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Permutation entropy and its associated frameworks are remarkable examples of physics-inspired techniques adept at processing complex and extensive datasets. Despite substantial progress in developing and applying these tools, their use has been predominantly limited to structured datasets such as time series or images. Here, we introduce the k -nearest neighbor permutation entropy, an innovative extension of the permutation entropy tailored for unstructured data, irrespective of their spatial or temporal configuration and dimensionality. Our approach builds upon nearest neighbor graphs to establish neighborhood relations and uses random walks to extract ordinal patterns and their distribution, thereby defining the k -nearest neighbor permutation entropy. This tool not only adeptly identifies variations in patterns of unstructured data but also does so with a precision that significantly surpasses conventional measures such as spatial autocorrelation. Additionally, it provides a natural approach for incorporating amplitude information and time gaps when analyzing time series or images, thus significantly enhancing its noise resilience and predictive capabilities compared to the usual permutation entropy. Our research substantially expands the applicability of ordinal methods to more general data types, opening promising research avenues for extending the permutation entropy toolkit for unstructured data. [ABSTRACT FROM AUTHOR]

Since Bandt and Pompe's seminal work, permutation entropy has been used in several applications and is now an essential tool for time series analysis. Beyond becoming a popular and successful technique, permutation entropy inspired a framework for mapping time series into symbolic sequences that triggered the development of many other tools, including an approach for creating networks from time series known as ordinal networks. Despite increasing popularity, the computational development of these methods is fragmented, and there were still no efforts focusing on creating a unified software package. Here, we present ordpy (http://github.com/arthurpessa/ordpy), a simple and open-source Python module that implements permutation entropy and several of the principal methods related to Bandt and Pompe's framework to analyze time series and two-dimensional data. In particular, ordpy implements permutation entropy, Tsallis and Rényi permutation entropies, complexity–entropy plane, complexity–entropy curves, missing ordinal patterns, ordinal networks, and missing ordinal transitions for one-dimensional (time series) and two-dimensional (images) data as well as their multiscale generalizations. We review some theoretical aspects of these tools and illustrate the use of ordpy by replicating several literature results. [ABSTRACT FROM AUTHOR]

Approaches for mapping time series to networks have become essential tools for dealing with the increasing challenges of characterizing data from complex systems. Among the different algorithms, the recently proposed ordinal networks stand out due to their simplicity and computational efficiency. However, applications of ordinal networks have been mainly focused on time series arising from nonlinear dynamical systems, while basic properties of ordinal networks related to simple stochastic processes remain poorly understood. Here, we investigate several properties of ordinal networks emerging from random time series, noisy periodic signals, fractional Brownian motion, and earthquake magnitude series. For ordinal networks of random series, we present an approach for building the exact form of the adjacency matrix, which in turn is useful for detecting nonrandom behavior in time series and the existence of missing transitions among ordinal patterns. We find that the average value of a local entropy, estimated from transition probabilities among neighboring nodes of ordinal networks, is more robust against noise addition than the standard permutation entropy. We show that ordinal networks can be used for estimating the Hurst exponent of time series with accuracy comparable with state-of-the-art methods. Finally, we argue that ordinal networks can detect sudden changes in Earth's seismic activity caused by large earthquakes. [ABSTRACT FROM AUTHOR]

An increasing abstraction has marked some recent investigations in network science. Examples include the development of algorithms that map time series data into networks whose vertices and edges can have different interpretations, beyond the classical idea of parts and interactions of a complex system. These approaches have proven useful for dealing with the growing complexity and volume of diverse data sets. However, the use of such algorithms is mostly limited to one-dimensional data, and there has been little effort towards extending these methods to higher-dimensional data such as images. Here we propose a generalization for the ordinal network algorithm for mapping images into networks. We investigate the emergence of connectivity constraints inherited from the symbolization process used for defining the network nodes and links, which in turn allows us to derive the exact structure of ordinal networks obtained from random images. We illustrate the use of this new algorithm in a series of applications involving randomization of periodic ornaments, images generated by two-dimensional fractional Brownian motion and the Ising model, and a data set of natural textures. These examples show that measures obtained from ordinal networks (such as average shortest path and global node entropy) extract important image properties related to roughness and symmetry, are robust against noise, and can achieve higher accuracy than traditional texture descriptors extracted from gray-level co-occurrence matrices in simple image classification tasks.