r"""

Bivariate Gaussian: Robust Parameter Estimation

-----------------------------------------------

Figure 3.23.

An example of computing the components of a bivariate Gaussian using a sample

with 1000 data values (points), with two levels of contamination. The core of

the distribution is a bivariate Gaussian with

:math:`(\mu_x, \mu_y, \sigma_1, \sigma_2, \alpha) = (10, 10, 2, 1, 45^\odot)`

The "contaminating" subsample contributes 5% (left) and 15% (right) of points

centered on the same :math:`(\mu_x, \mu_y)`,

and with :math:`\sigma_1 = \sigma_2 = 5`.

Ellipses show the 1- and 3-sigma contours. The solid lines correspond to the

input distribution. The thin dotted lines show the nonrobust estimate, and the

dashed lines show the robust estimate of the best-fit distribution parameters

(see Section 3.5.3 for details).

*ADDENDUM, December 19, 2013 by gully

I edited the figure to demonstrate some features and try my hand at matplotlib.

Specifically, I wanted to know more about the "robust estimate" and its limits.

Exactly how much better is the robust compared to the nonrobust.

"""

# Author:Jake VanderPlas

# Edited: Michael Gully-Santiago

# December 19, 2013

# License: BSD

# The figure produced by this code is published in the textbook

# "Statistics, Data Mining, and Machine Learning in Astronomy" (2013)

# For more information, see http://astroML.github.com

# To report a bug or issue, use the following forum:

# https://groups.google.com/forum/#!forum/astroml-general

import numpy as np

from scipy import stats

from matplotlib import pyplot as plt

from matplotlib.patches import Ellipse

from astroML.stats import fit_bivariate_normal

from astroML.stats.random import bivariate_normal

# percent sign needs to be escaped if usetex is activated

import matplotlib

if matplotlib.rcParams.get('text.usetex'):

pct = r'\%'

else:

pct = r'%'

#----------------------------------------------------------------------

# This function adjusts matplotlib settings for a uniform feel in the textbook.

# Note that with usetex=True, fonts are rendered with LaTeX. This may

# result in an error if LaTeX is not installed on your system. In that case,

# you can set usetex to False.

# from astroML.plotting import setup_text_plots

# setup_text_plots(fontsize=16, usetex=True)

N = 1000

sigma1 = 2.0

sigma2 = 0.2

mu = [10, 10]

# it's easier to visualize the x and y sigmas with alpha=0.0

alpha_deg = 0.0 #used to be 45 deg...

alpha = alpha_deg * np.pi / 180

#------------------------------------------------------------

# Draw N points from a multivariate normal distribution

#

# we use the bivariate_normal function from astroML. A more

# general function for this is numpy.random.multivariate_normal(),

# which requires the user to specify the full covariance matrix.

# bivariate_normal() generates this covariance matrix for the

# given inputs

np.random.seed(0)

X = bivariate_normal(mu, sigma1, sigma2, alpha, N)

#------------------------------------------------------------

# Create the figure showing the fits

fig = plt.figure(figsize=(8, 8))

fig.subplots_adjust(left=0.1, right=0.90, wspace=0.25,

bottom=0.1, top=0.9, hspace=0.3)

ax = fig.add_subplot(2, 2, 0)

# add outliers distributed using a bivariate normal.

x, y = X.T

# scatter the points

ax.scatter(x, y, s=2, lw=0, c='k', alpha=0.5)

# Draw elipses showing the fits

for Nsig in [1, 3]:

# True fit

E = Ellipse((10, 10), sigma1 * Nsig, sigma2 * Nsig, alpha_deg,

ec='k', fc='none')

ax.add_patch(E)

ax.set_xlim(5.5, 14.5)

ax.set_ylim(5.5, 14.5)

ax.set_xlabel('$x$')

ax.set_ylabel('$y$')

# We'll create ten levels of contamination, but not show each one.

# Instead, we will show the fit sigma versus contamination level.

cont_frac = [0.0001, 0.001, 0.003, 0.006, 0.01, 0.03, 0.05, 0.07, 0.10,

0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7]

n_els = len(cont_frac)

sigfit_rob = np.ones(n_els)

sigfit_nonrob = np.ones(n_els)

sigfit2_rob = np.ones(n_els)

sigfit2_nonrob = np.ones(n_els)

alpfit_rob = np.ones(n_els)

alpfit_nonrob = np.ones(n_els)

ax1 = fig.add_subplot(2, 2, 1)

ax2 = fig.add_subplot(2, 2, 3)

ax3 = fig.add_subplot(2, 2, 2)

for i, f in enumerate(cont_frac):

# add outliers distributed using a bivariate normal.

X[:int(f * N)] = bivariate_normal((10, 10), 2, 4, 45 * np.pi / 180., int(f * N))

x, y = X.T

# compute the non-robust statistics

(mu_nr, sigma1_nr, sigma2_nr, alpha_nr) = fit_bivariate_normal(x, y, robust=False)

sigfit_nonrob[i] = sigma1_nr

sigfit2_nonrob[i] = sigma2_nr

alpfit_nonrob[i] = alpha_nr

# compute the robust statistics

(mu_r, sigma1_r, sigma2_r, alpha_r) = fit_bivariate_normal(x, y, robust=True)

sigfit_rob[i] = sigma1_r

sigfit2_rob[i] = sigma2_r

alpfit_rob[i] = alpha_r

# scatter the points

for i in range(n_els):

print "%g\t%g\t%g" %(cont_frac[i], sigfit_rob[i], alpfit_rob[i])

ax1.plot(cont_frac, sigfit_rob, '-k', label='Robust')

ax1.plot(cont_frac, sigfit_nonrob, ':k', label='Nonrobust')

ax3.plot(cont_frac, sigfit2_rob, '-b')

ax3.plot(cont_frac, sigfit2_nonrob,':b')

ax1.plot(cont_frac, sigma1*np.ones(n_els), '--r', label='Actual')

ax3.plot(cont_frac, sigma2*np.ones(n_els), '--g')

ax1.set_xlabel('Contamination Fraction')

ax1.set_ylabel('$\sigma_x$')

ax3.set_xlabel('Contamination Fraction')

ax3.set_ylabel('$\sigma_y$')

leg = ax1.legend(loc='best', fancybox=True)

leg.get_frame().set_alpha(0.5)

ax2.plot(cont_frac, alpfit_rob, '-k', label='Robust')

ax2.plot(cont_frac, alpfit_nonrob, ':k', label='Nonrobust')

ax2.plot(cont_frac, alpha*np.ones(n_els), '--r', label='Actual')

ax2.set_xlabel('Contamination Fraction')

ax2.set_ylabel(r'$\alpha$')

leg = ax2.legend(loc='best', fancybox=True)

leg.get_frame().set_alpha(0.5)

plt.show()