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.     

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。     

Genes,culture, and the human niche: An overview

The sharp distinction between biological traits and culturally based traits, which had long been standard in evolutionary approaches to behavior, was blurred in the early 1980s by mathematical models that allowed a co‐dependent evolution of genetic transmission and cultural information. Niche‐construction theory has since added another contrast to standard evolutionary theory, in that it views niche construction as a cause of evolutionary change rather than simply a product of selection. While offering a new understanding of the coevolution of genes, culture, and human behavior, niche‐construction models also invoke multivariate causality, which require multiple time series to resolve. The empirical challenge lies in obtaining time‐series data on causal pathways involved in the coevolution of genes, culture, and behavior. This is a significant issue in archeology, where time series are often sparse and causal behaviors are represented only by proxies in the material record.     

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.     

On the Evaluation of Information Flow in Multivariate Systems by the Directed Transfer Function

The directed transfer function (DTF) has been proposed as a measure of information flow between the components of multivariate time series. In this paper, we discuss the interpretation of the DTF and compare it with other measures for directed relationships. In particular, we show that the DTF does not indicate multivariate or bivariate Granger causality, but that it is closely related to the concept of impulse response function and can be viewed as a spectral measure for the total causal influence from one component to another. Furthermore, we investigate the statistical properties of the DTF and establish a simple significance level for testing for the null hypothesis of no information flow.     

A graphical approach for evaluating effective connectivity in neural systems

The identification of effective connectivity from time-series data such as electroencephalogram (EEG) or time-resolved function magnetic resonance imaging (fMRI) recordings is an important problem in brain imaging. One commonly used approach to inference effective connectivity is based on vector autoregressive models and the concept of Granger causality. However, this probabilistic concept of causality can lead to spurious causalities in the presence of latent variables. Recently, graphical models have been used to discuss problems of causal inference for multivariate data. In this paper, we extend these concepts to the case of time-series and present a graphical approach for discussing Granger-causal relationships among multiple time-series. In particular, we propose a new graphical representation that allows the characterization of spurious causality and, thus, can be used to investigate spurious causality. The method is demonstrated with concurrent EEG and fMRI recordings which are used to investigate the interrelations between the alpha rhythm in the EEG and blood oxygenation level dependent (BOLD) responses in the fMRI. The results confirm previous findings on the location of the source of the EEG alpha rhythm.     

Forensic Ecotoxicology: Establishing Causality between Contaminants and Biological Effects in Field Studies

Twelve papers in this series were derived from two conference sessions focusing on causality in field studies. Eight of these papers involve case studies examining biological effects of chemical contaminants in field situations. Using a weight-of-evidence approach, these case studies were evaluated against seven proposed criteria for establishing causality. The seven criteria were: strength of association; consistency of association; specificity of association; time order; biological gradient; experimental evidence; and biological plausibility. One of these seven criteria, ‘specificity of association’ was found to be of little utility for establishing causality in these field studies. The case studies are presented in approximate order of increasing levels of biological organization (i.e., going from endpoints at the suborganismal level to endpoints at the population or community level). In case studies examining higher levels of biological organization, it appears that the ‘biological gradient’ criterion was also not useful in establishing causality. These results, together with suggestions from other papers in the series, are used to recommend a set of modified criteria for establishing causality in field studies of the biological effects of chemical contaminants.     

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.     

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.     

Causality from palaeontological time series

As custodians of deep time, palaeontologists have an obligation to seek the causes and consequences of long‐term evolutionary trajectories and the processes of ecosystem assembly and collapse. Building explicit process models on the relevant scales can be fraught with difficulties, and causal inference is typically limited to patterns of association. In this review, we discuss some of the ways in which causal connections can be extracted from palaeontological time series and provide an overview of three recently developed analytical frameworks that have been applied to palaeontological questions, namely linear stochastic differential equations, convergent cross mapping and transfer entropy. We outline how these methods differ conceptually, and in practice, and point to available software and worked examples. We end by discussing why a paradigm of dynamical causality is needed to decipher the messages encrypted in palaeontological patterns.     

A Weight-of-Evidence Framework for Assessing Sediment (Or Other) Contamination: Improving Certainty in the Decision-Making Process

A basic framework is presented for the ecological weight-of-evidence (WOE) process for sediment assessment that clearly defines its essential elements and will improve the certainty of conclusions about whether or not impairment exists due to sediment contamination, and, if so, which stressors and biological species (or ecological responses) are of greatest concern. The essential “Certainty Elements” are addressed in a transparent best professional judgment (BPJ) process with multiple lines-of-evidence (LOE) ultimately quantitatively integrated (but not necessarily combined into a single value). The WOE Certainty Elements include: (1) Development of a conceptual model (showing linkages of critical receptors and ecosystem quality characteristics); (2) Explanation of linkages between measurement endpoint responses (direct and indirect with associated spatial/temporal dynamics) and conceptual model components; (3) Identification of possible natural and anthropogenic stressors with associated exposure dynamics; (4) Evaluation of appropriate and quantitatively based reference (background) comparison methods; (5) Consideration of advantages and limitations of quantification methods used to integrate LOE; (6) Consideration of advantages and limitations of each LOE used; (7) Evaluation of causality criteria used for each LOE during output verification and how they were implemented; and (8) Combining the LOE into a WOE matrix for interpretation, showing causality linkages in the conceptual model. The framework identifies several statistical approaches for integrating within LOE, the suitability of which depends on physical characteristics of the system and the scale/nature of impairment. The quantification approaches include: (1) Gradient (regression methods); (2) Paired reference/test (before/after control impact and ANOVA methods); (3) Multiple reference (ANOVA and multivariate methods); and 4) Gradient with reference (regression, ANOVA and multivariate methods). This WOE framework can be used for any environmental assessment and is most effective when incorporated into the initial and final study design stages (e.g., the Problem Formulation and Risk Characterization stages of a risk assessment) with reassessment throughout the project and decision-making process, rather than in a retrospective data analysis approach where key certainty elements cannot be adequately addressed.     

Extended causal modeling to assess Partial Directed Coherence in multiple time series with significant instantaneous interactions

The Partial Directed Coherence (PDC) and its generalized formulation (gPDC) are popular tools for investigating, in the frequency domain, the concept of Granger causality among multivariate (MV) time series. PDC and gPDC are formalized in terms of the coefficients of an MV autoregressive (MVAR) model which describes only the lagged effects among the time series and forsakes instantaneous effects. However, instantaneous effects are known to affect linear parametric modeling, and are likely to occur in experimental time series. In this study, we investigate the impact on the assessment of frequency domain causality of excluding instantaneous effects from the model underlying PDC evaluation. Moreover, we propose the utilization of an extended MVAR model including both instantaneous and lagged effects. This model is used to assess PDC either in accordance with the definition of Granger causality when considering only lagged effects (iPDC), or with an extended form of causality, when we consider both instantaneous and lagged effects (ePDC). The approach is first evaluated on three theoretical examples of MVAR processes, which show that the presence of instantaneous correlations may produce misleading profiles of PDC and gPDC, while ePDC and iPDC derived from the extended model provide here a correct interpretation of extended and lagged causality. It is then applied to representative examples of cardiorespiratory and EEG MV time series. They suggest that ePDC and iPDC are better interpretable than PDC and gPDC in terms of the known cardiovascular and neural physiologies.     

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.     

Identifying Neural Drivers with Functional MRI: An Electrophysiological Validation

Whether functional magnetic resonance imaging (fMRI) allows the identification of neural drivers remains an open question of particular importance to refine physiological and neuropsychological models of the brain, and/or to understand neurophysiopathology. Here, in a rat model of absence epilepsy showing spontaneous spike-and-wave discharges originating from the first somatosensory cortex (S1BF), we performed simultaneous electroencephalographic (EEG) and fMRI measurements, and subsequent intracerebral EEG (iEEG) recordings in regions strongly activated in fMRI (S1BF, thalamus, and striatum). fMRI connectivity was determined from fMRI time series directly and from hidden state variables using a measure of Granger causality and Dynamic Causal Modelling that relates synaptic activity to fMRI. fMRI connectivity was compared to directed functional coupling estimated from iEEG using asymmetry in generalised synchronisation metrics. The neural driver of spike-and-wave discharges was estimated in S1BF from iEEG, and from fMRI only when hemodynamic effects were explicitly removed. Functional connectivity analysis applied directly on fMRI signals failed because hemodynamics varied between regions, rendering temporal precedence irrelevant. This paper provides the first experimental substantiation of the theoretical possibility to improve interregional coupling estimation from hidden neural states of fMRI. As such, it has important implications for future studies on brain connectivity using functional neuroimaging.     

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.     

Partial directed coherence: a new concept in neural structure determination

 This paper introduces a new frequency-domain approach to describe the relationships (direction of information flow) between multivariate time series based on the decomposition of multivariate partial coherences computed from multivariate autoregressive models. We discuss its application and compare its performance to other approaches to the problem of determining neural structure relations from the simultaneous measurement of neural electrophysiological signals. The new concept is shown to reflect a frequency-domain representation of the concept of Granger causality. Received: 25 April 2000 / Accepted in revised form: 13 November 2000     

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.     

Establishing Causality between Environmental Stressors and Effects on Aquatic Ecosystems

Establishing causal relationships between environmental stressors and observed effects in natural systems is difficult due to the many intrinsic environmental factors that can hinder this process and because there are no widely accepted and proven approaches for determining such relationships. Several types of approaches or combinations of approaches, each with their own sets of advantages and limitations, have been applied in a variety of ecological systems to investigate possible causal relationships between stressors and effects. These include controlled laboratory studies (including acute and chronic bioassays), experimental field manipulations, field studies based on synoptic field surveys, mathematical simulation modeling, statistical associations, various combinations of laboratory, experimental, and field studies, and the ecoepidemiological (weight or evidence) approach. The use of ecoepidemiological (“forensic toxicology”) principles is becoming increasingly attractive as a method to help establish causality because it does not involve the same limitations of other approaches and it can also be used to integrate disparate information within a logical framework so that scientifically and defensible regulatory decisions can be made. The objective of this Commentary series of papers on the issue on causality is to demonstrate the application of the ecoepidemiology approach, using a variety of case history studies, for establishing causal relationships between specific stressors and biological effects. For each case history provided in the following series of papers, the authors describe their study situation, summarize the results supporting a causal relationship, and then compare their study results against seven standard causal criteria.     

Wavelet-based estimation of hemodynamic response function from fMRI data

We present a new algorithm to estimate hemodynamic response function (HRF) and drift components of fMRI data in wavelet domain. The HRF is modeled by both parametric and nonparametric models. The functional Magnetic resonance Image (fMRI) noise is modeled as a fractional brownian motion (fBm). The HRF parameters are estimated in wavelet domain by exploiting the property that wavelet transforms with a sufficient number of vanishing moments decorrelates a fBm process. Using this property, the noise covariance matrix in wavelet domain can be assumed to be diagonal whose entries are estimated using the sample variance estimator at each scale. We study the influence of the sampling rate of fMRI time series and shape assumption of HRF on the estimation performance. Results are presented by adding synthetic HRFs on simulated and null fMRI data. We also compare these methods with an existing method,(1) where correlated fMRI noise is modeled by a second order polynomial functions.