Scipy curve_fit 给出错误答案

Question

我有一个如下图所示的振荡数据，并希望对其拟合正弦曲线。但是，我的结果不正确。

我想拟合这条曲线的函数是：

def radius (z,phi, a0, k0,):

    Z = z.reshape(z.shape[0],1)

    k = np.array([k0,])
    a = np.array([a0,])
    r0 = 110
    rs  = r0 + np.sum(a*np.sin(k*Z +phi), axis=1)
    return rs

正确的解决方案可能如下所示：

r_fit = radius(z, phi=np.pi/.8, a0=10,k0=0.017)
plt.plot(z, r,  label='data')
plt.plot(z,  r_fit, label='fitted curve')
plt.legend()

然而，我拟合曲线的结果是：

from scipy.optimize import curve_fit
popt, pcov = curve_fit(radius, xdata=z, ydata=r)

r_fit = radius(z, *popt)
plt.plot(z, r,  label='data')
plt.plot(z,  r_fit, label='fitted curve')
plt.legend()

我的数据也如下：

r = np.array([100.09061214, 100.17932773, 100.45526772, 102.27891728,
       113.12440802, 119.30644014, 119.86570527, 119.75184665,
       117.12160143, 101.55081608, 100.07280857, 100.12880236,
       100.39251753, 103.05404178, 117.15257288, 119.74048706,
       119.86955437, 119.37452005, 112.83384329, 101.0507198 ,
       100.05521567])

z = np.array([-407.90074345, -360.38004677, -312.99221012, -266.36934609,
       -224.36240585, -188.55933945, -155.21242348, -122.02778866,
        -87.84335638,  -47.0274899 ,    0.        ,   47.54559191,
         94.97469981,  141.33801462,  181.59490575,  215.77219256,
        248.95956379,  282.28027286,  318.16440024,  360.7246922 ,
        407.940799  ])

由于我的函数仅表示傅立叶级数，因此我也尝试了 scipy.fftpack.fft(r)，但无法重现与我计算 fft 的信号接近的信号。

Answer 1

这是一个带有正弦方程的图形 Python 拟合器，您的数据使用 scipy.optimize 差分进化遗传算法模块来确定 curve_fit 非线性求解器的初始参数估计值。该 scipy 模块使用拉丁超立方算法来确保彻底搜索需要搜索范围的参数空间。在本例中，这些界限取自数据的最大值和最小值。

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

r = numpy.array([100.09061214, 100.17932773, 100.45526772, 102.27891728,
       113.12440802, 119.30644014, 119.86570527, 119.75184665,
       117.12160143, 101.55081608, 100.07280857, 100.12880236,
       100.39251753, 103.05404178, 117.15257288, 119.74048706,
       119.86955437, 119.37452005, 112.83384329, 101.0507198 ,
       100.05521567])

z = numpy.array([-407.90074345, -360.38004677, -312.99221012, -266.36934609,
       -224.36240585, -188.55933945, -155.21242348, -122.02778866,
        -87.84335638,  -47.0274899 ,    0.        ,   47.54559191,
         94.97469981,  141.33801462,  181.59490575,  215.77219256,
        248.95956379,  282.28027286,  318.16440024,  360.7246922 ,
        407.940799  ])

 # rename data to match previous example code
xData = z
yData = r


def func (x, amplitude, center, width, offset): # equation sine[radians] + offset from zunzun.com
 return amplitude * numpy.sin(numpy.pi * (x - center) / width) + 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)

    diffY = maxY - minY
    diffX = maxX - minX

    parameterBounds = []
    parameterBounds.append([0.0, diffY]) # search bounds for amplitude
    parameterBounds.append([minX, maxX]) # search bounds for center
    parameterBounds.append([0.0, diffX]) # search bounds for width
    parameterBounds.append([minY, maxY]) # search bounds for offset

    # "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 2

问题在于，如果不提供初始猜测，则解决方案无法收敛。尝试添加一个合理的初始猜测：

p0 = [np.pi/.8, 10, 0.017]
popt, pcov = curve_fit(radius, xdata=z, ydata=r, p0=p0)

请注意，如果您要使用其他方法之一，例如 trf 或 dogbox，那么在没有初始猜测的情况下，由于参数无法收敛，这更有可能返回运行时错误。