Entry 5 (Honorable Mention)¶

"""
    Make a figure to visualize using MCMC (in particular, the Python package
    emcee) to infer 4 parameters from a parametrized model of the Milky Way's
    dark matter halo by using tracer stars from the Sagittarius Stream.
    
    If you're unfamiliar with the jargon here (walkers, Sgr, etc.), check out:
        - Law & Majewski 2010 
            http://iopscience.iop.org/0004-637X/714/1/229/pdf/apj_714_1_229.pdf
        - emcee
            http://dan.iel.fm/emcee/
        - Ensemble samplers with affine-invariance
            http://msp.org/camcos/2010/5-1/camcos-v5-n1-p04-p.pdf
        
"""

# Standard library
import cPickle as pickle

# Third-party dependencies
import astropy.units as u
import matplotlib
import matplotlib.pyplot as plt
import matplotlib.cm as cm
import matplotlib.gridspec as gridspec
import numpy as np

matplotlib.rcParams['font.family'] = "sans-serif"
font_color = "#dddddd"
tick_color = "#cdcdcd"

# Map the codified parameter names to their sexy latex equivalents
param_to_latex = dict(q1=r"$q_1$",
                      qz=r"$q_z$",
                      v_halo=r"$v_{halo}$",
                      phi=r"$\phi$")
                      
halo_params = ["q1", "qz", "v_halo", "phi"]
acceptance_fractions, flat_chain, chain = pickle.load(open("2013-03-12_05-03-29_q1_qz_v_halo_phi_w200_s400.pickle"))
chain = chain[(acceptance_fractions > 0.15) & (acceptance_fractions < 0.6)]

# Create a figure object with same aspect ratio as a sheet of paper...
fig = plt.figure(figsize=(16,20.6))

# I want the plot of individual walkers to span 2 columns
gs = gridspec.GridSpec(4, 3)

# The halo velocity (v_halo) parameter is stored in units of kpc/Myr, but I
#   want to plot it in km/s
for xx in range(chain.shape[0]):
    for yy in range(chain.shape[1]):
        chain[xx,yy,2] = (chain[xx,yy,2]*u.kpc/u.Myr).to(u.km/u.s).value

# I could compute this, but for now I just hard code it by looking at the 
#   plots by eye...
converged_idx = 150

# For each parameter, I want to plot each walker on one panel, and a histogram
#   of all links from all walkers past 150 steps (approximately when the chains
#   converged)
for ii,param in enumerate(halo_params):
    these_chains = chain[:,:,ii]
    
    # I'm going to color the walkers by their variance past the bulk convergence
    #   point, so here I compute the maximum variance to scale the others to 0-1
    max_var = max(np.var(these_chains[:,converged_idx:], axis=1))
    
    ax1 = plt.subplot(gs[ii, :2])
    ax1.set_axis_bgcolor("#333333")

    ax1.axvline(0, 
                color="#67A9CF", 
                alpha=0.7,
                linewidth=2)
    
    for walker in these_chains:
        ax1.plot(np.arange(len(walker))-converged_idx, walker,
                drawstyle="steps", 
                color=cm.bone_r(np.var(walker[converged_idx:]) / max_var),
                alpha=0.5)
                
    ax1.set_ylabel(param_to_latex[param], 
                   fontsize=36,
                   labelpad=18,
                   rotation="horizontal",
                   color=font_color)
    
    # Don't show ticks on the y-axis
    ax1.yaxis.set_ticks([])
    
    # For the plot on the bottom, add an x-axis label. Hide all others
    if ii == len(halo_params)-1:
        ax1.set_xlabel("step number", fontsize=24, labelpad=18, color=font_color)
    else:
        ax1.xaxis.set_visible(False)
    
    ax2 = plt.subplot(gs[ii, 2])
    ax2.set_axis_bgcolor("#555555")
    
    # Create a histogram of all values past the converged point. Make 100 bins
    #   between the y-axis bounds defined by the 'walkers' plot.
    ax2.hist(np.ravel(these_chains[:,converged_idx:]), 
             bins=np.linspace(ax1.get_ylim()[0],ax1.get_ylim()[1],100),
             orientation='horizontal',
             facecolor="#67A9CF",
             edgecolor="none")
    
    # Same y-bounds as the walkers plot, so they line up
    ax1.set_ylim(np.min(these_chains[:,0]), np.max(these_chains[:,0]))
    ax2.set_ylim(ax1.get_ylim())
    ax2.xaxis.set_visible(False)
    ax2.yaxis.tick_right()
    
    # For the first plot, add titles and shift them up a bit
    if ii == 0:
        t = ax1.set_title("Walkers", fontsize=30, color=font_color)
        t.set_y(1.01) 
        t = ax2.set_title("Posterior", fontsize=30, color=font_color)
        t.set_y(1.01) 
    
    if param == "v_halo":
        ax2.set_ylabel("km/s", 
                       fontsize=20,
                       rotation="horizontal",
                       color=font_color,
                       labelpad=16)
    elif param == "phi":
        ax2.set_ylabel("rad",
                       fontsize=20,
                       rotation="horizontal",
                       color=font_color,
                       labelpad=16)
    ax2.yaxis.set_label_position("right")
    
    # Adjust axis ticks, e.g. make them appear on the outside of the plots and
    #   change the padding / color.
    ax1.tick_params(axis='x', pad=2, direction='out', colors=tick_color, labelsize=14)
    ax2.tick_params(axis='y', pad=2, direction='out', colors=tick_color, labelsize=14)
    
    # Removes the top tick marks
    ax1.get_xaxis().tick_bottom()
    
    # Hack because the tick labels for phi are wonky... but this removed the 
    #   first and last tick labels so I can squash the plots right up against
    #   each other
    if param == "phi":
        ax2.set_yticks(ax2.get_yticks()[1:-2])
    else:
        ax2.set_yticks(ax2.get_yticks()[1:-1])

# Effing long equation for the logarithmic halo potential with a possible 
#   rotation relative to the MW disk.
fig.suptitle(r"$\Phi_{halo} = v_{halo}^2 \ln[(\frac{\cos^2\phi}{q_1^2}+\frac{\sin^2\phi}{q_2^2})x^2 + (\frac{\sin^2\phi}{q_1^2}+\frac{\cos^2\phi}{q_2^2})y^2 + 2\sin\phi\cos\phi(q_1^{-2}-q_2^{-2})xy + \frac{z^2}{q_z^2} + r_{halo}^2]$", fontsize=26, color=font_color)
fig.subplots_adjust(hspace=0.0, wspace=0.0, bottom=0.075, top=0.9, left=0.12, right=0.88)
plt.savefig("walkers.pdf", facecolor='#222222')