Plotting cumulative distributions#
This example shows how to plot the empirical cumulative distribution function (ECDF) of a sample. We also show the theoretical CDF.
In engineering, ECDFs are sometimes called "non-exceedance" curves: the y-value for a given x-value gives probability that an observation from the sample is below that x-value. For example, the value of 220 on the x-axis corresponds to about 0.80 on the y-axis, so there is an 80% chance that an observation in the sample does not exceed 220. Conversely, the empirical complementary cumulative distribution function (the ECCDF, or "exceedance" curve) shows the probability y that an observation from the sample is above a value x.
A direct method to plot ECDFs is
results in an ECCDF instead.
Alternatively, one can use
ax.hist(data, density=True, cumulative=True) to
first bin the data, as if plotting a histogram, and then compute and plot the
cumulative sums of the frequencies of entries in each bin. Here, to plot the
cumulative=-1. Note that this approach results in an
approximation of the E(C)CDF, whereas
Axes.ecdf is exact.
import matplotlib.pyplot as plt import numpy as np np.random.seed(19680801) mu = 200 sigma = 25 n_bins = 25 data = np.random.normal(mu, sigma, size=100) fig = plt.figure(figsize=(9, 4), layout="constrained") axs = fig.subplots(1, 2, sharex=True, sharey=True) # Cumulative distributions. axs.ecdf(data, label="CDF") n, bins, patches = axs.hist(data, n_bins, density=True, histtype="step", cumulative=True, label="Cumulative histogram") x = np.linspace(data.min(), data.max()) y = ((1 / (np.sqrt(2 * np.pi) * sigma)) * np.exp(-0.5 * (1 / sigma * (x - mu))**2)) y = y.cumsum() y /= y[-1] axs.plot(x, y, "k--", linewidth=1.5, label="Theory") # Complementary cumulative distributions. axs.ecdf(data, complementary=True, label="CCDF") axs.hist(data, bins=bins, density=True, histtype="step", cumulative=-1, label="Reversed cumulative histogram") axs.plot(x, 1 - y, "k--", linewidth=1.5, label="Theory") # Label the figure. fig.suptitle("Cumulative distributions") for ax in axs: ax.grid(True) ax.legend() ax.set_xlabel("Annual rainfall (mm)") ax.set_ylabel("Probability of occurrence") ax.label_outer() plt.show()
The use of the following functions, methods, classes and modules is shown in this example: