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This book features 13 papers presented at the Fifth International Symposium on Recurrence Plots, held August 2013 in Chicago, IL. It examines recent applications and developments in recurrence plots and recurrence quantification analysis (RQA) with special emphasis on biological and cognitive systems and the analysis of coupled systems using cross-recurrence methods.

Readers will discover new applications and insights into a range of systems provided by recurrence plot analysis and new theoretical and mathematical developments in recurrence plots. Recurrence plot based analysis is a powerful tool that operates on real-world complex systems that are nonlinear, non-stationary, noisy, of any statistical distribution, free of any particular model type and not particularly long. Quantitative analyses promote the detection of system state changes, synchronized dynamical regimes or classification of system states.

The book will be of interest to an interdisciplinary audience of recurrence plot users and researchers interested in time series analysis of complex systems in general.

A Recurrence Plot-Based Distance Measure

Abstract
Given a set of time series, our goal is to identify prototypes that cover the maximum possible amount of occurring subsequences regardless of their order. This scenario appears in the context of the automotive industry, where the goal is to determine operational profiles that comprise frequently recurring driving behavior patterns. This problem can be solved by clustering, however, standard distance measures such as the dynamic time warping distance might not be suitable for this task, because they aim at capturing the cost of aligning two time series rather than rewarding pairwise recurring patterns. In this contribution, we propose a novel time series distance measure, based on the notion of recurrence plots, which enables us to determine the (dis)similarity of multivariate time series that contain segments of similar trajectories at arbitrary positions. We use recurrence quantification analysis to measure the structures observed in recurrence plots and to investigate dynamical properties, such as determinism, which reflect the pairwise (dis)similarity of time series. In experiments on real-life test drives from Volkswagen, we demonstrate that clustering multivariate time series using the proposed recurrence plot-based distance measure results in prototypical test drives that cover significantly more recurring patterns than using the same clustering algorithm with dynamic time warping distance.
Stephan Spiegel, Johannes-Brijnesh Jain, Sahin Albayrak

Fast Computation of Recurrences in Long Time Series

Abstract
We present an approach to recurrence quantification analysis (RQA) that allows to process very long time series fast. To do so, it utilizes the paradigm Divide and Recombine. We divide the underlying matrix of a recurrence plot (RP) into sub matrices. The processing of the sub matrices is distributed across multiple graphics processing unit (GPU) devices. GPU devices perform RQA computations very fast since they match the problem very well. The individual results of the sub matrices are recombined into a global RQA solution. To address the specific challenges of subdividing the recurrence matrix, we introduce means of synchronization as well as additional data structures. Outperforming existing implementations dramatically, our GPU implementation of RQA processes time series consisting of $$N\approx$$ 1,000,000 data points in about 5 min.
Tobias Rawald, Mike Sips, Norbert Marwan, Doris Dransch

Unthresholded Recurrence Plots for Complex-Valued Representations of Narrow Band Signals

Abstract
We address the information content of unthresholded recurrence plots for complex-valued signals admitting a Fourier series representation (including periodic and sampled signals). Unthresholded recurrence plots of complex-valued signals contain the information of two real-valued signals simultaneously and can therefore be used to study the relationship between these signals. The graph theoretic procedure in our recent work [1], which was developed to characterize the uniqueness conditions for real-valued signals, is extended to the class of complex-valued signals. The special properties of complex signal representations provide alternative ways to employ unthresholded recurrence plots on narrow band signals. Examples and an application from EEG analysis clarify the results.
Aloys Sipers, Paul Borm, Ralf Peeters

Quantifying Redundancy and Information Content of Lines in Recurrence Plots Using the Theory of Framework Rigidity

Abstract
We address redundancy in the information content of unthresholded recurrence plots (URPs). The theory of framework rigidity is employed to explain and analyze this redundancy geometrically. First we show that the domain of a URP can be restricted to just a finite number of vertical or horizontal lines without loss of information. Then we construct a globally rigid framework to demonstrate a similar property for diagonal lines. This result gives theoretical support to recurrence quantification analysis (RQA), which analyzes and extracts features from an RP along such lines. Third, we construct a finite set of curves, one of which is a contour line, for which it again holds that the URP contains all information along them. This links the information content of lossy (thresholded) recurrence plots to that of URPs. This study is also a starting point in employing redundancy to improve existing recurrence plots based methods and algorithms, and to develop new ones. Several examples clarify the methods and an application from EEG artifact detection shows some of their practical potential.
Aloys Sipers, Paul Borm, Ralf Peeters

Recent Advances in Non-stationary Signal Processing Based on the Concept of Recurrence Plot Analysis

Abstract
This work concerns the analysis of non-stationary signals using Recurrence Plot Analysis concept. Non-stationary signals are present in real-life phenomena such as underwater mammal’s vocalizations, human speech, ultrasonic monitoring, detection of electrical discharges, transients, wireless communications, etc. This is why a large number of approaches for non-stationary signal analysis are developed such as wavelet analysis, higher order statistics, or quadratic time-frequency analysis. Following the context, the methods defined around the concept of Recurrence Plot Analysis (RPA) constitute an interesting way of analyzing non-stationary signals and, particularly, the transient ones. Starting from the phase space and the recurrence matrix, new approaches [the angular distance, recurrence-based autocorrelation function (ACF), average-magnitude difference function (AMDF) and time-distributed recurrence (TDR)] are introduced in order to extract information about the non-stationary signals, specific to different applications. Comparisons with existing analysis methods are presented, proving the interest and the potential of the RPA-based approaches.
Cornel Ioana, Angela Digulescu, Alexandru Serbanescu, Ion Candel, Florin-Marian Birleanu

A Recurrence-Based Approach for Feature Extraction in Brain-Computer Interface Systems

Abstract
The feature extraction stage is one of the main tasks underlying pattern recognition, and, is particularly important for designing Brain-Computer Interfaces (BCIs), i.e. structures capable of mapping brain signals in commands for external devices. Within one of the most used BCIs paradigms, that based on Steady State Visual Evoked Potentials (SSVEP), such task is classically performed in the spectral domain, albeit it does not necessarily provide the best achievable performance. The aim of this work is to use recurrence-based measures in an attempt to improve the classification performance obtained with a classical spectral approaches for a five-command SSVEP-BCI system. For both recurrence and spectral spaces, features were selected using a cluster measure defined by the Davies-Bouldin index and the classification stage was based on linear discriminant analysis. As the main result, it was found that the threshold $$\varepsilon$$ of the recurrence plot, chosen so as to yield a recurrence rate of 2.5 %, defined the key discriminant feature, typically providing a mean classification error of less than 2 % when information from 4 electrodes was used. Such classification performance was significantly better than that attained using spectral features, which strongly indicates that RQA is an efficient feature extraction technique for BCI.
Luisa F. S. Uribe, Filipe I. Fazanaro, Gabriela Castellano, Ricardo Suyama, Romis Attux, Eleri Cardozo, Diogo C. Soriano

Response to Active Standing of Heart Beat Interval, Systolic Blood Volume and Systolic Blood Pressure: Recurrence Plot Analysis

Abstract
Recurrence quantitative analysis (RQA) indexes of beat-to-beat heart-beat interval and systolic blood pressure (SBP) have helped to understand the dynamical response to active standing. The peripheral blood volume is another variable of the cardiovascular control system with a crucial role during active standing since re-distribution of blood volume is necessary to counteract the gravity force and to provide enough blood supply to vital organs. Beat-to-beat photoplethysmographic systolic blood volume (SBV) oscillations may be useful to study the cardiovascular control if it is considered as a regulatory system with relevant local differences compared to blood pressure regulation. There are no previous reports of the SBV dynamical response to active standing. In this work we study simultaneously the dynamical response of heart-beat interval, SBP and SBV to active standing through comparison of RQA indexes evaluated during supine position and during active standing in 19 healthy volunteers. We show that in response to orthostatic stress, SBV oscillations have dynamic changes similar, but not identical, to SBP and the heart-beat interval. This suggests that these three variables are complementary for a better evaluation of the cardiovascular dynamics.
Hortensia González, Oscar Infante, Claudia Lerma

Recurrence Quantification Analysis as a Tool for Discrimination Among Different Dynamics Classes: The Heart Rate Variability Associated to Different Age Groups

Abstract
We propose a classification method based on recurrence quantification analysis (RQA) combined with support vector machines (SVM). This method combines in an effective way various quantitative descriptors to allow a refined discrimination among dynamical non linear systems that presents dynamics which are very similar to each other. To show how effective this methodology is, firstly, based on synthetic data, it is applied on time series generated from the logistic map with nearby parameter values and in the chaotic regime. Next, it is applied to human biosignals, namely, heart rate variability (HRV) time series obtained from four groups of individuals (premature newborns, full-term newborns, healthy young adults, and adults with severe coronary disease). Roughly the proposed methodology works as follows: The signals are transformed into recurrence plots (RP) and a set of RQA statistical features (recurrence rate, determinism, averaged and maximal diagonal line lengths, entropy, laminarity, trapping time, and length of longest vertical line) are extracted to form the input vector for a SVM classifier. Results show that the method discriminates groups of different ages with classification accuracy better than $$75\,\%$$. Given that heart rate continuously fluctuates over time and reflects different mechanisms to maintain cardiovascular homeostasis of an individual, the results obtained may allow to draw important information on the autonomic control of circulation in normal and diseased conditions.
Laurita dos Santos, Joaquim J. Barroso, Moacir F. de Godoy, Elbert E. N. Macau, Ubiratan S. Freitas

Analyzing Social Interactions: The Promises and Challenges of Using Cross Recurrence Quantification Analysis

Abstract
The scientific investigation of social interactions presents substantial challenges: interacting agents engage each other at many different levels and timescales (motor and physiological coordination, joint attention, linguistic exchanges, etc.), often making their behaviors interdependent in non-linear ways. In this paper we review the current use of Cross Recurrence Quantification Analysis (CRQA) in the analysis of social interactions, and assess its potential and challenges. We argue that the method can sensitively grasp the dynamics of human interactions, and that it has started producing valuable knowledge about them. However, much work is still necessary: more systematic analyses and interpretation of the recurrence indexes and more consistent reporting of the results,more emphasis on theory-driven studies, exploring interactions involving more than 2 agents and multiple aspects of coordination,and assessing and quantifying complementary coordinative mechanisms. These challenges are discussed and operationalized in recommendations to further develop the field.
Riccardo Fusaroli, Ivana Konvalinka, Sebastian Wallot

Cross-Recurrence Quantification Analysis of the Influence of Coupling Constraints on Interpersonal Coordination and Communication

Abstract
This chapter describes a methodological strategy for studying the influence of coupling constraints on interpersonal coordination using cross-recurrence quantification analysis (CRQA). In Study 1, we investigated interpersonal coordination during conversation in virtual-reality (VR) and real-world environments. Consistent with previous studies, we found enhanced coordination when participants were talking to each other compared to when they were talking to experimenters. In doing so we also demonstrated the utility of VR in studying interpersonal coordination involved in cooperative conversation. In Study 2, we investigated the influence of mechanical coupling on interpersonal coordination and communication, in which conversing pairs were coupled mechanically (standing on the same balance board) or not (they stood on individual balance boards). We found a relationship between movement coordination and performance in a conversational task in the coupled condition, suggesting a functional link between coordination and communication. We offer these studies as methodological examples of how CRQA can be used to study the relation between interpersonal coordination and conversation.
Michael Tolston, Kris Ariyabuddhiphongs, Michael A. Riley, Kevin Shockley

Recurrence Quantification as an Analysis of Temporal Coordination with Complex Signals

Abstract
Ample past research demonstrates that human rhythmic behavior and rhythmic coordination reveal complex dynamics. More recently, researchers have begun to examine the dynamics of coordination with complex, fractal signals. Here, we present preliminary research investigating how recurrence quantification techniques might be applied to study temporal coordination with complex signals. Participants attempted to synchronize their rhythmic finger tapping behavior with metronomes with varying fractal scaling properties. The results demonstrated that coordination, as assessed by recurrence analyses, differed with the fractal scaling of the metronome stimulus. Overall, these results suggest that recurrence analyses may aid in understanding temporal coordination between complex systems.
Charles A. Coey, Auriel Washburn, Michael J. Richardson

Synchronicity Assessment Using a Non-parametric Dynamic Dissimilarity Measure

Abstract
In this paper, we introduce a non-parametric dynamic dissimilarity measure (DDM) of synchronicity based on recurrence plots, which is particularly suited to use in small samples. The measure attempts to capture the dissimilarity of the topology of the dynamics of time series, based on an epoch analysis of the cumulative sums of data series. The measure is applied to US State macroeconomic data and is used to assess how synchronous US State business cycle variables are with US aggregates.
Patrick Crowley, Christopher Trombley

Understanding the Interrelationship Between Commodity and Stock Indices Daily Movement Using ACE and Recurrence Analysis

Abstract
The relationship between the temporal evolution of the commodity market and the stock market has long term implications for policy makers, and particularly in the case of emerging markets, the economy as a whole. We analyze the complex dynamics of the daily variation of two indices of stock and commodity exchange respectively of India. To understand whether there is any difference between emerging markets and developed markets in terms of a dynamic correlation between the two market indices, we also examine the complex dynamics of stock and commodity indices of the US market. We compare the daily variation of the commodity and stock prices in the two countries separately. For this purpose we have considered commodity India along with Dow Jones Industrial Average (DJIA) and Dow Jones-AIG Commodity (DJ-AIGCI) indices for stock and commodities, USA, from June 2005 to August 2008. To analyse the dynamics of the time variation of the indices we use a set of analytical methods based on recurrence plots. Our studies show that the dynamics of the Indian stock and commodity exchanges have a lagged correlation while those of US market have a lead correlation and a weaker correlation.
Kousik Guhathakurta, Norbert Marwan, Basabi Bhattacharya, A. Roy Chowdhury