"""
==================================================
Using histograms to plot a cumulative distribution
==================================================

This shows how to plot a cumulative, normalized histogram as a
step function in order to visualize the empirical cumulative
distribution function (CDF) of a sample. We also show the theoretical CDF.

A couple of other options to the ``hist`` function are demonstrated.
Namely, we use the ``normed`` parameter to normalize the histogram and
a couple of different options to the ``cumulative`` parameter.
The ``normed`` parameter takes a boolean value. When ``True``, the bin
heights are scaled such that the total area of the histogram is 1. The
``cumulative`` kwarg is a little more nuanced. Like ``normed``, you
can pass it True or False, but you can also pass it -1 to reverse the
distribution.

Since we're showing a normalized and cumulative histogram, these curves
are effectively the cumulative distribution functions (CDFs) of the
samples. In engineering, empirical CDFs are sometimes called
"non-exceedance" curves. In other words, you can look at the
y-value for a given-x-value to get the probability of and observation
from the sample not exceeding that x-value. For example, the value of
225 on the x-axis corresponds to about 0.85 on the y-axis, so there's an
85% chance that an observation in the sample does not exceed 225.
Conversely, setting, ``cumulative`` to -1 as is done in the
last series for this example, creates a "exceedance" curve.

Selecting different bin counts and sizes can significantly affect the
shape of a histogram. The Astropy docs have a great section on how to
select these parameters:
http://docs.astropy.org/en/stable/visualization/histogram.html

"""

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(19680801)

mu = 200
sigma = 25
n_bins = 50
x = np.random.normal(mu, sigma, size=100)

fig, ax = plt.subplots(figsize=(8, 4))

# plot the cumulative histogram
n, bins, patches = ax.hist(x, n_bins, density=True, histtype='step',
                           cumulative=True, label='Empirical')

# Add a line showing the expected distribution.
y = ((1 / (np.sqrt(2 * np.pi) * sigma)) *
     np.exp(-0.5 * (1 / sigma * (bins - mu))**2))
y = y.cumsum()
y /= y[-1]

ax.plot(bins, y, 'k--', linewidth=1.5, label='Theoretical')

# Overlay a reversed cumulative histogram.
ax.hist(x, bins=bins, density=True, histtype='step', cumulative=-1,
        label='Reversed emp.')

# tidy up the figure
ax.grid(True)
ax.legend(loc='right')
ax.set_title('Cumulative step histograms')
ax.set_xlabel('Annual rainfall (mm)')
ax.set_ylabel('Likelihood of occurrence')

plt.show()
