A cellular automaton's smooth density histogram can show the clustering of points in its time series and enables recognition of one of the four classes of behavior for cellular automata. The histogram in the lower left of the graphic shows the original density (fraction of black cells) as a function of the number of steps. The yellow cloud in the right panel represents the distribution obtained using SmoothDensityHistogram in Mathematica.
Different clouds show either more random behavior (classes III and IV) or more periodic behavior (classes I and II), with a thinner, smaller, and smoother cloud indicating repetition. Adjusting the bandwidth can highlight the points within the clusters.