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
in the Software without restriction, including without limitation the rights
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The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.

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
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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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".')

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))