Divergence Measures on Distributions of Ordinal Patterns in Time Series

Music City Center 

Permutation entropy has emerged as a widely used statistical measure for assessing the complexity of time
series. Recent work has applied divergence measures, including Kullback-Leibler divergence and Jensen-Shannon divergence, to quantify
the similarity between time series based on ordinal patterns, and to measure the deviation from generative
models. This approach also allows for creating meaningful embeddings of the data. While early approaches
assumed a uniform null model for the underlying data generation process, many real-world applications involve
more complex distributions and random walk models. In this study, we present an analysis of several
null models for generating time series data, from both theoretical perspectives. We successfully
derive theorems describing the behavior of random walks with uniformly and normally distributed steps respectively,
as well as introducing a novel random walk null model based on transition matrices inferred from
real-world data using Markov chains. We apply these methods to real-world datasets from economics and other fields, showcasing how divergence measures effectively capture the complexity of empirical time series data.

Permutations entropy

Time Series

Kull-Leibler divergence

Permutation Jensen-Shannon distance

Markov chain 

Section on Statistical Computing