# Exponential curve fitting in SciPy

## Question:

I have two NumPy arrays `x` and `y`. When I try to fit my data using exponential function and `curve_fit` (SciPy) with this simple code

``````#!/usr/bin/env python
from pylab import *
from scipy.optimize import curve_fit

x = np.array([399.75, 989.25, 1578.75, 2168.25, 2757.75, 3347.25, 3936.75, 4526.25, 5115.75, 5705.25])
y = np.array([109,62,39,13,10,4,2,0,1,2])

def func(x, a, b, c, d):
return a*np.exp(b-c*x)+d

popt, pcov = curve_fit(func, x, y)
``````

I get wrong coefficients `popt`

``````[a,b,c,d] = [1., 1., 1., 24.19999988]
``````

What is the problem?

Firstly I would recommend modifying your equation to `a*np.exp(-c*(x-b))+d`, otherwise the exponential will always be centered on `x=0` which may not always be the case.
You also need to specify reasonable initial conditions (the 4th argument to `curve_fit` specifies initial conditions for `[a,b,c,d]`).

This code fits nicely:

``````from pylab import *
from scipy.optimize import curve_fit

x = np.array([399.75, 989.25, 1578.75, 2168.25, 2757.75, 3347.25, 3936.75, 4526.25, 5115.75, 5705.25])
y = np.array([109,62,39,13,10,4,2,0,1,2])

def func(x, a, b, c, d):
return a*np.exp(-c*(x-b))+d

popt, pcov = curve_fit(func, x, y, [100,400,0.001,0])
print popt

plot(x,y)
x=linspace(400,6000,10000)
plot(x,func(x,*popt))
show()
``````

First comment: since `a*exp(b - c*x) = (a*exp(b))*exp(-c*x) = A*exp(-c*x)`, `a` or `b` is redundant. I’ll drop `b` and use:

``````def func(x, a, c, d):
return a*np.exp(-c*x)+d
``````

That isn’t the main issue. The problem is simply that `curve_fit` fails to converge to a solution to this problem when you use the default initial guess (which is all 1s). Check `pcov`; you’ll see that it is `inf`. This is not surprising, because if `c` is 1, most of the values of `exp(-c*x)` underflow to 0:

``````In [32]: np.exp(-x)
Out[32]:
array([  2.45912644e-174,   0.00000000e+000,   0.00000000e+000,
0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
0.00000000e+000])
``````

This suggests that `c` should be small. A better initial guess is, say, `p0 = (1, 1e-6, 1)`. Then I get:

``````In [36]: popt, pcov = curve_fit(func, x, y, p0=(1, 1e-6, 1))

In [37]: popt
Out[37]: array([  1.63561656e+02,   9.71142196e-04,  -1.16854450e+00])
``````

This looks reasonable:

``````In [42]: xx = np.linspace(300, 6000, 1000)

In [43]: yy = func(xx, *popt)

In [44]: plot(x, y, 'ko')