  Permutation entropy has become a standard tool for time series analysis that
exploits the temporal properties of these data sets. Many current applications
use an approach based on Shannon entropy, which implicitly assumes an
underlying uniform distribution of patterns. In this paper, we analyze random
walk null models for time series and determine the corresponding permutation
distributions. These new techniques allow us to explicitly describe the
behavior of real world data in terms of more complex generative processes.
Additionally, building on recent results of Martinez, we define a validation
measure that allows us to determine when a random walk is an appropriate model
for a time series. We demonstrate the usefulness of our methods using empirical
data drawn from a variety of fields.
