A framework for assessing frequency domain causality in physiological time series with instantaneous effects

Abstract

We present an approach for the quantification of directional relations in multiple time series exhibiting significant zero-lag interactions. To overcome the limitations of the traditional multivariate autoregressive (MVAR) modelling of multiple series, we introduce an extended MVAR (eMVAR) framework allowing either exclusive consideration of time-lagged effects according to the classic notion of Granger causality, or consideration of combined instantaneous and lagged effects according to an extended causality definition. The spectral representation of the eMVAR model is exploited to derive novel frequency domain causality measures that generalize to the case of instantaneous effects the known directed coherence (DC) and partial DC measures. The new measures are illustrated in theoretical examples showing that they reduce to the known measures in the absence of instantaneous causality, and describe peculiar aspects of directional interaction among multiple series when instantaneous causality is non-negligible. Then, the issue of estimating eMVAR models from time-series data is faced, proposing two approaches for model identification and discussing problems related to the underlying model assumptions. Finally, applications of the framework on cardiovascular variability series and multichannel EEG recordings are presented, showing how it allows one to highlight patterns of frequency domain causality consistent with well-interpretable physiological interaction mechanisms.

Compensated Transfer Entropy as a Tool for Reliably Estimating Information Transfer in Physiological Time Series

“…Instantaneous effects are the practical evidence of the concept of instantaneous causality, which is a known issue in causal analysis [16,19]. In practice, instantaneous causality between two time series may either have a proper causal meaning, when the time resolution of the measurements is lower than the time scale of the lagged causal influences between the underlying processes, or be void of such causal meaning, in the case of common driving effects occurring when an unmeasured process simultaneously affects the two processes under analysis [17]. In either case, instantaneous causality has an impact on the estimation of the TE: if it is causally meaningful, the analysis misses the zero-lag effect x n →y n , if not, the analysis includes potential spurious effects taking the form x 1:n-1 →x n →y n ; these misleading detections may impair respectively the sensitivity and the specificity of TE estimation.…”

“…Since an instantaneous correlation arises between two observed variables which are affected by latent variables with the same time delay, in the context of model-based analysis attempts have been made to counteract this problem by accounting for residual correlations which reflect zero-lag effects. Indeed, recent studies have proposed to incorporate terms from the covariance matrix of the model residuals into the so-called partial Granger causality measures [21,22], or to express the residual correlation in terms of model coefficients and exploit the resulting new model structure for defining extended Granger causality measures [17,18]. However, as similar approaches cannot be followed in the model-free context of TE analysis, instantaneous effects are usually not considered in the computation of TE on experimental data.…”

“…These effects may reflect fast (within sample) physiologically meaningful interactions, or be void of physiological meaning (e.g., may be due to unobserved confounders). In either case, instantaneous effects have an impact on the computation of any causality measure [17,18]. In particular, the presence of unmeasured exogenous inputs or latent variables which cannot be included in the observed data set (e.g., because they are not accessible) is a critical issue when investigating Granger causality in experimental data, as it may easily lead to the detection of spurious causalities [19][20][21].…”

“…This is a crucial point in TE analysis because, even though it is now well recognized that instantaneous causality plays a key role in Granger causality analysis, instantaneous effects are commonly disregarded in the computation of TE on experimental data. While recent studies have started to unravel the issue of instantaneous causality in the linear parametric framework of multivariate autoregressive models [17,18,23,24], there is a paucity of works addressing the consequences of excluding instantaneous effects from the computation of model-free causality measures. In this paper, the issue of instantaneous causality is faced allowing for the possibility of zero-lag effects in TE computation, according to two alternative empirical procedures: if instantaneous effects are deemed as causally meaningful, the zero-lag term is assimilated to the lagged terms to make it causally relevant; if not, the zero-lag term is incorporated in both CE computations to obtain a compensation of its confounding effects.…”

Assessing causality in brain dynamics and cardiovascular control

“…Among the contributions covering the issue of assessing causality in the frequency domain, Baccalá et al [17] derive a unified asymptotic theory for all the partial directed coherence estimators, thus leading to a formal derivation of confidence intervals and threshold for testing the null hypothesis of absence of a causal link as a function of the frequency. Faes et al [18] address in the frequency domain the theoretically challenging issue of the dependence of causality on the canonical form of the multivariate model necessary to interpret instantaneous links resulting from the inadequate temporal resolution in relation to the latencies among signals. Wen et al [19] propose an efficient method for estimating Granger causality among a subgroup of signals present in Ω starting from the spectral density matrix describing all the causal interactions in Ω.…”