Title: Granger Components Analysis: Unsupervised learning of latent temporal dependencies
Venue: NeurIPS 2023
Accepted: yes
Keywords: components analysis

unsupervised learning

Granger Causality
Abstract: A new technique for unsupervised learning of time series data based on the notion of Granger causality is presented. The technique learns pairs of projections of a multivariate data set such that the resulting components -- "driving" and "driven" -- maximize the strength of the Granger causality between the latent time series (how strongly the past of the driving signal predicts the present of the driven signal). A coordinate descent algorithm that learns pairs of coefficient vectors in an alternating fashion is developed and shown to blindly identify the underlying sources (up to scale) on simulated vector autoregressive (VAR) data. The technique is tested on scalp electroencephalography (EEG) data from a motor imagery experiment where the resulting components lateralize with the side of the cued hand, and also on functional magnetic resonance imaging (fMRI) data, where the recovered components express previously reported resting-state networks.

Granger Components Analysis: Unsupervised learning of latent temporal dependencies

Abstract

A new technique for unsupervised learning of time series data based on the notion of Granger causality is presented. The technique learns pairs of projections of a multivariate data set such that the resulting components - 'driving' and 'driven' maximize the strength of the Granger causality between the latent time series (how strongly the past of the driving signal predicts the present of the driven signal). A coordinate descent algorithm that learns pairs of coefficient vectors in an alternating fashion is developed and shown to blindly identify the underlying sources (up to scale) on simulated vector autoregressive (VAR) data. The technique is tested on scalp electroencephalography (EEG) data from a motor imagery experiment where the resulting components lateralize with the side of the cued hand, and also on functional magnetic resonance imaging (fMRI) data, where the recovered components express previously reported resting-state networks.

1 Introduction

Unsupervised component analysis techniques such as Independent Components Analysis (ICA) and Canonical Correlation Analysis (CCA) are routinely employed for dimensionality reduction, blind source separation, and feature selection [1, 2]. In particular, these methods are often applied to data from complex systems such as the brain and financial markets, as their interpretability facilitates insight into the structure and dynamics of the system. Nevertheless, the sources underlying complex systems often violate the assumptions made by these classical tools. For example, the sources of activity in the brain are highly dependent due to functional connectivity, and their dynamics exhibit lagged (in addition to instantaneous) temporal correlations.

Granger causality [3] is a well-known technique for measuring a form of dependence rooted in the temporal precedence effect. Time series x ( t ) is said to 'Granger-cause' y ( t ) if the past of x improves the prediction of the present value of y above that of its own past. The technique has many generalizations [4, 5, 6, 7, 8, 9, 10, 11] and is conventionally utilized as a statistical test. One specifies the two time series being probed and their assumed directionality, and then carries out hypothesis testing on the measured relationship. The time series are typically defined in the sensor space, meaning that if the sources of interest are mixed in the observations, Granger causality may not optimally identify the true relationship. Relatedly, statistically independent sources that are linearly mixed at the sensors may spuriously indicate Granger causality [12, 13, 14].

Here the concept of Granger causality is employed to propose a new criterion for unsupervised learning that is appropriate in the case of temporally-dependent source signals. The basic idea is to identify two projections of a multivariate time series such that the Granger causality among the resulting pair of components is maximized. A coordinate descent algorithm that decomposes a multivariate dataset into pairs of components is proposed such that each pair consists of a 'driving'

(Granger-causing) and a 'driven' (Granger-caused) signal. To demonstrate proof-of-concept, the proposed method is tested on simulated data following the vector autoregressive (V AR) model that is frequently employed in studies of Granger causality [15]. Experiments on recordings from the human brain - previously collected scalp electroencephalogram (EEG) and functional magnetic resonance imaging (fMRI) datasets - are then performed. The technique is evaluated by analyzing and interpreting the recovered components, in particular their relation to what is known about the underlying systems from the human neuroscience literature.

The contributions of this work are:

A new approach to blind source separation that is appropriate when the desired signals exhibit temporal dependencies.

A grouped coordinate descent algorithm that learns pairs of signals with a Granger causal relationship and is shown to identify the underlying sources (up to scale) on simulated data.

Experimental findings from the human brain demonstrating that the technique recovers components whose structure is consistent with earlier findings from neuroscience.

2 Related work

The classical approach to inferring latent temporal relationships is CCA [16] and its extensions, for example kernel [17] and Deep CCA [18]. Both CCA and the proposed approach yield pairs of latent variables exhibiting temporal dependencies. In both cases, there is a natural ordering of the component pairs (i.e., magnitude of correlation in CCA, strength of Granger causality in the present work). The key differences between CCA and the proposed method are that: (i) whereas CCA operates on two pre-defined 'views' of the data, the proposed method learns latent relationships underlying a single multivariate record, and (ii) the proposed method extracts signals related by Granger causality as opposed to instantaneous correlation (Fig 1).

Multivariate Granger causality [9] is an approach that extends the measure of Granger causality to two sets of variables. As with CCA, the two sets (e.g. regions of interest in the human brain) are first defined, followed by a statistical procedure that assays the relationship between the multivariate records. Several previous works have combined Granger causality with CCA [19, 20, 21]. These techniques employ the CCA framework to eliminate the need for explicitly computing autoregressive models. In all cases, the two sets of variables (or views of the data) are first defined, followed by the assessment of Granger causality. Approaches that combine VAR modeling with PCA [22] and ICA [23] have also been proposed.

3 Problem formulation

The following linear signal model is assumed:

where x ( t ) ∈ R D is an observed, wide-sense stationary multivariate time series, A ∈ R D × K is a mixing matrix relating the latent sources s ( t ) = [ s 1 ( t ) . . . s K ( t ) ] ∈ R K to the observations, and e ( t ) ∈ R D is an additive error term representing sensor noise as well as deviations from the assumed signal model.

The latent sources are assumed to exhibit a temporal dependence according to:

where → denotes 'Granger-causality', meaning that the past of s I i ( t ) is linearly predictive of the present of s J i ( t ) beyond that which can be explained by the past of s J i ( t ) , and P is the number of temporally dependent latent signal pairs. The mappings I i and J i yield the indices of the driving and driven signals, respectively, of the i th pair of latent sources.

The goal is to recover pairs of signals y i ( t ) = w T i x ( t ) and z i ( t ) = v T i x ( t ) such that:

Below, an objective based on the Granger causality between pairs of component signals is proposed.

(a)

Canonical Correlation Analysis (CCA)

A

x

Wihlmm

(b) Granger Components Analysis (GCA)

MMM

X

4 Granger Components Analysis

Throughout, a history length of L samples is assumed. Denote ϵ z as the residual of a linear regression from z p ( t ) = [ z ( t -1) , . . . , z ( t -L )] onto z ( t ) , and ϵ zy as the residual of a regression from [ z p ( t ) , y p ( t )] onto z ( t ) , where y p ( t ) = [ y ( t -1) , . . . , y ( t -L )] . The goal is to identify latent component time series y ( t ) and z ( t ) such that the Granger causality from y to z is maximized:

The objective function (3) is closely related to the logarithm of the F-statistic between the 'full' and 'reduced' regression models inherent to Granger causality [15]. The smaller the ratio of residual powers, the more information is contained in the past of driving signal y about the present of driven signal z .

To relate the objective function to the coefficient vectors w and v , the mean residuals are written as (see Supplementary Material ):

where the following covariance vectors and covariance matrices have been defined:

and where ⊗ denotes the Kronecker product, 1 M is a column vector of M ones, and I M is the M -byM identity matrix. The LD -byLD block covariance matrices are given by:

where Σ ( τ ) = E { x ( t ) x T ( t -τ ) } is the lagged covariance of the observations.

The optimization problem (3) is non-convex because the objective function is invariant to scalings of w and v . Consequently, multiple pairs of coefficient vectors locally minimize the objective, and it is only possible to recover the latent sources up to a scaling factor.

A known property of Granger causality [9, 10] is that if y → z , then ay + bz → cz , where a , b , and c are arbitrary constants. This means that, without appropriate modifications to the objective, it is only possible to identify the driven signal z . To resolve this ambiguity, the concept of time-reversed Granger causality [24, 12] is utilized. Note that if y → z in forward time, then z → y in reversed time. Forward and reversed time are then combined into a single objective function according to:

where ˜ ϵ indicates that the mean residual is computed with time reversed.

5 Grouped coordinate descent algorithm

To minimize (4) with respect to the coefficient vectors w and v , a grouped coordinate descent algorithm that optimizes for v while holding w fixed and vice versa is proposed. This approach is commonly employed in problems where the independent variables partition naturally [25].

At each iteration of the algorithm, non-linear constraints of the form ∥ v ∥ = 1 (or ∥ w ∥ = 1 ) are added to yield unit-norm solutions. The resulting constrained optimization problem may be solved with conventional interior-point or sequential quadratic programming solvers. During development, MATLAB's 'fmincon' and Python's 'scipy.optimize' were both tested. An expression for the gradient of the objective function (4) with respect to v and w is derived in the Supplementary Material . However, it was found that the use of numerical differentiation reduced computational time and was thus employed for all empirical analyses.

As the cost function is non-convex, there are potentially several pairs of components that yield meaningful latent sources. In order to recover P pairs of components { y i ( t ) , z i ( t ) } , i = 1 , . . . , P , here it is proposed to repeat the optimization after the first iterate, but not before removing the contribution of the driving signal y 1 ( t ) from the data. This takes the form of a spatiotemporal regression such that any signals that are correlated with y 1 ( t ) or its lagged versions y 1 ( t -l ) , l = 1 , . . . , L are removed. This procedure is repeated until the desired number of component pairs P is obtained.

The proposed algorithm is described in Algorithm 1, where X is a D -byT matrix storing the observed data, Y p is an L -byT convolution matrix allowing the regression of y p ( t ) onto X , convmatrix is an operation that produces a convolution matrix, # denotes the Moore-Penrose pseudoinverse, and σ is a small positive constant that is used to randomly initialize the weights of the coefficient vectors.

6 Experiments

Illustrative example: Gaussian VAR system

As a first step, proof-of-concept simulations were performed to evaluate the method's ability to recover the latent structure embedded in mixed time series. A D = 10 -dimensional time series with K = 5 latent sources whose activity was linearly mixed in the observations (Fig 2a) was generated. The innovation process was distributed according to N ( 0 , I ) . White Gaussian noise with σ = 0 . 1 was also added to the sensors. The elements of the mixing matrix were drawn from the uniform distribution over [0 , 1] . Importantly, three of the sources exhibited temporal dependencies, with s 1 → s 2 and s 2 → s 3 , while s 4 and s 5 were independent from all other sources. For convenience, s 1 -s 3 are referred to as 'connected' and s 4 -s 5 as 'disconnected' sources. The dynamics of the connected sources followed a VAR(3) system with coefficients taken from Stokes and Purdon [26], a study that demonstrated the shortcomings of conventional Granger causality. The disconnected sources also followed a VAR(3) model but with the off-diagonal elements set to 0 (no cross-correlation with other sources). 100 runs of 5000 samples were simulated, and P = 3 pairs of Granger components were estimated.

The value of the objective function after convergence J ∗ is depicted in the vertical axis of Fig 2b (error bars indicate standard errors of the mean across 100 replicates). The true value of the objective (obtained by measuring the Granger causality between s 1 and s 2 and between s 2 and s 3 ) is indicated in red and blue markers for s 1 → s 2 and s 2 → s 3 , respectively. It is apparent that the method recovered latent dependencies whose magnitude matched the true strength of causality. As expected, the Granger causality between y 3 and z 3 was near zero. It was then evaluated whether the time series recovered in the first two Granger pairs ( y 1 , z 1 ) and ( y 2 , z 2 ) corresponded to those of the latent dependent sources ( s 1 , s 2 ) and ( s 2 , s 3 ) , respectively. Indeed, GCA recovered the dynamics of the connected sources with high reliability (mean r 2 > 0 . 96 across 100 replicates, Fig 2d-g). Note that both the s 1 → s 2 and s 2 → s 3 relationships were identified, with s 2 captured by both y 2 and z 1 .

As a comparison to classic techniques, PCA and ICA were also evaluated on this simulated dataset. In both cases, a significantly lower degree of fidelity (approximately 50% and 30% variance explained for PCA and ICA, respectively) was observed in the recovered signals with respect to ground-truth (Fig S1 in Supplementary Material ).

Control experiment: independent sources

As a null control, simulations with statistically independent sources ( s 1 ⊥ ⊥ s 2 ⊥ ⊥ s 3 ) were also conducted. The objective function values of the first three Granger pairs are shown in Fig 2c, where it is apparent that the algorithm did not recover meaningful temporal dependencies. As depicted in Fig 2h-k, the dynamics of the independent latent sources could not be recovered.

EEG during motor imagery

Next, the evaluation of GCA on real-world data was undertaken, concentrating on systems with a partially understood underlying structure, ensuring that the results could be interpreted meaningfully. The technique was first applied to publicly available EEG data recorded from n = 52 human participants performing a motor imagery task [27]. Subjects were asked to imagine moving either their left or right hand, and three seconds of brain activity following the cue were recorded. To recover components representative of the cohort, a 'group GCA' was performed where the covariance matrices Σ 1: L and ˜ Σ were aggregated across subjects prior to minimizing (4). Tikhonov regularization that reduced the condition number of the block covariance matrices was applied (see Supplementary Material for details), and the temporal aperture was set to L = 16 samples (0.5 seconds). It was asked whether GCA would recover signals exhibiting known properties of the motor imagery circuitry: (i) contralateralization of brain activity with respect to the cued hand, and (ii) the flow of information from pre-motor (planning) to motor (execution) regions [28].

P = 2 pairs of driving and driven signals were computed for each experimental condition: 'Left motor imagery' and 'Right motor imagery' (Fig 3). To identify the spatial distribution of the driving and driven signals, the 'forward models' corresponding to each component were computed [29]. In panels a-d, the spatial topographies of the driving and driven signals are shown on the top left and top right, respectively. The time courses of driving and driven signals are depicted below the scalp maps.

(a)

(b)

(c)

S,IS,S,

Si

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$→→S2→S

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S→S→→S

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For both GC1 and GC2, the components recovered by GCA for left motor imagery were reflected around the midline relative to right motor imagery. The driven components of GC1 and GC2 exhibited asymmetric spatial topographies with activation over frontocentral and central electrodes (Fig 3a-d, top right). This pattern is consistent with activation of the motor cortex in one hemisphere. On the other hand, the spatial topographies of the driving components were more symmetric, with activation over prefrontal and frontal electrodes (Fig 3a-d, top left), particularly for GC1. This is consistent with the location of anticipatory regions such as the prefrontal cortex and anterior cingulate [30, 31], as well as premotor regions such as the supplementary motor area (SMA) [32].

Interestingly, the components recovered by GC1 were ipsilateral to the cued hand, while those recovered in GC2 exhibited the classical contralateralization of the motor cortex. This ordering may reflect the inhibition of the uncued hand that is required by the task, as well as the desynchronization of EEG that is observed during motor activation [33]. For example, the time courses of the components

(a)

Left motor imagery

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Right motor imagery

GC1

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in GC1 exhibited a faster initial oscillation than those in GC2 (Fig 3a-d, bottom panels). From visual inspection of the time courses, the driving signal in each component pair appears to lead the driven signal. Note that while the spatial topographies are flipped with respect to the cued hand, the time courses are qualitatively similar between left and right motor imagery.

The power spectra of the recovered components, as well as those of several individual electrodes, are depicted in Figure S2. Of note, it is interesting that the driving signal of GC2 has relatively large 10 Hz 'mu' power [33], whereas the corresponding driven signal has relatively low 10 Hz power. This appears to be consistent with the driving signal being preparatory (10 Hz activity is synchronized) and the driven signal reflecting execution (10 Hz desynchronization).

The magnitude of Granger causality between all pairs of electrodes (represented here as J so that positive values indicate a strong relationship) is depicted in Fig 3e for the right motor imagery condition (see Fig S3 for the results from left motor imagery). The single strongest relationship is 0.12, and it is difficult to interpret the overall pattern of Granger causalities. The matrix of Granger causality strengths for the recovered components is shown in Fig 3f. In addition to larger magnitudes compared to those observed between individual electrodes (as high as 0.39 for y 1 → z 1 ), the bottom row of the matrix suggests that y 2 drives both z 1 and z 2 . Given the left frontal topography of y 2 and the more central topographies of z 1 (ipsilateral to cued hand) and z 2 (contralateral to cued hand), this is suggestive of an EEG source that projects to both the activated and inhibited sources.

For comparison to existing approaches, the spatial topographies of the components recovered by PCA, ICA, and MVARICA [23] are depicted in Fig S4. The components recovered by the proposed method were not evident in those yielded by these conventional approaches (compare Fig 3 with Fig S4). Moreover, it is notable that the lateralization of the forward models with respect to the side of the cued hand is not readily apparent in Fig S4.

Resting-state fMRI

At rest, the human brain exhibits stereotyped patterns of activity in the form of resting-state networks (RSN) [34, 35, 36]. Perhaps the most well-known of these is the default-mode network (DMN) [37], a collection of brain regions that appears to become active during introspection and mind-wandering. A previously collected fMRI dataset [38] from n = 20 healthy adults in the resting state (eyes open) was employed here to investigate whether GCA could recover components (i.e., sets of brain regions) that reflect the presence of these previously reported networks. As the original study investigated the effects of near-infrared brain stimulation on fMRI activity, only the first 10 minutes of the 30 minute recording (i.e., the segment occurring before the onset of stimulation) was analyzed here. The input data was parcellated into D = 150 regions-of-interest (ROIs) following a standard brain atlas [39]. The time series of each ROI was formed by computing the mean of all grey matter voxels in the ROI at each time sample. The temporal aperture was set to L = 4 samples ( ≈ 11 s) and the number of components to P = 6 . Block covariance matrices were pooled across participants prior to analysis, leading to a group analysis. The recovered components were then analyzed by examining the individual brain regions most strongly expressed by the forward model [29] of each driving and driven component.

The five regions with strongest expression in the forward model of each driving and driven component are listed in Table 1. Interestingly, the ROI most strongly expressed in the driving signal of GC1 was the right angular gyrus, one of the three hubs of the DMN [37] (Fig 4a, top row). The left angular gyrus was the fourth strongest driving signal in GC1. The driven regions of GC1 were bilateral visual areas and included the primary visual cortex (Fig 4a, bottom row). The finding of the DMN and visual networks in the first Granger pair is consistent with this eyes-open, resting dataset [40].

The remaining Granger components tended to cluster into networks that have also been previously reported in the literature. For example, the driving signal of GC2 consisted of the precuneus and posterior cingulate, the other two hubs of the DMN (the pericallosal sulcus borders the posterior cingulate; Fig 4b, top). The driven portion of GC2 was comprised of a contiguous region in the parietal cortex (Fig 4b, bottom), an area belonging to the 'sensorimotor' network [40].

The analysis also identified a network in the right frontal region that appeared in the driving portion of GC3 as well as the driven portion of GC4 (Fig 4c, top and Fig 4d, bottom). A set of ROIs in the left temporal cortex was identified in the driving signal of GC4, while the expression of the insular

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cortex and anterior cingulate in driving GC6 is consistent with another well-known RSN known as the 'salience network' [41].

Overall, GCA recovered components whose spatial distribution is consistent with multiple RSNs previously reported in the fMRI literature. Note that the GCs measured here tended to capture RSNs that interact with one another, as opposed to interacting brain regions from the same RSN.

7 Limitations

There are several limitations of the present work. The proposed method is limited to capturing components that are expressed linearly in the observations. In the high-dimensional case (large LD ), estimation of the block covariance matrices Σ 1: L and ˜ Σ is challenging, particularly in the case of low sample sizes. Regularization approaches more sophisticated than the Tikhonov approach employed here may alleviate this challenge [42].

Although the components recovered by GCA are ordered according to the strength of the Granger causal relationship, this work has not proposed a methodology for selecting the number of components P to estimate (or retain). One post hoc approach is to perform statistical significance testing on the recovered Granger component pairs to assess the likelihood of a non-spurious relationship. This may be accomplished in a non-parametric fashion by employing the method on surrogate data lacking any Granger causal structure, for example by randomizing the phase spectrum of the observed signals [43]. Furthermore, this work has not considered a methodology for selection of the autoregressive

model order L . Knowledge of the underlying systems was utilized in the two real-world examples here to select the history lengths (i.e., 500 ms for EEG and 4 samples for fMRI).

In defining the GCA objective (3), it has been assumed that the Granger causal relationships between latent variables are unidirectional: if y Granger-causes z , then z does not Granger-cause y . An extension of GCA that instead maximizes net Granger causality [12] and accounts for bidirectional relationships may facilitate the recovery of pairs of sources that exhibit temporal dependencies in both directions.

8 Conclusion

This work has proposed an approach to blind source separation that is applicable when the underlying sources are related by a Granger causal relationship (i.e., the past of one signal forecasts the present of another). In contrast to conventional approaches that require statistical independence for identification, the present approach exploits temporal dependencies to recover pairs of components with a directed relationship. A coordinate descent algorithm that decomposes a multivariate dataset into pairs of components, ranked by the strength of the temporal dependency, was then developed and evaluated on simulated and real datasets. Future efforts may extend this line of work to permit non-iterative solutions to the linear GCA problem, or propose modifications that allow one to recover nonlinear temporal dependencies.

The code that was employed to generate the experimental results is available at https://github. com/dmochow/gca .

Acknowledgments

The author would like to thank the anonymous Reviewers for providing many useful comments and critiques that helped to improve the quality of the paper.

References

David R Hardoon, Sandor Szedmak, and John Shawe-Taylor. Canonical correlation analysis: An overview with application to learning methods. Neural computation , 16(12):2639-2664, 2004.

Aapo Hyvärinen. Independent component analysis: recent advances. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences , 371(1984):20110534, 2013.

Clive WJ Granger. Investigating causal relations by econometric models and cross-spectral methods. Econometrica: journal of the Econometric Society , pages 424-438, 1969.

John Geweke. Measurement of linear dependence and feedback between multiple time series. Journal of the American statistical association , 77(378):304-313, 1982.

John F Geweke. Measures of conditional linear dependence and feedback between time series. Journal of the American Statistical Association , 79(388):907-915, 1984.

Craig Hiemstra and Jonathan D Jones. Testing for linear and nonlinear granger causality in the stock price-volume relation. The Journal of Finance , 49(5):1639-1664, 1994.

Nicola Ancona, Daniele Marinazzo, and Sebastiano Stramaglia. Radial basis function approach to nonlinear granger causality of time series. Physical Review E , 70(5):056221, 2004.

Daniele Marinazzo, Mario Pellicoro, and Sebastiano Stramaglia. Kernel method for nonlinear granger causality. Physical review letters , 100(14):144103, 2008.

Adam B Barrett, Lionel Barnett, and Anil K Seth. Multivariate granger causality and generalized variance. Physical Review E , 81(4):041907, 2010.

Lionel Barnett and Anil K Seth. The mvgc multivariate granger causality toolbox: a new approach to granger-causal inference. Journal of neuroscience methods , 223:50-68, 2014.

Alex Tank, Ian Covert, Nicholas Foti, Ali Shojaie, and Emily Fox. Neural granger causality. arXiv preprint arXiv:1802.05842 , 2018.

Irene Winkler, Danny Panknin, Daniel Bartz, Klaus-Robert Müller, and Stefan Haufe. Validity of time reversal for testing granger causality. IEEE Transactions on Signal Processing , 64(11):2746-2760, 2016.

Clemens Brunner, Martin Billinger, Martin Seeber, Timothy R Mullen, and Scott Makeig. Volume conduction influences scalp-based connectivity estimates. Frontiers in computational neuroscience , 10:121, 2016.

Frederik Van de Steen, Luca Faes, Esin Karahan, Jitkomut Songsiri, Pedro A Valdes-Sosa, and Daniele Marinazzo. Critical comments on eeg sensor space dynamical connectivity analysis. Brain topography , 32:643-654, 2019.

Helmut Lütkepohl. New introduction to multiple time series analysis . Springer Science & Business Media, 2005.

Harold Hotelling. Relations between two sets of variates. In Breakthroughs in statistics , pages 162-190. Springer, 1992.

Shotaro Akaho. A kernel method for canonical correlation analysis. arXiv preprint cs/0609071 , 2006.

Galen Andrew, Raman Arora, Jeff Bilmes, and Karen Livescu. Deep canonical correlation analysis. In International conference on machine learning , pages 1247-1255. PMLR, 2013.

Pieter W Otter. On wiener-granger causality, information and canonical correlation. Economics Letters , 35(2):187-191, 1991.

João R Sato, André Fujita, Elisson F Cardoso, Carlos E Thomaz, Michael J Brammer, and Edson Amaro Jr. Analyzing the connectivity between regions of interest: an approach based on cluster granger causality for fmri data analysis. Neuroimage , 52(4):1444-1455, 2010.

Guorong Wu, Xujun Duan, Wei Liao, Qing Gao, and Huafu Chen. Kernel canonical-correlation granger causality for multiple time series. Physical Review E , 83(4):041921, 2011.

MA Repucci, ND Schiff, and JD Victor. General strategy for hierarchical decomposition of multivariate time series: implications for temporal lobe seizures. Annals of Biomedical Engineering , 29(12):1135, 2001.

Germán Gómez-Herrero, Mercedes Atienza, Karen Egiazarian, and Jose L Cantero. Measuring directional coupling between eeg sources. Neuroimage , 43(3):497-508, 2008.

Stefan Haufe, Vadim V Nikulin, Klaus-Robert Müller, and Guido Nolte. A critical assessment of connectivity measures for eeg data: a simulation study. Neuroimage , 64:120-133, 2013.

JC Bezdek, RJ Hathaway, RE Howard, CA Wilson, and MP Windham. Local convergence analysis of a grouped variable version of coordinate descent. Journal of Optimization Theory and Applications , 54:471-477, 1987.

Patrick A Stokes and Patrick L Purdon. A study of problems encountered in granger causality analysis from a neuroscience perspective. Proceedings of the national academy of sciences , 114(34):E7063-E7072, 2017.

Hohyun Cho, Minkyu Ahn, Sangtae Ahn, Moonyoung Kwon, and Sung Chan Jun. Eeg datasets for motor imagery brain-computer interface. GigaScience , 6(7):gix034, 2017.

Martin Lotze and Ulrike Halsband. Motor imagery. Journal of Physiology-paris , 99(4-6):386395, 2006.

Stefan Haufe, Frank Meinecke, Kai Görgen, Sven Dähne, John-Dylan Haynes, Benjamin Blankertz, and Felix Bießmann. On the interpretation of weight vectors of linear models in multivariate neuroimaging. Neuroimage , 87:96-110, 2014.

Hualou Liang, Steven L Bressler, Mingzhou Ding, Wilson A Truccolo, and Richard Nakamura. Synchronized activity in prefrontal cortex during anticipation of visuomotor processing. Neuroreport , 13(16):2011-2015, 2002.

Myeong-Ho Sohn, Mark V Albert, Kwanjin Jung, Cameron S Carter, and John R Anderson. Anticipation of conflict monitoring in the anterior cingulate cortex and the prefrontal cortex. Proceedings of the National Academy of Sciences , 104(25):10330-10334, 2007.

Ignazio Puzzo, Nicholas R Cooper, Petra Vetter, and Riccardo Russo. Eeg activation differences in the pre-motor cortex and supplementary motor area between normal individuals with high and low traits of autism. Brain research , 1342:104-110, 2010.

Gert Pfurtscheller, Clemens Brunner, Alois Schlögl, and FH Lopes Da Silva. Mu rhythm (de) synchronization and eeg single-trial classification of different motor imagery tasks. NeuroImage , 31(1):153-159, 2006.

Bharat Biswal, F Zerrin Yetkin, Victor M Haughton, and James S Hyde. Functional connectivity in the motor cortex of resting human brain using echo-planar mri. Magnetic resonance in medicine , 34(4):537-541, 1995.

Michael D Greicius, Ben Krasnow, Allan L Reiss, and Vinod Menon. Functional connectivity in the resting brain: a network analysis of the default mode hypothesis. Proceedings of the national academy of sciences , 100(1):253-258, 2003.

Michael D Fox, Maurizio Corbetta, Abraham Z Snyder, Justin L Vincent, and Marcus E Raichle. Spontaneous neuronal activity distinguishes human dorsal and ventral attention systems. Proceedings of the National Academy of Sciences , 103(26):10046-10051, 2006.

Marcus E Raichle and Abraham Z Snyder. A default mode of brain function: a brief history of an evolving idea. Neuroimage , 37(4):1083-1090, 2007.

Grzegorz M Dmochowski, Ahmed Duke Shereen, Destiny Berisha, and Jacek P Dmochowski. Near-infrared light increases functional connectivity with a non-thermal mechanism. Cerebral Cortex Communications , 1(1):tgaa004, 2020.

Christophe Destrieux, Bruce Fischl, Anders Dale, and Eric Halgren. Automatic parcellation of human cortical gyri and sulci using standard anatomical nomenclature. Neuroimage , 53(1):1-15, 2010.

Malaak N Moussa, Matthew R Steen, Paul J Laurienti, and Satoru Hayasaka. Consistency of network modules in resting-state fmri connectome data. 2012.

Vinod Menon and Lucina Q Uddin. Saliency, switching, attention and control: a network model of insula function. Brain structure and function , 214:655-667, 2010.

Karl Werner, Magnus Jansson, and Petre Stoica. On estimation of covariance matrices with kronecker product structure. IEEE Transactions on Signal Processing , 56(2):478-491, 2008.

James Theiler, Stephen Eubank, André Longtin, Bryan Galdrikian, and J Doyne Farmer. Testing for nonlinearity in time series: the method of surrogate data. Physica D: Nonlinear Phenomena , 58(1-4):77-94, 1992.

Peer reviews:
comments: SUMMARY:
This paper presented a novel unsupervised learning approach using Granger Causality by identifying the driving/driven components. This method was demonstrated on EEG and fMRI data, in coincide with the neurophysiological facts.

STRENGTHS:
The paper proposed an algorithm to identify the pairwise causal structure between latent variables from a multivariate observational data set and the results are supprted by simulation data and empirical data.

WEAKNESSES:
1)  The number of latent variables is a key parameter for this unsupervised learning, however it is pre-defined without any adaptive mechanism. similarly, no adaptive mechanism for L. 
2) The literature is outdated and there is no performance comparison against similar algorithms for causal inference. PCA and ICA are not algorithms for causal inference.

QUESTIONS:
No further questions.

LIMITATIONS:
The simulation is limited to a specific model without variations, for example, the noise level, the model complexity, the length of the time series, etc. Makes it is difficult to assess the performance and the generaslibility of the proposed algorithm.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-05

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper formulates the problem of learning a pair of spatial projections that optimize a criterion based on the Granger causality between the resulting components, which is itself based on regressing the first, driving, component to the second, driven, component using a Wiener filter and the converse for the time-reversed signals. A block coordinate descent algorithm is proposed to solve for one spatial projection while the other is fixed and vice versa. Experimental results on EEG and resting state fMRI data illustrate the identification of meaningful spatial filters.\n\nSTRENGTHS:\nThe paper is well motivated, well written, and clear.\n\nSynthetic experiments are simple but results are convincing. \n\nThe method is applied to two different modalities of neuroimaging data and paradigms (EEG during motor imagery tasks and resting state fMRI) and the results are discussed in depth.\n\nWEAKNESSES:\nExplicit statement of the assumptions about the nature of the relationship (linear or non-linear), time-invariant, etc. is lacking. Standard Granger causality assumes a linear, time-invariant relationship. These assumptions should be stated when introducing the pairs in (2). \n  \nThe algorithm could be made more succinct by removing some of the redundancies. \n\nMinor: \nLine 79 should be clarified that $\\mathbf{y}_p(t)$ is also lagged form. \n\nLine 93 variables should be defined. I assume that there exists scalar $c$ for any choice of $a$ and $b$. \n\nOn line 114, the statement 'the driven signal is not explicitly removed.' is a bit misleading. While it is not removed the explainable variance associated this component is removed by the filtering (spatiotemporal regression).\n\nIn algorithm 1, the dependency between the cost function $J$ and $\\mathbf{X}$ is not explicit. \n\n*Nit picks:* I find the capitalization of methods beyond proper nouns a bit jarring. \"Granger Causlity\" -> \"Granger causality\". \"Kernel CCA\" -> \"kernel CCA\". \n\nLines 70, 81 closing double quotes are wrong direction.\n\nLines 132–133 and 160–161 \"We asked GCA\" seems odd phrasing.\n\nQUESTIONS:\nIs automatic differentiation an option over the numerical differentiation or the manually coded derivative? \n\nHow would the method perform on synthetic data where the sources are instantaneously mixed only (like the ideal case for ICA)? \n\nWas the use of subject versus group explored? Or was it a problem of lack of data? \n\nLine 194: Is each ROI summarized by its mean or principal component? This is not mentioned.\n\nLIMITATIONS:\nA stopping criterion for estimating the number of sources is missing.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a novel (blind) source separation method that extracts pairs of components from multivariate time series between which Granger causality (GC) is maximal. This can be a very useful tool to assess direction information flow between brain areas in an unsupervised way, without having to specify the areas a-priori. The method is very elegant and implemented in a straightforward way by setting up a corresponding optimization problem and solving it via block-coordinate descent alternating between updates of the projection vectors for the sending and receiving source. Analytic gradients are provided but also autograd is reported to work well. An interesting deflation scheme is also provided whereby the sending source is projected out in each step. Thus, the same sending source cannot be found in multiple GC component pairs, but the (residual) of a sending source can play a role as either sender of receiver in a subsequently extracted pair. The method also implements time-reversal, a method to robustify GC estimates with respect to artifacts of volume conduction. A small set of simulations illustrates the convincing properties of the method, and a convincing application to motor-imagery brain-computer interface data is also provided.\n\nSTRENGTHS:\nThe paper proposes an elegant and potentially useful method. The derivation of the method is easy to follow, apart from minor gaps. The technical parts are sound. The simulations and real data results are convincing and provide a good picture of the capabilities of the method. The writing is clear. Overall, a nice and self-contained paper.\n\nWEAKNESSES:\nThe simulations and real data analyses could be better developed. The simulations could be more quantitative, e.g. studying 100 systems instead of only one, and reporting distribution of reconstruction metrics. The impact of factors such as the SNR could be systematically studied, and different types of noise could be studied. More methods could be included in the empirical comparisons. For example, BSS methods like MVARICA [1] and SCSA [2] do not assume independent components but model the sources exactly by an MVAR model, from which GC between every pair of components can be assessed. In the BCI context, the extraction of class-specific sources using CSP [3] or SMR oscillations using SSD [4] could be compared to the proposed method. Although I understand that not for all methods working code may be found. \nTheoretically, it would be critical to also discuss the identifiability of the model. Linear mixtures of MVAR processes are again MVAR models and a valid question is why the maximization of GC should provide the “true” unmixing. This is especially critical as research has shown that even a mixing of independent sources can induce spurious GC [5-9]. Moreover, no non-Gaussianity of residuals as in [2] is assumed to guide the reconstruction.\n\n[1] https://www.sciencedirect.com/science/article/abs/pii/S1053811908008549\n[2] https://ieeexplore.ieee.org/abstract/document/5466024\n[3] https://ieeexplore.ieee.org/abstract/document/4408441\n[4] https://www.sciencedirect.com/science/article/abs/pii/S1053811914005503\n[5] https://www.sciencedirect.com/science/article/abs/pii/S1053811912009469\n[6] https://www.sciencedirect.com/science/article/abs/pii/S105381191401009X\n[7] https://ieeexplore.ieee.org/abstract/document/7412766\n[8] https://link.springer.com/article/10.1007/s10548-016-0538-7\n[9] https://www.frontiersin.org/articles/10.3389/fncom.2016.00121/full\n\nQUESTIONS:\n-\tIt would be interesting if the authors could provide more information about the extracted sources in the motor imagery real data example. Could you please plot power spectra of the sources compared to some of the relevant channels over the motor areas. Similarly, it would be interesting to see the GC or TRGC connectivity spectra of the extracted component look like in comparison to the power spectrum. For this, spectral G-causality as implemented in the MVGC toolbox could be used. Finally, the corresponding forward models could be mapped into the brain using some inverse modeling techniques, which could give further insight into the physiological relevance of the extracted sources. [10] analyze functional connectivity before and during motor imagery using undirected FC metrics, and it could be interesting to see if the sources are similar. \n-\tSome of the derivations are not immediately clear to me. The step from Eqs. (3) and (5) to (6) is not completely clear. Perhaps a few intermediate steps could be added? When expanding the square of the residual, should there be no mixed terms (products of z and z_p)? \n-\tz_p and y_p should be defined. These are the temporally embedded multivariate versions of z and  y. In Eqs (3) and (4) these should still depend on (t)\n-\tline 93: this is not the only possibly undesirable case, one can also show that spurious GC and emerge from mixtures of independent sources. E.g. x_1 = s + n_1 and x_2 = s + n_2 for independent noises n_1 and n_2 can lead to GC between x_1 and x_2 in either direction.\n-\tLine 95 (“note that”): this is not true in that strict sense. What can be shown is that the flow from y -> z is reduced compared to the original temporal order, but there is no guarantee that it would vanish or even reverse (see [7])\n-\tHowever, it is true that, for a given unidirectional flow x -> y, subtracting the time reversed GC affects both the direction x -> y and y -> x (essentially adding a negative term for the direction y -> x), it makes sense to not only consider (and optimize) time reversed differences GCR_(x->y) = GC_(x->y) – GC_(x_rev -> y_rev) but also to directly work with the net GC between both directions: TRGC_(x->y) = GCR_(x->y) - GCR_(y->x).\n-\tSupplement “The forward model expresses the level of correlation”: technically it is pretty much the covariance.\n- No reference for the origin of the fMRI dataset is given and no IRB approval for that study is mentioned.\n\nLIMITATIONS:\nThe paper is overall solid; however, a theoretical discussion of the model identifiability would be indicated, also considering the proposed deflation scheme. The simulations and real data analyses could be extended.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a factorization model to extract P pairs of latent components from a multivariate time series such that in each pair one of the time series Granger causes the other. The authors apply the approach in analysing EEG and fMRI data to show meaningful conclusion.\n\nSTRENGTHS:\nThe paper addresses an interesting problem of finding latent variables with Granger causal structure from a multivariate time series. The paper is generally well written.\n\nWEAKNESSES:\nAlthough the paper proposes an intriguing approach for finding latent causal structure, in general, it might be limiting since it only considers structure with pairwise time series demonstrating causal influence on each other while in practice we would expect multiple time series potentially affecting each other. Can the authors elaborate the effectiveness of the assumed latent structure a bit more?\n\nThe simulated data does not provide complete insight of the performance of the method. It might be useful to explore more realistic situation such as more time series, more intricate (conditional) causal influences among multiple time series, and/or situations where the proposed model might fail to capture causality, e.g., one time series driving more than one time series. This might be helpful in better assessing any false positive detections and false negative misses.\n\nIt will be useful to compare the proposed method to standard conditional Granger causality on the real dataset to extract the underlying causal structure and assess if it is similar to the one inferred by the proposed approach.\n\nQUESTIONS:\n- how are the maps in Figure 3 and 4 generated, using w and v vectors or using A matrix?\n- line 114 and line 142: if y_1 and z_1 are being explicitly removed, then how can s_2 -> s_3 be identified after s_1 -> s_2? can you please elaborate?\n\nLIMITATIONS:\nThe authors address limitation of this approach, e.g., in the context of driving time series driving only a single driven time serie\ns.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-09"}