#+PROPERTY: header-args :exports both :output-dir results :session xs :kernel python3
#+HTML_HEAD: <link rel="stylesheet" href="tufte.css" />
#+OPTIONS: html-style:nil
#+HTML_CONTAINER: section
#+TITLE: Investigaton of Monte-Carlo Methods
#+AUTHOR: Valentin Boettcher

* Init
** Required Modules
#+NAME: e988e3f2-ad1f-49a3-ad60-bedba3863283
#+begin_src jupyter-python :exports both :tangle tangled/xs.py
  import numpy as np
  import matplotlib.pyplot as plt
  import monte_carlo
#+end_src

#+RESULTS: e988e3f2-ad1f-49a3-ad60-bedba3863283

** Utilities
#+NAME: 53548778-a4c1-461a-9b1f-0f401df12b08
#+BEGIN_SRC jupyter-python :exports both
%run ../utility.py
%load_ext autoreload
%aimport monte_carlo
%autoreload 1
#+END_SRC

#+RESULTS: 53548778-a4c1-461a-9b1f-0f401df12b08

* Implementation
#+NAME: 777a013b-6c20-44bd-b58b-6a7690c21c0e
#+BEGIN_SRC jupyter-python :exports both :results raw drawer :exports code :tangle tangled/xs.py
  """
  Implementation of the analytical cross section for q q_bar ->
  gamma gamma

  Author: Valentin Boettcher <hiro@protagon.space>
  """

  import numpy as np

  # NOTE: a more elegant solution would be a decorator
  def energy_factor(charge, esp):
      """
      Calculates the factor common to all other values in this module

      Arguments:
      esp -- center of momentum energy in GeV
      charge -- charge of the particle in units of the elementary charge
      """

      return charge**4/(137.036*esp)**2/6


  def diff_xs(θ, charge, esp):
      """
      Calculates the differential cross section as a function of the
      azimuth angle θ in units of 1/GeV².

      Here dΩ=sinθdθdφ

      Arguments:
      θ -- azimuth angle
      esp -- center of momentum energy in GeV
      charge -- charge of the particle in units of the elementary charge
      """

      f = energy_factor(charge, esp)
      return f*((np.cos(θ)**2+1)/np.sin(θ)**2)

  def diff_xs_cosθ(cosθ, charge, esp):
      """
      Calculates the differential cross section as a function of the
      cosine of the azimuth angle θ in units of 1/GeV².

      Here dΩ=d(cosθ)dφ

      Arguments:
      cosθ -- cosine of the azimuth angle
      esp -- center of momentum energy in GeV
      charge -- charge of the particle in units of the elementary charge
      """

      f = energy_factor(charge, esp)
      return f*((cosθ**2+1)/(1-cosθ**2))


  def diff_xs_eta(η, charge, esp):
      """
      Calculates the differential cross section as a function of the
      pseudo rapidity of the photons in units of 1/GeV^2.

      This is actually the crossection dσ/(dφdη).

      Arguments:
      η -- pseudo rapidity
      esp -- center of momentum energy in GeV
      charge -- charge of the particle in units of the elementary charge
      """

      f = energy_factor(charge, esp)
      return f*(np.tanh(η)**2 + 1)


  def diff_xs_p_t(p_t, charge, esp):
      """
      Calculates the differential cross section as a function of the
      transverse momentum (p_t) of the photons in units of 1/GeV^2.

      This is actually the crossection dσ/(dφdp_t).

      Arguments:
      p_t -- transverse momentum in GeV
      esp -- center of momentum energy in GeV
      charge -- charge of the particle in units of the elementary charge
      """

      f = energy_factor(charge, esp)
      sqrt_fact = np.sqrt(1-(2*p_t/esp)**2)
      return f/p_t*(1/sqrt_fact + sqrt_fact)


  def total_xs_eta(η, charge, esp):
      """
      Calculates the total cross section as a function of the pseudo
      rapidity of the photons in units of 1/GeV^2.  If the rapditiy is
      specified as a tuple, it is interpreted as an interval.  Otherwise
      the interval [-η, η] will be used.

      Arguments:
      η -- pseudo rapidity (tuple or number)
      esp -- center of momentum energy in GeV
      charge -- charge of the particle in units of the elementar charge
      """

      f = energy_factor(charge, esp)
      if not isinstance(η, tuple):
          η = (-η, η)

      if len(η) != 2:
          raise ValueError('Invalid η cut.')

      def F(x):
          return np.tanh(x) - 2*x

      return 2*np.pi*f*(F(η[0]) - F(η[1]))
#+END_SRC

#+RESULTS: 777a013b-6c20-44bd-b58b-6a7690c21c0e

* Calculations
First, set up the input parameters.
#+BEGIN_SRC jupyter-python :exports both :results raw drawer
η = 2.5
charge = 1/3
esp = 200  # GeV
#+END_SRC

#+RESULTS:

Set up the integration and plot intervals.
#+begin_src jupyter-python :exports both :results raw drawer
interval_η = [-η, η]
interval = η_to_θ([-η, η])
interval_cosθ = np.cos(interval)
interval_pt = np.sort(η_to_pt([0, η], esp/2))
plot_interval = [0.1, np.pi-.1]
#+end_src

#+RESULTS:

#+begin_note
Note that we could utilize the symetry of the integrand throughout,
but that doen't reduce variance and would complicate things now.
#+end_note

** Analytical Integration
 And now calculate the cross section in picobarn.
 #+BEGIN_SRC jupyter-python :exports both :results raw file :file xs.tex
   xs_gev = total_xs_eta(η, charge, esp)
   xs_pb = gev_to_pb(xs_gev)
   tex_value(xs_pb, unit=r'\pico\barn', prefix=r'\sigma = ',
             prec=6, save=('results', 'xs.tex'))
 #+END_SRC

 #+RESULTS:
 : \(\sigma = \SI{0.053793}{\pico\barn}\)

 Lets plot the total xs as a function of η.
 #+begin_src jupyter-python :exports both :results raw drawer
   fig, ax = set_up_plot()
   η_s = np.linspace(0, 3, 1000)
   ax.plot(η_s, gev_to_pb(total_xs_eta(η_s, charge, esp)))
   ax.set_xlabel(r'$\eta$')
   ax.set_ylabel(r'$\sigma$ [pb]')
   ax.set_xlim([0, max(η_s)])
   ax.set_ylim(0)
   save_fig(fig, 'total_xs', 'xs', size=[2.5, 2.5])
 #+end_src

 #+RESULTS:
 [[file:./.ob-jupyter/4522eb3fbeaa14978f9838371acb0650910b8dbf.png]]


 Compared to sherpa, it's pretty close.
 #+NAME: 81b5ed93-0312-45dc-beec-e2ba92e22626
 #+BEGIN_SRC jupyter-python :exports both :results raw drawer
   sherpa = 0.05380
   xs_pb - sherpa
 #+END_SRC

 #+RESULTS: 81b5ed93-0312-45dc-beec-e2ba92e22626
 : -6.7112594623469635e-06

 I had to set the runcard option ~EW_SCHEME: alpha0~ to use the pure
 QED coupling constant.

** Numerical Integration
Plot our nice distribution:
#+begin_src jupyter-python :exports both :results raw drawer
  plot_points = np.linspace(*plot_interval, 1000)

  fig, ax = set_up_plot()
  ax.plot(plot_points, gev_to_pb(diff_xs(plot_points, charge=charge, esp=esp)))
  ax.set_xlabel(r'$\theta$')
  ax.set_ylabel(r'$d\sigma/d\Omega$ [pb]')
  ax.set_xlim([plot_points.min(), plot_points.max()])
  ax.axvline(interval[0], color='gray', linestyle='--')
  ax.axvline(interval[1], color='gray', linestyle='--', label=rf'$|\eta|={η}$')
  ax.legend()
  save_fig(fig, 'diff_xs', 'xs', size=[2.5, 2.5])
#+end_src

#+RESULTS:
[[file:./.ob-jupyter/3dd905e7608b91a9d89503cb41660152f3b4b55c.png]]

Define the integrand.
#+begin_src jupyter-python :exports both :results raw drawer
  def xs_pb_int(θ):
      return 2*np.pi*gev_to_pb(np.sin(θ)*diff_xs(θ, charge=charge, esp=esp))

  def xs_pb_int_η(η):
      return 2*np.pi*gev_to_pb(diff_xs_eta(η, charge, esp))
#+end_src

#+RESULTS:

Plot the integrand. # TODO: remove duplication
#+begin_src jupyter-python :exports both :results raw drawer
  fig, ax = set_up_plot()
  ax.plot(plot_points, xs_pb_int(plot_points))
  ax.set_xlabel(r'$\theta$')
  ax.set_ylabel(r'$2\pi\cdot d\sigma/d\theta [pb]')
  ax.set_xlim([plot_points.min(), plot_points.max()])
  ax.axvline(interval[0], color='gray', linestyle='--')
  ax.axvline(interval[1], color='gray', linestyle='--', label=rf'$|\eta|={η}$')
  save_fig(fig, 'xs_integrand', 'xs', size=[3, 2.2])
#+end_src

#+RESULTS:
[[file:./.ob-jupyter/ccb6653162c81c3f3e843225cb8d759178f497e0.png]]
*** Integral over θ
Intergrate σ with the mc method.
#+begin_src jupyter-python :exports both :results raw drawer
  xs_pb_res = monte_carlo.integrate(xs_pb_int, interval, epsilon=1e-3)
  xs_pb_res
#+end_src

#+RESULTS:
: IntegrationResult(result=0.05409249508068646, sigma=0.0007897018499681794, N=3286)

We gonna export that as tex.
#+begin_src jupyter-python :exports both :results raw drawer
  tex_value(*xs_pb_res.combined_result, unit=r'\pico\barn',
            prefix=r'\sigma = ', save=('results', 'xs_mc.tex'))
  tex_value(xs_pb_res.N, prefix=r'N = ', save=('results', 'xs_mc_N.tex'))
#+end_src

#+RESULTS:
: \(N = 3286\)

*** Integration over η
Plot the intgrand of the pseudo rap.
#+begin_src jupyter-python :exports both :results raw drawer
  fig, ax = set_up_plot()
  points = np.linspace(-4, 4, 1000)
  ax.set_xlim([-4, 4])
  ax.plot(points, xs_pb_int_η(points))
  ax.set_xlabel(r'$\eta$')
  ax.set_ylabel(r'$2\pi\cdot d\sigma/d\eta$ [pb]')
  ax.axvline(interval_η[0], color='gray', linestyle='--')
  ax.axvline(interval_η[1], color='gray', linestyle='--', label=rf'$|\eta|={η}$')
  save_fig(fig, 'xs_integrand_eta', 'xs', size=[3, 2])
#+end_src

#+RESULTS:
[[file:./.ob-jupyter/87a932866f779a2a07abed4ca251fa98113beca7.png]]

#+begin_src jupyter-python :exports both :results raw drawer
  xs_pb_η = monte_carlo.integrate(xs_pb_int_η,
                                  interval_η, epsilon=1e-3)
  xs_pb_η
#+end_src

#+RESULTS:
: IntegrationResult(result=0.05514934067591144, sigma=0.0009042241764415168, N=135)

As we see, the result is a little better if we use pseudo rapidity,
because the differential cross section does not difverge anymore.  But
becase our η interval is covering the range where all the variance is
occuring, the improvement is rather marginal.

And yet again export that as tex.
#+begin_src jupyter-python :exports both :results raw drawer
  tex_value(*xs_pb_η.combined_result, unit=r'\pico\barn', prefix=r'\sigma = ',
            save=('results', 'xs_mc_eta.tex'))
  tex_value(xs_pb_η.N, prefix=r'N = ', save=('results', 'xs_mc_eta_N.tex'))
#+end_src

#+RESULTS:
: \(N = 135\)

*** Using =VEGAS=
Now we use =VEGAS= on the θ parametrisation and see what happens.
#+begin_src jupyter-python :exports both :results raw drawer
  xs_pb_vegas, xs_pb_vegas_σ, xs_θ_intervals = \
      monte_carlo.integrate_vegas(xs_pb_int, interval,
                                  num_increments=20, alpha=4,
                                  point_density=1000, acumulate=True)
  xs_pb_vegas, xs_pb_vegas_σ
#+end_src

#+RESULTS:
| 0.0538027817458234 | 5.484354854125085e-05 |

This is pretty good, although the variance reduction may be achieved
partially by accumulating the results from all runns. The uncertainty
is being overestimated!

And export that as tex.
#+begin_src jupyter-python :exports both :results raw drawer
  tex_value(xs_pb_vegas, xs_pb_vegas_σ, unit=r'\pico\barn',
            prefix=r'\sigma = ', save=('results', 'xs_mc_θ_vegas.tex'))
#+end_src

#+RESULTS:
: \(\sigma = \SI{0.05380\pm 0.00005}{\pico\barn}\)

Surprisingly, without acumulation, the result ain't much different.
This depends, of course, on the iteration count.
#+begin_src jupyter-python :exports both :results raw drawer
  monte_carlo.integrate_vegas(xs_pb_int, interval, num_increments=20,
                              alpha=4, point_density=1000,
                              acumulate=False)[0:2]
#+end_src

#+RESULTS:
| 0.05374045987757417 | 8.173017257626389e-05 |

*** Testing the Statistics
Let's battle test the statistics.
#+begin_src jupyter-python :exports both :results raw drawer
  num_runs = 1000
  num_within = 0

  for _ in range(num_runs):
      val, err = \
          monte_carlo.integrate(xs_pb_int, interval, epsilon=1e-3).combined_result
      if abs(xs_pb - val) <= err:
          num_within += 1

  num_within/num_runs
#+end_src

#+RESULTS:
: 0.689

So we see: the standard deviation is sound.

Doing the same thing with =VEGAS= works as well.
#+begin_src jupyter-python :exports both :results raw drawer
  num_runs = 1000
  num_within = 0
  for _ in range(num_runs):
      val, err, _ = \
          monte_carlo.integrate_vegas(xs_pb_int, interval,
                                      num_increments=8, alpha=1,
                                      point_density=1000, acumulate=False)

      if abs(xs_pb - val) <= err:
          num_within += 1
  num_within/num_runs
#+end_src

#+RESULTS:
: 0.67

** Sampling and Analysis
Define the sample number.
#+begin_src jupyter-python :exports both :results raw drawer
  sample_num = 1000
#+end_src

#+RESULTS:
Let's define shortcuts for our distributions. The 2π are just there
for formal correctnes. Factors do not influecence the outcome.
#+begin_src jupyter-python :exports both :results raw drawer
  def dist_cosθ(x):
      return gev_to_pb(diff_xs_cosθ(x, charge, esp))*2*np.pi

  def dist_η(x):
      return gev_to_pb(diff_xs_eta(x, charge, esp))*2*np.pi
#+end_src

#+RESULTS:

*** Sampling the cosθ cross section

Now we monte-carlo sample our distribution. We observe that the efficiency his very bad!
#+begin_src jupyter-python :exports both :results raw drawer
  cosθ_sample, cosθ_efficiency = \
      monte_carlo.sample_unweighted_array(sample_num, dist_cosθ,
                                          interval_cosθ, report_efficiency=True)
  cosθ_efficiency
#+end_src

#+RESULTS:
: 0.028076221366308534

Our distribution has a lot of variance, as can be seen by plotting it.
#+begin_src jupyter-python :exports both :results raw drawer
  pts = np.linspace(*interval_cosθ, 100)
  fig, ax = set_up_plot()
  ax.plot(pts, dist_cosθ(pts))
  ax.set_xlabel(r'$\cos\theta$')
  ax.set_ylabel(r'$\frac{d\sigma}{d\Omega}$')
#+end_src

#+RESULTS:
:RESULTS:
: Text(0, 0.5, '$\\frac{d\\sigma}{d\\Omega}$')
[[file:./.ob-jupyter/6921725d93ce91ce1e0364e6f745d46f3a76b3f2.png]]
:END:

We define a friendly and easy to integrate upper limit function.
#+begin_src jupyter-python :exports both :results raw drawer
  upper_limit = dist_cosθ(interval_cosθ[0]) \
      /interval_cosθ[0]**2
  upper_base = dist_cosθ(0)

  def upper(x):
      return  upper_base + upper_limit*x**2

  def upper_int(x):
      return  upper_base*x + upper_limit*x**3/3

  ax.plot(pts, upper(pts), label='Upper bound')
  ax.legend()
  ax.set_xlabel(r'$\cos\theta$')
  ax.set_ylabel(r'$\frac{d\sigma}{d\Omega}$')
  save_fig(fig, 'upper_bound', 'xs_sampling', size=(4, 4))
  fig
#+end_src

#+RESULTS:
[[file:./.ob-jupyter/ddfcebac4157ce417e5b868a88731d554c726141.png]]


To increase our efficiency, we have to specify an upper bound. That is
at least a little bit better. The numeric inversion is horribly inefficent.
#+begin_src jupyter-python :exports both :results raw drawer
  cosθ_sample, cosθ_efficiency = \
      monte_carlo.sample_unweighted_array(sample_num, dist_cosθ,
                                          interval_cosθ, report_efficiency=True,
                                          upper_bound=[upper, upper_int])
  cosθ_efficiency
#+end_src

#+RESULTS:
: 0.07981958240421651
<<cosθ-bare-eff>>

Nice! And now draw some histograms.

We define an auxilliary method for convenience.
#+begin_src jupyter-python :exports both :results raw drawer :tangle tangled/plot_utils.py
  """
  Some shorthands for common plotting tasks related to the investigation
  of monte-carlo methods in one rimension.

  Author: Valentin Boettcher <hiro at protagon.space>
  """

  import matplotlib.pyplot as plt

  def draw_histo(points, xlabel, bins=20):
      heights, edges = np.histogram(points, bins)
      centers = (edges[1:] + edges[:-1])/2
      deviations = np.sqrt(heights)

      fig, ax = set_up_plot()
      ax.errorbar(centers, heights, deviations, linestyle='none', color='orange')
      ax.step(edges,  [heights[0], *heights], color='#1f77b4')

      ax.set_xlabel(xlabel)
      ax.set_xlim([points.min(), points.max()])
      return fig, ax
#+end_src

#+RESULTS:

The histogram for cosθ.
#+begin_src jupyter-python :exports both :results raw drawer
fig, _ = draw_histo(cosθ_sample, r'$\cos\theta$')
save_fig(fig, 'histo_cos_theta', 'xs', size=(4,3))
#+end_src

#+RESULTS:
[[file:./.ob-jupyter/57f63adafce401329082b5362df69e3bfd651738.png]]

*** Observables
Now we define some utilities to draw real 4-momentum samples.
#+begin_src jupyter-python :exports both :tangle tangled/xs.py
  def sample_momenta(sample_num, interval, charge, esp, seed=None):
      """Samples `sample_num` unweighted photon 4-momenta from the
      cross-section.

      :param sample_num: number of samples to take
      :param interval: cosθ interval to sample from
      :param charge: the charge of the quark
      :param esp: center of mass energy
      :param seed: the seed for the rng, optional, default is system
          time

      :returns: an array of 4 photon momenta

      :rtype: np.ndarray
      """
      cosθ_sample = \
          monte_carlo.sample_unweighted_array(sample_num,
                                              lambda x:
                                                diff_xs_cosθ(x, charge, esp),
                                             interval_cosθ)
      φ_sample = np.random.uniform(0, 1, sample_num)

      def make_momentum(esp, cosθ, φ):
          sinθ = np.sqrt(1-cosθ**2)
          return np.array([1, sinθ*np.cos(φ), sinθ*np.sin(φ), cosθ])*esp/2

      momenta = np.array([make_momentum(esp, cosθ, φ) \
                           for cosθ, φ in np.array([cosθ_sample, φ_sample]).T])
      return momenta
#+end_src

#+RESULTS:

To generate histograms of other obeservables, we have to define them
as functions on 4-impuleses. Using those to transform samples is
analogous to transforming the distribution itself.
#+begin_src jupyter-python :session obs :exports both :results raw drawer :tangle tangled/observables.py
  """This module defines some observables on arrays of 4-pulses."""
  import numpy as np

  def p_t(p):
      """Transverse momentum

      :param p: array of 4-momenta
      """

      return np.linalg.norm(p[:,1:3], axis=1)

  def η(p):
      """Pseudo rapidity.

      :param p: array of 4-momenta
      """

      return np.arccosh(np.linalg.norm(p[:,1:], axis=1)/p_t(p))*np.sign(p[:, 3])
#+end_src

#+RESULTS:

And import them.
#+begin_src jupyter-python :exports both :results raw drawer
  %aimport tangled.observables
  obs = tangled.observables
#+end_src

#+RESULTS:

Lets try it out.
#+begin_src jupyter-python :exports both :results raw drawer
  momentum_sample = sample_momenta(2000, interval_cosθ, charge, esp)
  momentum_sample
#+end_src

#+RESULTS:
: array([[100.        ,  48.96033819,  57.4112904 , -65.62643537],
:        [100.        ,  41.96222688,  20.41198939,  88.44502363],
:        [100.        ,  58.16555642,  25.62592984, -77.20155287],
:        ...,
:        [100.        ,  60.81855224,   5.52673184,  79.1868609 ],
:        [100.        ,  39.11244981,  43.57566968,  81.06403198],
:        [100.        ,  63.55704011,  12.99669204,  76.10248779]])

Now let's make a histogram of the η distribution.
#+begin_src jupyter-python :exports both :results raw drawer
  η_sample = obs.η(momentum_sample)
  draw_histo(η_sample, r'$\eta$')
#+end_src

#+RESULTS:
:RESULTS:
| <Figure | size | 432x288 | with | 1 | Axes> | <matplotlib.axes._subplots.AxesSubplot | at | 0x7f62fa6b4430> |
[[file:./.ob-jupyter/e04af38d1dcc52c09ded26b65234a071f2386835.png]]
:END:


And the same for the p_t (transverse momentum) distribution.
#+begin_src jupyter-python :exports both :results raw drawer
  p_t_sample = obs.p_t(momentum_sample)
  draw_histo(p_t_sample, r'$p_T$ [GeV]')
#+end_src

#+RESULTS:
:RESULTS:
| <Figure | size | 432x288 | with | 1 | Axes> | <matplotlib.axes._subplots.AxesSubplot | at | 0x7f62fa951b20> |
[[file:./.ob-jupyter/b9671fcec48bc029f5fe7e1cafb23a219045d90b.png]]
:END:

That looks somewhat fishy, but it isn't.
#+begin_src jupyter-python :exports both :results raw drawer
  fig, ax = set_up_plot()
  points = np.linspace(interval_pt[0], interval_pt[1] - .01, 1000)
  ax.plot(points, gev_to_pb(diff_xs_p_t(points, charge, esp)))
  ax.set_xlabel(r'$p_T$')
  ax.set_xlim(interval_pt[0], interval_pt[1] + 1)
  ax.set_ylim([0, gev_to_pb(diff_xs_p_t(interval_pt[1] -.01, charge, esp))])
  ax.set_ylabel(r'$\frac{d\sigma}{dp_t}$ [pb]')
  save_fig(fig, 'diff_xs_p_t', 'xs_sampling', size=[4, 3])
#+end_src

#+RESULTS:
[[file:./.ob-jupyter/e127df693158dd4194b53f0a6f66ca2fca18af41.png]]
this is strongly peaked at p_t=100GeV. (The jacobian goes like 1/x there!)

*** Sampling the η cross section
An again we see that the efficiency is way, way! better...
#+begin_src jupyter-python :exports both :results raw drawer
  η_sample, η_efficiency = \
      monte_carlo.sample_unweighted_array(sample_num, dist_η,
                                          interval_η, report_efficiency=True)
  η_efficiency
#+end_src

#+RESULTS:
: 0.40801
<<η-eff>>

Let's draw a histogram to compare with the previous results.
#+begin_src jupyter-python :exports both :results raw drawer
  draw_histo(η_sample, r'$\eta$')
#+end_src

#+RESULTS:
:RESULTS:
| <Figure | size | 432x288 | with | 1 | Axes> | <matplotlib.axes._subplots.AxesSubplot | at | 0x7f62fa6b4c10> |
[[file:./.ob-jupyter/e2afd8b997280a89d7bcbf617018797c57909f79.png]]
:END:


Looks good to me :).

*** Sampling with =VEGAS=
Let's define some little helpers.
#+begin_src jupyter-python :exports both :tangle tangled/plot_utils.py
  def plot_increments(ax, increment_borders, label=None, *args, **kwargs):
      """Plot the increment borders from a list.  The first and last one

      :param ax: the axis on which to draw
      :param list increment_borders: the borders of the increments
      :param str label: the label to apply to one of the vertical lines
      """

      ax.axvline(x=increment_borders[1], label=label, *args, **kwargs)

      for increment in increment_borders[2:-1]:
          ax.axvline(x=increment, *args, **kwargs)

  def plot_vegas_weighted_distribution(ax, points, dist,
                                       increment_borders, *args, **kwargs):
      """Plot the distribution with VEGAS weights applied.

      :param ax: axis
      :param points: points
      :param dist: distribution
      :param increment_borders: increment borders
      """

      num_increments = increment_borders.size
      weighted_dist = dist.copy()

      for left_border, right_border in zip(increment_borders[:-1],
                                          increment_borders[1:]):
          length = right_border - left_border
          mask = (left_border <= points) & (points <= right_border)
          weighted_dist[mask] = dist[mask]*num_increments*length

      ax.plot(points, weighted_dist, *args, **kwargs)
#+end_src

#+RESULTS:

To get the increments, we have to let =VEGAS= loose on our
distribution. We throw away the integral, but keep the increments.

#+begin_src jupyter-python :exports both :results raw drawer
  _, _, increments = monte_carlo.integrate_vegas(dist_cosθ,
                                                 interval_cosθ,
                                                 num_increments=10, alpha=1,
                                                 epsilon=.01)
  increments
#+end_src

#+RESULTS:
: array([-0.9866143 , -0.90976814, -0.80054812, -0.61044436, -0.33330198,
:         0.00274503,  0.33807643,  0.61258903,  0.80033334,  0.90956851,
:         0.9866143 ])

Visualizing the increment borders gives us the information we want.
#+begin_src jupyter-python :exports both :results raw drawer
  pts = np.linspace(*interval_cosθ, 100)
  fig, ax = set_up_plot()
  ax.plot(pts, dist_cosθ(pts))
  ax.set_xlabel(r'$\cos\theta$')
  ax.set_ylabel(r'$\frac{d\sigma}{d\Omega}$')
  ax.set_xlim(*interval_cosθ)
  plot_increments(ax, increments,
                  label='Increment Borderds', color='gray', linestyle='--')
  ax.legend()
#+end_src

#+RESULTS:
:RESULTS:
: <matplotlib.legend.Legend at 0x7f62fa1a18e0>
[[file:./.ob-jupyter/487a63dc470d0aa11e8ea6c8fedd3509fc81bb47.png]]
:END:

We can now plot the reweighted distribution to observe the variance
reduction visually.

#+begin_src jupyter-python :exports both :results raw drawer
  pts = np.linspace(*interval_cosθ, 1000)
  fig, ax = set_up_plot()
  plot_vegas_weighted_distribution(ax, pts, dist_cosθ(pts), increments)
  ax.set_xlabel(r'$\cos\theta$')
  ax.set_ylabel(r'$\frac{d\sigma}{d\Omega}$')
  ax.set_xlim(*interval_cosθ)
  plot_increments(ax, increments,
                  label='Increment Borderds', color='gray', linestyle='--')
  ax.legend()
#+end_src

#+RESULTS:
:RESULTS:
: <matplotlib.legend.Legend at 0x7f62fa15b550>
[[file:./.ob-jupyter/a4d7908401fba70cd0d8550f21c2151a0f3a2b8c.png]]
:END:


I am batman!
Now, draw a sample and look at the efficiency.

#+begin_src jupyter-python :exports both :results raw drawer
  cosθ_sample_strat, cosθ_efficiency_strat = \
      monte_carlo.sample_unweighted_array(sample_num, dist_cosθ,
                                          increment_borders=increments,
                                          report_efficiency=True)
  cosθ_efficiency_strat
#+end_src

#+RESULTS:
: 0.1365

If we compare that to [[cosθ-bare-eff]], we can see the improvement :P.
It is even better the [[η-eff]].  The histogram looks just the same.

#+begin_src jupyter-python :exports both :results raw drawer
fig, _ = draw_histo(cosθ_sample_strat, r'$\cos\theta$')
save_fig(fig, 'histo_cos_theta_strat', 'xs', size=(4,3))
#+end_src

#+RESULTS:
[[file:./.ob-jupyter/fe3cad135a6a8264071e025a7e8debdb15a79677.png]]