Influence analysis for highdimensional time series with an application to epileptic seizure onset zone detection.  
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PMID: 23354014 Owner: NLM Status: Publisher 
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Granger causality is a useful concept for studying causal relations in networks. However, numerical problems occur when applying the corresponding methodology to highdimensional time series showing comovement, e.g. EEG recordings or economic data. In order to deal with these shortcomings, we propose a novel method for the causal analysis of such multivariate time series based on Granger causality and factor models. We present the theoretical background, succesfully assess our methodology with the help of simulated data and show a potential application in EEG analysis of epileptic seizures. 
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Christoph Flamm; Andreas Graef; Susanne Pirker; Christoph Baumgartner; Manfred Deistler 
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Type: JOURNAL ARTICLE Date: 2013124 
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Title: Journal of neuroscience methods Volume:  ISSN: 1872678X ISO Abbreviation: J. Neurosci. Methods Publication Date: 2013 Jan 
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Created Date: 2013128 Completed Date:  Revised Date:  
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Copyright © 2012. Published by Elsevier B.V. 
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Institute for Mathematical Methods in Economics, Vienna University of Technology, Vienna, Austria. Electronic address: christoph.flamm@tuwien.ac.at. 
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Journal Information Journal ID (nlmta): J Neurosci Methods Journal ID (isoabbrev): J. Neurosci. Methods ISSN: 01650270 ISSN: 1872678X Publisher: Elsevier/NorthHolland Biomedical Press 
Article Information © 2013 Elsevier B.V. License: Received Day: 9 Month: 8 Year: 2012 Revision Received Day: 27 Month: 12 Year: 2012 Accepted Day: 29 Month: 12 Year: 2012 pmcrelease publication date: Day: 30 Month: 3 Year: 2013 Print publication date: Day: 30 Month: 3 Year: 2013 Volume: 214 Issue: 1 First Page: 80 Last Page: 90 PubMed Id: 23354014 ID: 3719213 Publisher Id: NSM6534 DOI: 10.1016/j.jneumeth.2012.12.025 
Influence analysis for highdimensional time series with an application to epileptic seizure onset zone detection  
Christoph Flamma⁎  Email: christoph.flamm@tuwien.ac.at 
Andreas Graefa  
Susanne Pirkerb  
Christoph Baumgartnerb  
Manfred Deistlera  
aInstitute for Mathematical Methods in Economics, Vienna University of Technology, Vienna, Austria 

bGeneral Hospital Hietzing with Neurological Center Rosenhügel and Karl Landsteiner Institute for Clinical Epilepsy Research and Cognitive Neurology, Vienna, Austria 

⁎Corresponding author at: Institute for Mathematical Methods in Economics, Vienna University of Technology, Argentinierstraße 8/2/1052, 1040 Vienna, Austria. Tel.: +43 1 58801 105273. christoph.flamm@tuwien.ac.at 
In many cases the problem of identification of the dependence structure in multivariate time series arises. This is important, for example, in biology and economics, and in particular for neuroscience data, e.g. electroencephalographic (EEG) data, where the connections between brain regions are analyzed.
Investigations of this kind have been conducted in several ways, which include (^{Formisano et al., 2008; Astolfi et al., 2005; Pereda et al., 2005; Möller et al., 2001; Cassidy and Brown, 2002; Gates et al., 2010}).
The focus of this paper will be on the detection of Granger causality in multivariate time series which show strong comovement, i.e. high correlation between the componentseries. In ^{Granger (1969)} the causality between two time series is analyzed. The idea of this causality concept is based on predictability: if the knowledge of the past of one time series improves the prediction of a second one, the first is said to be Granger causal for the second. Note that this specific definition is just one possible way among many of defining causality. We refer the interested reader to ^{Bressler and Seth (2011)} for background information on Granger causality and to ^{Pearl (2000)} for a historical overview of causality.
Multivariate extensions of this causality concept have been developed, for recent references see e.g. ^{Lütkepohl (2007)} or ^{Eichler (2007)}. Besides, various topics of Granger causality have been discussed, see e.g. ^{Sims (1972), Geweke (1982), Dhamala et al. (2008), Barnett and Seth (2011)} and ^{Marinazzo et al. (2011)}. For recent applications in neuroscience see e.g. ^{Guo et al. (2008), Liao et al. (2010), Sommerlade et al. (2012)} and ^{Flamm et al. (2012)}.
In practice we often encounter highdimensional time series, which show comovement. As this comovement normally generates numerical problems, the question arises how to investigate the causality structure of these data.
For the analysis of highly correlated data, such as EEG data, factor models are a useful tool, see e.g. ^{Molenaar and Nesselroade (2001)} and ^{Molenaar (1985)}. The idea behind factor models is the separation of the observations into latent variables (describing the comovement) and noise. The latent variables are described by a small number of factors. In this modeling approach, up to now causality was not considered.
In this paper we propose a methodology for the causal analysis of highdimensional comoving data, by combining factor models and Granger causality analysis. We will present the theoretical background as well as an application to simulated data and to EEG recordings.
This paper is structured as follows: in order to get a grasp of Granger causality and factor models, we give a short introduction to both topics in the remainder of this section. In Section 2 we apply Granger causality to factor models and discuss the challenges arising. In Section 3 we propose a methodology for this kind of analysis. We apply this methodology to simulated and EEG data and present the results in Section 4. This paper is concluded in Section 5.
In this paper we distinguish between two different types of processes. In the classic Granger causal analysis we investigate ndimensional stochastic processes (y(t))t∈ℤ generated by n components, as discussed in this section. In the factor model case we analyze ndimensional stochastic processes (x(t))t∈ℤ, whose latent variable process is generated by a small number of components, as discussed in Section 1.5. For notational purposes we simply write y when referring to the whole stochastic process (y(t))t∈ℤ, this also applies for all other processes.
For the classic Granger causal analysis, we consider an ndimensional stochastic process (y(t))t∈ℤ, y(t):Ω→ℝn, which is weakly stationary with mean zero. We refer to ^{Hannan and Deistler (2012)} and ^{Brockwell and Davis (1991)} for treatment of time series.
In this paper we only consider linearly regular processes, which admit a Wold representation, see ^{Rozanov (1967)} and ^{Hannan (1970)}. The covariance function of y is given by γ(s)=Ey(t+s)y(t)′. For the remainder of the paper we assume, that ∑∥ γ(s) ∥ < ∞ holds and that the spectral density
(1)
f(λ)=12π∑s=−∞∞γ(s)e−iλs 
(2)
c−1In≤fyy(λ)≤cInforallλ∈[−π,π] 
(3)
∑m=0∞A(m)y(t−m)=ɛ(t) 
We use z to denote the backshift operator on ℤ: z(y(t)t∈ℤ)=(y(t−1)t∈ℤ), as well as a complex variable. Using this notation we may rewrite Eq. (3) as
(4)
a(z)y(t)=ɛ(t), 
We additionally assume that the stability condition det a(z) ≠ 0 for z ≤ 1 holds.
By using a˜(z)=−∑m=1∞A(m)zm we rewrite (4) as
(5)
y(t)=a˜(z)y(t)+ɛ(t). 
The transfer functionk(z)=a−1(z)=∑m=0∞K(m)zm exists inside and on the unit circle. There is a unique weakly stationary solution of (3) of the form
(6)
y(t)=∑m=0∞K(m)ɛ(t−m)=k(z)ɛ(t) 
This solution (6) of the system (3) is called an autoregressive (∞) process. It corresponds to the Wold representation. For the sake of simplicity of notation we will skip the (∞) sign henceforth.
For a stationary process z, let z(t)¯=closure(span(z(s)s≤t)) denote the space spanned by the past and present of z in the Hilbert space of all square integrable random variables. Time t represents the present unless noted otherwise.
Note that, if (2) holds for the whole process, it also holds for all subprocesses, and therefore all subprocesses have AR representations.
Due to the nature of our application, we will often refer to their components as channels in this paper.
There have been long and thorough discussions about causality throughout the last decades, a brief summary can be found in ^{Pearl (2000)}. As already stated various ideas exist how to formalize causality. The definition we will use for our causal investigation is Granger causality, as introduced in ^{Granger (1969)}, based on a suggestion in ^{Wiener (1956)}.
According to the original definition in ^{Granger (1969)}, we say a time series y_{1} is causing another time series y_{2}, denoted by y_{1} → y_{2}, if we are able to predict y_{2} better using all available information in the universe than using all information apart from y_{1}.
Granger's definition is based on the decrease of the variance of the (linear least squares) prediction error. For a better understanding we present an equivalent definition of Granger causality based on the autoregressive coefficients for the bivariate case.
Let y(t) = (y_{1}(t), y_{2}(t))′ satisfy the assumptions of Section 1.2, then we consider the joint AR representation (5) at time point t + 1.
(7)
y1(t+1)y2(t+1)=a˜(z)y1(t+1)y2(t+1)︸+ɛ1(t+1)ɛ2(t+1) 
∑m=1∞A11(m)y1(t+1−m)+∑m=1∞A12(m)y2(t+1−m)∑m=1∞A21(m)y1(t+1−m)+∑m=1∞A22(m)y2(t+1−m). 
We say that y_{1}is Granger noncausal for y_{2} if A_{21}(m) = 0 ∀ m (i.e. a˜21(z)=0). In other words y1(t)¯ does not influence the prediction of y_{2}(t + 1).
Otherwise we say y_{1}is Granger causal for y_{2}. In this case the knowledge of the present and past of y_{1} improves the prediction of y_{2}(t + 1), i.e. the variance of the prediction error is smaller when using the past and present of both y_{1} and y_{2} compared to using only the past and present of y_{2} itself.
Because of the properties of AR representations, the lower line of Eq. (7) represents the orthogonal projection of (y_{1}(t + 1), y_{2}(t + 1))′ onto its past, which is its best linear (least squares) predictor.
In the last section we considered the bivariate case. For our purposes we use the multivariate extension of Granger's classic definition according to ^{Eichler (2007)}, restricted to the causality between two componentprocesses.
For the remainder of the paper we will focus on ndimensional stationary processes. Let V = {1, …, n}, and we use the notation y_{V} for y to stress the fact that the elements of V correspond to the onedimensional component processes of y. To refer to a subprocess corresponding to I ⊂ V we write y_{I}(t) = (y_{i}(t)i ∈ I)′, for a (univariate) component process we simply write y_{i}. With these notations we are able to state the following definition.
Let i, j ∈ V (i ≠ j). Then the process y_{i}is Granger noncausal for y_{j}with respect to y_{V}, denoted by y_{i} ↛ y_{j}y_{V}, if
(8)
a˜ji(z)=0, 
In other words, the past and the present of y_{i} do not influence the linear prediction of y_{j}(t + 1) in the AR representation of y_{V}.
Otherwise we say the process y_{i} is Granger causal for y_{j}with respect to y_{V}, which we will denote as y_{i} → y_{j}y_{V}. If the relation has not been derived yet we will sloppily write yi→?yjyV.
This definition of causality is sometimes referred to as conditional Granger causality, because the causal effect of y_{i} on y_{j} conditioned on y_{V∖{i,j}} is analyzed.
For the interested reader we note that for a complete causal analysis of a process y_{V} all causality relations for arbitrary subprocesses have to be considered, not only the causality relations between two componentprocesses with respect to the whole process, see ^{Eichler (2006)}. However, the causality relations between two componentprocesses with respect to the whole process are sufficient for our purposes.
Note that up to now we have only considered regular AR systems, i.e. AR systems where Cov(ɛ(t)) = Σ is regular, because the presented definition of Granger causality is based on them. In Section 2 we will loosen this restriction.
Factor models are useful for the analysis and forecasting of highdimensional time series, when the single time series show similarities or a kind of comovement. The idea of factor models is to separate the ndimensional observations x(t) into a part representing the comovement χ(t) (also referred to as latent variables) and a part representing the noise of the data η(t):
(9)
x(t)=χ(t)+η(t). 
Hereby the ndimensional latent variables χ are generated by a qdimensional process, where q << n. These q driving processes are called factors, there from the term factor model. We assume that x is stationary with mean zero.
The comovement χ(t) describes the common movement of the data and η(t) the residual part of the data, for ECoG data we describe this in detail in Section 4.2.
For our purposes we will separate the latent variables χ(t) from the noise by means of Principal Component Analysis (PCA) and derive the static factors z(t). See ^{Pearson (1901), Hotelling (1933)} and ^{Jackson (2004)} for background information on PCA.
In the second step we will model the static factors z(t) as a regular AR process.
The described modeling approach is sometimes referred to as a quasistatic factor model, see e.g. ^{Deistler and Zinner (2007)}.
In detail we proceed as follows:
In order to separate x(t) = χ(t) + η(t), we apply the PCA: we calculate the eigenvalue decomposition of the covariance matrix Ω of the observed variables x(t)
Ω=Cov(x(t))=OΓO′=(O1 O2)Γ100Γ2O1′O2′=O1Γ1O1′+O2Γ2O2′, 
Let
z(t)=O1′x(t) 
(10)
χ(t)=O1z(t)=Λz(t), 
χ(t)=Λz(t)=ΛU−1︸Λ˜Uz(t)︸z˜(t)=Λ˜z˜(t). 
However, this will not impair our causality analysis in Section 2, because the spaces spanned by the original static factor z(t) and the rotated ones z˜(t) are equal.
Proceeding, we assume the static factors z(t) to satisfy the requirements of Section 1.2, in particular assumption (2), and we model them as a regular AR process
(11)
a(z)z(t)=ɛ(t), 
Eqs. (10) and (11) and the causal invertability of a(z) together yield
(12)
χ(t)=Λz(t)=Λa−1(z)ɛ(t). 
In other words the qdimensional white noise process ɛ generates the ndimensional latent variables χ.
In this section we discuss the application of Granger causality to factor models.
What is the challenge in using Granger causality for the analysis of high dimensional comoving data? The naïve approach would be to fit an ndimensional AR model to the ndimensional observations x(t). In practice we typically encounter two problems:
First, Granger causality analysis is usually considered in the case of regular AR systems, i.e. where the error covariance matrix Σ is regular (compare Section 1.4). As EEG data are highly correlated and show strong comovement, regular AR models lead to a poor estimation and misleading results of the subsequent causality analysis. A visual analysis of EEG data quickly confirms this comovement.
Second, fitting of a ndimensional AR(p) model requires the estimation of n^{2}p parameters. In order to obtain reliable estimators a large sample size is required, in particular for large crosssectional dimension n. However, as neurological signals (in particular EEG) show a highly nonstationary behavior, such large sample sizes impair the estimation quality. Again, this leads to poor results of the causality analysis.
In order to avoid these problems we consider factor models. Naturally the question arises, which causalities can be reasonably analyzed in this context. In this paper we assume that the dependence of the latent variables χ(t) properly reflects the causal structure of the observations x(t). This assumption seems meaningful despite the separation (9) into noise and latent variables, as will be discussed in Section 5.1. This assumption is not necessarily satisfied, see ^{Anderson and Deistler (1984)}.
The first idea for a causal analysis in the factor model case would be to consider relations of the form
(13)
χi→?χjχV. 
However the usage of the exhaustive set V leads to problems: relation (13) is derived from the projection of χ_{j}(t + 1) onto χV(t)¯, compare criterion (8). Although the projection itself is unique, the projection coefficients are not, i.e. the contribution of the single χ_{i}(t), χ_{i}(t− 1), … to the explanation of χ_{j}(t + 1) is not uniquely determined. This is due to the fact that χ_{V}(t) contains linear dependent elements. As these contributions are not unique, the application of criterion (8) is not reasonable. Thus an analysis involving relations of the form (13) is not meaningful.
Note that in special cases the above problem might not arise.
Therefore we restrict the conditioning set, instead of V we use channel selections I ⊂ V, # I = q < n.^{2}
We consider relations of the form
(14)
χi→?χjχIi,j∈I. 
In the following we will discuss the choice of the channel selection I, such that relations of the type (14) yield reasonable results.
For a channel selection I ⊃ i, j we consider the corresponding subsystem of (12)
(15)
χI(t)=ΛIz(t)=ΛIa−1(z)ɛ(t), 
In order to yield reasonable causal relations of the form (14) we only consider channel selections I where the corresponding Λ_{I} is regular. In the remainder of this paper we will call such I admissible.
By rewriting (15) as an AR representation we obtain
(16)
ΛIa(z)ΛI−1︸a˘(z)χI(t)=ΛIɛ(t)︸ɛ˘(t) 
The Granger causality relations of this representation (16) can easily be checked by criterion (8), which is of the form
χi↛χjχI⇔a˘ji(z)=0i,j∈I;i≠j. 
However, it is important to note that the causality relations do depend on the channel selection I. This can be seen from the following example. Consider the following system
(17)
χ1χ2χ3χ4(t)=100010001111︷Λ10αz010001︷a(z)−1ɛ1ɛ2ɛ3(t)==10−αz010001111−αzɛ1ɛ2ɛ3(t) 
(18)
10−αz010001χ1χ2χ3(t)=ɛ1ɛ2ɛ3(t), 
(19)
1−αz−αzαz010−αz−αz1+αzχ1χ2χ4(t)=ɛ˘1ɛ˘2ɛ˘3(t), 
(20)
Cov(ɛ˘(t))=101011113. 
According to criterion (8), in subsystem (18), χ_{2} is Granger noncausal for χ_{1} due to a_{12}(z) = 0. In subsystem (19)a_{12}(z) ≠ 0, thus χ_{2} is Granger causal for χ_{1}. Compare Fig. 1 for a graphical representation of these two subsystems.
For the interested reader we note that it is also possible to use generalized dynamic factor models, as discussed in ^{Deistler et al. (2010)}, as a basis for the Granger causal analysis.
We apply the method discussed above to EEG recordings. As we have seen in the previous section, a strict Granger causality investigation between channel i and j depends on the chosen channel set I. Different channel selections I and I˜ may lead to different results. However, we are interested in a single overall statement, whether channel i influences channel j or not.
For this purpose we propose a workable methodology, which is based on the ideas of Section 2.2. We will apply it to simulated data and to EEG recordings taken from an epilepsy patient.
Our methodology consists of three steps. First, we use PCA to separate the observations into the latent variables (explaining the comovement) and the noise, see Section 2.2. As discussed in Section 2.1, we assume that the causal structure of the observations is reflected in the causality structure of the latent variables.
Second, for fixed i and j we analyze the conditional Granger causality relation χi→?χjχI, given a fixed channel selection I ⊃ i, j.
Third, we perform this analysis for all admissible channel selections I˜⊃i,j and derive a heuristic statement for the influence from χ_{i} to χ_{j}, condensing the information of all subsystems.
In detail we proceed as follows:
First we perform a PCA on the observations x in order to obtain the factor loading matrix Λ and the static factors z. The dimension of the static factors q is determined via a Scree plot, see ^{Cattell (1966)}. In this graphical method the principal components are sorted according to their explanation of the total variance of the data in descending order. Usually some kind of bending point can be observed in this graphical representation, which divides the principal components into important and unimportant ones, see Fig. 5.
Now let i, j, I be fixed. The straightforward application of the approach of Section 2.2 yields two problems:
While in theory we can easily distinguish regular and singular matrices Λ_{I} in Eq. (16) by considering the determinant, the estimator ΛˆI will typically yield det(ΛˆI)≠0. The causality relations drawn from systems with very small values of det(ΛˆI) are not meaningful, which is due to the fact that a˘(z) in (16) cannot be computed reliably. The term det(ΛˆI) is a measure for the similarity of the selected channels. Therefore we only consider channel selections I with det(ΛˆI) exceeding a threshold τ. This threshold is chosen empirically in order to yield reasonable results.
A similar challenge arises in the estimation of a˘ˆji(z) (which has a finite order now). In theory A˘ji(m)=0 ∀m signifies that χ_{i} is Granger noncausal for χ_{j}. However, in estimation we typically have A˘ˆji(m)≠0, so we have to statistically test whether the polynomial coefficients A˘ˆji(m) (for all lags m) are significantly jointly different from zero.
For this purpose we use an Ftest (H0:A˘ji(m)=0 ∀ m), which is implemented in the GCCA toolbox and described in ^{Seth (2010)}. We consider the pvalue of the test as a measure for Granger causality: rejection of H0 (p < 0.03) signifies Granger causality, acceptance means noncausality.
In order to sum up, for each channel selection I (for fixed i, j) we obtain two values: det(ΛˆI) as a similarity measure of the channels in I and the pvalue as an indicator for the causality from χ_{i} to χ_{j}.
As a global influence statement from χ_{i} to χ_{j} is our goal, we want to condense the different conditional causality statements based on distinct channel selections I into a single one. For this purpose we propose an intuitive rule: if all statements for distinct channel selections match, we conclude a global influence statement.
In other words: if χ_{i} → χ_{j}χ_{I} for all I with det(ΛˆI)>τ, we say that χ_{i}influences χ_{j}. On the other hand, if χ_{i} ↛ χ_{j}χ_{I} for all I with det(ΛˆI)>τ, we say that χ_{i}does not influence χ_{j}. In case of nonconclusive Granger causality statements we do not derive any global influence statement.
Finally, as the causality structures of the observations and the latent variables are equal by assumption (compare Section 2.1), we say x_{i}influences x_{j} if χ_{i} influences χ_{j}. The analog reasoning holds in case of noninfluence.
For a better understanding we want to visualize the described methodology: for i, j fixed we plot a point for each distinct channel selection I ⊃ i, j into the plane spanned by det(ΛˆI) on the xaxis and the pvalue on the yaxis. This procedure yields graphs such as shown in Fig. 2. In such a plot we only consider points with det(ΛˆI)>τ, which are located to the right of the dashed vertical threshold line. Points to the left of this determinant threshold line are ignored, because the corresponding pvalues are not meaningful due to numerical instabilities.
A point situated below the dotted line represents a pvalue <0.03 and therefore indicates Granger causality. Consequently a point lying above the dotted line indicates Granger noncausality.
Fig. 2, where each plot is constructed as described above, illustrates the three cases we distinguish:
In plot (a) all relevant points are situated below the dotted line, i.e. each point individually indicates causality (H0 of noncausality rejected due to p < 0.03), thus we have global influence. We observe the opposite situation in plot (b), where all relevant points are above the dotted line, i.e. each point individually indicates noncausality, so we speak of global noninfluence.
Plot (c) illustrates a situation where distinct channel selections lead to Granger causality as well as Granger noncausality statements. In this case, we refrain from concluding on global influence.
In the final application we distinguish only if χ_{i} influences χ_{j} (and therefore x_{i} influences x_{j}) or not. Bayesian inference could be used in this context to distinguish between these two cases. However, the calculation of prior distributions and a Bayes factor would be difficult.
In order to assess the proposed methodology we apply it to simulated data where we know the imposed dependence structure. Consider the following signal model
(21)
x(t)=Λz(t)+η(t) 
a(z)z(t)=ɛ(t). 
First we simulate the 3dimensional static factors z(t) as an AR(2) process with
a(z)=1−0.2z00−0.3z21−0.5z0−0.7z201−0.5z 
Λ=100010001100010001100010001 
Cov(η(t))=diag(0.15,0.15,0.61,1.37,0.61,0.15,1.37,1.37,0.61). 
The Granger causality structure of the simulated 3dimensional static factors (z1*,z2*,z3*)′ is depicted in Fig. 3(a), the resulting influence structure of the 9dimensional comoving system (x_{1}, …, x_{9})′ in Fig. 3(b). Note that the simple design of Λ yields the clear graph in plot (b).
In order to demonstrate the practicability of our methodology we apply it to invasive EEG data taken from a patient suffering from epilepsy.
Epilepsy is defined as a neurological disorder with recurring unprovoked seizures, it affects 0.7% of the general population (see ^{Hirtz et al., 2007}). Seizures are characterized by abnormal hypersynchronous neuronal activity, which can affect both hemispheres of the brain (generalized seizure) or a circumscribed area in one hemisphere (focal seizure). About one third of patients with focal epilepsies develop drug resistant seizures, which cannot be cured with a medical therapy. If a clear identification of the seizure onset zone, i.e. the area where the seizure starts, is possible, a resective surgical intervention is a valuable treatment option (see ^{Schuele and Lüders, 2008}). This identification is done by a visual inspection of the EEG data by clinicians, which is a difficult and time consuming task. For a better spatial resolution (in particular if EEG is insufficient for the localization) an invasive recording using subdural strip electrodes (electrocorticography, ECoG), which are placed directly on the brain surface, is performed (see ^{PondalSordo et al., 2007}).
While the visual inspection of the ECoG data is still regarded as gold standard, see ^{GötzTrabert et al. (2008)} and ^{Jenssen et al. (2011)} for two recent studies, there has been an increasing interest in a mathematical analysis of these data. In particular causality measures, aiming at the quantification of the aforementioned synchronous activity, have been used to detect the seizure onset zone, see e.g. ^{Kim et al. (2010)} and ^{Wilke et al. (2008)}.
By applying our proposed methodology to ictal ECoG data, i.e. recordings during an epileptic seizure, we follow this technical approach of locating the seizure onset zone.
The ECoG data used in this exemplary study are taken from a patient (male, 43 years) suffering from therapyresistant focal epilepsy. The patient underwent a presurgical longterm video EEG monitoring at the Hospital Hietzing with Neurological Center Rosenhügel. Three subdural strip electrodes with a total of 25 channels were implanted, and the electrode B1 (outside the seizure focus) was chosen as reference. Compare the MRI (magnetic resonance imaging) scan in Fig. 8 for details. Recording was performed using a Micromed^{®} system at a sampling frequency of 1024 Hz. After recording, the ECoG data were preprocessed in Matlab^{®}: line interference was removed using a notch filter at 50 Hz, and a highpass filter at 1 Hz was applied in order to get rid of physiologically irrelevant lowfrequency contributions. The signals were lowpass filtered at 64 Hz in order to avoid aliasing and then downsampled to 128 Hz sampling rate.
In this paper we analyze the beginning of the first seizure which occurred during the recording time. For the identification of the seizure onset zone we use data from a 4 s time window (as indicated by the clinicians) consisting of 24 recorded channels. Fig. 4 shows 6 selected channels of the ECoG recordings during the investigated time interval.
As can be seen, the data seem to be nonstationary, this point will be discussed thoroughly in Section 5.2.
We apply our methodology to the simulated data from the signal model (21).
For the initial calculation of the PCA, we determine the number of static factors q by considering the Scree plot, see Fig. 5. This figure shows the percentage of the explained variance per factor. We observe that three factors explain the majority of the variance, thus we choose q = 3. Furthermore, by application of the BIC criterion we obtain an ARmodel order of p = 2 (matching the imposed model order).
Proceeding according to our methodology, for fixed source i and target j we obtain causality relations for all channel selections I ⊃ i, j. They are represented as points in a graph as described in Section 3.1. Compare Fig. 2 for the idea. Hereby points with pvalues >0.4 are displayed with pvalue = 0.4, because this does not change the results of the analysis (highly nonsignificant anyway) and facilitates the visualization.
In Fig. 6 all these plots are arranged in a 9 × 9 matrix plot, where the columns indicate the source channels x_{i} and the rows the target channels x_{j}. Clearly, the (j, i)subplot quantifies the influence from x_{i} to x_{j}. Obviously, diagonal elements are not displayed.
Let us consider the interpretation of three selected subplots in Fig. 6 in detail:
In subplot (3, 1) all points to the right of the determinant threshold line are located below the dotted line, and therefore represent pvalues smaller than 0.03 (i.e. the null hypothesis of noncausality is rejected). This means that for all admissible channel selections I, we have χ_{1} → χ_{3}χ_{I}. Thus we say x_{1} influences x_{3}.
In subplot (3, 2) all points to the right of the determinant threshold line are located above the dotted line. Thus we say x_{2} does not influence x_{3}.
In subplot (4, 1) all points are located to the left of the determinant threshold line, therefore we do not draw any conclusions. The reason for this behavior is that x_{1} and x_{4} are both generated by z1* and therefore are highly correlated.
Thus we have shown that we successfully retrieve the imposed dependence structure of (21) by interpreting each of the subplots in the described way. Channel x_{1} influences channels x_{2}, x_{3}, x_{5}, x_{6}, x_{8}, x_{9}, so do channels x_{4} and x_{7}. This is illustrated in Fig. 3(b) where each influence relation is symbolized by an arrow.
We apply our methodology (see Section 3.1) to ictal invasive EEG (ECoG) data described in Section 3.3. We process a 4s segment from the initial phase of a seizure in order to locate the seizure onset zone. Fig. 4 shows 6 out of the 24 processed channels.
The number of static factors is determined by a Scree plot as before in Section 4.1. In order to achieve an explained variance greater than 80% we choose q = 5.
The choice of the ARmodel order for the application was challenging. The problem is that filtering (in the preprocessing step of the data) could affect the ARmodel order (and the Granger causal analysis of the data), see ^{Barnett and Seth (2011)}. The application of AIC and BIC criteria in a sliding data window revealed optimal ARorders in the range from 4 to 40. In the rhythmic epochs of the data ARmodel orders of 6 or 8 dominate. Hence, in accordance with the survey paper (^{Tseng et al., 1995}), we choose the ARmodel order for the Granger causal analysis p = 8. This chosen model order corresponds to a time lag of 62.5 ms.
Furthermore, we choose τ = 0.05 in order to discard points of the causality analysis which yield unreasonable results due to numerical problems caused by channel similarity.
In the application of our methodology to ECoG data the comovement χ(t) from Eq. (9) represents the common movement of the data. This common movement is the synchronized epileptic activity (we are interested in) and is characterized by high amplitude wave forms. On a theoretical level η(t) is noise, but in the application it is simply the rest after the important parts of the data (common movement) have been removed. According to our key assumption that the causality structure of the latent variables reflects the causality structure of the data, no important causal information is found in these low amplitude residuals. This assumption holds in the case of the observed ECoG data, as we discuss in Section 5.1.
In analogy to the analysis of the signal model (21) in Section 4.1 we obtain a 24 × 24 matrix plot. Fig. 7 shows a 6 × 6 submatrix, corresponding to the 6 channels displayed in Fig. 4.
Here we briefly want to discuss 4 subplots of Fig. 7 in detail:
Subplot (2, 3) describes the causality relations from B8 to A12. All points to the right of the determinant threshold are located below the dotted line, thus we say channel B8 influences channel A12.
In subplot (5, 3) all points located to the right of the determinant threshold are above the dotted line. Therefore B8 does not influence C5.
An interesting case occurs in subplot (3, 2). We have admissible points above and below the dotted line. In this case we refrain from any influence statement.
Finally in subplot (4, 2) all points are located to the left of the determinant threshold. We do not draw any conclusion in this case, as there are no admissible channel selections.
By interpreting each subplot in the 24 × 24 matrix in this way, we obtain all influence relations. In particular we are interested in the (source, target) channel pairs, where the source does influence the target channel. Fig. 8 shows a MRI scan of the patient's brain together with the electrode positions, the calculated influences are represented by arrows.
Channels B8, A12, A11 and A10 have the highest number of outgoing arrows, see Fig. 9 for a quantification of the outdegree per channel. This observation suggests that the seizure onset zone comprises these four electrodes, which is in good accordance with the visual analysis of the clinicians. In Section 5.2 we will discuss the neurophysiological aspects of this result.
In this paper we propose a procedure for deriving influence relations in highdimensional and comoving time series using PCA and Granger causality. In this section we discuss various aspects of our approach. We start with a discussion of the methodology proposed in Section 3.1, followed by a discussion of its application to simulated data and ECoG recordings.
A key assumption of this paper is that the latent variables reflect the causality structure of the observations. In other words, that we can infer the dependencies between the channels from the latent variables, and that the noise is assumed not to contain any causality information. Although this is a strong assumption, it seems reasonable to us: this assumption is very similar to the one that the causal dependencies are reflected by distinct, highamplitude wave forms in the signal, e.g. high amplitudes in ictal ECoG recordings. We believe that in particular in neurophysiological applications this is meaningful, as we expect these highamplitude oscillations to carry substantial information about the causality structure of the generating cortical mechanisms.
In our approach we investigate all dependencies in all qdimensional subsystems corresponding to the latent variables. By the application of the PCA we perform this analysis in a computationally efficient way, as we only have to compute one PCA and estimate one single ARmodel. Of course it would also be possible to refrain from the usage of factor models. In this case one would fit an ARmodel for each qdimensional subprocess of the observations, which would result in a higher computational effort.
In contrast, the proposed factor modelbased methodology yields a simple mathematical method for the causal analysis of the whole multivariate system. Another advantage of our method lies in the simple measure for the channel similarity of a channel selection I, det(ΛˆI), which allows for a straightforward comparison for different channel selections.
As mentioned in Section 3.1, our methodology consists of three steps: PCA, Granger causality analysis for fixed channel selection I and derivation of an influence statement. This modular design allows for an easy adaptation of each single step, i.e. alternative methods could be used in each step independently of the others.
First, the usage of sparse PCA would enforce additional zeros in the loading matrix Λ. ^{Thurstone (1947)} suggested five criteria for a simple structure and (^{d’Aspremont et al., 2007}) gave a direct formulation for sparse PCA.
Second, here we use two central indicators in the Granger causality analysis for a fixed channel selection: det(Λˆ)I as a measure of channel similarity (as mentioned above) and the pvalue of an Ftest as an indicator for Granger causality. We employ the latter due to its wellestablished theory. Note that in neuroscience literature various other directed dependency measures are used. Prominent ones include the linear measure of conditional dependence introduced by ^{Geweke (1984)}, the Partial Directed Coherence (PDC) introduced by ^{Baccala and Sameshima (2001)}, its numerous modifications and the Directed Transfer Function (DTF) proposed by ^{Kaminski and Blinowska (1991)}. Compare ^{Flamm et al. (2012)} for an overview and a discussion of these measures with regard to Granger causality.
Other measures for the channel similarity would also be conceivable, e.g. the determinant of the covariance matrix of the errors det(Σ(ɛ˘(t))) in (16). However, we note that det(ΛˆI) revealed good numerical properties.
Third, in this paper we propose a workable rule for the derivation of influence statements: If χ_{i} → χ_{j}χ_{I} for all admissible I, we say that x_{i} influences x_{j}. One could imagine other rules depending on specific applications. For example only the channel selection with the largest det(Λˆ)I could be taken into account for the influence statement. Another possible rule for influence statements might be based on the comparison of the number of points above and below the dotted line, e.g. the majority or a certain percentage.
A naturally arising question is whether and to which extent our definition of influence is meaningful. In our opinion it is a workable procedure for causality analysis in highdimensional comoving systems: Intuitively one expects a certain kind of dependence if χ_{i} is causal for χ_{j} for all (admissible) channel selections.
A potential weakness of our definition of influence is the fact that in practical applications one is often confronted with the case where no influence statement can be inferred. Compare subplot (3, 2) in Fig. 7, where causality as well as noncausality relations are symbolized. In such cases we recommend a more precise investigation which particular channel selections yield causality and which do not. If, in applications, the conditioning on channels of a specific region yields noncausality between channels i and j, but the conditioning on all other regions indicates causality, further investigation could be performed to explain this. Due to the existence of such undecidable cases we avoid the term causality and refer to the derived dependence statement as influence.
We want to conclude this first part of the discussion with a side remark.
The problem of highdimensional comoving data often occurs in practical applications. We propose a methodology connecting Granger causal analysis and factor models. Other ideas in order to cope with the comovement include the analysis of manually selected subsystems (where the number of observations equals the number of driving components) or the causal analysis of socalled extracted atoms, which represent condensed information of the system, see ^{Eichler (2005)}.
Our methodology correctly identifies the dependence structure of the signal model (21) (see Section 3.2), as we have discussed in Section 4.1. During this analysis we encountered the challenges discussed in the previous section.
The determination of q, i.e. the dimension of the static (and the dynamic) factors, is very important for the simulated data as well as the ECoG recordings and potentially has great influence on the analysis. In both applications we chose q based on Scree plots in a reasonable way. The assumption that the dimension of the dynamic and the static factors match is very strong. If the dimension of the dynamic factors is lower than the one of the static factors, the AR modeling of the static factors (11) does not hold any longer. The resulting model class is richer but the proposed approach would have to be generalized.
A major problem of the presented causal analysis of the ECoG data is the stationarity. As can be seen in Fig. 4 the data do not seem to be stationary. During the main phase of the epileptic seizure the channels show distinct rhythmic behavior, which is stationary in nature. However, we are interested in the beginning of the epileptic seizure where the rhythmic activity starts to spread. Due to this evolution our period of interest is nonstationary.
There are two main approaches to cope with the problem of causal analysis of nonstationary data: the first is to use nonlinear methods, see e.g. ^{Marinazzo et al. (2011)}, the second is to employ the shortest possible time window for the analysis. We focus on the second approach and use a short time window:
The segment length of the analyzed data is bounded below with 4 s (at a sampling rate of 128 Hz) in order to yield a reasonable result in the causal analysis. This is the lowest possible reasonable window length for the used methods (first PCA extraction of 5 important components and second ARmodeling with order p = 8) due to parameter estimation, compare Section 2.1. Reasonable in this sense means stability of the results: a slight temporal shift of the analyzed data window will only cause small changes in the causal analysis; for shorter segments the window shift will cause significant changes in the causal analysis.
The application of our methodology to ictal ECoG data shows promising results. We successfully localize the seizure onset zone of the analyzed patient by identifying the zone with the highest number of outgoing arrows. In this manner we define the area comprising the electrodes B8, A12, A11 and A10 as the seizure onset zone, compare the MRI scan in Fig. 8 and the outdegree histogram in Fig. 9. This result correlates well with the visual inspection of the raw ECoG data by clinicians. They mark the beginning of the epileptic activity on each channel, and by the temporal delay between channels the propagation and seizure onset zone are determined. Table 1 summarizes the findings of three clinical experts who independently marked the electrodes initially involved in the epileptic activity. For each of the three investigations the electrodes identified by our methodology are comprised in the set of initial and followup electrodes. Electrodes B8, A12 and A11 are specified as initial in two out of three cases and as followup in the third case. Inversively, electrode A10 is indicated as initial in one case and as followup in two cases.
The visual analysis investigates each channel on its own and then compares all channels. In contrast our proposed methodology analyzes the whole multivariate set of channels at once. So the two underlying ideas are different, and therefore it is remarkable that the results correlate so well.
In our opinion the reason for this good correlation between our results and the clinical findings is the following:
In case of focal epilepsy, the pathological synchronous activity (characterizing the epileptic seizure) starts at a circumscribed brain area. From this seizure onset zone ictal activity spreads to its immediate vicinity recruiting more and more parts of the neural network. This leads to distinct comovement of the observations. One could imagine a “focus” located in the seizure onset zone driving the surrounding channels by imposing its oscillatory frequency in the course of the recruiting process. This could be interpreted as a kind of information transfer or causal interaction: the electrodes in the focus influence the behavior of the surrounding electrodes in the initial phase of the seizure.
Therefore we expect to obtain indications for the seizure onset zone by applying a Grangercausal analysis to factor models within the initial seconds of the seizure. We think that the aforementioned results strengthen this hypothesis. However, for a confirmation of this hypothesis the application of the proposed methodology to a broader data basis is essential.
In the course of this recruiting process we obviously expect feedback mechanisms between the channels (besides unidirectional dependence). This can be observed in Fig. 8, consider e.g. channels A9 ↔ A10, A10 ↔ A12 and B6 ↔ A12. However, in the seizure onset zone the departing arrows predominate, i.e. we have channels with a high outdegree. This situation is reflected in the outdegree histogram in Fig. 9. Channels with the highest outdegrees coincide with the seizure onset zone, and with increasing distance to the seizure onset zone the respective outdegree decreases. Channel A8 with an outdegree of zero is an exception, as we cannot infer any influence statement for this source channel (only nonadmissible channel selections for all target channels).
Finally we want to address a general problem of EEG/ECoG analysis: coverage, i.e. cortical activity is only measured at certain points. Usually the focus of a seizure is located in deeper brain regions or is not directly covered by an electrode. Therefore, the presented methodology as well as the visual analysis only detect a circumscribed area where the epileptic activity is noticed first.
In case of incomplete coverage our algorithm would detect the area which influences the other areas to the greatest extent. However, this is not necessarily the seizure onset zone.
Concluding we think that our methodology might have the potential to assist clinicians in the presurgical evaluation by objectivating their visual ECoG examination (e.g. as an Addon to clinical recording software).
In this paper we propose a causal analysis of highdimensional comoving data, connecting the topics of Granger causality and factor models. The application of Granger causality to factor models is not straightforward, and we propose a natural extension termed influence. The latter allows for an investigation of the dependence structure in certain cases.
This methodology might often allow an insight into the dependence structure of highly correlated data in neuroscience such as EEG.
Notes
1In this context A < B means that B − A is a positive definite matrix.
2In this context # denotes the number of elements in a set.
3Simulation is done using the function arsim of the Matlab^{®} package arfit, described in ^{Schneider and Neumaier (2001)}.
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This work was supported by the Austrian Science Fund FWF (Grants P22961 and L630).
We wish to thank Prof. Czech from the Vienna General Hospital, University Clinic for Neurosurgery, for providing the MR image with the subdural electrode strip positions.
We are grateful to Sandra Zeckl from the Hospital Hietzing, Neurological Center Rosenhügel, for the visual inspection of the ECoG data.
Furthermore, we want to thank our colleague Elisabeth Felsenstein for helpful discussions on factor models.
Article Categories:
Keywords: Keywords Granger causality, Factor model, Principal Component Analysis, EEG, Epilepsy, Onset zone detection. 
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