python curve_fit does not give reasonable fitting result

Question

I am trying to fit gaussian to a spectrum and the y values are on the order of 10^(-19). Curve_fit gives me poor fitting result, both before and after I multiply my whole data by 10^(-19). Attached is my code, it is fairly simple set of data except that the values are very small. If I want to keep my original values, how would I get a reasonable gaussian fit that would give me the correct parameters?

#get fits data
aaa=pyfits.getdata('p1.cal.fits')

aaa=np.matrix(aaa)
nrow=np.shape(aaa)[0]
ncol=np.shape(aaa)[1]

ylo=79
yhi=90
xlo=0
xhi=1023
glo=430
ghi=470

#sum all the rows to get spectrum
ysum=[]
for x in range(xlo,xhi):
sum=np.sum(aaa[ylo:yhi,x])
ysum.append(sum)

wavelen_pix=range(xhi-xlo)
max=np.max(ysum)
print "maximum is at x=", np.where(ysum==max)

##fit gaussian
#fit only part of my data in the chosen range [glo:ghi]
x=wavelen_pix[glo:ghi]
y=ysum[glo:ghi]
def func(x, a, x0, sigma):
    return a*np.exp(-(x-x0)**2/float((2*sigma**2)))

sig=np.std(ysum[500:1000]) #std of background noise

popt, pcov = curve_fit(func, x, sig)
print popt  
#this gives me [1.,1.,1.], which is obviously wrong
gaus=func(x,popt[0],popt[1],popt[2])

aaa is a 153 by 1024 image matrix, partly looks like this:

matrix([[ -8.99793629e-20,   8.57133275e-21,   4.83523386e-20, ...,
-1.54811004e-20,   5.22941515e-20,   1.71179195e-20],
[  2.75769318e-20,   1.03177243e-20,  -3.19634928e-21, ...,
1.66583803e-20,  -9.88712568e-22,  -2.56897725e-20],
[  2.88121935e-20,   8.57964252e-21,  -2.60784327e-20, ...,
1.72335180e-20,  -7.61189937e-21,  -3.45333075e-20],
..., 
[  1.04006903e-20,   1.61200683e-20,   7.04195205e-20, ...,
1.72459645e-20,   4.29404029e-20,   1.99889374e-20],
[  3.22315752e-21,  -5.61394194e-21,   3.28763096e-20, ...,
1.99063583e-20,   2.12989880e-20,  -1.23250648e-21],
[  3.66591810e-20,  -8.08647455e-22,  -6.22773168e-20, ...,
-4.06145681e-21,   4.92453132e-21,   4.23689309e-20]], dtype=float32)

Answer 1

Forget about rescaling, or making linear changes, or using the p0 parameter, which usually don't work! Try using the bounds parameter in the curve_fit for n parameters like this:

a0=np.array([a01,...,a0n])
af=np.array([af1,...,afn])
method="trf",bounds=(a0,af)

Hope it works! ;)

Answer 2

You are calling curve_fit incorrectly, here is the usage

curve_fit(f, xdata, ydata, p0=None, sigma=None, absolute_sigma=False, check_finite=True, **kw)

• f is your function whose first arg is an array of independent variables, and whose subsequent args are the function parameters (such as amplitude, center, etc)
• xdata are the independent variables
• ydata are the dependedent variable
• p0 is an initial guess at the function parameters (for Guassian this is amplitude, width, center)

By default p0 is set to a list of ones [1,1,...], which is probably why you get that as a result, the fit just never executed because you called it incorrectly.

Try estimating the amplitude, center, and width from the data, then make a p0 object (see below for details)

init_guess = ( a_i, x0_i, sig_i) # same order as they are supplied to your function
popt, pcov = curve_fit(func, xdata=x,ydata=y,p0=init_guess)

Here is a short example

xdata = np.linspace(0, 4, 50)
mygauss = ( 10,2,0.5) #( amp, center, width)
y     = func(xdata, *mygauss  ) # using your func defined above    
ydata = y + 2*(np.random.random(50)- 0.5) # add some noise to create fake data

Now I can guess the fit params

ai    = np.max( ydata) # guess the amplitude
xi    = xdata[ np.argmax( ydata)] # guess the position of center

Guessing the width is tricky, I would first find where the half max is located (there are two, but you only need to find one, as the Gaussian is symmetric):

pos_half = argmin( np.abs( ydata-ao/2 ) ) # subtract half the amplitude and find the minimum

Now evaluate how far this is from the center of the gaussian (xi) :

sig_i = np.abs( xi - xdata[ pos_half] ) # estimate the width

Now you can make make the initial guess

init_guess = (ai, xi sig_i)

and fit

params, variance = curve_fit( func, xdata=xdata, ydata=ydata, p0=init_guess)
print params
#array([ 9.99457443,  2.01992858,  0.49599629])

which is very close to mygauss. Hope it helps.