Python:具有非线性最小二乘的双曲线高斯拟合

Question

我对数学的了解有限,这就是我可能被困的原因.我有一个光谱,我试图适应两个高斯峰.我可以适应最大的峰值,但我无法适应最小的峰值.我知道我需要对两个峰值的高斯函数求和,但我不知道哪里出了问题.显示当前输出的图像:

蓝线是我的数据,绿线是我目前适合的.在我的数据中主峰左侧有一个肩膀,我目前正在尝试使用以下代码:

import matplotlib.pyplot as pt
import numpy as np
from scipy.optimize import leastsq
from pylab import *

time = []
counts = []


for i in open('/some/folder/to/file.txt', 'r'):
    segs = i.split()
    time.append(float(segs[0]))
    counts.append(segs[1])

time_array = arange(len(time), dtype=float)
counts_array = arange(len(counts))
time_array[0:] = time
counts_array[0:] = counts


def model(time_array0, coeffs0):
    a = coeffs0[0] + coeffs0[1] * np.exp( - ((time_array0-coeffs0[2])/coeffs0[3])**2 )
    b = coeffs0[4] + coeffs0[5] * np.exp( - ((time_array0-coeffs0[6])/coeffs0[7])**2 ) 
    c = a+b
    return c


def residuals(coeffs, counts_array, time_array):
    return counts_array - model(time_array, coeffs)

# 0 = baseline, 1 = amplitude, 2 = centre, 3 = width
peak1 = np.array([0,6337,16.2,4.47,0,2300,13.5,2], dtype=float)
#peak2 = np.array([0,2300,13.5,2], dtype=float)

x, flag = leastsq(residuals, peak1, args=(counts_array, time_array))
#z, flag = leastsq(residuals, peak2, args=(counts_array, time_array))

plt.plot(time_array, counts_array)
plt.plot(time_array, model(time_array, x), color = 'g') 
#plt.plot(time_array, model(time_array, z), color = 'r')
plt.show()

Answer 1

这段代码对我有用,前提是你只是拟合一个两个高斯分布组合的函数.

我刚刚创建了一个残差函数,它增加了两个高斯函数,然后从实际数据中减去它们.

我传给Numpy最小二乘函数的参数(p)包括:第一高斯函数的平均值(m),第一和第二高斯函数的平均值差(dm,即水平位移),标准差第一个(sd1)和第二个(sd2)的标准偏差.

import numpy as np
from scipy.optimize import leastsq
import matplotlib.pyplot as plt

######################################
# Setting up test data
def norm(x, mean, sd):
  norm = []
  for i in range(x.size):
    norm += [1.0/(sd*np.sqrt(2*np.pi))*np.exp(-(x[i] - mean)**2/(2*sd**2))]
  return np.array(norm)

mean1, mean2 = 0, -2
std1, std2 = 0.5, 1 

x = np.linspace(-20, 20, 500)
y_real = norm(x, mean1, std1) + norm(x, mean2, std2)

######################################
# Solving
m, dm, sd1, sd2 = [5, 10, 1, 1]
p = [m, dm, sd1, sd2] # Initial guesses for leastsq
y_init = norm(x, m, sd1) + norm(x, m + dm, sd2) # For final comparison plot

def res(p, y, x):
  m, dm, sd1, sd2 = p
  m1 = m
  m2 = m1 + dm
  y_fit = norm(x, m1, sd1) + norm(x, m2, sd2)
  err = y - y_fit
  return err

plsq = leastsq(res, p, args = (y_real, x))

y_est = norm(x, plsq[0][0], plsq[0][2]) + norm(x, plsq[0][0] + plsq[0][1], plsq[0][3])

plt.plot(x, y_real, label='Real Data')
plt.plot(x, y_init, 'r.', label='Starting Guess')
plt.plot(x, y_est, 'g.', label='Fitted')
plt.legend()
plt.show()

Answer 2

您可以使用scikit-learn中的高斯混合模型:

from sklearn import mixture
import matplotlib.pyplot
import matplotlib.mlab
import numpy as np
clf = mixture.GMM(n_components=2, covariance_type='full')
clf.fit(yourdata)
m1, m2 = clf.means_
w1, w2 = clf.weights_
c1, c2 = clf.covars_
histdist = matplotlib.pyplot.hist(yourdata, 100, normed=True)
plotgauss1 = lambda x: plot(x,w1*matplotlib.mlab.normpdf(x,m1,np.sqrt(c1))[0], linewidth=3)
plotgauss2 = lambda x: plot(x,w2*matplotlib.mlab.normpdf(x,m2,np.sqrt(c2))[0], linewidth=3)
plotgauss1(histdist[1])
plotgauss2(histdist[1])

您还可以使用下面的函数来使用ncomp参数拟合所需的高斯数:

from sklearn import mixture
%pylab

def fit_mixture(data, ncomp=2, doplot=False):
    clf = mixture.GMM(n_components=ncomp, covariance_type='full')
    clf.fit(data)
    ml = clf.means_
    wl = clf.weights_
    cl = clf.covars_
    ms = [m[0] for m in ml]
    cs = [numpy.sqrt(c[0][0]) for c in cl]
    ws = [w for w in wl]
    if doplot == True:
        histo = hist(data, 200, normed=True)
        for w, m, c in zip(ws, ms, cs):
            plot(histo[1],w*matplotlib.mlab.normpdf(histo[1],m,np.sqrt(c)), linewidth=3)
    return ms, cs, ws