Q:

Fitting data with integral function

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:

def f(x, a, b):
    return a*x**2+b*x

And then proceeds with the fitting popt, pcov = curve_fit(f,x,y)

But the question is now, how does one go about defining the function in point 1. if the function contains an integral (or a discrete sum), e.g.:

The experimental data is still given for x and f(x), so point 2. would be similar I imagine once I can define f(x) in python. By the way I forgot to say that it is assumed that g(t) has a well known form here, and contains the fitting parameters, i.e. parameters like a and b given in the polynomial example. Any help is much appreciated. The question is really supposed to be a generic one, and the functions used in the post are just random examples.

A:

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.

A:

Sometimes you can be lucky and you're able to evaluate the integral analytically. In the following example the product of h(t)=exp(-(t-x)**2/2) and a second degree polynomial g(t) is integrated from 0 to infinity. Sympy is used to evaluate the Integral and generate a function usable for curve_fit():
import sympy as sy
sy.init_printing()  # LaTeX-like pretty printing of IPython

t, x = sy.symbols("t, x", real=True)

h = sy.exp(-(t-x)**2/2)

a0, a1, a2 = sy.symbols('a:3', real=True)  # unknown coefficients
g = a0 + a1*t + a2*t**2

gh = (g*h).simplify()  # the intgrand
G = sy.integrate(gh, (t, 0, sy.oo)).simplify()  # integrate from 0 to infinty

# Generate numeric function to be usable by curve_fit()
G_opt = sy.lambdify((x, t, a0, a1, a2), G)

print(G_opt(1, 2, 3, 4, 5))  # example usage

Note that in general the problem is often ill-posed since the integral does not neccesarily converge in a large enough neighborhood of the solution (which is assumed by curve_fit()).

