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-valuefor a given x-value gives probability that an observation from the sample isbelow that x-value. For example, the value of 220 on the x-axis corresponds toabout 0.80 on the y-axis, so there is an 80% chance that an observation in thesample does not exceed 220. Conversely, the empiricalcomplementarycumulative distribution function (the ECCDF, or "exceedance" curve) shows theprobability y that an observation from the sample is above a value x.

A direct method to plot ECDFs isAxes.ecdf. Passingcomplementary=Trueresults in an ECCDF instead.

Alternatively, one can useax.hist(data,density=True,cumulative=True) tofirst bin the data, as if plotting a histogram, and then compute and plot thecumulative sums of the frequencies of entries in each bin. Here, to plot theECCDF, passcumulative=-1. Note that this approach results in anapproximation of the E(C)CDF, whereasAxes.ecdf is exact.

importmatplotlib.pyplotaspltimportnumpyasnpnp.random.seed(19680801)mu=200sigma=25n_bins=25data=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[0].ecdf(data,label="CDF")n,bins,patches=axs[0].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[0].plot(x,y,"k--",linewidth=1.5,label="Theory")# Complementary cumulative distributions.axs[1].ecdf(data,complementary=True,label="CCDF")axs[1].hist(data,bins=bins,density=True,histtype="step",cumulative=-1,label="Reversed cumulative histogram")axs[1].plot(x,1-y,"k--",linewidth=1.5,label="Theory")# Label the figure.fig.suptitle("Cumulative distributions")foraxinaxs:ax.grid(True)ax.legend()ax.set_xlabel("Annual rainfall (mm)")ax.set_ylabel("Probability of occurrence")ax.label_outer()plt.show()

Tags:plot-type: ecdfplot-type: histogramdomain: statistics

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