使用 scipy curve_fit 拟合指数曲线（拟合曲线与真实曲线匹配）

Question

我正在尝试使用 curve_fit (scipy.optimize) 拟合指数曲线，但拟合曲线与真实曲线完全不同。现在我正在使用以下代码：

X=[0.0, 9.0, 18.0, 27.0, 36.0, 45.0, 54.0]
Y=[0.090316199, -0.078157925, -0.350137315, -0.695193468, -1.106773689, -1.60467115, -2.196169408]

#plot Y against X
fig = plt.figure(num=None, figsize=(9, 7),facecolor='w', edgecolor='k')
ax=fig.add_subplot(111)
ax.scatter(X,Y)

#fit using curve_fit
popt, pcov = curve_fit(func, X, Y,maxfev=10000)

#compute Y_estiamted using fitted parameters 
Y_estimated=[popt[0]*np.exp(i+popt[1])+popt[2] for i in X]

#plot Y_estiamted against X
ax.scatter(X,Y_estimated, c='r')

def func(x,a,b,c):
    return a*(np.exp(x+b))+c

蓝色曲线为真实曲线，红色曲线为拟合曲线。

如您所见，拟合的红色曲线与真实的蓝色曲线完全不匹配。如有任何帮助，我们将不胜感激！

Answer 1

我认为问题出在模型功能上。如果将其更改为类似以下的函数：

def func(x, a, b, c, d):
    return a * (np.exp(d*(x + b))) + c

然后它找到了一个不错的选择：

我更改了代码中的一些内容：

def func(x, a, b, c, d):
    return a * (np.exp(d*(x + b))) + c


X = [0.0, 9.0, 18.0, 27.0, 36.0, 45.0, 54.0]
Y = [0.090316199, -0.078157925, -0.350137315, -0.695193468, -1.106773689, -1.60467115, -2.196169408]


# plot Y against X
fig = plt.figure(num=None, figsize=(9, 7), facecolor='w', edgecolor='k')
ax = fig.add_subplot(111)
ax.scatter(X, Y)

# fit using curve_fit
popt, pcov = curve_fit(func, X, Y, maxfev=10000)

# compute Y_estiamted using fitted parameters
x = np.linspace(min(X), max(X), 100)
Y_estimated = func(x, *popt)

# plot Y_estiamted against X
ax.plot(x, Y_estimated, c='r')

Answer 2

我非常适合具有单个形状参数和小偏移量“1.0 - pow(a, x) + b”的渐近指数型方程。这是一个图形化的 Python 拟合器，它使用这个等式处理您的数据。

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

# ignore warnings within curve_fit() routine
import warnings
warnings.filterwarnings("ignore")

X=[0.0, 9.0, 18.0, 27.0, 36.0, 45.0, 54.0]
Y=[0.090316199, -0.078157925, -0.350137315, -0.695193468, -1.106773689, -1.60467115, -2.196169408]

# alias data to match previous example
xData = numpy.array(X, dtype=float)
yData = numpy.array(Y, dtype=float)

def func(x, a, b): # Asymptotic Exponential A equation with offset from zunzun.com
    return 1.0 - numpy.power(a, x) + b

# these are the same as the scipy defaults
initialParameters = numpy.array([1.0, 1.0])

# curve fit the test data
fittedParameters, pcov = curve_fit(func, xData, yData, initialParameters)

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('Parameters:', fittedParameters)
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)

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