More discussions for granger causality and new causality measures

Granger causality (GC) has been widely applied in economics and neuroscience to reveal causality influence of time series. In our previous paper (Hu et al., in IEEE Trans on Neural Netw, 22(6), pp. 829–844, 2011), we proposed new causalities in time and frequency domains and particularly focused on new causality in frequency domain by pointing out the shortcomings/limitations of GC or Granger-alike causality metrics and the advantages of new causality. In this paper we continue our previous discussions and focus on new causality and GC or Granger-alike causality metrics in time domain. Although one strong motivation was introduced in our previous paper (Hu et al., in IEEE Trans on Neural Netw, 22(6), pp. 829–844, 2011) we here present additional motivation for the proposed new causality metric and restate the previous motivation for completeness. We point out one property of conditional GC in time domain and the shortcomings/limitations of conditional GC which cannot reveal the real strength of the directional causality among three time series. We also show the shortcomings/limitations of directed causality (DC) or normalize DC for multivariate time series and demonstrate it cannot reveal real causality at all. By calculating GC and new causality values for an example we demonstrate the influence of one of the time series on the other is linearly increased as the coupling strength is linearly increased. This fact further supports reasonability of new causality metric. We point out that larger instantaneous correlation does not necessarily mean larger true causality (e.g., GC and new causality), or vice versa. Finally we conduct analysis of statistical test for significance and asymptotic distribution property of new causality metric by illustrative examples.     

Linear and nonlinear causality between signals: methods, examples and neurophysiological applications

In this paper, we will present and review the most usual methods to detect linear and nonlinear causality between signals: linear Granger causality test (Geweke in J Am Stat Assoc 77:304–313, 1982) extended to direct causality in multivariate case (LGC), directed coherence (DCOH, Saito and Harashima in Recent advances in EEG and EMG data processing, Elsevier, Amsterdam, 1981), partial directed coherence (PDC, Sameshima and Baccala 1999) and nonlinear Granger causality test of Baek and Brock (in Working Paper University of Iowa, 1992) extended to direct causality in multivariate case (partial nonlinear Granger causality, PNGC). All these methods are tested and compared on several ARX, Poisson and nonlinear models, and on neurophysiological data (depth EEG). The results show that LGC, DCOH and PDC are not very robust in relation to nonlinear linkages but they seem to correctly find linear linkages if only the autoregressive parts are nonlinear. PNGC is extremely dependent on the choice of parameters. Moreover, LGC and PNGC may give misleading results in the case of causality on a spectral band, which is illustrated by our neurophysiological database.Electronic Supplementary Material Supplementary material is available to authorised users in the online version of this article at .     

Granger causality between multiple interdependent neurobiological time series: blockwise versus pairwise methods

Granger causality is becoming an important tool for determining causal relations between neurobiological time series. For multivariate data, there is often the need to examine causal relations between two blocks of time series, where each block could represent a brain region of interest. Two alternative methods are available. In the pairwise method, bivariate autoregressive models are fit to all pairwise combinations involving one time series from the first block and one from the second. The total Granger causality between the two blocks is then derived by summing pairwise causality values from each of these models. This approach is intuitive but computationally cumbersome. Theoretically, a more concise method can be derived, which we term the blockwise Granger causality method. In this method, a single multivariate model is fit to all the time series, and the causality between the two blocks is then computed from this model. We compare these two methods by applying them to cortical local field potential recordings from monkeys performing a sensorimotor task. The obtained results demonstrate consistency between the two methods and point to the significance potential of utilizing Granger causality analysis in understanding coupled neural systems.     

Measuring frequency domain granger causality for multiple blocks of interacting time series

In the past years, several frequency-domain causality measures based on vector autoregressive time series modeling have been suggested to assess directional connectivity in neural systems. The most followed approaches are based on representing the considered set of multiple time series as a realization of two or three vector-valued processes, yielding the so-called Geweke linear feedback measures, or as a realization of multiple scalar-valued processes, yielding popular measures like the directed coherence (DC) and the partial DC (PDC). In the present study, these two approaches are unified and generalized by proposing novel frequency-domain causality measures which extend the existing measures to the analysis of multiple blocks of time series. Specifically, the block DC (bDC) and block PDC (bPDC) extend DC and PDC to vector-valued processes, while their logarithmic counterparts, denoted as multivariate total feedback $f^\mathrm{m}$ and direct feedback $g^\mathrm{m}$ , represent into a full multivariate framework the Geweke’s measures. Theoretical analysis of the proposed measures shows that they: (i) possess desirable properties of causality measures; (ii) are able to reflect either direct causality (bPDC, $g^\mathrm{m})$ or total (direct + indirect) causality (bDC, $f^\mathrm{m})$ between time series blocks; (iii) reduce to the DC and PDC measures for scalar-valued processes, and to the Geweke’s measures for pairs of processes; (iv) are able to capture internal dependencies between the scalar constituents of the analyzed vector processes. Numerical analysis showed that the proposed measures can be efficiently estimated from short time series, allow to represent in an objective, compact way the information derived from the causal analysis of several pairs of time series, and may detect frequency domain causality more accurately than existing measures. The proposed measures find their natural application in the evaluation of directional interactions in neurophysiological settings where several brain activity signals are simultaneously recorded from multiple regions of interest.     

On the spectral formulation of Granger causality

Spectral measures of causality are used to explore the role of different rhythms in the causal connectivity between brain regions. We study several spectral measures related to Granger causality, comprising the bivariate and conditional Geweke measures, the directed transfer function, and the partial directed coherence. We derive the formulation of dependence and causality in the spectral domain from the more general formulation in the information-theory framework. We argue that the transfer entropy, the most general measure derived from the concept of Granger causality, lacks a spectral representation in terms of only the processes associated with the recorded signals. For all the spectral measures we show how they are related to mutual information rates when explicitly considering the parametric autoregressive representation of the processes. In this way we express the conditional Geweke spectral measure in terms of a multiple coherence involving innovation variables inherent to the autoregressive representation. We also link partial directed coherence with Sims' criterion of causality. Given our results, we discuss the causal interpretation of the spectral measures related to Granger causality and stress the necessity to explicitly consider their specific formulation based on modeling the signals as linear Gaussian stationary autoregressive processes.     

Manifest Variable Granger Causality Models for Developmental Research: A Taxonomy

Granger models are popular when it comes to testing hypotheses that relate series of measures causally to each other. In this article, we propose a taxonomy of Granger causality models. The taxonomy results from crossing the four variables Order of Lag, Type of (Contemporaneous) Effect, Direction of Effect, and Segment of Dependent Series Targeted. Among the uses of such a taxonomy are that existing models can be embedded in the context of possible other models, new models can be derived, models can be compared, and the relation of statistical models to theories of causality can be specified. Sample models are depicted, and parameters of interest are indicated. For two new models, empirical data examples are provided from research on the development of aggression in adolescents.     

Listen to Genes: Dealing with Microarray Data in the Frequency Domain

Background

We present a novel and systematic approach to analyze temporal microarray data. The approach includes normalization, clustering and network analysis of genes.

Methodology

Genes are normalized using an error model based uniform normalization method aimed at identifying and estimating the sources of variations. The model minimizes the correlation among error terms across replicates. The normalized gene expressions are then clustered in terms of their power spectrum density. The method of complex Granger causality is introduced to reveal interactions between sets of genes. Complex Granger causality along with partial Granger causality is applied in both time and frequency domains to selected as well as all the genes to reveal the interesting networks of interactions. The approach is successfully applied to Arabidopsis leaf microarray data generated from 31,000 genes observed over 22 time points over 22 days. Three circuits: a circadian gene circuit, an ethylene circuit and a new global circuit showing a hierarchical structure to determine the initiators of leaf senescence are analyzed in detail.

Conclusions

We use a totally data-driven approach to form biological hypothesis. Clustering using the power-spectrum analysis helps us identify genes of potential interest. Their dynamics can be captured accurately in the time and frequency domain using the methods of complex and partial Granger causality. With the rise in availability of temporal microarray data, such methods can be useful tools in uncovering the hidden biological interactions. We show our method in a step by step manner with help of toy models as well as a real biological dataset. We also analyse three distinct gene circuits of potential interest to Arabidopsis researchers.     

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.     

A novel method for the identification of synchronization effects in multichannel ECoG with an application to epilepsy

In this paper, we present a novel method for the identification of synchronization effects in multichannel electrocorticograms (ECoG). Based on autoregressive modeling, we define a dependency measure termed extrinsic-to-intrinsic power ratio (EIPR) which quantifies directed coupling effects in the time domain. Hereby, a dynamic input channel selection algorithm assures the estimation of the model parameters despite the strong spatial correlation among the high number of involved ECoG channels. We compare EIPR to the partial directed coherence, show its ability to indicate Granger causality and successfully validate a signal model. Applying EIPR to ictal ECoG data of patients suffering from temporal lobe epilepsy allows us to identify the electrodes of the seizure onset zone. The results obtained by the proposed method are in good accordance with the clinical findings.     

BOLD Granger Causality Reflects Vascular Anatomy

A number of studies have tried to exploit subtle phase differences in BOLD time series to resolve the order of sequential activation of brain regions, or more generally the ability of signal in one region to predict subsequent signal in another region. More recently, such lag-based measures have been applied to investigate directed functional connectivity, although this application has been controversial. We attempted to use large publicly available datasets (FCON 1000, ADHD 200, Human Connectome Project) to determine whether consistent spatial patterns of Granger Causality are observed in typical fMRI data. For BOLD datasets from 1,240 typically developing subjects ages 7–40, we measured Granger causality between time series for every pair of 7,266 spherical ROIs covering the gray matter and 264 seed ROIs at hubs of the brain’s functional network architecture. Granger causality estimates were strongly reproducible for connections in a test and replication sample (n=620 subjects for each group), as well as in data from a single subject scanned repeatedly, both during resting and passive video viewing. The same effect was even stronger in high temporal resolution fMRI data from the Human Connectome Project, and was observed independently in data collected during performance of 7 task paradigms. The spatial distribution of Granger causality reflected vascular anatomy with a progression from Granger causality sources, in Circle of Willis arterial inflow distributions, to sinks, near large venous vascular structures such as dural venous sinuses and at the periphery of the brain. Attempts to resolve BOLD phase differences with Granger causality should consider the possibility of reproducible vascular confounds, a problem that is independent of the known regional variability of the hemodynamic response.     

Analyzing multiple spike trains with nonparametric granger causality

Simultaneous recordings of spike trains from multiple single neurons are becoming commonplace. Understanding the interaction patterns among these spike trains remains a key research area. A question of interest is the evaluation of information flow between neurons through the analysis of whether one spike train exerts causal influence on another. For continuous-valued time series data, Granger causality has proven an effective method for this purpose. However, the basis for Granger causality estimation is autoregressive data modeling, which is not directly applicable to spike trains. Various filtering options distort the properties of spike trains as point processes. Here we propose a new nonparametric approach to estimate Granger causality directly from the Fourier transforms of spike train data. We validate the method on synthetic spike trains generated by model networks of neurons with known connectivity patterns and then apply it to neurons simultaneously recorded from the thalamus and the primary somatosensory cortex of a squirrel monkey undergoing tactile stimulation.     

Causality and pathway search in microarray time series experiment

MOTIVATION: Interaction among time series can be explored in many ways. All the approach has the usual problem of low power and high dimensional model. Here we attempted to build a causality network among a set of time series. The causality has been established by Granger causality, and then constructing the pathway has been implemented by finding the Minimal Spanning Tree within each connected component of the inferred network. False discovery rate measurement has been used to identify the most significant causalities. RESULTS: Simulation shows good convergence and accuracy of the algorithm. Robustness of the procedure has been demonstrated by applying the algorithm in a non-stationary time series setup. Application of the algorithm in a real dataset identified many causalities, with some overlap with previously known ones. Assembled network of the genes reveals features of the network that are common wisdom about naturally occurring networks.     

Assessment of Interaction Between Cardio-Respiratory Signals Using Directed Coherence on Healthy Subjects During Postural Change

PurposeTo detect and quantify the directional interaction changes between cardio-respiratory system during postural change.MethodTraditional frequency domain analysis based on power spectrum and coherence are insufficient to quantify nonlinear structures and complexity of physiological subsystems. Recently, Granger causality is found as preferable method for evaluation of causality i.e., directional interaction. Frequency domain Granger causality based on directed coherence has been used in this study to identify directional interaction between cardiac and respiratory signal during postural change from supine to standing for healthy subjects.ResultECG and respiration signal are recorded for this study. The beat-to-beat variability series from ECG provides heart rate (RR) and the respiration amplitude corresponds to RESP time series. It was observed that respiration is responsible for the changes in ECG signal during supine position as compared to standing. The outflow of information from RESP to RR increases during supine results in stronger interaction but reduces during standing result in reduction of interaction. Similarly, the effect of RR on RESP is found significant only during standing.ConclusionThe proposed directed coherence approach detects the cardio-respiratory regulation during postural change and provide information about coupling changes during this transition.     

Akaike causality in state space

We present a new approach of explaining instantaneous causality in multivariate fMRI time series by a state space model. A given single time series can be divided into two noise-driven processes, a common process shared among multivariate time series and a specific process refining the common process. By assuming that noises are independent, a causality map is drawn using Akaike noise contribution ratio theory. The method is illustrated by an application to fMRI data recorded under visual stimulation.     

Causal measures of structure and plasticity in simulated and living neural networks

A major goal of neuroscience is to understand the relationship between neural structures and their function. Recording of neural activity with arrays of electrodes is a primary tool employed toward this goal. However, the relationships among the neural activity recorded by these arrays are often highly complex making it problematic to accurately quantify a network's structural information and then relate that structure to its function. Current statistical methods including cross correlation and coherence have achieved only modest success in characterizing the structural connectivity. Over the last decade an alternative technique known as Granger causality is emerging within neuroscience. This technique, borrowed from the field of economics, provides a strong mathematical foundation based on linear auto-regression to detect and quantify "causal" relationships among different time series. This paper presents a combination of three Granger based analytical methods that can quickly provide a relatively complete representation of the causal structure within a neural network. These are a simple pairwise Granger causality metric, a conditional metric, and a little known computationally inexpensive subtractive conditional method. Each causal metric is first described and evaluated in a series of biologically plausible neural simulations. We then demonstrate how Granger causality can detect and quantify changes in the strength of those relationships during plasticity using 60 channel spike train data from an in vitro cortical network measured on a microelectrode array. We show that these metrics can not only detect the presence of causal relationships, they also provide crucial information about the strength and direction of that relationship, particularly when that relationship maybe changing during plasticity. Although we focus on the analysis of multichannel spike train data the metrics we describe are applicable to any stationary time series in which causal relationships among multiple measures is desired. These techniques can be especially useful when the interactions among those measures are highly complex, difficult to untangle, and maybe changing over time.     

收敛交叉映射方法及其在生态学中的应用

收敛交叉映射(CCM)是一种分析非线性系统中时间序列变量间因果关系的方法。其不同于传统的线性系统分析方法,是通过对变量进行状态空间重构来获取变量的历史信息,随着时间序列不断增长,当其估计性能呈现收敛的性质时,可以判断因果关系的存在。本文介绍了CCM的发展史及其较传统的格兰杰因果检验的优点,详细阐明了CCM的原理、算法过程和实现途径。CCM作为一种针对变量间具有弱到中等强度耦合关系的系统分析方法,可以用来有效地解决非线性生态系统多变量间复杂的因果关系问题。将该方法应用于具有空间信息的多点位时间序列变量间因果分析时,应充分考虑点位间的空间自相关性,与可以去除变量及序列间空间相关性的方法相结合,从而确保CCM对变量因果关系的分析更加准确,结果也更具有信服力。     

Does energy intensity contribute to CO2 emissions? A trivariate analysis in selected African countries

The present study investigates the dynamic relationship between energy intensity and CO₂ emissions by incorporating economic growth in environment CO₂ emissions function using data of Sub Saharan African countries. For this purpose, we applied panel cointegration to examine the long run relationship between the series. We employed the VECM Granger causality to test the direction of causality amid the variables.At panel level, our results validate the existence of cointegration among the series. The long run panel results show that energy intensity has positive and statistically significant impact on CO₂ emissions. There is also positive and negative link of non-linear and linear terms of real GDP per capita with CO₂ emissions supporting the presence of environmental Kuznets curve (EKC). The causality analysis reveals the bidirectional causality between economic growth and CO₂ emissions while energy intensity Granger causes economic growth and hence CO₂ emissions, while across the individual countries, the results differ. This paper opens up new insights for policy makers to design comprehensive economic, energy and environmental policy for sustainable long run economic growth.     

Attention-Dependent Modulation of Cortical Taste Circuits Revealed by Granger Causality with Signal-Dependent Noise

We show, for the first time, that in cortical areas, for example the insular, orbitofrontal, and lateral prefrontal cortex, there is signal-dependent noise in the fMRI blood-oxygen level dependent (BOLD) time series, with the variance of the noise increasing approximately linearly with the square of the signal. Classical Granger causal models are based on autoregressive models with time invariant covariance structure, and thus do not take this signal-dependent noise into account. To address this limitation, here we describe a Granger causal model with signal-dependent noise, and a novel, likelihood ratio test for causal inferences. We apply this approach to the data from an fMRI study to investigate the source of the top-down attentional control of taste intensity and taste pleasantness processing. The Granger causality with signal-dependent noise analysis reveals effects not identified by classical Granger causal analysis. In particular, there is a top-down effect from the posterior lateral prefrontal cortex to the insular taste cortex during attention to intensity but not to pleasantness, and there is a top-down effect from the anterior and posterior lateral prefrontal cortex to the orbitofrontal cortex during attention to pleasantness but not to intensity. In addition, there is stronger forward effective connectivity from the insular taste cortex to the orbitofrontal cortex during attention to pleasantness than during attention to intensity. These findings indicate the importance of explicitly modeling signal-dependent noise in functional neuroimaging, and reveal some of the processes involved in a biased activation theory of selective attention.     

Granger mediation analysis of multiple time series with an application to functional magnetic resonance imaging

This paper presents Granger mediation analysis, a new framework for causal mediation analysis of multiple time series. This framework is motivated by a functional magnetic resonance imaging (fMRI) experiment where we are interested in estimating the mediation effects between a randomized stimulus time series and brain activity time series from two brain regions. The independent observation assumption is thus unrealistic for this type of time‐series data. To address this challenge, our framework integrates two types of models: causal mediation analysis across the mediation variables, and vector autoregressive (VAR) models across the temporal observations. We use “Granger” to refer to VAR correlations modeled in this paper. We further extend this framework to handle multilevel data, in order to model individual variability and correlated errors between the mediator and the outcome variables. Using Rubin's potential outcome framework, we show that the causal mediation effects are identifiable under our time‐series model. We further develop computationally efficient algorithms to maximize our likelihood‐based estimation criteria. Simulation studies show that our method reduces the estimation bias and improves statistical power, compared with existing approaches. On a real fMRI data set, our approach quantifies the causal effects through a brain pathway, while capturing the dynamic dependence between two brain regions.