在Python中拟合具有固定参数的函数之和

Question

我要适合的信号是多个正弦函数（和噪声）的叠加，我想同时适合所有频率。 这里是一个示例数据文件，它以240d ^ -1和261.8181d ^ -1两个频率生成： https ://owncloud.gwdg.de/index.php/s/JZQTJ3VMYZH8qNB和时间序列图（节选）

到目前为止，我可以将一个正弦函数设置为另一个正弦函数，同时将频率固定为一个值。 我从周期图获得频率，最后我对拟合的幅度和相位感兴趣。

import numpy as np
from scipy import optimize
import bottleneck as bn

def f_sinus0(x,a,b,c,d):
    return a*np.sin(b*x+c)+d

def fit_single(t, flux, flux_err, freq_model, c0 = 0.):

    # initial guess for the parameter
    d0 = bn.nanmean(flux)
    a0 = 3*np.std(flux)/np.sqrt(2.)

    # fit function with fixed frequency "freq_model"
    popt, pcov = optimize.curve_fit(lambda x, a, c, d:
        f_sinus0(x, a, freq_model*2*np.pi, c, d),
        t, flux, sigma = flux_err, p0 = (a0,c0,d0),
        bounds=([a0-0.5*abs(a0),-np.inf,d0-0.25*abs(d0)],
        [a0+0.5*abs(a0),np.inf,d0+0.25*abs(d0)]),
        absolute_sigma=True)
    perr = np.sqrt(np.diag(pcov))

    return popt, perr

filename = 'data-test.csv'

data = np.loadtxt(filename)
time = data[0]
flux = data[1]
flux_err = data[2]

freq_model = 260 #d^-1

popt, perr = fit_single(time, flux, flux_err, freq_model, c0 = 0.)

现在，我想同时调整两个频率。 我定义了一个函数，该函数根据像这样的input-parameter-list的长度返回拟合函数之和

def f_multiple_sin(x, *params):
    y = np.zeros_like(x)
    for i in range(0, len(params), 4): #4=amplitude, freq, phase, offset
        amplitude = params[i]
        freq = params[i+1]
        phase = params[i+2]
        offset = params[i+3]
        y = y + amplitude*np.sin(np.multiply(freq, x)+phase)+offset
    return y

进行拟合

def fit_multiple(t, flux, flux_err, guess):
    popt, pcov = optimize.curve_fit(
        f_multiple_sin, t, flux, sigma=flux_err, p0=guess,
        bounds=(guess-np.multiply(guess,0.1),guess+np.multiply(guess,0.1)),
        absolute_sigma=True
        )

    perr = np.sqrt(np.diag(pcov))

    return popt, perr

guess = [4.50148944e-03, 2.40000040e+02, 3.01766641e-03, 8.99996136e-01, 3.14546648e-03, 2.61818207e+02, 2.94282247e-03, 5.56770657e-06]
popt, perr = fit_multiple(time, flux, flux_err, guess)

使用来自各个拟合的结果作为初始参数guess = [amplitude1, frequency1, phase1, offset1, amplitude2,...]

但是，我该如何拟合多个具有固定频率的正弦函数？ 在这种情况下， lambda方法对我而言似乎不是那么直接。

Answer 1

这是使用scipy.optimize.leastsq的解决方案，它为我提供了更多的自由。 但是，在错误评估中，您必须格外小心。 在另一方面，它是没有那么严格的curve_fit有关参数的数量。 在这个解决方案适合我基本上三个列表，幅度，频率和相位。 At似乎很方便地将其排序给函数。 最后，您可以修复任何频率子集。 不过，我给人的印象是，收敛对起始参数非常敏感。

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


def multisine(x, ampList, freqList, phaseList):
    assert len( ampList ) == len( freqList )
    assert len( ampList ) == len( phaseList )
    out=0
    for a, f, p in zip( ampList, freqList, phaseList ):
        out += a * np.sin( x * f + p )
    return out


### FixedFrequencies is a list of values and positions in the list to pass to multisine....remember counting from zero
def multisine_fixed_fs(x, params, n, FixedFrequencies=None):
    if FixedFrequencies is None:
        assert len( params ) == 3 *  n
        ampList = params[ : n]
        freqList = params[ n : 2* n] 
        phaseList = params[ 2 * n : ]
    else:
        assert len( params ) + len( FixedFrequencies ) == 3 *  n
        ampList = params[ : n]
        freqList = list(params[ n : -n ])
        phaseList = params[ -n : ]
        sortedList = sorted( list(FixedFrequencies), key=lambda x: x[-1] )
        for fixed in sortedList:
            freqList.insert(fixed[-1], fixed[0] )

    return multisine(x, ampList, freqList, phaseList)


def residuals(params, data, n, FixedFrequencies=None):
    xList, yList = zip( *data )
    thyList = [ multisine_fixed_fs( x, params, n , FixedFrequencies=FixedFrequencies ) for x in xList ]
    d = [ y1- y2 for y1, y2 in zip( yList, thyList ) ]
    return d



xList = np.linspace( 0, 100, 100 )
yList = np.fromiter( ( multisine(x, [ 1, .3 ], [ .4, .42 ],[ 0, .1] ) for x in xList ), np.float )

data = zip( xList, yList )

fit, err = so.leastsq( residuals,  x0=[ 1.2, .32 ] + [ .42, .43 ] + [ 0.1, 0.12 ], args=( data, 2 ) )
print fit

fit, err = so.leastsq( residuals,  x0=[ 1.2, .32 ] + [ .42 ] + [ 0.1, 0.12 ], args=( data, 2 , [ [ .45, 1 ] ]) )
print fit
y2List = np.fromiter( ( multisine(x, [ fit[0], fit[1] ], [ fit[2], .45 ],[ fit[-2], fit[-1] ] ) for x in xList ), np.float )

fit, err = so.leastsq( residuals,  x0=[ 1.2, .32 ]  + [ 0.1, 0.12 ], args=( data, 2 , [ [ .39, 0 ],[ .45, 1 ] ]) )
print fit
y3List = np.fromiter( ( multisine(x, [ fit[0], fit[1] ], [ .39, .45 ],[ fit[-2], fit[-1] ] ) for x in xList ), np.float )

fig = plt.figure(1)
ax = fig.add_subplot( 1, 1, 1 )
ax.plot(xList,yList)
ax.plot(xList,y2List)
ax.plot(xList,y3List)

plt.show()

提供：

>> [ 1.00000006e+00   2.99999889e-01   3.99999999e-01   4.20000009e-01 1.47117910e-07   6.38318486e+00 ]
>> [ 1.12714624  0.12278804  0.40198029  0.08039605 -1.08564396 ]
>> [ 1.05124097 -0.32600116  0.6633511   1.18400026 ]