Objective comparison of methods to decode anomalous diffusion

Deviations from Brownian motion leading to anomalous diffusion are found in transport dynamics from quantum physics to life sciences. The characterization of anomalous diffusion from the measurement of an individual trajectory is a challenging task, which traditionally relies on calculating the trajectory mean squared displacement. However, this approach breaks down for cases of practical interest, e.g., short or noisy trajectories, heterogeneous behaviour, or non-ergodic processes. Recently, several new approaches have been proposed, mostly building on the ongoing machine-learning revolution. To perform an objective comparison of methods, we gathered the community and organized an open competition, the Anomalous Diffusion challenge (AnDi). Participating teams applied their algorithms to a commonly-defined dataset including diverse conditions. Although no single method performed best across all scenarios, machine-learning-based approaches achieved superior performance for all tasks. The discussion of the challenge results provides practical advice for users and a benchmark for developers.

Continuous-Time Stochastic Processes

Chapter 19 contains the theory of continuous-time stochastic processes, including their mathematical presentation in the time and frequency domains. The typical processes, including Gaussian, white noise, binary, and harmonic processes, are presented. A comprehensive analysis of stationary and ergodic processes and linear-time-invariant systems with stochastic inputs is presented. The processes are analysed in terms of their autocorrelation functions and power spectral densities, which are related via the Wiener–Khintchine theorem. This chapter is important for understanding the theory of digital communication systems. The notation used in this chapter complies with the notation used in other chapters of the book, which makes the book self-sufficient. For readers who are not familiar with continuous-time stochastic processes, it is highly advisable to read this chapter and become familiar with its notation, due to its importance for understanding the content of Chapters 3 to 9.

Discrete-Time Stochastic Processes

Chapter 3 focuses on the theory of discrete-time-stochastic processes, including their mathematical presentation in time and frequency domains. Typical discrete processes, including the Gaussian process, white noise and binary and harmonic processes, are presented. A comprehensive analysis of discrete-time stationary and ergodic processes and linear-time-invariant (LTI) systems with discrete stochastic inputs is presented. The processes are analysed in terms of their autocorrelation functions and power spectral densities that are related by the Wiener–Khintchine theorem. This chapter is placed at the beginning of the book because its content is a prerequisite for the chapters that follow, in particular, the chapter related to the theory of discrete communication systems. The unique notation used in this chapter will be used in the rest of the book. For readers of the book, it is highly advisable to read this chapter first and acquire its notation.

Narrative Economics and Qualitative Studies of the Russian Innovation System

Narrative economics holds a vast potential since it can expand the array of questions and data covered by the analysis of economic change. The ideas that narratives contain reflect the actors’ perceptions of the rules structuring recurring interactions. Therefore, analysis of popular narratives can provide valuable insights into the relevant rules and behaviour patterns in their relation to the social context. In this study, we identified the most significant sources containing narratives and selected the most relevant narratives with biographical elements. The biographical method implies that narratives about events and phenomena are examined by looking at the actors’ biographical stories. Our research aims to test the applicability of the biographical method within the narrative economics approach to study the Russian innovation system. The analysis has revealed the most widely spread ideas associated with rules and institutions, such as researchers’ dependence on the funds that the government or companies are willing to invest in their studies. Another problem is the production of the next generation of academic workforce in Russia. We systematized the problem situations linked to the spreading of narratives and found that they are related to the setbacks in the state management of the Russian innovation system. We conducted a qualitative study of narratives combined with quantitative content analysis. The changing frequency of the key words and phrases reflects the dynamics of narratives and shows their ‘viral’ quality. Further analysis and systematization of qualitative data on innovation systems will create a better understanding of complex non-ergodic processes. It would also be productive to analyze official documents on the current rules and institutional development and compare the results with the evidence obtained through the narrative approach.

Lack of group-to-individual generalizability is a threat to human subjects research

Only for ergodic processes will inferences based on group-level data generalize to individual experience or behavior. Because human social and psychological processes typically have an individually variable and time-varying nature, they are unlikely to be ergodic. In this paper, six studies with a repeated-measure design were used for symmetric comparisons of interindividual and intraindividual variation. Our results delineate the potential scope and impact of nonergodic data in human subjects research. Analyses across six samples (with 87–94 participants and an equal number of assessments per participant) showed some degree of agreement in central tendency estimates (mean) between groups and individuals across constructs and data collection paradigms. However, the variance around the expected value was two to four times larger within individuals than within groups. This suggests that literatures in social and medical sciences may overestimate the accuracy of aggregated statistical estimates. This observation could have serious consequences for how we understand the consistency between group and individual correlations, and the generalizability of conclusions between domains. Researchers should explicitly test for equivalence of processes at the individual and group level across the social and medical sciences.

Flow complexity in open systems: interlacing complexity index based on mutual information

Flow complexity is related to a number of phenomena in science and engineering and has been approached from the perspective of chaotic dynamical systems, ergodic processes or mixing of fluids, just to name a few. To the best of our knowledge, all existing methods to quantify flow complexity are only valid for infinite time evolution, for closed systems or for mixing of two substances. We introduce an index of flow complexity coined interlacing complexity index (ICI), valid for a single-phase flow in an open system with inlet and outlet regions, involving finite times. ICI is based on Shannon’s mutual information (MI), and inspired by an analogy between inlet–outlet open flow systems and communication systems in communication theory. The roles of transmitter, receiver and communication channel are played, respectively, by the inlet, the outlet and the flow transport between them. A perfectly laminar flow in a straight tube can be compared to an ideal communication channel where the transmitted and received messages are identical and hence the MI between input and output is maximal. For more complex flows, generated by more intricate conditions or geometries, the ability to discriminate the outlet position by knowing the inlet position is decreased, reducing the corresponding MI. The behaviour of the ICI has been tested with numerical experiments on diverse flows cases. The results indicate that the ICI provides a sensitive complexity measure with intuitive interpretation in a diversity of conditions and in agreement with other observations, such as Dean vortices and subjective visual assessments. As a crucial component of the ICI formulation, we also introduce the natural distribution of streamlines and the natural distribution of world-lines, with invariance properties with respect to the cross-section used to parameterize them, valid for any type of mass-preserving flow.