Reducing the bias of causality measures

Discovering the mesoscale for chains of conflict

Conflicts, like many social processes, are related events that span multiple scales in time, from the instantaneous to multi-year development, and in space, from one neighborhood to continents. Yet, there is little systematic work on connecting the multiple scales, formal treatment of causality between events, and measures of uncertainty for how events are related to one another. We develop a method for extracting causally related chains of events that addresses these limitations with armed conflict. Our method explicitly accounts for an adjustable spatial and temporal scale of interaction for clustering individual events from a detailed data set, the Armed Conflict Event & Location Data Project. With it, we discover a mesoscale ranging from a week to a few months and tens to hundreds of kilometers, where long-range correlations and nontrivial dynamics relating conflict events emerge. Importantly, clusters in the mesoscale, while extracted from conflict statistics, are identifiable with mechanism cited in field studies. We leverage our technique to identify zones of causal interaction around conflict hotspots that naturally incorporate uncertainties. Thus, we show how a systematic, data-driven, and scalable procedure extracts social objects for study, providing a scope for scrutinizing and predicting conflict and other processes.

Measuring the Rate of Information Exchange in Point-Process Data With Application to Cardiovascular Variability

The amount of information exchanged per unit of time between two dynamic processes is an important concept for the analysis of complex systems. Theoretical formulations and data-efficient estimators have been recently introduced for this quantity, known as the mutual information rate (MIR), allowing its continuous-time computation for event-based data sets measured as realizations of coupled point processes. This work presents the implementation of MIR for point process applications in Network Physiology and cardiovascular variability, which typically feature short and noisy experimental time series. We assess the bias of MIR estimated for uncoupled point processes in the frame of surrogate data, and we compensate it by introducing a corrected MIR (cMIR) measure designed to return zero values when the two processes do not exchange information. The method is first tested extensively in synthetic point processes including a physiologically-based model of the heartbeat dynamics and the blood pressure propagation times, where we show the ability of cMIR to compensate the negative bias of MIR and return statistically significant values even for weakly coupled processes. The method is then assessed in real point-process data measured from healthy subjects during different physiological conditions, showing that cMIR between heartbeat and pressure propagation times increases significantly during postural stress, though not during mental stress. These results document that cMIR reflects physiological mechanisms of cardiovascular variability related to the joint neural autonomic modulation of heart rate and arterial compliance.

Alpha band frontal connectivity is a state-specific electroencephalographic correlate of unresponsiveness during exposure to dexmedetomidine and propofol

BACKGROUND

Coherent alpha electroencephalogram (EEG) rhythms in the frontal cortex have been correlated with the hypnotic effects of propofol and dexmedetomidine, but less is known about frontal connectivity as a state-specific correlate of unresponsiveness as compared with long-range connectivity. We aimed to distinguish dose- and state-dependent effects of dexmedetomidine and propofol on EEG connectivity.

METHODS

Forty-seven healthy males received either dexmedetomidine (n=23) or propofol (n=24) as target-controlled infusion with stepwise increments until loss of responsiveness (LOR). We attempted to arouse participants during constant dosing (return of responsiveness [ROR]), and the target concentration was then increased 50% to achieve presumed loss of consciousness. We collected 64-channel EEG data and prefrontal-frontal and anterior-posterior functional connectivity in the alpha band (8-14 Hz) was measured using coherence and weighted phase lag index (wPLI). Directed connectivity was measured with directed phase lag index (dPLI).

RESULTS

Prefrontal-frontal EEG-based connectivity discriminated the states at the different drug concentrations. At ROR, prefrontal-frontal connectivity reversed to the level observed before LOR, indicating that connectivity changes were related to unresponsiveness rather than drug concentration. Unresponsiveness was associated with emergence of frontal-to-prefrontal dominance (dPLI: -0.13 to -0.40) in contrast to baseline (dPLI: 0.01-0.02). Coherence, wPLI, and dPLI had similar capability to discriminate the states that differed in terms of responsiveness and drug concentration. In contrast, anterior-posterior connectivity in the alpha band did not differentiate LOR and ROR.

CONCLUSIONS

Local prefrontal-frontal EEG-based connectivity reflects unresponsiveness induced by propofol or dexmedetomidine, suggesting its utility in monitoring the anaesthetised state with these agents.

CLINICAL TRIAL REGISTRATION

NCT01889004.

Network Analysis of Multivariate Transfer Entropy of Cryptocurrencies in Times of Turbulence

We investigate the effects of the recent financial turbulence of 2020 on the market of cryptocurrencies taking into account the hourly price and volume of transactions from December 2019 to April 2020. The data were subdivided into time frames and analyzed the directed network generated by the estimation of the multivariate transfer entropy. The approach followed here is based on a greedy algorithm and multiple hypothesis testing. Then, we explored the clustering coefficient and the degree distributions of nodes for each subperiod. It is found the clustering coefficient increases dramatically in March and coincides with the most severe fall of the recent worldwide stock markets crash. Further, the log-likelihood in all cases bent over a power law distribution, with a higher estimated power during the period of major financial contraction. Our results suggest the financial turbulence induce a higher flow of information on the cryptocurrency market in the sense of a higher clustering coefficient and complexity of the network. Hence, the complex properties of the multivariate transfer entropy network may provide early warning signals of increasing systematic risk in turbulence times of the cryptocurrency markets.

Dynamic Cortical Connectivity during General Anesthesia in Surgical Patients

BACKGROUND

Functional connectivity across the cortex has been posited to be important for consciousness and anesthesia, but functional connectivity patterns during the course of surgery and general anesthesia are unknown. The authors tested the hypothesis that disrupted cortical connectivity patterns would correlate with surgical anesthesia.

METHODS

Surgical patients (n = 53) were recruited for study participation. Whole-scalp (16-channel) wireless electroencephalographic data were prospectively collected throughout the perioperative period. Functional connectivity was assessed using weighted phase lag index. During anesthetic maintenance, the temporal dynamics of connectivity states were characterized via Markov chain analysis, and state transition probabilities were quantified.

RESULTS

Compared to baseline (weighted phase lag index, 0.163, ± 0.091), alpha frontal-parietal connectivity was not significantly different across the remaining anesthetic and perioperative epochs, ranging from 0.100 (± 0.041) to 0.218 (± 0.136) (P > 0.05 for all time periods). In contrast, there were significant increases in alpha prefrontal-frontal connectivity (peak = 0.201 [0.154, 0.248]; P < 0.001), theta prefrontal-frontal connectivity (peak = 0.137 [0.091, 0.182]; P < 0.001), and theta frontal-parietal connectivity (peak = 0.128 [0.084, 0.173]; P < 0.001) during anesthetic maintenance. Additionally, shifts occurred between states of high prefrontal-frontal connectivity (alpha, beta) with suppressed frontal-parietal connectivity, and high frontal-parietal connectivity (alpha, theta) with reduced prefrontal-frontal connectivity. These shifts occurred in a nonrandom manner (P < 0.05 compared to random transitions), suggesting structured transitions of connectivity during general anesthesia.

CONCLUSIONS

Functional connectivity patterns dynamically shift during surgery and general anesthesia but do so in a structured way. Thus, a single measure of functional connectivity will likely not be a reliable correlate of surgical anesthesia.

Transfer entropy as a variable selection methodology of cryptocurrencies in the framework of a high dimensional predictive model

We determine the number of statistically significant factors in a high dimensional predictive model of cryptocurrencies using a random matrix test. The applied predictive model is of the reduced rank regression (RRR) type; in particular, we choose a flavor that can be regarded as canonical correlation analysis (CCA). A variable selection of hourly cryptocurrencies is performed using the Symbolic estimation of Transfer Entropy (STE) measure from information theory. In simulated studies, STE shows better performance compared to the Granger causality approach when considering a nonlinear system and a linear system with many drivers. In the application to cryptocurrencies, the directed graph associated to the variable selection shows a robust pattern of predictor and response clusters, where the community detection was contrasted with the modularity approach. Also, the centralities of the network discriminate between the two main types of cryptocurrencies, i.e., coins and tokens. On the factor determination of the predictive model, the result supports retaining more factors contrary to the usual visual inspection, with the additional advantage that the subjective element is avoided. In particular, it is observed that the dynamic behavior of the number of factors is moderately anticorrelated with the dynamics of the constructed composite index of predictor and response cryptocurrencies. This finding opens up new insights for anticipating possible declines in cryptocurrency prices on exchanges. Furthermore, our study suggests the existence of specific-predictor and specific-response factors, where only a small number of currencies are predominant.

Evaluation of Granger Causality Measures for Constructing Networks from Multivariate Time Series

Granger causality and variants of this concept allow the study of complex dynamical systems as networks constructed from multivariate time series. In this work, a large number of Granger causality measures used to form causality networks from multivariate time series are assessed. These measures are in the time domain, such as model-based and information measures, the frequency domain, and the phase domain. The study aims also to compare bivariate and multivariate measures, linear and nonlinear measures, as well as the use of dimension reduction in linear model-based measures and information measures. The latter is particular relevant in the study of high-dimensional time series. For the performance of the multivariate causality measures, low and high dimensional coupled dynamical systems are considered in discrete and continuous time, as well as deterministic and stochastic. The measures are evaluated and ranked according to their ability to provide causality networks that match the original coupling structure. The simulation study concludes that the Granger causality measures using dimension reduction are superior and should be preferred particularly in studies involving many observed variables, such as multi-channel electroencephalograms and financial markets.

Parallelism Strategies for Big Data Delayed Transfer Entropy Evaluation

Generated and collected data have been rising with the popularization of technologies such as Internet of Things, social media, and smartphone, leading big data term creation. One class of big data hidden information is causality. Among the tools to infer causal relationships, there is Delay Transfer Entropy (DTE); however, it has a high demanding processing power. Many approaches were proposed to overcome DTE performance issues such as GPU and FPGA implementations. Our study compared different parallel strategies to calculate DTE from big data series using a heterogeneous Beowulf cluster. Task Parallelism was significantly faster in comparison to Data Parallelism. With big data trend in sight, these results may enable bigger datasets analysis or better statistical evidence.

Neurophysiologic Correlates of Ketamine Sedation and Anesthesia: A High-density Electroencephalography Study in Healthy Volunteers

BACKGROUND

Previous studies have demonstrated inconsistent neurophysiologic effects of ketamine, although discrepant findings might relate to differences in doses studied, brain regions analyzed, coadministration of other anesthetic medications, and resolution of the electroencephalograph. The objective of this study was to characterize the dose-dependent effects of ketamine on cortical oscillations and functional connectivity.

METHODS

Ten healthy human volunteers were recruited for study participation. The data were recorded using a 128-channel electroencephalograph during baseline consciousness, subanesthetic dosing (0.5 mg/kg over 40 min), anesthetic dosing (1.5 mg/kg bolus), and recovery. No other sedative or anesthetic medications were administered. Spectrograms, topomaps, and functional connectivity (weighted and directed phase lag index) were computed and analyzed.

RESULTS

Frontal theta bandwidth power increased most dramatically during ketamine anesthesia (mean power ± SD, 4.25 ± 1.90 dB) compared to the baseline (0.64 ± 0.28 dB), subanesthetic (0.60 ± 0.30 dB), and recovery (0.68 ± 0.41 dB) states; P < 0.001. Gamma power also increased during ketamine anesthesia. Weighted phase lag index demonstrated theta phase locking within anterior regions (0.2349 ± 0.1170, P < 0.001) and between anterior and posterior regions (0.2159 ± 0.1538, P < 0.01) during ketamine anesthesia. Alpha power gradually decreased with subanesthetic ketamine, and anterior-to-posterior directed connectivity was maximally reduced (0.0282 ± 0.0772) during ketamine anesthesia compared to all other states (P < 0.05).

CONCLUSIONS

Ketamine anesthesia correlates most clearly with distinct changes in the theta bandwidth, including increased power and functional connectivity. Anterior-to-posterior connectivity in the alpha bandwidth becomes maximally depressed with anesthetic ketamine administration, suggesting a dose-dependent effect.

Assessment of resampling methods for causality testing: A note on the US inflation behavior

Different resampling methods for the null hypothesis of no Granger causality are assessed in the setting of multivariate time series, taking into account that the driving-response coupling is conditioned on the other observed variables. As appropriate test statistic for this setting, the partial transfer entropy (PTE), an information and model-free measure, is used. Two resampling techniques, time-shifted surrogates and the stationary bootstrap, are combined with three independence settings (giving a total of six resampling methods), all approximating the null hypothesis of no Granger causality. In these three settings, the level of dependence is changed, while the conditioning variables remain intact. The empirical null distribution of the PTE, as the surrogate and bootstrapped time series become more independent, is examined along with the size and power of the respective tests. Additionally, we consider a seventh resampling method by contemporaneously resampling the driving and the response time series using the stationary bootstrap. Although this case does not comply with the no causality hypothesis, one can obtain an accurate sampling distribution for the mean of the test statistic since its value is zero under H0. Results indicate that as the resampling setting gets more independent, the test becomes more conservative. Finally, we conclude with a real application. More specifically, we investigate the causal links among the growth rates for the US CPI, money supply and crude oil. Based on the PTE and the seven resampling methods, we consistently find that changes in crude oil cause inflation conditioning on money supply in the post-1986 period. However this relationship cannot be explained on the basis of traditional cost-push mechanisms.

Horizontal visibility graph transfer entropy (HVG-TE): A novel metric to characterize directed connectivity in large-scale brain networks

We propose a new measure, horizontal visibility graph transfer entropy (HVG-TE), to estimate the direction of information flow between pairs of time series. HVG-TE quantifies the transfer entropy between the degree sequences of horizontal visibility graphs derived from original time series. Twenty-one Rössler attractors unidirectionally coupled in the posterior-to-anterior direction were used to simulate 21-channel Electroencephalography (EEG) brain networks and validate the performance of the HVG-TE. We showed that the HVG-TE is robust to different levels of coupling strengths between the coupled Rössler attractors, a wide range of time delays, different sample sizes, the effects of noise and linear mixing, and the choice of reference for EEG data. We also applied HVG-TE to EEG data in 20 healthy controls and compared its performance to a recently introduces phase-based TE measure (PTE). We found that compared with PTE, HVG-TE consistently detected stronger posterior-to-anterior information flow patterns in the alpha-band (8-13Hz) EEG brain networks for three different references. Moreover, in contrast to PTE, HVG-TE does not require an assumption on the periodicity of input signals, therefore it can be more widely applicable, even for non-periodic signals. This study shows that the HVG-TE is a directed connectivity measure to characterise the direction of information flow in large-scale brain networks.

Accelerated Recovery of Consciousness after General Anesthesia Is Associated with Increased Functional Brain Connectivity in the High-Gamma Bandwidth

Recent data from our laboratory demonstrate that high-frequency gamma connectivity across the cortex is present during consciousness and depressed during unconsciousness. However, these data were derived from static and well-defined states of arousal rather than during transitions that would suggest functional relevance. We also recently found that subanesthetic ketamine administered during isoflurane anesthesia accelerates recovery upon discontinuation of the primary anesthetic and increases gamma power during emergence. In the current study we re-analyzed electroencephalogram (EEG) data to test the hypothesis that functional cortical connectivity between anterior and posterior cortical regions would be increased during accelerated recovery induced by ketamine when compared to saline-treated controls. Rodents were instrumented with intracranial EEG electrodes and general anesthesia was induced with isoflurane anesthesia. After 37.5 min of continuous isoflurane anesthesia, a subanesthetic dose of ketamine (25 mg/kg intraperitoneal) was administered, with evidence of a 44% reduction in emergence time. In this study, we analyzed gamma and theta coherence (measure of undirected functional connectivity) and normalized symbolic transfer entropy (measure of directed functional connectivity) between frontal and parietal cortices during various levels of consciousness, with a focus on emergence from isoflurane anesthesia. During accelerated emergence in the ketamine-treated group, there was increased frontal-parietal coherence {p = 0.005, 0.05-0.23 [95% confidence interval (CI)]} and normalized symbolic transfer entropy [frontal to parietal: p < 0.001, 0.010-0.026 (95% CI); parietal to frontal: p < 0.001, 0.009-0.025 (95% CI)] in high-frequency gamma bandwidth as compared with the saline-treated group. Surrogates of cortical information exchange in high-frequency gamma are increased in association with accelerated recovery from anesthesia. This finding adds evidence suggesting a functional significance of high-gamma information transfer in consciousness.

Identifying delayed directional couplings with symbolic transfer entropy

We propose a straightforward extension of symbolic transfer entropy to enable the investigation of delayed directional relationships between coupled dynamical systems from time series. Analyzing time series from chaotic model systems, we demonstrate the applicability and limitations of our approach. Our findings obtained from applying our method to infer delayed directed interactions in the human epileptic brain underline the importance of our approach for improving the construction of functional network structures from data.

Directional connectivity between frontal and posterior brain regions is altered with increasing concentrations of propofol

Recent studies using electroencephalography (EEG) suggest that alteration of coherent activity between the anterior and posterior brain regions might be used as a neurophysiologic correlate of anesthetic-induced unconsciousness. One way to assess causal relationships between brain regions is given by renormalized partial directed coherence (rPDC). Importantly, directional connectivity is evaluated in the frequency domain by taking into account the whole multichannel EEG, as opposed to time domain or two channel approaches. rPDC was applied here in order to investigate propofol induced changes in causal connectivity between four states of consciousness: awake (AWA), deep sedation (SED), loss (LOC) and return of consciousness (ROC) by gathering full 10/20 system human EEG data in ten healthy male subjects. The target-controlled drug infusion was started at low rate with subsequent gradual stepwise increases at 10 min intervals in order to carefully approach LOC (defined as loss of motor responsiveness to a verbal stimulus). The direction of the causal EEG-network connections clearly changed from AWA to SED and LOC. Propofol induced a decrease (p = 0.002-0.004) in occipital-to-frontal rPDC of 8-16 Hz EEG activity and an increase (p = 0.001-0.040) in frontal-to-occipital rPDC of 10-20 Hz activity on both sides of the brain during SED and LOC. In addition, frontal-to-parietal rPDC within 1-12 Hz increased in the left hemisphere at LOC compared to AWA (p = 0.003). However, no significant changes were detected between the SED and the LOC states. The observed decrease in back-to-front EEG connectivity appears compatible with impaired information flow from the posterior sensory and association cortices to the executive prefrontal areas, possibly related to decreased ability to perceive the surrounding world during sedation. The observed increase in the opposite (front-to-back) connectivity suggests a propofol concentration dependent association and is not directly related to the level of consciousness per se.

Quantifying 'causality' in complex systems: understanding transfer entropy

‘Causal’ direction is of great importance when dealing with complex systems. Often big volumes of data in the form of time series are available and it is important to develop methods that can inform about possible causal connections between the different observables. Here we investigate the ability of the Transfer Entropy measure to identify causal relations embedded in emergent coherent correlations. We do this by firstly applying Transfer Entropy to an amended Ising model. In addition we use a simple Random Transition model to test the reliability of Transfer Entropy as a measure of ‘causal’ direction in the presence of stochastic fluctuations. In particular we systematically study the effect of the finite size of data sets.

Information directionality in coupled time series using transcripts

In ordinal symbolic dynamics, transcripts describe the algebraic relationship between ordinal patterns. Using the concept of transcript, we exploit the mathematical structure of the group of permutations to derive properties and relations among information measures of the symbolic representations of time series. These theoretical results are then applied for the assessment of coupling directionality in dynamical systems, where suitable coupling directionality measures are introduced depending only on transcripts. These measures improve the reliability of the information flow estimates and reduce to well-established coupling directionality quantifiers when some general conditions are satisfied. Furthermore, by generalizing the definition of transcript to ordinal patterns of different lengths, several of the commonly used information directionality measures can be encompassed within the same framework.

Direct-coupling information measure from nonuniform embedding

A measure to estimate the direct and directional coupling in multivariate time series is proposed. The measure is an extension of a recently published measure of conditional mutual information from mixed embedding (MIME) for bivariate time series. In the proposed measure of partial MIME (PMIME), the embedding is on all observed variables and it is optimized in explaining the response variable. It is shown that PMIME detects correctly direct coupling and outperforms the (linear) conditional Granger causality and the partial transfer entropy. We demonstrate that PMIME does not rely on significance test and embedding parameters and the number of observed variables has no effect on its statistical accuracy; it may only slow the computations. The importance of these points is shown in simulations and in an application to epileptic multichannel scalp electroencephalograms.

Measuring connectivity in linear multivariate processes: definitions, interpretation, and practical analysis

This tutorial paper introduces a common framework for the evaluation of widely used frequency-domain measures of coupling (coherence, partial coherence) and causality (directed coherence, partial directed coherence) from the parametric representation of linear multivariate (MV) processes. After providing a comprehensive time-domain definition of the various forms of connectivity observed in MV processes, we particularize them to MV autoregressive (MVAR) processes and derive the corresponding frequency-domain measures. Then, we discuss the theoretical interpretation of these MVAR-based connectivity measures, showing that each of them reflects a specific time-domain connectivity definition and how this results in the description of peculiar aspects of the information transfer in MV processes. Furthermore, issues related to the practical utilization of these measures on real-time series are pointed out, including MVAR model estimation and significance assessment. Finally, limitations and pitfalls arising from model mis-specification are discussed, indicating possible solutions and providing practical recommendations for a safe computation of the connectivity measures. An example of estimation of the presented measures from multiple EEG signals recorded during a combined visuomotor task is also reported, showing how evaluation of coupling and causality in the frequency domain may help describing specific neurophysiological mechanisms.

Causality and persistence in ecological systems: a nonparametric spectral granger causality approach

Abstract Directionality in coupling, defined as the linkage relating causes to their effects at a later time, can be used to explain the core dynamics of ecological systems by untangling direct and feedback relationships between the different components of the systems. Inferring causality from measured ecological variables sampled through time remains a formidable challenge further made difficult by the action of periodic drivers overlapping the natural dynamics of the system. Periodicity in the drivers can often mask the self-sustained oscillations originating from the autonomous dynamics. While linear and direct causal relationships are commonly addressed in the time domain, using the well-established machinery of Granger causality (G-causality), the presence of periodic forcing requires frequency-based statistics (e.g., the Fourier transform), able to distinguish coupling induced by oscillations in external drivers from genuine endogenous interactions. Recent nonparametric spectral extensions of G-causality to the frequency domain pave the way for the scale-by-scale decomposition of causality, which can improve our ability to link oscillatory behaviors of ecological networks to causal mechanisms. The performance of both spectral G-causality and its conditional extension for multivariate systems is explored in quantifying causal interactions within ecological networks. Through two case studies involving synthetic and actual time series, it is demonstrated that conditional G-causality outperforms standard G-causality in identifying causal links and their concomitant timescales.

Assessing directed interactions from neurophysiological signals--an overview

The study of synchronization phenomena in coupled dynamical systems is an active field of research in many scientific disciplines including the neurosciences. Over the last decades, a number of time series analysis techniques have been proposed to capture both linear and nonlinear aspects of interactions. While most of these techniques allow one to quantify the strength of interactions, developments that resulted from advances in nonlinear dynamics and in information and synchronization theory aim at assessing directed interactions. Most of these techniques, however, assume the underlying systems to be at least approximately stationary and require a large number of data points to robustly assess directed interactions. Recent extensions allow assessing directed interactions from short and transient signals and are particularly suited for the analysis of evoked and event-related activity.