Recap example: Monte-Carlo¶

%%bash
mkdir -p DiffusionExample

%%writefile DiffusionExample/MonteCarlo.py
import matplotlib.pyplot as plt
from numpy import sum, array
from numpy.random import randint, choice



class MonteCarlo(object):
    """ A simple Monte Carlo implementation """

    def __init__(self, energy, density, temperature=1, itermax=1000):
        from numpy import any, array
        density = array(density)
        self.itermax = itermax

        if temperature == 0:
            raise NotImplementedError(
                "Zero temperature not implemented")
        if temperature < 0e0:
            raise ValueError(
                "Negative temperature makes no sense")

        if len(density) < 2:
            raise ValueError("Density is too short")
        # of the right kind (integer). Unless it is zero length,
        # in which case type does not matter.
        if density.dtype.kind != 'i' and len(density) > 0:
            raise TypeError("Density should be an array of *integers*.")
        # and the right values (positive or null)
        if any(density < 0):
            raise ValueError("Density should be an array of" +
                             "*positive* integers.")
        if density.ndim != 1:
            raise ValueError("Density should be an a *1-dimensional*" +
                             "array of positive integers.")
        if sum(density) == 0:
            raise ValueError("Density is empty.")

        self.current_energy = energy(density)
        self.temperature = temperature
        self.density = density

    def random_direction(self): return choice([-1, 1])

    def random_agent(self, density):
        # Particle index
        particle = randint(sum(density))
        current = 0
        for location, n in enumerate(density):
            current += n
            if current > particle:
                break
        return location

    def change_density(self, density):
        """ Move one particle left or right. """

        location = self.random_agent(density)

        # Move direction
        if(density[location]-1 < 0):
            return array(density)
        if location == 0:
            direction = 1
        elif location == len(density) - 1:
            direction = -1
        else:
            direction = self.random_direction()

        # Now make change
        result = array(density)
        result[location] -= 1
        result[location + direction] += 1
        return result

    def accept_change(self, prior, successor):
        """ Returns true if should accept change. """
        from numpy import exp
        from numpy.random import uniform
        if successor <= prior:
            return True
        else:
            return exp(-(successor - prior) / self.temperature) > uniform()

    def step(self):
        iteration = 0
        while iteration < self.itermax:
            new_density = self.change_density(self.density)
            new_energy = energy(new_density)

            accept = self.accept_change(self.current_energy, new_energy)
            if accept:
                self.density, self.current_energy = new_density, new_energy
            iteration += 1

        return self.current_energy, self.density


def energy(density, coefficient=1):
    """ Energy associated with the diffusion model
        :Parameters:
        density: array of positive integers
        Number of particles at each position i in the array/geometry
    """
    from numpy import array, any, sum

    # Make sure input is an array
    density = array(density)

    # of the right kind (integer). Unless it is zero length, in which case type does not matter.
    if density.dtype.kind != 'i' and len(density) > 0:
        raise TypeError("Density should be an array of *integers*.")
    # and the right values (positive or null)
    if any(density < 0):
        raise ValueError("Density should be an array" +
                         "of *positive* integers.")
    if density.ndim != 1:
        raise ValueError("Density should be an a *1-dimensional*" +
                         "array of positive integers.")

    return coefficient * 0.5 * sum(density * (density - 1))

import sys
sys.path.append('DiffusionExample')
from MonteCarlo import MonteCarlo, energy
import numpy as np
import numpy.random as random
from matplotlib import animation
from matplotlib import pyplot as plt
from IPython.display import HTML


Temperature = 0.1
density = [np.sin(i) for i in np.linspace(0.1, 3, 100)]
density = np.array(density)*100
density = density.astype(int)

fig = plt.figure()
ax = plt.axes(xlim=(-1, len(density)), ylim=(0, np.max(density)+1))
image = ax.scatter(range(len(density)), density)

txt_energy = plt.text(0, 100, 'Energy = 0')
plt.xlabel('Temperature = 0.1')
plt.ylabel('Energy Density')


mc = MonteCarlo(energy, density, temperature=Temperature)


def simulate(step):
    energy, density = mc.step()
    image.set_offsets(np.vstack((range(len(density)), density)).T)
    txt_energy.set_text('Energy = {}'.format(energy))


anim = animation.FuncAnimation(fig, simulate, frames=200,
                               interval=50)
HTML(anim.to_jshtml())

%%writefile DiffusionExample/test_model.py
from MonteCarlo import MonteCarlo
from unittest.mock import MagicMock
from pytest import raises, approx


def test_input_sanity():
    """ Check incorrect input do fail """
    energy = MagicMock()

    with raises(NotImplementedError) as exception:
        MonteCarlo(sum, [1, 1, 1], 0e0)
    with raises(ValueError) as exception:
        MonteCarlo(energy, [1, 1, 1], temperature=-1e0)

    with raises(TypeError) as exception:
        MonteCarlo(energy, [1.0, 2, 3])
    with raises(ValueError) as exception:
        MonteCarlo(energy, [-1, 2, 3])
    with raises(ValueError) as exception:
        MonteCarlo(energy, [[1, 2, 3], [3, 4, 5]])
    with raises(ValueError) as exception:
        MonteCarlo(energy, [3])
    with raises(ValueError) as exception:
        MonteCarlo(energy, [0, 0])


def test_move_particle_one_over():
    """ Check density is change by a particle hopping left or right. """
    from numpy import nonzero, multiply
    from numpy.random import randint

    energy = MagicMock()

    for i in range(100):
        # Do this n times, to avoid
        # issues with random numbers
        # Create density

        density = randint(50, size=randint(2, 6))
        mc = MonteCarlo(energy, density)
        # Change it
        new_density = mc.change_density(density)

        # Make sure any movement is by one
        indices = nonzero(density - new_density)[0]
        assert len(indices) == 2, "densities differ in two places"
        assert \
            multiply.reduce((density - new_density)[indices]) == -1, \
            "densities differ by + and - 1"


def test_equal_probability():
    """ Check particles have equal probability of movement. """
    from numpy import array, sqrt, count_nonzero

    energy = MagicMock()

    density = array([1, 0, 99])
    mc = MonteCarlo(energy, density)
    changes_at_zero = [
        (density - mc.change_density(density))[0] != 0 for i in range(10000)]
    assert count_nonzero(changes_at_zero) \
            == approx(0.01 * len(changes_at_zero), 0.5 * sqrt(len(changes_at_zero)))


def test_accept_change():
    """ Check that move is accepted if second energy is lower """
    from numpy import sqrt, count_nonzero, exp

    energy = MagicMock
    mc = MonteCarlo(energy, [1, 1, 1], temperature=100.0)
    # Should always be true.
    # But do more than one draw,
    # in case randomness incorrectly crept into
    # implementation
    for i in range(10):
        assert mc.accept_change(0.5, 0.4)
        assert mc.accept_change(0.5, 0.5)

    # This should be accepted only part of the time,
    # depending on exponential distribution
    prior, successor = 0.4, 0.5
    accepted = [mc.accept_change(prior, successor) for i in range(10000)]
    assert count_nonzero(accepted) / float(len(accepted)) \
        == approx(exp(-(successor - prior) / mc.temperature), 3e0 / sqrt(len(accepted)))



def test_main_algorithm():
    import numpy as np
    from numpy import testing
    from unittest.mock import Mock

    density=[1, 1, 1, 1, 1]
    energy=MagicMock()
    mc=MonteCarlo(energy, density, itermax = 5)

    acceptance=[True, True, True, True, True]
    mc.accept_change=Mock(side_effect = acceptance)
    mc.random_agent=Mock(side_effect = [0, 1, 2, 3, 4])
    mc.random_direction=Mock(side_effect = [1, 1, 1, 1, -1])
    np.testing.assert_equal(mc.step()[1], [0, 1, 1, 2, 1])

%%bash
cd DiffusionExample
py.test