.38876852 1.39927921 2.98341838]

Answer 2

这是一个包含您的数据和方程的图形拟合器，使用 scipy 的差分进化遗传算法进行初始参数估计。 scipy 实现使用拉丁超立方算法来确保对参数空间的彻底搜索，这需要搜索范围 - 正如您从代码中看到的那样，这些范围可以很宽，并且更容易为初始参数估计而不是给出具体值。

import numpy, scipy, matplotlib
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy.optimize import differential_evolution
import warnings


xData = numpy.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0])
yData = numpy.array([0.073, 2.521, 15.879, 48.365, 72.68, 90.298, 92.111, 93.44, 93.439, 93.389, 93.381, 93.367, 93.94, 93.269, 96.376])


def func(x, a, b, c, d):
    return a / (1.0 + numpy.exp(-c * (x - d))) + b


# function for genetic algorithm to minimize (sum of squared error)
def sumOfSquaredError(parameterTuple):
    warnings.filterwarnings("ignore") # do not print warnings by genetic algorithm
    val = func(xData, *parameterTuple)
    return numpy.sum((yData - val) ** 2.0)


def generate_Initial_Parameters():
    parameterBounds = []
    parameterBounds.append([0.0, 100.0]) # search bounds for a
    parameterBounds.append([-10.0, 0.0]) # search bounds for b
    parameterBounds.append([0.0, 10.0]) # search bounds for c
    parameterBounds.append([0.0, 10.0]) # search bounds for d

    # "seed" the numpy random number generator for repeatable results
    result = differential_evolution(sumOfSquaredError, parameterBounds, seed=3)
    return result.x

# by default, differential_evolution completes by calling curve_fit() using parameter bounds
geneticParameters = generate_Initial_Parameters()

# now call curve_fit without passing bounds from the genetic algorithm,
# just in case the best fit parameters are aoutside those bounds
fittedParameters, pcov = curve_fit(func, xData, yData, geneticParameters)
print('Fitted parameters:', fittedParameters)
print()

modelPredictions = func(xData, *fittedParameters) 

absError = modelPredictions - yData

SE = numpy.square(absError) # squared errors
MSE = numpy.mean(SE) # mean squared errors
RMSE = numpy.sqrt(MSE) # Root Mean Squared Error, RMSE
Rsquared = 1.0 - (numpy.var(absError) / numpy.var(yData))

print()
print('RMSE:', RMSE)
print('R-squared:', Rsquared)

print()


##########################################################
# graphics output section
def ModelAndScatterPlot(graphWidth, graphHeight):
    f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)
    axes = f.add_subplot(111)

    # first the raw data as a scatter plot
    axes.plot(xData, yData,  'D')

    # create data for the fitted equation plot
    xModel = numpy.linspace(min(xData), max(xData))
    yModel = func(xModel, *fittedParameters)

    # now the model as a line plot
    axes.plot(xModel, yModel)

    axes.set_xlabel('X Data') # X axis data label
    axes.set_ylabel('Y Data') # Y axis data label

    plt.show()
    plt.close('all') # clean up after using pyplot

graphWidth = 800
graphHeight = 600
ModelAndScatterPlot(graphWidth, graphHeight)

Answer 3

我在 Python2.7 内核下尝试了你的代码。我没有收到你提到的错误。对于 x 的所有值，唯一的情况是 y_fit=71.50186844。