Ordinal Random Processes

Abstract

Ordinal patterns have proven to be a valuable tool in many fields. Here, we address the need for theoretical models. A paradigmatic example shows that a model for frequencies of ordinal patterns can be determined without any numerical values. We specify the important concept of stationary order and the fundamental problems to be solved in order to establish a genuine statistical methodology for ordinal time series.

Keywords: ordinal pattern; permutation entropy; stochastic process; time series.

Figure 3

Probabilities of patterns of length 4 (brown) on top of those of length 3 (blue) for the Brownian motion (left) and coin-tossing order with $d=1$ (right). For length 3, the probabilities coincide.

Figure 1

Histograms of pattern frequencies of a simulated time series for $d=1$ and $m=2$ (red), 3 (blue), and 4 (brown). Each pattern $\pi$ of length m is represented by a subinterval of $[0,1]$ with length $1/m!.$ The corresponding bar has area $p_{\pi }\left(d\right),$ so the height is $m!p_{\pi }\left(d\right).$ The permutations are arranged in hierarchical order so that each histogram is a refinement of the previous one. For details, see Section 3.4.

Figure 2

Global and local rank numbers obtained from coin tossing.

Figure 4

Probabilities of patterns of length 4 for the coin-tossing order with $d=2$ and $d=3.$ They are clearly different from $d=1$ while the changes from $d=2$ to $d=3$ are small.