Fitting data with integral function

Question

When using curve_fit from scipy.optimize to fit a some data in python, one first defines the fitting function (e.g. a 2nd order polynomial) as follows:

1. def f

Answer 1

Here's an example of fitting a curve defined in terms of an integral. The curve is the integral of sin(t*w)/t+p over t from 0 to Pi. Our x data points correspond to w, and we're adjusting the p parameter to to get the data to fit.

import math, numpy, scipy.optimize, scipy.integrate

def integrand(t, args):
    w, p = args
    return math.sin(t * w)/t + p

def curve(w, p):
    res = scipy.integrate.quad(integrand, 0.0, math.pi, [w, p])
    return res[0]

vcurve = numpy.vectorize(curve, excluded=set([1]))

truexdata = numpy.asarray([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0])
trueydata = vcurve(truexdata, 1.0)

xdata = truexdata + 0.1 * numpy.random.randn(8)
ydata = trueydata + 0.1 * numpy.random.randn(8)

popt, pcov = scipy.optimize.curve_fit(vcurve,
                                      xdata, ydata,
                                      p0=[2.0])
print popt

That'll print something out fairly close to 1.0, which is what we used as p when we created the trueydata.

Note that we use numpy.vectorize on the curve function to produce a vectorized version compatible with scipy.optimize.curve_fit.