Data fitting using fmin

We have seen already how to find the minimum of a function using fmin, in this example we will see how use it to fit a set of data with a curve minimizing an error function:

from pylab import *
from numpy import *
from numpy.random import normal
from scipy.optimize import fmin

# parametric function, x is the independent variable
# and c are the parameters.
# it's a polynomial of degree 2
fp = lambda c, x: c[0]+c[1]*x+c[2]*x*x
real_p = rand(3)

# error function to minimize
e = lambda p, x, y: (abs((fp(p,x)-y))).sum()

# generating data with noise
n = 30
x = linspace(0,1,n)
y = fp(real_p,x) + normal(0,0.05,n)

# fitting the data with fmin
p0 = rand(3) # initial parameter value
p = fmin(e, p0, args=(x,y))

print 'estimater parameters: ', p
print 'real parameters: ', real_p

xx = linspace(0,1,n*3)
plot(x,y,'bo', xx,fp(real_p,xx),'g', xx, fp(p,xx),'r')

show()

The following figure will be showed, in green the original curve used to generate the noisy data, in blue the noisy data and in red the curve found in the minimization process:

The parameters will be printed also:

Optimization terminated successfully.
         Current function value: 0.861885
         Iterations: 77
         Function evaluations: 146
estimater parameters:  [ 0.92504602  0.87328979  0.64051926]
real parameters:  [ 0.86284356  0.95994753  0.67643758]