python高斯噪声函数_Python高斯拟合模拟高斯噪声数据

I need to interpolate data coming from an instrument using a gaussian fit. To this end I thought about using the curve_fit function from scipy.

Since I'd like to test this functionality on fake data before trying it on the instrument I wrote the following code to generate noisy gaussian data and to fit it:

from scipy.optimize import curve_fit

import numpy

import pylab

# Create a gaussian function

def gaussian(x, a, b, c):

val = a * numpy.exp(-(x - b)**2 / (2*c**2))

return val

# Generate fake data.

zMinEntry = 80.0*1E-06

zMaxEntry = 180.0*1E-06

zStepEntry = 0.2*1E-06

x = numpy.arange(zMinEntry,

zMaxEntry,

zStepEntry,

dtype = numpy.float64)

n = len(x)

meanY = zMinEntry + (zMaxEntry - zMinEntry)/2

sigmaY = 10.0E-06

a = 1.0/(sigmaY*numpy.sqrt(2*numpy.pi))

y = gaussian(x, a, meanY, sigmaY) + a*0.1*numpy.random.normal(0, 1, size=len(x))

# Fit

popt, pcov = curve_fit(gaussian, x, y)

# Print results

print("Scale = %.3f +/- %.3f" % (popt[0], numpy.sqrt(pcov[0, 0])))

print("Offset = %.3f +/- %.3f" % (popt[1], numpy.sqrt(pcov[1, 1])))

print("Sigma = %.3f +/- %.3f" % (popt[2], numpy.sqrt(pcov[2, 2])))

pylab.plot(x, y, 'ro')

pylab.plot(x, gaussian(x, popt[0], popt[1], popt[2]))

pylab.grid(True)

pylab.show()

Unfortunately this does not work properly, the output of the code is the following:

Scale = 6174.816 +/- 7114424813.672

Offset = 429.319 +/- 3919751917.830

Sigma = 1602.869 +/- 17923909301.176

And the plotted result is (blue is the fit function, red dots is the noisy input data):

I also tried to look at this answer, but couldn't figure out where my problem is.

Am I missing something here? Or am I using the curve_fit function in the wrong way? Thanks in advance!

解决方案

I agree with Olaf in so far as it is a question of scale. The optimal parameters differ by many orders of magnitude. However, scaling the parameters with which you generated your toy data does not seem to solve the problem for your actual application. curve_fit uses lestsq, which numerically approximates the Jacobian, where numerical problems arise because of the differences in scale (try to use the full_output keyword in curve_fit).

In my experience it is often best to use fmin which does not rely on approximated derivatives but uses only function values. You now have to write your own least-squares function that is to be optimized.

Starting values are still important. In your case you can make sufficiently good guesses by taking the maximum amplitude for a and the corresponding x-values for band c.

In code, it looks like this:

from scipy.optimize import curve_fit,fmin

import numpy

import pylab

# Create a gaussian function

def gaussian(x, a, b, c):

val = a * numpy.exp(-(x - b)**2 / (2*c**2))

return val

# Generate fake data.

zMinEntry = 80.0*1E-06

zMaxEntry = 180.0*1E-06

zStepEntry = 0.2*1E-06

x = numpy.arange(zMinEntry,

zMaxEntry,

zStepEntry,

dtype = numpy.float64)

n = len(x)

meanY = zMinEntry + (zMaxEntry - zMinEntry)/2

sigmaY = 10.0E-06

a = 1.0/(sigmaY*numpy.sqrt(2*numpy.pi))

y = gaussian(x, a, meanY, sigmaY) + a*0.1*numpy.random.normal(0, 1, size=len(x))

print a, meanY, sigmaY

# estimate starting values from the data

a = y.max()

b = x[numpy.argmax(a)]

c = b

# define a least squares function to optimize

def minfunc(params):

return sum((y-gaussian(x,params[0],params[1],params[2]))**2)

# fit

popt = fmin(minfunc,[a,b,c])

# Print results

print("Scale = %.3f" % (popt[0]))

print("Offset = %.3f" % (popt[1]))

print("Sigma = %.3f" % (popt[2]))

pylab.plot(x, y, 'ro')

pylab.plot(x, gaussian(x, popt[0], popt[1], popt[2]),lw = 2)

pylab.xlim(x.min(),x.max())

pylab.grid(True)

pylab.show()