高斯的2D拟合将不起作用

Question

I am trying to fit a 2d data set. I can plot it without any problem at all but when I try to fit it I always get the same error message that I do not understand:

def twod_Gaussian(x,y,Amp,x0,y0,sigma_x,sigma_y,Offset):
    return Amp*np.exp(-((x-x0)/(2*sigma_x**2)+(y-y0)/(2*sigma_y**2)))+Offset

# get data
filename = '20180503-1455-43_confocal_xy_data.dat'
data = np.genfromtxt(filename, comments='#')
values = open(filename)
val = values.readlines(500)
res = float(val[11][-4:-1])
values.close()
x = [float(i) for i in data[:, 0]]
y = [float(i) for i in data[:, 1]]
count_rate = [float(i) for i in data[:, 3]]



#reshape data
xmax = max(x)
xmin = min(x)
ymax = max(y)
ymin = min(y)
cmax = max(count_rate)
cmin = min(count_rate)
np.percentile(count_rate, 99)
resX = (xmax-xmin)/res
rexY = (ymax-ymin)/res
res2 = len(x)/res
xx = np.array(x).reshape((int(res2), int(res)))
yy = np.array(y).reshape((int(res2), int(res)))



#plot data
count_matrix = np.array(count_rate).reshape((int(res2), int(res)))
fig = plt.figure()
ax = fig.gca(projection='3d')
CS = ax.plot_surface((xx)/1e-6, (yy)/1e-6, count_matrix, cmap='plasma')
plt.ticklabel_format(style='plain', axis='both', scilimits=(0, 0))
ax.set_xlabel(r'X position ($\mathrm{\mu}$m)')
ax.set_ylabel(r'Y position ($\mathrm{\mu}$m)')
fig.colorbar(CS, ax=ax, extend='max', format='%.0e')
plt.show()

#initial guess and trying to fit it
p0=(98e-06,81.5e-06,50000,0.5e-06,0.5e-06,20000)
popt, pcov = curve_fit(twod_Gaussian, x, y, count_rate, p0)

I always get the error message

ValueError: sigma has incorrect shape.

If I change the last line to

popt, pcov = curve_fit(twod_Gaussian, (x, y), count_rate, p0)

I get as error message:

twod_Gaussian() missing 1 required positional argument: 'Offset'

I dont have any idea why the shape should be wrong?

Answer 1

First, I believe the problem is that your model function definition has an extra argument because you are using two separate arguments for your two dimensions ( x and y ), instead of the one argument for the independent variable that curve_fit requires. Changing your model function to be

def twod_Gaussian(xy, Amp,x0,y0,sigma_x,sigma_y,Offset):
    x, y = xy
    return Amp*np.exp(-((x-x0)/(2*sigma_x**2)+(y-y0)/(2*sigma_y**2)))+Offset

and then calling curve_fit with

xy = x, y
popt, pcov = curve_fit(twod_Gaussian, xy, count_rate, p0)

should do what you want.

Second, you might also consider using lmfit ( https://lmfit.github.io/lmfit-py ). This library has a slightly higher level interface to curve-fitting, still wrapped around the scipy.optimize solvers. For example, it treats parameters as first-class, named objects, and easily supports placing bounds and constraints on parameters outside of the model function. For your case, it also supports defining multiple independent variables for a fitting function, not forcing you to assume that only the first argument of the model function is the independent variable, and all others are meant to be variables in the fit. With lmfit your problem would look approximately like (I didn't run this because I don't have your data):

from lmfit import Model
def twod_Gaussian(x, y, Amp,x0,y0,sigma_x,sigma_y,Offset):
    return Amp*np.exp(-((x-x0)/(2*sigma_x**2)+(y-y0)/(2*sigma_y**2)))+Offset

# turn your model function into a Model, specifying independent variables
g2model = Model(twod_Gaussian, independent_vars=['x', 'y'])

# create Parameters object -- ordered dict of *named parameters, 
# giving initial values here:
params = g2model.make_params(Amp=98e-6, x0=81e-6, y0=50000., sigma_x=5e-7, sigma_y=5e-7, Offset=20000)

# Note that using names is *way* better than using an ordered list. 
# for example, you may have mixed up the order and really meant:
params = g2model.make_params(x0=98e-6, y0=81e-6, Amp=50000., sigma_x=5e-7, sigma_y=5e-7, Offset=20000)

# you can place bounds on parameters, perhaps as
params['sigma_x'].min = 0
params['sigma_y'].min = 0

# do fit, passing independent vars explicitly
result = g2model.fit(count_rate, params, x=x, y=y)

# print out full report of statistics, best-fit values and stderrs:
print(result.fit_report())

# array of best-fit data == result.best_fit

There are lots more features, described in the docs.

Finally, to make a 2-D Gaussian with your function, y should be a row-wise array ( x is column-wise). Otherwise your twod_Gaussian() will return a 1-d array of the same length as x and y . So, I think you want to add

y = np.vstack(y)

just after your creation of y from your data array.