此示例代码使用具有两个形状参数(a和b)和偏移项(不影响曲率)的表达式。方程为“y=1.0/(1.0+exp(-a(x-b)))+Offset”，参数值a=2.1540318329369712E-01，b=-6.6744890642157646E+00，Offset=-3.524129985969645e-01，R平方为0.988，RMSE为0.0085。

该示例包含您用Python代码发布的数据，用于拟合和绘制，并使用scipy.optimize.differential_evolution遗传算法自动估计初始参数。差分进化的scipy实现使用拉丁超立方体算法来确保对参数空间的彻底搜索，这需要搜索的范围-在本示例代码中，这些范围基于最大和最小数据值。

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([19.1647, 18.0189, 16.9550, 15.7683, 14.7044, 13.6269, 12.6040, 11.4309, 10.2987, 9.23465, 8.18440, 7.89789, 7.62498, 7.36571, 7.01106, 6.71094, 6.46548, 6.27436, 6.16543, 6.05569, 5.91904, 5.78247, 5.53661, 4.85425, 4.29468, 3.74888, 3.16206, 2.58882, 1.93371, 1.52426, 1.14211, 0.719035, 0.377708, 0.0226971, -0.223181, -0.537231, -0.878491, -1.27484, -1.45266, -1.57583, -1.61717])

yData = numpy.array([0.644557, 0.641059, 0.637555, 0.634059, 0.634135, 0.631825, 0.631899, 0.627209, 0.622516, 0.617818, 0.616103, 0.613736, 0.610175, 0.606613, 0.605445, 0.603676, 0.604887, 0.600127, 0.604909, 0.588207, 0.581056, 0.576292, 0.566761, 0.555472, 0.545367, 0.538842, 0.529336, 0.518635, 0.506747, 0.499018, 0.491885, 0.484754, 0.475230, 0.464514, 0.454387, 0.444861, 0.437128, 0.415076, 0.401363, 0.390034, 0.378698])

def func(x, a, b, Offset): # Sigmoid A With Offset from zunzun.com

return 1.0 / (1.0 + numpy.exp(-a * (x-b))) + Offset

# 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():

# min and max used for bounds

maxX = max(xData)

minX = min(xData)

maxY = max(yData)

minY = min(yData)

parameterBounds = []

parameterBounds.append([minX, maxX]) # seach bounds for a

parameterBounds.append([minX, maxX]) # seach bounds for b

parameterBounds.append([0.0, maxY]) # seach bounds for Offset

# "seed" the numpy random number generator for repeatable results

result = differential_evolution(sumOfSquaredError, parameterBounds, seed=3)

return result.x

# generate initial parameter values

geneticParameters = generate_Initial_Parameters()

# curve fit the test data

fittedParameters, pcov = curve_fit(func, xData, yData, geneticParameters)

print('Parameters', fittedParameters)

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('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)