Fitting an ODE
==============

Bumps can fit black-box functions, such as odeint from scipy.

The following example is adapted from:

    https://people.duke.edu/~ccc14/sta-663/CalibratingODEs.html.

| Instructor: Cliburn Chan cliburn.chan@duke..edu
| Instructor: Janice McCarthy janice.mccarthy@duke.edu

::

    from bumps.names import *
    import numpy as np
    from scipy.integrate import odeint
    
Define the ODE

::

    def g(t, x0, a, b):
        """
        Solution to the ODE x'(t) = f(t,x,k) with initial condition x(0) = x0
        """
        return odeint(dfdt, x0, t, args=(a, b)).flatten()
    
    def dfdt(x, t, a, b):
        """Receptor synthesis-internalization model."""
        return a - b*x
    
Simulate some data.

Note that the function :func:`bumps.util.push_seed` is to set the random
number generator to a known state so that this function will create the
same data every time the simulation is run.  If not, then you wouldn't
be able to resume a fit since each time you resumed you would be fitting
different data.

::

    def simulate():
        from bumps.util import push_seed
    
        # Fake some data
        a = 2.0
        b = 0.5
        x0 = 10.0
        t = np.linspace(0, 10, 10)
        dy = 0.2*np.ones_like(t)
        with push_seed(1):
            y = g(t, x0, a, b) + dy*np.random.normal(size=t.shape)
        #print(a, b, x0, t, dt, gt)
        return t, y, dy
    
    t, y, dy = simulate()
    
Define the fit problem.

In this case :class:`bumps.curve.Curve` is initialized with *plot_x*
as a vector of length 1000.  This is so that a smooth curve is drawn between
the ten data points that were simulated in the fit.

::

    M = Curve(g, t, y, dy, x0=1., a=1., b=1.,
              plot_x=np.linspace(t[0], t[-1], 1000))
    M.x0.range(0, 100)
    M.a.range(0, 10)
    M.b.range(0, 10)
    
    problem = FitProblem(M)


.. only:: html

   Download: :download:`ode.py <ode.py>`.
