We Keep Coding

curve_fit() got multiple values for argument 'p0' - What does this mean for the curve fit?

So, I wrote this script to interpret and graph data from a dynamic light scattering experiment. However, when I run the program I get this curvefit error for p0. Does any one have any idea what the problem could be? I've tried various things without any success. I need all values, so it doesn't really make much sense. Here is the script:
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
def exp_fun(x, A, tau,beta):
res = A * np.exp(-(x / tau) ** beta)
return res
def exp_fun2(x, A, tau, tau2):
res = (1-A) * np.exp(-(x / tau)) + ( A) * np.exp(-(x / tau2))
return res
def exp_fun3(x, A1, tau, tau2, tau3,A2):
res = (1 - A1-A2) * np.exp(-(x / tau)) + (A1) * np.exp(-(x / tau2))+ (A2) * np.exp(-(x / tau3))
return res
def exp_fun2_str1(x, A, tau, tau2, beta):
res = (1-A) * np.exp(-(x / tau)) + ( A) * np.exp(-(x / tau2) ** beta)
return res
def exp_fun2_str2(x, A, tau, tau2, beta, beta2):
res = (1-A)*np.exp(-(x / tau) ** beta) + ( A) * np.exp(-(x / tau2) ** beta2)
return res
filepath = "C:\\ref-bsa-10\\"
filename = 'BSA1000'
filenamestatic='bsa10table.txt'
numfiles = 24
# load the static data
static = np.loadtxt(filepath + filenamestatic , skiprows=1, usecols=[0, 1,8])
#following code changed due to unicode errors: probably because of text files
#static = pd.read_csv(filepath + filenamestatic , delimiter='\t',skiprows=1)
#staticdata= np.array(static.values)
#angles=staticdata[:,0]
q=np.sqrt(static[:,1])*10**-6
#load and analyse dynamics data
tau1 = np.zeros([numfiles])
tau2 = np.zeros([numfiles])
A = np.zeros([numfiles])
beta1 = np.zeros([numfiles])
beta2 = np.zeros([numfiles])
dbeta1 = np.zeros([numfiles])
dbeta2 = np.zeros([numfiles])
dtau2 = np.zeros([numfiles])
dtau1 = np.zeros([numfiles])
dA = np.zeros([numfiles])
colore = plt.cm.jet(np.linspace(0, 1, numfiles))
plt.figure()
ave_n=5
for i in range(numfiles):
data = pd.read_csv(filepath + filename + "{:02d}_averaged_corr.txt".format(i), delimiter='\t', skiprows=2)
print(data)
a = np.array(data.values)
g2=a[:, 1] / np.mean(a[1:10, 1])
dg2=a[:, 2] / np.mean(a[1:10, 1])
t=a[:, 0]
plt.errorbar(t, g2, dg2, color=colore[i])
popt, pcov = curve_fit(exp_fun, t, g2, dg2,
# A tau beta
bounds=([0.9, 0, 0, ],
[1.1, 10 ** 9, 2, ]),
p0 =([1, 100, 1, ]) )
plt.plot(t, exp_fun(t, popt[0], popt[1], popt[2]), color='red',
linestyle='dashed')
A[i] = popt[0]
dA[i] = np.sqrt(np.diag(pcov))[0]
tau1[i] = popt[1]
dtau1[i] = np.sqrt(np.diag(pcov))[1]
# tau2[i] = popt[2]
# dtau2[i] = np.sqrt(np.diag(pcov))[2]
beta1[i] = popt[2]
dbeta1[i] = np.sqrt(np.diag(pcov))[2]
# beta2[i] = popt[4]
# dbeta2[i] = np.sqrt(np.diag(pcov))[4]
plt.xlabel('$t$ (ms)')
plt.xscale('log')
plt.ylabel(r'$g_2$-1')
plt.grid('on')
plt.figure()
plt.errorbar(q,A, dA,marker='o',linestyle='none')
plt.xlabel('q ($\mathrm{\mu}$m$^{-1}$)')
plt.ylabel('A')
plt.grid('on')
#plt.figure()
#plt.errorbar(q**4,1/tau2, 1/(tau2**2)*dtau2,marker='o',linestyle='none')
#plt.xlabel('q$^4$ ($\mathrm{\mu}$m$^{-4}$)')
#plt.ylabel('$\Gamma_2$ (s$^{-1}$)')
#plt.grid('on')
plt.figure()
plt.errorbar(q**2,1/tau1, 1/(tau1**2)*dtau1,marker='o',linestyle='none')
poptDp, pcovDp = curve_fit(lambda x, *p: x * p[0], q ** 2,1/tau1, p0=[1, 0], bounds=(0, 1000),
sigma=1/(tau1**2)*dtau1, ftol=1e-14, xtol=1e-14) # old: 1e-15
plt.plot(q** 2 , q** 2 * poptDp[0] , color='magenta')
D=poptDp[0]
dD= np.sqrt(np.diag(pcovDp))[0]
plt.xlabel('q$^2$ ($\mathrm{\mu}$m$^{-2}$)')
plt.ylabel('$\Gamma_1$ (s$^{-1}$)')
plt.grid('on')
R=10*10**-9
Dms=D*10**3*(10**-6)**2
kb=1.380649*10**(-23)
eta=kb *(273.15+21)/(6*np.pi*R*Dms)
#comparing two functions
#plt.figure()
#plt.errorbar(q,1/tau2, 1/(tau2**2)*dtau2,marker='o',linestyle='none')
#plt.errorbar(q,1/tau1, 1/(tau1**2)*dtau1,marker='o',linestyle='none')
#plt.xlabel('q ($\mathrm{\mu}$m$^{-1}$)')
#plt.ylabel('$\Gamma$ (s$^{-1}$)')
#plt.grid('on')
#plt.xscale('log')
#plt.yscale('log')
plt.figure()
plt.errorbar(q,beta1, dbeta1,marker='o',linestyle='none')
#plt.errorbar(q,beta2, dbeta2,marker='o',linestyle='none')
plt.grid('on')
plt.xlabel('q ($\mathrm{\mu}$m$^{-1}$)')
plt.ylabel('KWW')

Scipy odeint. Gravitational motion

I solve the motion in gravitational field around the sun with scipy and mathplotlib and have a problem. My solution is not correct. It is not like in example. Formulas that I used.
from scipy.integrate import odeint
import scipy.constants as constants
import numpy as np
import matplotlib.pyplot as plt
M = 1.989 * (10 ** 30)
G = constants.G
alpha = 30
alpha0 = (alpha / 180) * np.pi
v00 = 0.7
v0 = v00 * 1000
def dqdt(q, t):
x = q[0]
y = q[2]
ax = - G * M * (x / ((x ** 2 + y ** 2) ** 1.5))
ay = - G * M * (y / ((x ** 2 + y ** 2) ** 1.5))
return [q[1], ax, q[3], ay]
vx0 = v0 * np.cos(alpha0)
vy0 = v0 * np.sin(alpha0)
x0 = -150 * (10 ** 11)
y0 = 0 * (10 ** 11)
q0 = [x0, vx0, y0, vy0]
N = 1000000
t = np.linspace(0.0, 100000000000.0, N)
pos = odeint(dqdt, q0, t)
x1 = pos[:, 0]
y1 = pos[:, 2]
plt.plot(x1, y1, 0, 0, 'ro')
plt.ylabel('y')
plt.xlabel('x')
plt.grid(True)
plt.show()
How can I fix this?
Maybe you can tell me solution with another method for example with Euler's formula or with using other library.
I will be very greatful if you help me.

Plot a ball rolling down a curve

I want to "animate" a circle rolling over the sin graph, I made a code where the circle moves rapidly down a straight line, now I want the same but the acceleration will be changing.
My previous code:
import numpy as np
import matplotlib.pyplot as plt
theta = np.arange(0, np.pi * 2, (0.01 * np.pi))
x = np.arange(-50, 1, 1)
y = x - 7
plt.figure()
for t in np.arange(0, 4, 0.1):
plt.plot(x, y)
xc = ((-9.81 * t**2 * np.sin(np.pi / 2)) / 3) + (5 * np.cos(theta))
yc = ((-9.81 * t**2 * np.sin(np.pi / 2)) / 3) + (5 * np.sin(theta))
plt.plot(xc, yc, 'r')
xp = ((-9.81 * t**2 * np.sin(np.pi / 2)) / 3) + (5 * np.cos(np.pi * t))
yp = ((-9.81 * t**2 * np.sin(np.pi / 2)) / 3) + (5 * np.sin(np.pi * t))
plt.plot(xp, yp, 'bo')
plt.pause(0.01)
plt.cla()
plt.show()

You can do this by numerically integrating:
dt = 0.01
lst_x = []
lst_y = []
t = 0
while t < 10: #for instance
t += dt
a = get_acceleration(function, x)
x += v * dt + 0.5 * a * dt * dt
v += a * dt
y = get_position(fuction, x)
lst_x.append(x)
lst_y.append(y)
This is assuming the ball never leaves your slope! If it does, you'll also have to integrate in y in a similar way as done in x!!
Where your acceleration is going to be equal to g * cos(slope).

Still having trouble with curve fitting

I already opened a question on this topic, but I wasn't sure, if I should post it there, so I opened a new question here.
I have trouble again when fitting two or more peaks. First problem occurs with a calculated example function.
xg = np.random.uniform(0,1000,500)
mu1 = 200
sigma1 = 20
I1 = -2
mu2 = 800
sigma2 = 20
I2 = -1
yg3 = 0.0001*xg
yg1 = (I1 / (sigma1 * np.sqrt(2 * np.pi))) * np.exp( - (xg - mu1)**2 / (2 * sigma1**2) )
yg2 = (I2 / (sigma2 * np.sqrt(2 * np.pi))) * np.exp( - (xg - mu2)**2 / (2 * sigma2**2) )
yg=yg1+yg2+yg3
plt.figure(0, figsize=(8,8))
plt.plot(xg, yg, 'r.')
I tried two different approaches, I found in the documentation, which are shown below (modified for my data), but both give me wrong fitting data and a messy chaos of graphs (I guess one line per fitting step).
1st attempt:
import numpy as np
from lmfit.models import PseudoVoigtModel, LinearModel, GaussianModel, LorentzianModel
import sys
import matplotlib.pyplot as plt
gauss1 = PseudoVoigtModel(prefix='g1_')
pars.update(gauss1.make_params())
pars['g1_center'].set(200)
pars['g1_sigma'].set(15, min=3)
pars['g1_amplitude'].set(-0.5)
pars['g1_fwhm'].set(20, vary=True)
#pars['g1_fraction'].set(0, vary=True)
gauss2 = PseudoVoigtModel(prefix='g2_')
pars.update(gauss2.make_params())
pars['g2_center'].set(800)
pars['g2_sigma'].set(15)
pars['g2_amplitude'].set(-0.4)
pars['g2_fwhm'].set(20, vary=True)
#pars['g2_fraction'].set(0, vary=True)
mod = gauss1 + gauss2 + LinearModel()
pars.add('intercept', value=0, vary=True)
pars.add('slope', value=0.0001, vary=True)
init = mod.eval(pars, x=xg)
out = mod.fit(yg, pars, x=xg)
print(out.fit_report(min_correl=0.5))
plt.figure(5, figsize=(8,8))
out.plot_fit()
When I include the 'fraction'-parameter, I often get
'NameError: name 'pv1_fraction' is not defined in expr='<_ast.Module object at 0x00000000165E03C8>'.
although it should be defined. I get this Error for real data with this approach, too.
2nd attempt:
import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
import lmfit
def gauss(x, sigma, mu, A):
return A*np.exp(-(x-mu)**2/(2*sigma**2))
def linear(x, m, n):
return m*x + n
peak1 = lmfit.model.Model(gauss, prefix='p1_')
peak2 = lmfit.model.Model(gauss, prefix='p2_')
lin = lmfit.model.Model(linear, prefix='l_')
model = peak1 + lin + peak2
params = model.make_params()
params['p1_mu'] = lmfit.Parameter(value=200, min=100, max=250)
params['p2_mu'] = lmfit.Parameter(value=800, min=100, max=1000)
params['p1_sigma'] = lmfit.Parameter(value=15, min=0.01)
params['p2_sigma'] = lmfit.Parameter(value=20, min=0.01)
params['p1_A'] = lmfit.Parameter(value=-2, min=-3)
params['p2_A'] = lmfit.Parameter(value=-2, min=-3)
params['l_m'] = lmfit.Parameter(value=0)
params['l_n'] = lmfit.Parameter(value=0)
out = model.fit(yg, params, x=xg)
print out.fit_report()
plt.figure(8, figsize=(8,8))
out.plot_fit()
So the result looks like this, in both cases. It seems to plot all fitting attempts, but never solves it correctly. The best fitted parameters are in the range that I gave it.
Anyone knows this type of error? Or has any solutions for this? And does anyone know how to avoid the NameError when calling a model function from lmfit with those approaches?

I have a somewhat tolerable solution for you. Since I don't know how variable your data is, I cannot say that it will work in a general sense but should get you started. If your data is along 0-1000 and has two peaks or dips along a line as you showed, then it should work.
I used the scipy curve_fit and put all of the components of the function together into one function. One can pass starting locations into the curve_fit function. (you can probably do this with the lib you're using but I'm not familiar with it) There is a loop in loop where I vary the mu parameters to find the ones with the lowest squared error. If you are needing to fit your data many times or in some real-time scenario then this is not for you but if you just need to fit some data, launch this code and grab a coffee.
from scipy.optimize import curve_fit
import numpy as np
import matplotlib.pyplot as plt
import pylab
from matplotlib import cm as cm
import time
def my_function_big(x, m, n, #lin vars
sigma1, mu1, I1, #gaussian 1
sigma2, mu2, I2): #gaussian 2
y = m * x + n + (I1 / (sigma1 * np.sqrt(2 * np.pi))) * np.exp( - (x - mu1)**2 / (2 * sigma1**2) ) + (I2 / (sigma2 * np.sqrt(2 * np.pi))) * np.exp( - (x - mu2)**2 / (2 * sigma2**2) )
return y
#make some data
xs = np.random.uniform(0,1000,500)
mu1 = 200
sigma1 = 20
I1 = -2
mu2 = 800
sigma2 = 20
I2 = -1
yg3 = 0.0001 * xs
yg1 = (I1 / (sigma1 * np.sqrt(2 * np.pi))) * np.exp( - (xs - mu1)**2 / (2 * sigma1**2) )
yg2 = (I2 / (sigma2 * np.sqrt(2 * np.pi))) * np.exp( - (xs - mu2)**2 / (2 * sigma2**2) )
ys = yg1 + yg2 + yg3
xs = np.array(xs)
ys = np.array(ys)
#done making data
#start a double loop...very expensive but this is quick and dirty
#it would seem that the regular optimizer has trouble finding the minima so i
#found that having the near proper mu values helped it zero in much better
start = time.time()
serr = []
_x = []
_y = []
for x in np.linspace(0, 1000, 61):
for y in np.linspace(0, 1000, 61):
cfiti = curve_fit(my_function_big, xs, ys, p0=[0, 0, 1, x, 1, 1, y, 1], maxfev=20000000)
serr.append(np.sum((my_function_big(xs, *cfiti[0]) - ys) ** 2))
_x.append(x)
_y.append(y)
serr = np.array(serr)
_x = np.array(_x)
_y = np.array(_y)
print 'done loop in loop fitting'
print 'time: %0.1f' % (time.time() - start)
gridsize=20
plt.subplot(111)
plt.hexbin(_x, _y, C=serr, gridsize=gridsize, cmap=cm.jet, bins=None)
plt.axis([_x.min(), _x.max(), _y.min(), _y.max()])
cb = plt.colorbar()
cb.set_label('SE')
plt.show()
ix = np.argmin(serr.ravel())
mustart1 = _x.ravel()[ix]
mustart2 = _y.ravel()[ix]
print mustart1
print mustart2
cfit = curve_fit(my_function_big, xs, ys, p0=[0, 0, 1, mustart1, 1, 1, mustart2, 1], maxfev=2000000000)
xp = np.linspace(0, 1000, 1001)
plt.figure()
plt.scatter(xs, ys) #plot synthetic dat
plt.plot(xp, my_function_big(xp, *cfit[0]), '-', label='fit function') #plot data evaluated along 0-1000
plt.legend(loc=3, numpoints=1, prop={'size':12})
plt.show()
pylab.close()
Good luck!

In your first attempt:
pars['g1_fraction'].set(0, vary=True)
The fraction must be a value between 0 and 1, but I believe that cannot be zero. Try to put something like 0.000001, and it will work.

Create scipy curve fitting definitions for fourier series dynamically

I'd like to achieve a fourier series development for a x-y-dataset using numpy and scipy.
At first I want to fit my data with the first 8 cosines and plot additionally only the first harmonic. So I wrote the following two function defintions:
# fourier series defintions
tau = 0.045
def fourier8(x, a1, a2, a3, a4, a5, a6, a7, a8):
return a1 * np.cos(1 * np.pi / tau * x) + \
a2 * np.cos(2 * np.pi / tau * x) + \
a3 * np.cos(3 * np.pi / tau * x) + \
a4 * np.cos(4 * np.pi / tau * x) + \
a5 * np.cos(5 * np.pi / tau * x) + \
a6 * np.cos(6 * np.pi / tau * x) + \
a7 * np.cos(7 * np.pi / tau * x) + \
a8 * np.cos(8 * np.pi / tau * x)
def fourier1(x, a1):
return a1 * np.cos(1 * np.pi / tau * x)
Then I use them to fit my data:
# import and filename
filename = 'data.txt'
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
z, Ua = np.loadtxt(filename,delimiter=',', unpack=True)
tau = 0.045
# plot data
fig = plt.figure()
ax1 = fig.add_subplot(111)
p1, = plt.plot(z,Ua)
# fits
popt, pcov = curve_fit(fourier8, z, Ua)
# further plots
Ua_fit8 = fourier8(z,*popt)
Ua_fit1 = fourier1(z,popt[0])
p2, = plt.plot(z,Ua_fit8)
p3, = plt.plot(z,Ua_fit1)
plt.show()
which works as desired:
But know I got stuck making it generic for arbitary orders of harmonics, e.g. I want to fit my data with the first fifteen harmonics and plot the first three harmonics.
How could I achieve that without defining fourier1, fourier2, fourier3 ... , fourier15?
Here is the example text file data.txt to play with it:
1.000000000000000021e-03,2.794682735905079767e+02
2.000000000000000042e-03,2.792294526290349950e+02
2.999999999999999629e-03,2.779794770690260179e+02
4.000000000000000083e-03,2.757183469104809888e+02
5.000000000000000104e-03,2.734572167519349932e+02
5.999999999999999258e-03,2.711960865933900209e+02
7.000000000000000146e-03,2.689349564348440254e+02
8.000000000000000167e-03,2.714324329982829909e+02
8.999999999999999320e-03,2.739299095617229796e+02
1.000000000000000021e-02,2.764273861251620019e+02
1.100000000000000110e-02,2.789248626886010243e+02
1.199999999999999852e-02,2.799669443683339978e+02
1.299999999999999940e-02,2.795536311643609793e+02
1.400000000000000029e-02,2.791403179603880176e+02
1.499999999999999944e-02,2.787270047564149991e+02
1.600000000000000033e-02,2.783136915524419805e+02
1.699999999999999775e-02,2.673604939531239779e+02
1.799999999999999864e-02,2.564072963538059753e+02
1.899999999999999953e-02,2.454540987544889958e+02
2.000000000000000042e-02,2.345009011551709932e+02
2.099999999999999784e-02,1.781413355804160119e+02
2.200000000000000219e-02,7.637540203022649621e+01
2.299999999999999961e-02,-2.539053151996269975e+01
2.399999999999999703e-02,-1.271564650701519952e+02
2.500000000000000139e-02,-2.289223986203409993e+02
2.599999999999999881e-02,-2.399383538664330047e+02
2.700000000000000316e-02,-2.509543091125239869e+02
2.800000000000000058e-02,-2.619702643586149975e+02
2.899999999999999800e-02,-2.729862196047059797e+02
2.999999999999999889e-02,-2.786861050144170235e+02
3.099999999999999978e-02,-2.790699205877460258e+02
3.200000000000000067e-02,-2.794537361610759945e+02
3.300000000000000155e-02,-2.798375517344049968e+02
3.399999999999999550e-02,-2.802213673077350222e+02
3.500000000000000333e-02,-2.776516459805940258e+02
3.599999999999999728e-02,-2.750819246534539957e+02
3.700000000000000511e-02,-2.725122033263129993e+02
3.799999999999999906e-02,-2.699424819991720028e+02
3.899999999999999994e-02,-2.698567311502329744e+02
4.000000000000000083e-02,-2.722549507794930150e+02
4.100000000000000172e-02,-2.746531704087540220e+02
4.199999999999999567e-02,-2.770513900380149721e+02
4.299999999999999656e-02,-2.794496096672759791e+02
4.400000000000000439e-02,-2.800761105821779893e+02
4.499999999999999833e-02,-2.800761105821779893e+02
4.599999999999999922e-02,-2.800761105821779893e+02
4.700000000000000011e-02,-2.800761105821779893e+02
4.799999999999999406e-02,-2.788333722531979788e+02
4.900000000000000189e-02,-2.763478955952380147e+02
5.000000000000000278e-02,-2.738624189372779938e+02
5.100000000000000366e-02,-2.713769422793179729e+02
5.199999999999999761e-02,-2.688914656213580088e+02
5.299999999999999850e-02,-2.715383673199499981e+02
5.400000000000000633e-02,-2.741852690185419874e+02
5.499999999999999334e-02,-2.768321707171339767e+02
5.600000000000000117e-02,-2.794790724157260229e+02
5.700000000000000205e-02,-2.804776351435970128e+02
5.799999999999999600e-02,-2.798278589007459800e+02
5.899999999999999689e-02,-2.791780826578950041e+02
5.999999999999999778e-02,-2.785283064150449945e+02
6.100000000000000561e-02,-2.778785301721940186e+02
6.199999999999999956e-02,-2.670252067497989970e+02
6.300000000000000044e-02,-2.561718833274049985e+02
6.400000000000000133e-02,-2.453185599050100052e+02
6.500000000000000222e-02,-2.344652364826150119e+02
6.600000000000000311e-02,-1.780224826854309867e+02
6.700000000000000400e-02,-7.599029851345700592e+01
6.799999999999999101e-02,2.604188565851649884e+01
6.900000000000000577e-02,1.280740698304900036e+02
7.000000000000000666e-02,2.301062540024639986e+02
7.100000000000000755e-02,2.404921248105050040e+02
7.199999999999999456e-02,2.508779956185460094e+02
7.299999999999999545e-02,2.612638664265870148e+02
7.400000000000001021e-02,2.716497372346279917e+02
7.499999999999999722e-02,2.773051723900500178e+02
7.599999999999999811e-02,2.782301718928520131e+02
7.699999999999999900e-02,2.791551713956549747e+02
7.799999999999999989e-02,2.800801708984579932e+02
7.900000000000001465e-02,2.810051704012610116e+02
8.000000000000000167e-02,2.785107135689390248e+02
8.099999999999998868e-02,2.760162567366169810e+02
8.200000000000000344e-02,2.735217999042949941e+02
8.300000000000000433e-02,2.710273430719730072e+02
8.399999999999999134e-02,2.706544464035359852e+02
8.500000000000000611e-02,2.724031098989830184e+02
8.599999999999999312e-02,2.741517733944299948e+02
8.699999999999999400e-02,2.759004368898779944e+02
8.800000000000000877e-02,2.776491003853250277e+02
8.899999999999999578e-02,2.783445666445250026e+02
8.999999999999999667e-02,2.790400329037249776e+02

I would approach this by defining a single function fourier that accepts a variable number of arguments, using a loop through the cosines to evaluate the function for each of your input points:
def fourier(x, *a):
ret = a[0] * np.cos(np.pi / tau * x)
for deg in range(1, len(a)):
ret += a[deg] * np.cos((deg+1) * np.pi / tau * x)
return ret
Because this function takes a variable number of arguments, you need to provide a starting position of the correct dimensionality as a hint to the curve_fit function so it knows the number of cosines to use. In the example you provide with 8 cosines:
popt, pcov = curve_fit(fourier, z, Ua, [1.0] * 8)
Now, the use case you're asking about (fitting with 15 harmonics and plotting the first three) can be accomplished with:
# Fit with 15 harmonics
popt, pcov = curve_fit(fourier, z, Ua, [1.0] * 15)
# Plot data, 15 harmonics, and first 3 harmonics
fig = plt.figure()
ax1 = fig.add_subplot(111)
p1, = plt.plot(z,Ua)
p2, = plt.plot(z, fourier(z, *popt))
p3, = plt.plot(z, fourier(z, popt[0], popt[1], popt[2]))
plt.show()

based on this thread.
Passing additional arguments using scipy.optimize.curve_fit?
# import and filename
filename = 'data.txt'
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
z, Ua = np.loadtxt(filename,delimiter=',', unpack=True)
tau = 0.045
def make_fourier(na, nb):
def fourier(x, *a):
ret = 0.0
for deg in range(0, na):
ret += a[deg] * np.cos((deg+1) * np.pi / tau * x)
for deg in range(na, na+nb):
ret += a[deg] * np.sin((deg+1) * np.pi / tau * x)
return ret
return fourier
popt, pcov = curve_fit(make_fourier(15,15), z, Ua, [0.0]*30)
# Plot data, 15 harmonics, and first 3 harmonics
fig = plt.figure()
ax1 = fig.add_subplot(111)
p1, = plt.plot(z,Ua)
p2, = plt.plot(z, (make_fourier(15,15))(z, *popt))
#p3, = plt.plot(z, (make_fourier(8,8))(z, popt[0], popt[1], popt[2]))
plt.show()