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
True, the bin heights are scaled such that the total area of
the histogram is 1. The cumulative keyword argument 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.
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()
The use of the following functions, methods, classes and modules is shown in this example: