path: root/utils/plot-energy

#!/usr/bin/env python
# Copyright (c) 2019, Anthony Latorre <tlatorre at uchicago>
#
# This program is free software: you can redistribute it and/or modify it
# under the terms of the GNU General Public License as published by the Free
# Software Foundation, either version 3 of the License, or (at your option)
# any later version.
#
# This program is distributed in the hope that it will be useful, but WITHOUT
# ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
# FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
# more details.
#
# You should have received a copy of the GNU General Public License along with
# this program. If not, see <https://www.gnu.org/licenses/>.
"""
Script to plot final fit results along with sidebands for the dark matter
analysis. To run it just run:

    $ ./plot-energy [list of fit results]

Currently it will plot energy distributions for external muons, michel
electrons, atmospheric events with neutron followers, and prompt signal like
events. Each of these plots will have a different subplot for the particle ID
of the best fit, i.e. single electron, single muon, double electron, electron +
muon, or double muon.

When run with the --dc command line argument it instead produces corner plots
showing the distribution of the high level variables used in the contamination
analysis for all the different instrumental backgrounds and external muons.
"""
from __future__ import print_function, division
import numpy as np
from scipy.stats import iqr, poisson
from matplotlib.lines import Line2D
from scipy.stats import iqr, norm, beta
from scipy.special import spence
from itertools import izip_longest

PSUP_RADIUS = 840.0

# from https://stackoverflow.com/questions/287871/how-to-print-colored-text-in-terminal-in-python
class bcolors:
    HEADER = '\033[95m'
    OKBLUE = '\033[94m'
    OKGREEN = '\033[92m'
    WARNING = '\033[93m'
    FAIL = '\033[91m'
    ENDC = '\033[0m'
    BOLD = '\033[1m'
    UNDERLINE = '\033[4m'

# on retina screens, the default plots are way too small
# by using Qt5 and setting QT_AUTO_SCREEN_SCALE_FACTOR=1
# Qt5 will scale everything using the dpi in ~/.Xresources
import matplotlib
matplotlib.use("Qt5Agg")

font = {'family':'serif', 'serif': ['computer modern roman']}
matplotlib.rc('font',**font)

matplotlib.rc('text', usetex=True)

SNOMAN_MASS = {
    20: 0.511,
    21: 0.511,
    22: 105.658,
    23: 105.658
}

AV_RADIUS = 600.0

# Data cleaning bitmasks.
DC_MUON           = 0x1
DC_JUNK           = 0x2
DC_CRATE_ISOTROPY = 0x4
DC_QVNHIT         = 0x8
DC_NECK           = 0x10
DC_FLASHER        = 0x20
DC_ESUM           = 0x40
DC_OWL            = 0x80
DC_OWL_TRIGGER    = 0x100
DC_FTS            = 0x200
DC_ITC            = 0x400
DC_BREAKDOWN      = 0x800

particle_id = {20: 'e', 22: r'\mu'}

def grouper(iterable, n, fillvalue=None):
    "Collect data into fixed-length chunks or blocks"
    # grouper('ABCDEFG', 3, 'x') --> ABC DEF Gxx
    args = [iter(iterable)] * n
    return izip_longest(fillvalue=fillvalue, *args)

def plot_hist2(df, muons=False):
    for id, df_id in sorted(df.groupby('id')):
        if id == 20:
            plt.subplot(2,3,1)
        elif id == 22:
            plt.subplot(2,3,2)
        elif id == 2020:
            plt.subplot(2,3,4)
        elif id == 2022:
            plt.subplot(2,3,5)
        elif id == 2222:
            plt.subplot(2,3,6)

        if muons:
            plt.hist(np.log10(df_id.ke.values/1000), bins=np.linspace(0,4.5,100), histtype='step')
            plt.xlabel("log10(Energy (GeV))")
        else:
            plt.hist(df_id.ke.values, bins=np.linspace(20,10e3,100), histtype='step')
            plt.xlabel("Energy (MeV)")
        plt.title('$' + ''.join([particle_id[int(''.join(x))] for x in grouper(str(id),2)]) + '$')

    if len(df):
        plt.tight_layout()

def plot_hist(df, muons=False):
    for id, df_id in sorted(df.groupby('id')):
        if id == 20:
            plt.subplot(3,4,1)
        elif id == 22:
            plt.subplot(3,4,2)
        elif id == 2020:
            plt.subplot(3,4,5)
        elif id == 2022:
            plt.subplot(3,4,6)
        elif id == 2222:
            plt.subplot(3,4,7)
        elif id == 202020:
            plt.subplot(3,4,9)
        elif id == 202022:
            plt.subplot(3,4,10)
        elif id == 202222:
            plt.subplot(3,4,11)
        elif id == 222222:
            plt.subplot(3,4,12)

        if muons:
            plt.hist(np.log10(df_id.ke.values/1000), bins=np.linspace(0,4.5,100), histtype='step')
            plt.xlabel("log10(Energy (GeV))")
        else:
            plt.hist(df_id.ke.values, bins=np.linspace(20,10e3,100), histtype='step')
            plt.xlabel("Energy (MeV)")
        plt.title(str(id))

    if len(df):
        plt.tight_layout()

def chunks(l, n):
    """Yield successive n-sized chunks from l."""
    for i in range(0, len(l), n):
        yield l[i:i + n]

def print_warning(msg):
    print(bcolors.FAIL + msg + bcolors.ENDC,file=sys.stderr)

def unwrap(p, delta, axis=-1):
    """
    A modified version of np.unwrap() useful for unwrapping the 50 MHz clock.
    It unwraps discontinuities bigger than delta/2 by delta.

    Example:

        >>> a = np.arange(10) % 5
        >>> a
        array([0, 1, 2, 3, 4, 0, 1, 2, 3, 4])
        >>> unwrap(a,5)
        array([ 0.,  1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.,  9.])

    In the case of the 50 MHz clock delta should be 0x7ffffffffff*20.0.
    """
    p = np.asarray(p)
    nd = p.ndim
    dd = np.diff(p, axis=axis)
    slice1 = [slice(None, None)]*nd     # full slices
    slice1[axis] = slice(1, None)
    slice1 = tuple(slice1)
    ddmod = np.mod(dd + delta/2, delta) - delta/2
    np.copyto(ddmod, delta/2, where=(ddmod == -delta/2) & (dd > 0))
    ph_correct = ddmod - dd
    np.copyto(ph_correct, 0, where=abs(dd) < delta/2)
    up = np.array(p, copy=True, dtype='d')
    up[slice1] = p[slice1] + ph_correct.cumsum(axis)
    return up

def unwrap_50_mhz_clock(gtr):
    """
    Unwrap an array with 50 MHz clock times. These times should all be in
    nanoseconds and come from the KEV_GTR variable in the EV bank.

    Note: We assume here that the events are already ordered contiguously by
    GTID, so you shouldn't pass an array with multiple runs!
    """
    return unwrap(gtr,0x7ffffffffff*20.0)

def retrigger_cut(ev):
    """
    Cuts all retrigger events.
    """
    return ev[ev.dt > 500]

def breakdown_follower_cut(ev):
    """
    Cuts all events within 1 second of breakdown events.
    """
    breakdowns = ev[ev.dc & DC_BREAKDOWN != 0]
    return ev[~np.any((ev.gtr.values > breakdowns.gtr.values[:,np.newaxis]) & \
                      (ev.gtr.values < breakdowns.gtr.values[:,np.newaxis] + 1e9),axis=0)]

def flasher_follower_cut(ev):
    """
    Cuts all events within 200 microseconds of flasher events.
    """
    flashers = ev[ev.dc & DC_FLASHER != 0]
    return ev[~np.any((ev.gtr.values > flashers.gtr.values[:,np.newaxis]) & \
                      (ev.gtr.values < flashers.gtr.values[:,np.newaxis] + 200e3),axis=0)]

def muon_follower_cut(ev):
    """
    Cuts all events 200 microseconds after a muon.
    """
    muons = ev[ev.dc & DC_MUON != 0]
    return ev[~np.any((ev.gtr.values > muons.gtr.values[:,np.newaxis]) & \
                      (ev.gtr.values < muons.gtr.values[:,np.newaxis] + 200e3),axis=0)]

def michel_cut(ev):
    """
    Looks for Michel electrons after muons.
    """
    prompt_plus_muons = ev[ev.prompt | ((ev.dc & DC_MUON) != 0)]

    # Michel electrons and neutrons can be any event which is not a prompt
    # event
    follower = ev[~ev.prompt]

    # require Michel events to pass more of the SNO data cleaning cuts
    michel = follower[follower.dc & (DC_JUNK | DC_CRATE_ISOTROPY | DC_QVNHIT | DC_FLASHER | DC_NECK | DC_ESUM | DC_OWL | DC_OWL_TRIGGER | DC_FTS) == 0]

    michel = michel[michel.nhit >= 100]

    # Accept events which had a muon more than 800 nanoseconds but less than 20
    # microseconds before them. The 800 nanoseconds cut comes from Richie's
    # thesis. He also mentions that the In Time Channel Spread Cut is very
    # effective at cutting electron events caused by muons, so I should
    # implement that.
    #
    # Note: We currently don't look across run boundaries. This should be a
    # *very* small effect, and the logic to do so would be very complicated
    # since I would have to deal with 50 MHz clock rollovers, etc.
    if prompt_plus_muons.size and michel.size:
        mask = (michel.gtr.values > prompt_plus_muons.gtr.values[:,np.newaxis] + 800) & \
               (michel.gtr.values < prompt_plus_muons.gtr.values[:,np.newaxis] + 20e3)
        michel = michel.iloc[np.any(mask,axis=0)]
        michel['muon_gtid'] = pd.Series(prompt_plus_muons['gtid'].iloc[np.argmax(mask[:,np.any(mask,axis=0)],axis=0)].values,
                                        index=michel.index.values,
                                        dtype=np.int32)
        return michel
    else:
        # Return an empty slice since we need it to have the same datatype as
        # the other dataframes
        michel = ev[:0]
        michel['muon_gtid'] = -1
        return michel

def atmospheric_events(ev):
    """
    Tags atmospheric events which have a neutron follower.
    """
    prompt = ev[ev.prompt]

    # Michel electrons and neutrons can be any event which is not a prompt
    # event
    follower = ev[~ev.prompt]

    ev['atm'] = np.zeros(len(ev),dtype=np.bool)

    if prompt.size and follower.size:
        # neutron followers have to obey stricter set of data cleaning cuts
        neutron = follower[follower.dc & (DC_JUNK | DC_CRATE_ISOTROPY | DC_QVNHIT | DC_FLASHER | DC_NECK | DC_ESUM | DC_OWL | DC_OWL_TRIGGER | DC_FTS) == 0]
        neutron = neutron[~np.isnan(neutron.ftp_x) & ~np.isnan(neutron.rsp_energy)]
        # FIXME: What should the radius cut be here? AV? (r/r_psup)^3 < 0.9?
        neutron = neutron[neutron.ftp_r < AV_RADIUS]
        neutron = neutron[neutron.rsp_energy > 4.0]

        # neutron events accepted after 20 microseconds and before 250 ms (50 ms during salt)
        ev.loc[ev.prompt,'atm'] = np.any((neutron.gtr.values > prompt.gtr.values[:,np.newaxis] + 20e3) & \
                                         (neutron.gtr.values < prompt.gtr.values[:,np.newaxis] + 250e6),axis=1)

    return ev

def gtid_sort(ev, first_gtid):
    """
    Adds 0x1000000 to the gtid_sort column for all gtids before the first gtid
    in a run, which should be passed as a dictionary. This column can then be
    used to sort the events sequentially.

    This function should be passed to ev.groupby('run').apply(). We use this
    idiom instead of just looping over the groupby results since groupby()
    makes a copy of the dataframe, i.e.

        for run, ev_run in ev.groupby('run'):
            ev_run.loc[ev_run.gtid < first_gtid[run],'gtid_sort'] += 0x1000000

    would produce a SettingWithCopyWarning, so instead we use:

        ev = ev.groupby('run',as_index=False).apply(gtid_sort,first_gtid=first_gtid)

    which doesn't have this problem.
    """
    # see https://stackoverflow.com/questions/32460593/including-the-group-name-in-the-apply-function-pandas-python
    run = ev.name

    if run not in first_gtid:
        print_warning("No RHDR bank for run %i! Assuming first event is the first GTID." % run)
        first_gtid[run] = ev.gtid.iloc[0]

    ev.loc[ev.gtid < first_gtid[run],'gtid_sort'] += 0x1000000

    return ev

def prompt_event(ev):
    ev['prompt'] = (ev.nhit >= 100)
    ev.loc[ev.prompt,'prompt'] &= np.concatenate(([True],np.diff(ev[ev.prompt].gtr.values) > 250e6))
    return ev

# Taken from https://raw.githubusercontent.com/mwaskom/seaborn/c73055b2a9d9830c6fbbace07127c370389d04dd/seaborn/utils.py
def despine(fig=None, ax=None, top=True, right=True, left=False,
            bottom=False, offset=None, trim=False):
    """Remove the top and right spines from plot(s).

    fig : matplotlib figure, optional
        Figure to despine all axes of, default uses current figure.
    ax : matplotlib axes, optional
        Specific axes object to despine.
    top, right, left, bottom : boolean, optional
        If True, remove that spine.
    offset : int or dict, optional
        Absolute distance, in points, spines should be moved away
        from the axes (negative values move spines inward). A single value
        applies to all spines; a dict can be used to set offset values per
        side.
    trim : bool, optional
        If True, limit spines to the smallest and largest major tick
        on each non-despined axis.

    Returns
    -------
    None

    """
    # Get references to the axes we want
    if fig is None and ax is None:
        axes = plt.gcf().axes
    elif fig is not None:
        axes = fig.axes
    elif ax is not None:
        axes = [ax]

    for ax_i in axes:
        for side in ["top", "right", "left", "bottom"]:
            # Toggle the spine objects
            is_visible = not locals()[side]
            ax_i.spines[side].set_visible(is_visible)
            if offset is not None and is_visible:
                try:
                    val = offset.get(side, 0)
                except AttributeError:
                    val = offset
                _set_spine_position(ax_i.spines[side], ('outward', val))

        # Potentially move the ticks
        if left and not right:
            maj_on = any(
                t.tick1line.get_visible()
                for t in ax_i.yaxis.majorTicks
            )
            min_on = any(
                t.tick1line.get_visible()
                for t in ax_i.yaxis.minorTicks
            )
            ax_i.yaxis.set_ticks_position("right")
            for t in ax_i.yaxis.majorTicks:
                t.tick2line.set_visible(maj_on)
            for t in ax_i.yaxis.minorTicks:
                t.tick2line.set_visible(min_on)

        if bottom and not top:
            maj_on = any(
                t.tick1line.get_visible()
                for t in ax_i.xaxis.majorTicks
            )
            min_on = any(
                t.tick1line.get_visible()
                for t in ax_i.xaxis.minorTicks
            )
            ax_i.xaxis.set_ticks_position("top")
            for t in ax_i.xaxis.majorTicks:
                t.tick2line.set_visible(maj_on)
            for t in ax_i.xaxis.minorTicks:
                t.tick2line.set_visible(min_on)

        if trim:
            # clip off the parts of the spines that extend past major ticks
            xticks = ax_i.get_xticks()
            if xticks.size:
                firsttick = np.compress(xticks >= min(ax_i.get_xlim()),
                                        xticks)[0]
                lasttick = np.compress(xticks <= max(ax_i.get_xlim()),
                                       xticks)[-1]
                ax_i.spines['bottom'].set_bounds(firsttick, lasttick)
                ax_i.spines['top'].set_bounds(firsttick, lasttick)
                newticks = xticks.compress(xticks <= lasttick)
                newticks = newticks.compress(newticks >= firsttick)
                ax_i.set_xticks(newticks)

            yticks = ax_i.get_yticks()
            if yticks.size:
                firsttick = np.compress(yticks >= min(ax_i.get_ylim()),
                                        yticks)[0]
                lasttick = np.compress(yticks <= max(ax_i.get_ylim()),
                                       yticks)[-1]
                ax_i.spines['left'].set_bounds(firsttick, lasttick)
                ax_i.spines['right'].set_bounds(firsttick, lasttick)
                newticks = yticks.compress(yticks <= lasttick)
                newticks = newticks.compress(newticks >= firsttick)
                ax_i.set_yticks(newticks)

def plot_corner_plot(ev, title, save=None):
    variables = ['r_psup','psi','z','udotr']
    labels = [r'$(r/r_\mathrm{PSUP})^3$',r'$\psi$','z',r'$\vec{u}\cdot\vec{r}$']
    limits = [(0,1),(0,10),(-840,840),(-1,1)]
    cuts = [0.9,6,0,-0.5]

    ev = ev.dropna(subset=variables)

    fig = plt.figure(figsize=(FIGSIZE[0],FIGSIZE[0]))
    despine(fig,trim=True)
    for i in range(len(variables)):
        for j in range(len(variables)):
            if j > i:
                continue
            ax = plt.subplot(len(variables),len(variables),i*len(variables)+j+1)
            if i == j:
                plt.hist(ev[variables[i]],bins=np.linspace(limits[i][0],limits[i][1],100),histtype='step')
                plt.gca().set_xlim(limits[i])
            else:
                plt.scatter(ev[variables[j]],ev[variables[i]],s=0.5)
                plt.gca().set_xlim(limits[j])
                plt.gca().set_ylim(limits[i])
                n = len(ev)
                if n:
                    p_i_lo = np.count_nonzero(ev[variables[i]] < cuts[i])/n
                    p_j_lo = np.count_nonzero(ev[variables[j]] < cuts[j])/n
                    p_lolo = p_i_lo*p_j_lo
                    p_lohi = p_i_lo*(1-p_j_lo)
                    p_hilo = (1-p_i_lo)*p_j_lo
                    p_hihi = (1-p_i_lo)*(1-p_j_lo)
                    n_lolo = np.count_nonzero((ev[variables[i]] < cuts[i]) & (ev[variables[j]] < cuts[j]))
                    n_lohi = np.count_nonzero((ev[variables[i]] < cuts[i]) & (ev[variables[j]] >= cuts[j]))
                    n_hilo = np.count_nonzero((ev[variables[i]] >= cuts[i]) & (ev[variables[j]] < cuts[j]))
                    n_hihi = np.count_nonzero((ev[variables[i]] >= cuts[i]) & (ev[variables[j]] >= cuts[j]))
                    observed = np.array([n_lolo,n_lohi,n_hilo,n_hihi])
                    expected = n*np.array([p_lolo,p_lohi,p_hilo,p_hihi])
                    psi = -poisson.logpmf(observed,expected).sum() + poisson.logpmf(observed,observed).sum()
                    psi /= np.std(-poisson.logpmf(np.random.poisson(observed,size=(10000,4)),observed).sum(axis=1) + poisson.logpmf(observed,observed).sum())
                    plt.title(r"$\psi = %.1f$" % psi)
            if i == len(variables) - 1:
                plt.xlabel(labels[j])
            else:
                plt.setp(ax.get_xticklabels(),visible=False)
            if j == 0:
                plt.ylabel(labels[i])
            else:
                plt.setp(ax.get_yticklabels(),visible=False)
            plt.axvline(cuts[j],color='k',ls='--',alpha=0.5)
            if i != j:
                plt.axhline(cuts[i],color='k',ls='--',alpha=0.5)

    plt.tight_layout()

    if save:
        plt.savefig(save + ".pdf")
        plt.savefig(save + ".eps")

    plt.suptitle(title)

def intersect_sphere(pos, dir, R):
    """
    Compute the first intersection of a ray starting at `pos` with direction
    `dir` and a sphere centered at the origin with radius `R`. The distance to
    the intersection is returned.
    
    Example:
    
        pos = np.array([0,0,0])
        dir = np.array([1,0,0])
    
        l = intersect_sphere(pos,dir,PSUP_RADIUS):
        if l is not None:
            hit = pos + l*dir
            print("ray intersects sphere at %.2f %.2f %.2f", hit[0], hit[1], hit[2])
        else:
            print("ray didn't intersect sphere")
    """

    b = 2*np.dot(dir,pos)
    c = np.dot(pos,pos) - R*R

    if b*b - 4*c <= 0:
        # Ray doesn't intersect the sphere.
        return None

    # First, check the shorter solution.
    l = (-b - np.sqrt(b*b - 4*c))/2

    # If the shorter solution is less than 0, check the second solution.
    if l < 0:
        l = (-b + np.sqrt(b*b - 4*c))/2

    # If the distance is still negative, we didn't intersect the sphere.
    if l < 0:
        return None

    return l

def get_dx(row):
    pos = np.array([row.x,row.y,row.z])
    dir = np.array([np.sin(row.theta1)*np.cos(row.phi1),
                    np.sin(row.theta1)*np.sin(row.phi1),
                    np.cos(row.theta1)])
    l = intersect_sphere(pos,-dir,PSUP_RADIUS)
    if l is not None:
        pos -= dir*l
        michel_pos = np.array([row.x_michel,row.y_michel,row.z_michel])
        return np.linalg.norm(michel_pos-pos)
    else:
        return 0

def dx_to_energy(dx):
    lines = []
    with open("../src/muE_water_liquid.txt") as f:
        for i, line in enumerate(f):
            if i < 10:
                continue
            if 'Minimum ionization' in line:
                continue
            if 'Muon critical energy' in line:
                continue
            lines.append(line)
    data = np.genfromtxt(lines)
    return np.interp(dx,data[:,8],data[:,0])

def iqr_std_err(x):
    """
    Returns the approximate standard deviation assuming the central part of the
    distribution is gaussian.
    """
    x = x.dropna()
    n = len(x)
    if n == 0:
        return np.nan
    # see https://stats.stackexchange.com/questions/110902/error-on-interquartile-range
    std = iqr(x.values)/1.3489795
    return 1.573*std/np.sqrt(n)

def iqr_std(x):
    """
    Returns the approximate standard deviation assuming the central part of the
    distribution is gaussian.
    """
    x = x.dropna()
    n = len(x)
    if n == 0:
        return np.nan
    return iqr(x.values)/1.3489795

def quantile_error(x,q):
    """
    Returns the standard error for the qth quantile of `x`. The error is
    computed using the Maritz-Jarrett method described here:
    https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/quantse.htm.
    """
    x = np.sort(x)
    n = len(x)
    m = int(q*n+0.5)
    A = m - 1
    B = n - m
    i = np.arange(1,len(x)+1)
    w = beta.cdf(i/n,A,B) - beta.cdf((i-1)/n,A,B)
    return np.sqrt(np.sum(w*x**2)-np.sum(w*x)**2)

def q90_err(x):
    """
    Returns the error on the 90th percentile for all the non NaN values in a
    Series `x`.
    """
    x = x.dropna()
    n = len(x)
    if n == 0:
        return np.nan
    return quantile_error(x.values,0.9)

def q90(x):
    """
    Returns the 90th percentile for all the non NaN values in a Series `x`.
    """
    x = x.dropna()
    n = len(x)
    if n == 0:
        return np.nan
    return np.percentile(x.values,90.0)

def median(x):
    """
    Returns the median for all the non NaN values in a Series `x`.
    """
    x = x.dropna()
    n = len(x)
    if n == 0:
        return np.nan
    return np.median(x.values)

def median_err(x):
    """
    Returns the approximate error on the median for all the non NaN values in a
    Series `x`. The error on the median is approximated assuming the central
    part of the distribution is gaussian.
    """
    x = x.dropna()
    n = len(x)
    if n == 0:
        return np.nan
    # First we estimate the standard deviation using the interquartile range.
    # Here we are essentially assuming the central part of the distribution is
    # gaussian.
    std = iqr(x.values)/1.3489795
    median = np.median(x.values)
    # Now we estimate the error on the median for a gaussian
    # See https://stats.stackexchange.com/questions/45124/central-limit-theorem-for-sample-medians.
    return 1/(2*np.sqrt(n)*norm.pdf(median,median,std))

def std_err(x):
    x = x.dropna()
    mean = np.mean(x)
    std = np.std(x)
    n = len(x)
    if n == 0:
        return np.nan
    elif n == 1:
        return 0.0
    u4 = np.mean((x-mean)**4)
    error = np.sqrt((u4-(n-3)*std**4/(n-1))/n)/(2*std)
    return error

# Fermi constant
GF                      = 1.16637887e-5 # 1/MeV^2
ELECTRON_MASS           = 0.5109989461  # MeV
MUON_MASS               = 105.6583745   # MeV
PROTON_MASS             = 938.272081    # MeV
FINE_STRUCTURE_CONSTANT = 7.297352566417e-3

def f(x):
    y = (5/(3*x**2) + 16*x/3 + 4/x + (12-8*x)*np.log(1/x-1) - 8)*np.log(MUON_MASS/ELECTRON_MASS)
    y += (6-4*x)*(2*spence(x) - 2*np.log(x)**2 + np.log(x) + np.log(1-x)*(3*np.log(x)-1/x-1) - np.pi**2/3-2)
    y += (1-x)*(34*x**2+(5-34*x**2+17*x)*np.log(x) - 22*x)/(3*x**2)
    y += 6*(1-x)*np.log(x)
    return y

def michel_spectrum(T):
    """
    Michel electron energy spectrum for a free muon. `T` should be the kinetic
    energy of the electron or positron in MeV.

    Note: The result is not normalized.

    From https://arxiv.org/abs/1406.3575.
    """
    E = T + ELECTRON_MASS
    x = 2*E/MUON_MASS
    mask = (x > 0) & (x < 1)
    y = np.zeros_like(x,dtype=np.double)
    y[mask] = GF**2*MUON_MASS**5*x[mask]**2*(6-4*x[mask]+FINE_STRUCTURE_CONSTANT*f(x[mask])/np.pi)/(192*np.pi**3)
    y *= 2*MUON_MASS
    return y

if __name__ == '__main__':
    import argparse
    import matplotlib.pyplot as plt
    import numpy as np
    import pandas as pd
    import sys
    import h5py

    parser = argparse.ArgumentParser("plot fit results")
    parser.add_argument("filenames", nargs='+', help="input files")
    parser.add_argument("--dc", action='store_true', default=False, help="plot corner plots for backgrounds")
    parser.add_argument("--save", action='store_true', default=False, help="save corner plots for backgrounds")
    args = parser.parse_args()

    ev = pd.concat([pd.read_hdf(filename, "ev") for filename in args.filenames],ignore_index=True)
    fits = pd.concat([pd.read_hdf(filename, "fits") for filename in args.filenames],ignore_index=True)
    rhdr = pd.concat([pd.read_hdf(filename, "rhdr") for filename in args.filenames],ignore_index=True)

    first_gtid = rhdr.set_index('run').to_dict()['first_gtid']

    # First, remove junk events since orphans won't have a 50 MHz clock and so
    # could screw up the 50 MHz clock unwrapping
    ev = ev[ev.dc & DC_JUNK == 0]

    # We need the events to be in time order here in order to calculate the
    # delta t between events. It's not obvious exactly how to do this. You
    # could sort by GTID, but that wraps around. Similarly we can't sort by the
    # 50 MHz clock because it also wraps around. Finally, I'm hesitant to sort
    # by the 10 MHz clock since it can be unreliable.
    #
    # Update: Phil proposed a clever way to get the events in order using the
    # GTID:
    #
    # > The GTID rollover should be easy to handle because there should never
    # > be two identical GTID's in a run. So if you order the events by GTID,
    # > you can assume that events with GTID's that come logically before the
    # > first GTID in the run must have occurred after the other events.
    #
    # Therefore, we can just add 0x1000000 to all GTIDs before the first GTID
    # in the event and sort on that. We get the first GTID from the RHDR bank.
    ev['gtid_sort'] = ev['gtid'].copy()

    ev = ev.groupby('run',as_index=False).apply(gtid_sort,first_gtid=first_gtid).reset_index(level=0,drop=True)

    ev = ev.sort_values(by=['run','gtid_sort'],kind='mergesort')

    for run, ev_run in ev.groupby('run'):
        # Warn about 50 MHz clock jumps since they could indicate that the
        # events aren't in order.
        dt = np.diff(ev_run.gtr)
        if np.count_nonzero((np.abs(dt) > 1e9) & (dt > -0x7ffffffffff*20.0/2)):
            print_warning("Warning: %i 50 MHz clock jumps in run %i. Are the events in order?" % \
                (np.count_nonzero((np.abs(dt) > 1e9) & (dt > -0x7ffffffffff*20.0/2)),run))

    # unwrap the 50 MHz clock within each run
    ev.gtr = ev.groupby(['run'],group_keys=False)['gtr'].transform(unwrap_50_mhz_clock)

    for run, ev_run in ev.groupby('run'):
        # Warn about GTID jumps since we could be missing a potential flasher
        # and/or breakdown, and we need all the events in order to do a
        # retrigger cut
        if np.count_nonzero(np.diff(ev_run.gtid) != 1):
            print_warning("Warning: %i GTID jumps in run %i" % (np.count_nonzero(np.diff(ev_run.gtid) != 1),run))

    # calculate the time difference between each event and the previous event
    # so we can tag retrigger events
    ev['dt'] = ev.groupby(['run'],group_keys=False)['gtr'].transform(lambda x: np.concatenate(([1e9],np.diff(x.values))))

    # This is a bit of a hack. It appears that many times the fit will
    # actually do much better by including a very low energy electron or
    # muon. I believe the reason for this is that of course my likelihood
    # function is not perfect (for example, I don't include the correct
    # angular distribution for Rayleigh scattered light), and so the fitter
    # often wants to add a very low energy electron or muon to fix things.
    #
    # Ideally I would fix the likelihood function, but for now we just
    # discard any fit results which have a very low energy electron or
    # muon.
    #
    # FIXME: Test this since query() is new to pandas
    fits = fits.query('not (n > 1 and ((id1 == 20 and energy1 < 20)  or (id2 == 20 and energy2 < 20)  or (id3 == 20 and energy3 < 20)))')
    fits = fits.query('not (n > 1 and ((id2 == 22 and energy1 < 200) or (id2 == 22 and energy2 < 200) or (id3 == 22 and energy3 < 200)))')

    # Calculate the approximate Ockham factor.
    # See Chapter 20 in "Probability Theory: The Logic of Science" by Jaynes
    #
    # Note: This is a really approximate form by assuming that the shape of
    # the likelihood space is equal to the average uncertainty in the
    # different parameters.
    fits['w'] = fits['n']*np.log(0.1*0.001) + np.log(fits['energy1']) + fits['n']*np.log(1e-4/(4*np.pi))

    # Apply a fudge factor to the Ockham factor of 100 for each extra particle
    # FIXME: I chose 100 a while ago but didn't really investigate what the
    # optimal value was or exactly why it was needed. Should do this.
    fits['w'] -= fits['n']*100

    # Note: we index on the left hand site with loc to avoid a copy error
    #
    # See https://www.dataquest.io/blog/settingwithcopywarning/
    fits.loc[fits['n'] > 1, 'w'] += np.log(fits[fits['n'] > 1]['energy2'])
    fits.loc[fits['n'] > 2, 'w'] += np.log(fits[fits['n'] > 2]['energy3'])

    fits['fmin'] = fits['fmin'] - fits['w']

    # See https://stackoverflow.com/questions/11976503/how-to-keep-index-when-using-pandas-merge
    # for how to properly divide the psi column by nhit_cal which is in the ev
    # dataframe before we actually merge
    fits['psi'] /= fits.reset_index().merge(ev,on=['run','gtid']).set_index('index')['nhit_cal']
    fits.loc[fits['n'] == 1,'ke'] = fits['energy1']
    fits.loc[fits['n'] == 2,'ke'] = fits['energy1'] + fits['energy2']
    fits.loc[fits['n'] == 3,'ke'] = fits['energy1'] + fits['energy2'] + fits['energy3']
    fits['id'] = fits['id1']
    fits.loc[fits['n'] == 2, 'id'] = fits['id1']*100 + fits['id2']
    fits.loc[fits['n'] == 3, 'id'] = fits['id1']*10000 + fits['id2']*100 + fits['id3']
    fits['theta'] = fits['theta1']
    fits['r'] = np.sqrt(fits.x**2 + fits.y**2 + fits.z**2)
    fits['r_psup'] = (fits['r']/PSUP_RADIUS)**3

    ev['ftp_r'] = np.sqrt(ev.ftp_x**2 + ev.ftp_y**2 + ev.ftp_z**2)
    ev['ftp_r_psup'] = (ev['ftp_r']/PSUP_RADIUS)**3

    print("number of events = %i" % len(ev))

    # Now, select prompt events.
    #
    # We define a prompt event here as any event with an NHIT > 100 and whose
    # previous > 100 nhit event was more than 250 ms ago
    #
    # Note: It's important we do this *before* applying the data cleaning cuts
    # since otherwise we may have a prompt event identified only after the
    # cuts.
    #
    # For example, suppose there was a breakdown and for whatever reason
    # the *second* event after the breakdown didn't get tagged correctly. If we
    # apply the data cleaning cuts first and then tag prompt events then this
    # event will get tagged as a prompt event.
    ev = ev.groupby('run',group_keys=False).apply(prompt_event)

    print("number of events after prompt nhit cut = %i" % np.count_nonzero(ev.prompt))

    # flasher follower cut
    ev = ev.groupby('run',group_keys=False).apply(flasher_follower_cut)

    # breakdown follower cut
    ev = ev.groupby('run',group_keys=False).apply(breakdown_follower_cut)

    # retrigger cut
    ev = ev.groupby('run',group_keys=False).apply(retrigger_cut)

    if args.save:
        # default \textwidth for a fullpage article in Latex is 16.50764 cm.
        # You can figure this out by compiling the following TeX document:
        #
        #    \documentclass{article}
        #    \usepackage{fullpage}
        #    \usepackage{layouts}
        #    \begin{document}
        #    textwidth in cm: \printinunitsof{cm}\prntlen{\textwidth}
        #    \end{document}

        width = 16.50764
        width /= 2.54 # cm -> inches
        # According to this page:
        # http://www-personal.umich.edu/~jpboyd/eng403_chap2_tuftegospel.pdf,
        # Tufte suggests an aspect ratio of 1.5 - 1.6.
        height = width/1.5
        FIGSIZE = (width,height)

        import matplotlib.pyplot as plt

        font = {'family':'serif', 'serif': ['computer modern roman']}
        plt.rc('font',**font)

        plt.rc('text', usetex=True)
    else:
        # on retina screens, the default plots are way too small
        # by using Qt5 and setting QT_AUTO_SCREEN_SCALE_FACTOR=1
        # Qt5 will scale everything using the dpi in ~/.Xresources
        import matplotlib
        matplotlib.use("Qt5Agg")

        import matplotlib.pyplot as plt

        # Default figure size. Currently set to my monitor width and height so that
        # things are properly formatted
        FIGSIZE = (13.78,7.48)

        # Make the defalt font bigger
        plt.rc('font', size=22)

    if args.dc:
        ev = ev[ev.prompt]

        ev.set_index(['run','gtid'])

        ev = pd.merge(fits,ev,how='inner',on=['run','gtid'])
        ev_single_particle = ev[(ev.id2 == 0) & (ev.id3 == 0)]
        ev_single_particle = ev_single_particle.sort_values('fmin').groupby(['run','gtid']).nth(0)
        ev = ev.sort_values('fmin').groupby(['run','gtid']).nth(0)

        ev['cos_theta'] = np.cos(ev['theta1'])
        ev['udotr'] = np.sin(ev_single_particle.theta1)*np.cos(ev_single_particle.phi1)*ev_single_particle.x + \
                      np.sin(ev_single_particle.theta1)*np.sin(ev_single_particle.phi1)*ev_single_particle.y + \
                      np.cos(ev_single_particle.theta1)*ev_single_particle.z
        ev['udotr'] /= ev.r

        flashers = ev[ev.dc & (DC_JUNK | DC_CRATE_ISOTROPY | DC_QVNHIT | DC_FLASHER | DC_NECK | DC_ITC | DC_BREAKDOWN) == DC_FLASHER]
        muon = ev[ev.dc & (DC_JUNK | DC_CRATE_ISOTROPY | DC_QVNHIT | DC_FLASHER | DC_NECK | DC_ITC | DC_BREAKDOWN | DC_MUON) == DC_MUON]
        neck = ev[(ev.dc & (DC_JUNK | DC_CRATE_ISOTROPY | DC_QVNHIT | DC_NECK)) == DC_NECK]
        noise = ev[(ev.dc & (DC_ITC | DC_QVNHIT | DC_JUNK | DC_CRATE_ISOTROPY)) != 0]
        breakdown = ev[ev.nhit >= 1000]
        breakdown = breakdown[breakdown.dc & (DC_JUNK | DC_CRATE_ISOTROPY | DC_QVNHIT | DC_NECK | DC_ITC) == 0]
        breakdown = breakdown[breakdown.dc & (DC_FLASHER | DC_BREAKDOWN) != 0]
        signal = ev[ev.dc & (DC_JUNK | DC_CRATE_ISOTROPY | DC_QVNHIT | DC_FLASHER | DC_NECK | DC_ITC | DC_BREAKDOWN | DC_MUON) == 0]

        with pd.option_context('display.max_rows', None, 'display.max_columns', None):
            print("Noise events")
            print(noise[['psi','x','y','z','id1','id2']])
            print("Muons")
            print(muon[['psi','r','id1','id2','id3','energy1','energy2','energy3']])
            print("Neck")
            print(neck[neck.psi < 6][['psi','r','id1','cos_theta']])
            print("Flashers")
            print(flashers[flashers.udotr > 0])
            print("Signal")
            print(signal)

        # save as PDF b/c EPS doesn't support alpha values
        if args.save:
            plot_corner_plot(breakdown,"Breakdowns",save="breakdown_corner_plot")
            plot_corner_plot(muon,"Muons",save="muon_corner_plot")
            plot_corner_plot(flashers,"Flashers",save="flashers_corner_plot")
            plot_corner_plot(neck,"Neck",save="neck_corner_plot")
            plot_corner_plot(noise,"Noise",save="noise_corner_plot")
            plot_corner_plot(signal,"Signal",save="signal_corner_plot")
        else:
            plot_corner_plot(breakdown,"Breakdowns")
            plot_corner_plot(muon,"Muons")
            plot_corner_plot(flashers,"Flashers")
            plot_corner_plot(neck,"Neck")
            plot_corner_plot(noise,"Noise")
            plot_corner_plot(signal,"Signal")

        fig = plt.figure(figsize=FIGSIZE)
        plot_hist2(flashers)
        despine(fig,trim=True)
        plt.suptitle("Flashers")
        fig = plt.figure(figsize=FIGSIZE)
        plot_hist2(muon,muons=True)
        despine(fig,trim=True)
        plt.suptitle("Muons")
        plt.show()
        sys.exit(0)

    # First, do basic data cleaning which is done for all events.
    ev = ev[ev.dc & (DC_JUNK | DC_CRATE_ISOTROPY | DC_QVNHIT | DC_FLASHER | DC_NECK | DC_ITC | DC_BREAKDOWN) == 0]

    # 00-orphan cut
    ev = ev[(ev.gtid & 0xff) != 0]

    print("number of events after data cleaning = %i" % np.count_nonzero(ev.prompt))

    # Now, we select events tagged by the muon tag which should tag only
    # external muons. We keep the sample of muons since it's needed later to
    # identify Michel electrons and to apply the muon follower cut
    muons = ev[(ev.dc & DC_MUON) != 0]

    print("number of muons = %i" % len(muons))

    # Try to identify Michel electrons. Currently, the event selection is based
    # on Richie's thesis. Here, we do the following:
    #
    # 1. Apply more data cleaning cuts to potential Michel electrons
    # 2. Nhit >= 100
    # 3. It must be > 800 ns and less than 20 microseconds from a prompt event
    #    or a muon
    michel = ev.groupby('run',group_keys=False).apply(michel_cut)

    print("number of michel events = %i" % len(michel))

    # Tag atmospheric events.
    #
    # Note: We don't cut atmospheric events or muons yet because we still need
    # all the events in order to apply the muon follower cut.
    ev = ev.groupby('run',group_keys=False).apply(atmospheric_events)

    print("number of events after neutron follower cut = %i" % np.count_nonzero(ev.prompt & (~ev.atm)))

    # remove events 200 microseconds after a muon
    ev = ev.groupby('run',group_keys=False).apply(muon_follower_cut)

    # Get rid of muon events in our main event sample
    ev = ev[(ev.dc & DC_MUON) == 0]

    prompt = ev[ev.prompt & ~ev.atm]

    atm = ev[ev.atm]

    print("number of events after muon cut = %i" % len(prompt))

    # Check to see if there are any events with missing fit information
    atm_ra = atm[['run','gtid']].to_records(index=False)
    muons_ra = muons[['run','gtid']].to_records(index=False)
    prompt_ra = prompt[['run','gtid']].to_records(index=False)
    michel_ra = michel[['run','gtid']].to_records(index=False)
    fits_ra = fits[['run','gtid']].to_records(index=False)

    if len(atm_ra) and np.count_nonzero(~np.isin(atm_ra,fits_ra)):
        print_warning("skipping %i atmospheric events because they are missing fit information!" % np.count_nonzero(~np.isin(atm_ra,fits_ra)))

    if len(muons_ra) and np.count_nonzero(~np.isin(muons_ra,fits_ra)):
        print_warning("skipping %i muon events because they are missing fit information!" % np.count_nonzero(~np.isin(muons_ra,fits_ra)))

    if len(prompt_ra) and np.count_nonzero(~np.isin(prompt_ra,fits_ra)):
        print_warning("skipping %i signal events because they are missing fit information!" % np.count_nonzero(~np.isin(prompt_ra,fits_ra)))

    if len(michel_ra) and np.count_nonzero(~np.isin(michel_ra,fits_ra)):
        print_warning("skipping %i Michel events because they are missing fit information!" % np.count_nonzero(~np.isin(michel_ra,fits_ra)))

    # Now, we merge the event info with the fitter info.
    #
    # Note: This means that the dataframe now contains multiple rows for each
    # event, one for each fit hypothesis.
    atm = pd.merge(fits,atm,how='inner',on=['run','gtid'])
    muons = pd.merge(fits,muons,how='inner',on=['run','gtid'])
    michel = pd.merge(fits,michel,how='inner',on=['run','gtid'])
    prompt = pd.merge(fits,prompt,how='inner',on=['run','gtid'])

    # get rid of events which don't have a fit
    nan = np.isnan(prompt.fmin.values)

    if np.count_nonzero(nan):
        print_warning("skipping %i signal events because the negative log likelihood is nan!" % len(prompt[nan].groupby(['run','gtid'])))

    prompt = prompt[~nan]

    nan_atm = np.isnan(atm.fmin.values)

    if np.count_nonzero(nan_atm):
        print_warning("skipping %i atmospheric events because the negative log likelihood is nan!" % len(atm[nan_atm].groupby(['run','gtid'])))

    atm = atm[~nan_atm]

    nan_muon = np.isnan(muons.fmin.values)

    if np.count_nonzero(nan_muon):
        print_warning("skipping %i muons because the negative log likelihood is nan!" % len(muons[nan_muon].groupby(['run','gtid'])))

    muons = muons[~nan_muon]

    nan_michel = np.isnan(michel.fmin.values)

    if np.count_nonzero(nan_michel):
        print_warning("skipping %i michel electron events because the negative log likelihood is nan!" % len(michel[nan_michel].groupby(['run','gtid'])))

    michel = michel[~nan_michel]

    # get the best fit
    prompt = prompt.sort_values('fmin').groupby(['run','gtid']).nth(0)
    atm = atm.sort_values('fmin').groupby(['run','gtid']).nth(0)
    michel_best_fit = michel.sort_values('fmin').groupby(['run','gtid']).nth(0)
    muon_best_fit = muons.sort_values('fmin').groupby(['run','gtid']).nth(0)
    muons = muons[muons.id == 22]

    # require r < 6 meters
    prompt = prompt[prompt.r_psup < 0.9]
    atm = atm[atm.r_psup < 0.9]

    print("number of events after radius cut = %i" % len(prompt))

    # Note: Need to design and apply a psi based cut here

    fig = plt.figure(figsize=FIGSIZE)
    plot_hist2(prompt)
    despine(fig,trim=True)
    if args.save:
        plt.savefig("prompt.pdf")
        plt.savefig("prompt.eps")
    else:
        plt.suptitle("Without Neutron Follower")
    fig = plt.figure(figsize=FIGSIZE)
    plot_hist2(atm)
    despine(fig,trim=True)
    if args.save:
        plt.savefig("atm.pdf")
        plt.savefig("atm.eps")
    else:
        plt.suptitle("With Neutron Follower")
    fig = plt.figure(figsize=FIGSIZE)
    plot_hist2(michel_best_fit)
    despine(fig,trim=True)
    if args.save:
        plt.savefig("michel_electrons.pdf")
        plt.savefig("michel_electrons.eps")
    else:
        plt.suptitle("Michel Electrons")
    fig = plt.figure(figsize=FIGSIZE)
    plot_hist2(muon_best_fit,muons=True)
    despine(fig,trim=True)
    if len(muon_best_fit):
        plt.tight_layout()
    if args.save:
        plt.savefig("external_muons.pdf")
        plt.savefig("external_muons.eps")
    else:
        plt.suptitle("External Muons")

    # Plot the energy and angular distribution for external muons
    fig = plt.figure(figsize=FIGSIZE)
    plt.subplot(2,1,1)
    plt.hist(muons.ke.values, bins=np.logspace(3,7,100), histtype='step')
    plt.xlabel("Energy (MeV)")
    plt.gca().set_xscale("log")
    plt.subplot(2,1,2)
    plt.hist(np.cos(muons.theta.values), bins=np.linspace(-1,1,100), histtype='step')
    despine(fig,trim=True)
    plt.xlabel(r"$\cos(\theta)$")
    plt.tight_layout()
    if args.save:
        plt.savefig("muon_energy_cos_theta.pdf")
        plt.savefig("muon_energy_cos_theta.eps")
    else:
        plt.suptitle("Muons")

    # For the Michel energy plot, we only look at the single particle electron
    # fit
    michel = michel[michel.id == 20]

    stopping_muons = pd.merge(muons,michel,left_on=['run','gtid'],right_on=['run','muon_gtid'],suffixes=('','_michel'))

    if len(stopping_muons):
        # project muon to PSUP
        stopping_muons['dx'] = stopping_muons.apply(get_dx,axis=1)
        # energy based on distance travelled
        stopping_muons['T_dx'] = dx_to_energy(stopping_muons.dx)
        stopping_muons['dT'] = stopping_muons['energy1'] - stopping_muons['T_dx']

        fig = plt.figure(figsize=FIGSIZE)
        plt.hist((stopping_muons['energy1']-stopping_muons['T_dx'])*100/stopping_muons['T_dx'], bins=np.linspace(-100,100,200), histtype='step')
        despine(fig,trim=True)
        plt.xlabel("Fractional energy difference (\%)")
        plt.title("Fractional energy difference for Stopping Muons")
        plt.tight_layout()
        if args.save:
            plt.savefig("stopping_muon_fractional_energy_difference.pdf")
            plt.savefig("stopping_muon_fractional_energy_difference.eps")
        else:
            plt.title("Stopping Muon Fractional Energy Difference")

        # 100 bins between 50 MeV and 10 GeV
        bins = np.arange(50,10000,1000)

        pd_bins = pd.cut(stopping_muons['energy1'],bins)

        T = (bins[1:] + bins[:-1])/2

        dT = stopping_muons.groupby(pd_bins)['dT'].agg(['mean','sem','std',std_err,median,median_err,iqr_std,iqr_std_err])

        fig = plt.figure(figsize=FIGSIZE)
        plt.errorbar(T,dT['median']*100/T,yerr=dT['median_err']*100/T)
        despine(fig,trim=True)
        plt.xlabel("Kinetic Energy (MeV)")
        plt.ylabel(r"Energy bias (\%)")
        plt.tight_layout()
        if args.save:
            plt.savefig("stopping_muon_energy_bias.pdf")
            plt.savefig("stopping_muon_energy_bias.eps")
        else:
            plt.title("Stopping Muon Energy Bias")

        fig = plt.figure(figsize=FIGSIZE)
        plt.errorbar(T,dT['iqr_std']*100/T,yerr=dT['iqr_std_err']*100/T)
        despine(fig,trim=True)
        plt.xlabel("Kinetic Energy (MeV)")
        plt.ylabel(r"Energy resolution (\%)")
        plt.tight_layout()
        if args.save:
            plt.savefig("stopping_muon_energy_resolution.pdf")
            plt.savefig("stopping_muon_energy_resolution.eps")
        else:
            plt.title("Stopping Muon Energy Resolution")

    fig = plt.figure(figsize=FIGSIZE)
    bins=np.linspace(0,100,100)
    plt.hist(michel.ke.values, bins=bins, histtype='step', label="Dark Matter Fitter")
    if michel.size:
        plt.hist(michel[~np.isnan(michel.rsp_energy.values)].rsp_energy.values, bins=np.linspace(20,100,100), histtype='step',label="RSP")
    x = np.linspace(0,100,1000)
    y = michel_spectrum(x)
    y /= np.trapz(y,x=x)
    N = len(michel)
    plt.plot(x, N*y*(bins[1]-bins[0]), ls='--', color='k', label="Michel Spectrum")
    despine(fig,trim=True)
    plt.xlabel("Energy (MeV)")
    plt.tight_layout()
    plt.legend()
    if args.save:
        plt.savefig("michel_electrons_ke.pdf")
        plt.savefig("michel_electrons_ke.eps")
    else:
        plt.title("Michel Electrons")
    plt.show()