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# tboggs/dirichlet_plots.png

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A script to generate contour plots of Dirichlet distributions
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 '''Functions for drawing contours of Dirichlet distributions.''' # Author: Thomas Boggs 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".')

### valo commented Mar 25, 2014

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'


### tboggs commented Jul 2, 2014

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)


### aloctavodia commented Mar 1, 2016

Hi,

For those using Python 3. You should do:

from functools import reduce

### mlandnlp commented Jun 16, 2017

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

### tboggs commented Apr 18, 2020

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.

### willfrew commented May 14, 2022

Really useful - thanks!

### kwahcarioca commented Aug 29, 2023 • edited

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))