根据评论,这里是使用scipy的differential_evolution模块确定有界参数估计的示例。该模块内部使用拉丁超立方算法来确保对参数空间进行彻底搜索,并需要在其中搜索的边界,虽然这些边界可以很宽松。默认情况下,differential_evolution模块将在使用边界后内部以curve_fit()的调用结束-这可以禁用-并确保最终拟合的参数没有被限制,此示例通过稍后调用curve_fit而不传递边界来实现。从打印结果中可以看出,对differential_evolution的调用显示第一个参数受到-0.185的限制,而这不适用于稍后对curve_fit()的调用。在您的情况下,您可以使下限为零,以便参数不为负数,但如果代码导致参数达到或非常接近边界,则这不是最佳选择,如此示例所示。
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(t, n_0, L, offset):
return n_0*numpy.exp(-L*t) + offset
def sumOfSquaredError(parameterTuple):
warnings.filterwarnings("ignore")
val = func(xData, *parameterTuple)
return numpy.sum((yData - val) ** 2.0)
def generate_Initial_Parameters():
maxX = max(xData)
minX = min(xData)
maxY = max(yData)
minY = min(yData)
parameterBounds = []
parameterBounds.append([-0.185, maxX])
parameterBounds.append([minX, maxX])
parameterBounds.append([0.0, maxY])
result = differential_evolution(sumOfSquaredError, parameterBounds, seed=3)
return result.x
geneticParameters = generate_Initial_Parameters()
print('fit with parameter bounds (note the -0.185)')
print(geneticParameters)
print()
fittedParameters, pcov = curve_fit(func, xData, yData, geneticParameters)
print('re-fit with no parameter bounds')
print(fittedParameters)
print()
modelPredictions = func(xData, *fittedParameters)
absError = modelPredictions - yData
SE = numpy.square(absError)
MSE = numpy.mean(SE)
RMSE = numpy.sqrt(MSE)
Rsquared = 1.0 - (numpy.var(absError) / numpy.var(yData))
print()
print('RMSE:', RMSE)
print('R-squared:', Rsquared)
print()
def ModelAndScatterPlot(graphWidth, graphHeight):
f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)
axes = f.add_subplot(111)
axes.plot(xData, yData, 'D')
xModel = numpy.linspace(min(xData), max(xData))
yModel = func(xModel, *fittedParameters)
axes.plot(xModel, yModel)
axes.set_xlabel('X Data')
axes.set_ylabel('Y Data')
plt.show()
plt.close('all')
graphWidth = 800
graphHeight = 600
ModelAndScatterPlot(graphWidth, graphHeight)