A script to generate contour plots of Dirichlet distributions

dirichlet_plots.png

simplex_plots.py

'''Functions for drawing contours of Dirichlet distributions.

MIT License

Copyright (c) 2014 Thomas Boggs

Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
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The above copyright notice and this permission notice shall be included in all
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THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
'''

from __future__ import division, print_function

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.tri as tri

_corners = np.array([[0, 0], [1, 0], [0.5, 0.75**0.5]])
_AREA = 0.5 * 1 * 0.75**0.5
_triangle = tri.Triangulation(_corners[:, 0], _corners[:, 1])

# For each corner of the triangle, the pair of other corners
_pairs = [_corners[np.roll(range(3), -i)[1:]] for i in range(3)]
# The area of the triangle formed by point xy and another pair or points
tri_area = lambda xy, pair: 0.5 * np.linalg.norm(np.cross(*(pair - xy)))

def xy2bc(xy, tol=1.e-4):
    '''Converts 2D Cartesian coordinates to barycentric.

    Arguments:

        `xy`: A length-2 sequence containing the x and y value.
    '''
    coords = np.array([tri_area(xy, p) for p in _pairs]) / _AREA
    return np.clip(coords, tol, 1.0 - tol)

class Dirichlet(object):
    def __init__(self, alpha):
        '''Creates Dirichlet distribution with parameter `alpha`.'''
        from math import gamma
        from operator import mul
        self._alpha = np.array(alpha)
        self._coef = gamma(np.sum(self._alpha)) / \
                     np.multiply.reduce([gamma(a) for a in self._alpha])
    def pdf(self, x):
        '''Returns pdf value for `x`.'''
        from operator import mul
        return self._coef * np.multiply.reduce([xx ** (aa - 1)
                                                for (xx, aa)in zip(x, self._alpha)])
    def sample(self, N):
        '''Generates a random sample of size `N`.'''
        return np.random.dirichlet(self._alpha, N)

def draw_pdf_contours(dist, border=False, nlevels=200, subdiv=8, **kwargs):
    '''Draws pdf contours over an equilateral triangle (2-simplex).

    Arguments:

        `dist`: A distribution instance with a `pdf` method.

        `border` (bool): If True, the simplex border is drawn.

        `nlevels` (int): Number of contours to draw.

        `subdiv` (int): Number of recursive mesh subdivisions to create.

        kwargs: Keyword args passed on to `plt.triplot`.
    '''
    from matplotlib import ticker, cm
    import math

    refiner = tri.UniformTriRefiner(_triangle)
    trimesh = refiner.refine_triangulation(subdiv=subdiv)
    pvals = [dist.pdf(xy2bc(xy)) for xy in zip(trimesh.x, trimesh.y)]

    plt.tricontourf(trimesh, pvals, nlevels, cmap='jet', **kwargs)
    plt.axis('equal')
    plt.xlim(0, 1)
    plt.ylim(0, 0.75**0.5)
    plt.axis('off')
    if border is True:
        plt.triplot(_triangle, linewidth=1)

def plot_points(X, barycentric=True, border=True, **kwargs):
    '''Plots a set of points in the simplex.

    Arguments:

        `X` (ndarray): A 2xN array (if in Cartesian coords) or 3xN array
                       (if in barycentric coords) of points to plot.

        `barycentric` (bool): Indicates if `X` is in barycentric coords.

        `border` (bool): If True, the simplex border is drawn.

        kwargs: Keyword args passed on to `plt.plot`.
    '''
    if barycentric is True:
        X = X.dot(_corners)
    plt.plot(X[:, 0], X[:, 1], 'k.', ms=1, **kwargs)
    plt.axis('equal')
    plt.xlim(0, 1)
    plt.ylim(0, 0.75**0.5)
    plt.axis('off')
    if border is True:
        plt.triplot(_triangle, linewidth=1)

if __name__ == '__main__':
    f = plt.figure(figsize=(8, 6))
    alphas = [[0.999] * 3,
              [5] * 3,
              [2, 5, 15]]
    for (i, alpha) in enumerate(alphas):
        plt.subplot(2, len(alphas), i + 1)
        dist = Dirichlet(alpha)
        draw_pdf_contours(dist)
        title = r'$\alpha$ = (%.3f, %.3f, %.3f)' % tuple(alpha)
        plt.title(title, fontdict={'fontsize': 8})
        plt.subplot(2, len(alphas), i + 1 + len(alphas))
        plot_points(dist.sample(5000))
    plt.savefig('dirichlet_plots.png')
    print('Wrote plots to "dirichlet_plots.png".')

Requires matplotlib (v1.3 or greater).

To generate the plot shown above, simply run:

$ python simplex_plot.py

Very strange. I have matplotlib 1.3.1 and I get this error:

Traceback (most recent call last):
  File "simplex_plots.py", line 106, in <module>
    draw_pdf_contours(dist)
  File "simplex_plots.py", line 60, in draw_pdf_contours
    refiner = tri.trirefine.UniformTriRefiner(_triangle)
AttributeError: 'module' object has no attribute 'trirefine'

This is a late reply because - for some reason - I never received notification of your comment. Did you edit the code before running? I ask because the line referenced in your error (line 60) doesn't match what is in my code. Your line is

refiner = tri.trirefine.UniformTriRefiner(_triangle)

which has an extra trirefine submodule referenced. The line in my code is just

refiner = tri.UniformTriRefiner(_triangle)

Hi,

For those using Python 3. You should do:

from functools import reduce

Thank you for creating the script and helping me build more intuition for the Dirichlet Distribution :-)

It came to my attention that the function xy2bc was incorrect, which resulted in varying inaccuracy over the simplex. While it didn't appear to make a difference for the tolerance used, I've updated this gist with a corrected implementation that uses fractional triangle areas to compute the barycentric coordinates. I also made some minor edits to account for python and matplotlib API changes since the original post.

Really useful - thanks!

Your code is very nice showing how to implement Dirichlet straight from the formula. I ve also tried to experiment it calling scipy.stats.dirichlet library instead. It worked well but we needed to change the tolerance of the xy2bc generator from 1e-4 to 1e-9. Otherwise the assertion of the library code _multivariate.py wont let us to run.
if (np.abs(np.sum(x, 0) - 1.0) > 10e-10).any():
raise ValueError("The input vector 'x' must lie within the normal "
"simplex. but np.sum(x, 0) = %s." % np.sum(x, 0))