Figure 2.

The “random walk”: the signal profile of a stationary time series may reveal self-affinity. (A) At each time step a walker moves randomly to the left (−1) or right (+1) with equal probability. At any time step the probability of being at a certain displacement from the origin depends on the number of different paths that could take the walker there. (B) The walker’s steps form a time series that is stationary as its value does not depend on time. (C) The signal profile can take arbitrarily large values as the time increases. (D) Looking at the walker time series on a longer time-scale the standard deviation does not change as the signal cannot take larger values. (E) The cumulative sum, or random walk process, on a longer time-scale shows larger variance than on the shorter time-scale (C) therefore the walker may exhibit self-affinity or scale-free behavior.