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

Abstract

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

Appendix

In this Appendix we prove the properties stated for the new proposed frequency domain measures of causality for multiple vector processes. First, we demonstrate the properties of the non-logarithmic measures (i.e., bDC and bPDC, see (14)); then, we exploit the relation existing between logarithmic and non-logarithmic measures to demonstrate the properties of the measures defined in (13).

Properties 6 and 7 To determine the bounds of the bDC and bPDC functions, we first reformulate their definitions as follows. From (12a), the spectral density matrix of the vector process $Y_{i}$ can be formulated as a sum of $M$ contributions, ${\mathbf{S}}_{ii} ( \omega )=\sum \nolimits _{m=1}^M {{\mathbf{S}}_{i\vert m} ( \omega )}$, with each contribution given by ${\mathbf{S}}_{i\vert m} ( \omega )={\mathbf{H}}_{im} ( \omega )\Sigma _{mm} {\mathbf{H}}_{im}^*( \omega )$. This decomposition can be expressed separating the contribution to S $_{ii}$($\omega$) coming from $Y_{j}$ from all other contributions: ${\mathbf{S}}_{ii} ( \omega )={\mathbf{S}}_{i\vert j} ( \omega )+{\mathbf{S}}_{i\vert -j} ( \omega )$, with ${\mathbf{S}}_{i\vert -j} ( \omega )=\sum \nolimits _{m\ne j} {{\mathbf{S}}_{i\vert m} ( \omega )}$. In a similar way, from (12b) the inverse spectral matrix of the vector process $Y_{j}$ can be decomposed as ${\mathbf{P}}_{jj} ( \omega )=\sum \nolimits _{m=1}^M {{\mathbf{P}}_{j\rightarrow m} ( \omega )}$, where ${\mathbf{P}}_{j\rightarrow m} ( \omega )={{\bar{\mathbf{A}}}}_{mj}^*( \omega ){{\varvec{\Sigma }}}_{mm}^{-1} {{\bar{\mathbf{A}}}}_{mj} ( \omega )$, and the sum can be further represented as ${\mathbf{P}}_{jj} ( \omega )={\mathbf{P}}_{j\rightarrow i} ( \omega )+{\mathbf{P}}_{j\rightarrow -i} ( \omega )$, with ${\mathbf{P}}_{j\rightarrow -i} ( \omega )=\sum \nolimits _{m\ne i} {{\mathbf{P}}_{j\rightarrow m} ( \omega )}$. With this notation, the block DC and block PDC defined in (14a) and (14b) can be expressed as

$$\begin{aligned}&\gamma _{ij}^{(\mathrm{b})} ( \omega )=\frac{\left| {{\mathbf{S}}_{ii} ( \omega )} \right|-\left| {{\mathbf{S}}_{i\vert -j} ( \omega )} \right|}{\left| {{\mathbf{S}}_{ii} ( \omega )} \right|},\nonumber \\&\pi _{ij}^{(\mathrm{b})} ( \omega )=\frac{\left| {{\mathbf{P}}_{jj} ( \omega )} \right|-\left| {{\mathbf{P}}_{j\rightarrow -i} ( \omega )} \right|}{\left| {{\mathbf{P}}_{jj} ( \omega )} \right|}. \end{aligned}$$

(17)

Now we show that all matrices involved in (17) are positive semidefinite. To this end, we recall two known matrix properties (Berman and Shaked Monderer 2003): given two Hermitian positive semidefinite $n\times n$ square matrices A and B, (a) the $m\times m$ matrix C = L*AL is positive semidefinite for any $n\times m$ matrix L, and (b) the sum $\mathbf{D} = \mathbf{A}+\mathbf{B}$ is positive semidefinite. These properties can be proven recalling the definition of a semidefinite positive matrix, i.e., A is positive semidefinite if $X$*A $X\ge 0$ for any $n$-dimensional complex vector $X$, and showing that (a) for any $m$-dimensional complex vector $Z$ we have that $Z$*C $Z=Z$*L*AL $Z=X$*A $X\ge 0$ with $X$ = L $Z$; (b) for any $n$-dimensional complex vector $X$, $X$*D $X=X$*A $X+X$*B $X\ge 0$. In our case, S $_{i\vert {m}}(\omega )$ and P $_{j\rightarrow { m}}(\omega )$ are positive semidefinite according to property (a) since for strictly causal VAR processes the diagonal matrices ${{\varvec{\Sigma }}}_{m{m}}$ and ${{\varvec{\Sigma }}}_{_{m{m}} }^{-1}$ have all positive diagonal elements, and thus are positive definite, for each $m={1},\ldots ,M$. It follows that, according to property (b), the matrix sums leading to ${\mathbf{S}}_{i _\vert {-j}} (\omega )$ and ${\mathbf{P}}_{j _\rightarrow {-i}} (\omega )$, as well as the total sums resulting in the matrices ${\mathbf{S}}_{ii} (\omega )$ and ${\mathbf{P}}_{jj} (\omega )$, are positive semidefinite.

Therefore, since the determinant of positive semidefinite matrices is always non-negative, we have $\vert {\mathbf{S}}_{ii} (\omega )\vert \ge 0,\vert {\mathbf{P}}_{jj} (\omega )\vert \ge 0,\vert {\mathbf{S}}_{i\vert -j} (\omega )\vert \ge 0,\vert {\mathbf{P}}_{j\rightarrow -i} (\omega )\vert \ge 0$. These conditions set the upper bound for the bDC and bPDC, i.e., $\gamma _{ij}^{(\mathrm{b})} ( \omega )\le 1,\,\pi _{ij}^{(\mathrm{b})} ( \omega )\le 1$. Moreover, exploiting the property $\left| {{\mathbf{A}}+{\mathbf{B}}} \right|\ge \left| {\mathbf{A}} \right|+\left| {\mathbf{B}} \right|$, valid for positive semidefinite matrices [derived from the Minkowski determinant theorem (Mirsky (1955)], we have also that $\left| {{\mathbf{S}}_{ii} ( \omega )} \right|\ge \left| {{\mathbf{S}}_{i\vert -j} ( \omega )} \right|$ and $\left| {{\mathbf{P}}_{jj} ( \omega )} \right|\ge \left| {{\mathbf{P}}_{j\rightarrow -i} ( \omega )} \right|$; these last conditions set the lower bound for bDC and bPDC, i.e., $\gamma _{ij}^{(\mathrm{b})} ( \omega )\ge 0,\pi _{ij}^{(\mathrm{b})} ( \omega )\ge 0$.

From the definitions in (14a) and (14b) it follows immediately that ${\mathbf{H}}_{ij} (\omega )=0$ entails $\gamma _{ij}^{(\mathrm{b})} ( \omega )=0$ and ${{\bar{\mathbf{A}}}}_{ij} (\omega )=0$ entails $\pi _{ij}^{(\mathrm{b})} ( \omega )=0$. Moreover, combining the definitions in (14) with the discussion above we see that the condition ${\mathbf{H}}_{im} (\omega )=0$ for each $m\ne j$ entails ${\mathbf{S}}_{i _\vert {-j}} (\omega )=0$, and thus $\gamma _{ij}^{(\mathrm{b})} ( \omega )=1$, and that the condition ${{\bar{\mathbf{A}}}}_{m{j}} (\omega )=0$ for each $m\ne i$ entails ${\mathbf{P}}_{j _\rightarrow {-i}} (\omega )=0$, and thus $\pi _{ij}^{(\mathrm{b})} ( \omega )=1$. This completes the proof of properties 6 and 7.

Properties 8, 9, and 10 These properties describe how the bDC and bPDC can be expressed in terms of the corresponding traditional scalar causality measures, i.e., the DC and PDC, when one or both the interacting processes $Y_{i}$ and $Y_{j}$ are scalar. In particular, when the destination process $Y_{i}$ is scalar, its dimension is $M_{i}= 1$, so that its spectral density is scalar (i.e., ${\mathbf{S}}_{ii} (\omega )=S_{ii} (\omega ))$ and the product ${\mathbf{H}}_{ij} (\omega )\sum _{jj} {\mathbf{H}}_{ij} ^*(\omega )$ becomes the scalar quantity $\sum \nolimits _{m=1}^{M_j } {\sigma _{j_{m} j_{m} }^2 \left| {H_{ij_m } ( \omega )} \right|^2}$, where $\sigma ^2_{j_{m} j_{m}}$ and $H_{ij_{m}} (\omega )$ denote here the innovation variance of the $m$-th scalar process composing the input vector process $Y_{j}$, and the transfer function from such scalar process to the scalar output process $Y_{i}$. In this case, comparing (14a) with (6) we see that the bDC reduces to the cumulative squared DC:

$$\begin{aligned} \gamma _{ij}^{(\mathrm{b})} ( \omega )=\frac{\sum \nolimits _{m=1}^{M_j } {\sigma _{j_m j_m }^2 \left| {H_{ij_m } ( \omega )} \right|^2} }{S_{ii} ( \omega )}=\sum \nolimits _{m=1}^{M_j } {\left| {\gamma _{ij_m } ( \omega )} \right|^2}.\nonumber \\ \end{aligned}$$

(18)

thus proving property 9. In a similar way, when the source process $Y_{j}$ is scalar (i.e., $M_{j}=1$), its inverse spectral density is also scalar (${\mathbf{P}}_{jj} (\omega )=P_{jj} (\omega ))$ and the product ${{\bar{\mathbf{A}}}}_{ij} ^*(\omega )\sum _{ij} ^{-{1}}{{\bar{\mathbf{A}}}}_{ij} (\omega )$ becomes the scalar quantity $\sum \nolimits _{m=1}^{M_i } {{\left| {\bar{A}_{i_m j} ( \omega )} \right|^2} \mathord {\left. {} \right. } {\sigma _{i_m i_m }^2 }}$. Hence, in this case the bPDC reduces to the cumulative squared PDC from $Y_{j}$ to the scalar processes that compose the output vector process $Y_{i}$:

$$\begin{aligned} \pi _{ij}^{(\mathrm{b})} ( \omega )\!=\!\frac{\sum \nolimits _{m=1}^{M_i } {{\left| {\bar{A}_{i_m j} ( \omega )} \right|^2}\! \mathord {\left. {} \right. } {\sigma _{i_m i_m }^2 }} }{P_{jj} ( \omega )}\!=\!\sum \nolimits _{m=1}^{M_i } {\left| {\pi _{i_m j} ( \omega )} \right|^2},\nonumber \\ \end{aligned}$$

(19)

thus proving property 10. In the case in which both the source process $Y_{j}$ and the destination process $Y_{i}$ are scalar (i.e., $M_{i}=M_{j}= 1$), it follows immediately from (18) and (19) that the bDC and bPDC reduce to the squared modulus of the traditional DC and PDC, i.e., $\gamma _{ij}^{(\mathrm{b})} ( \omega )=\left| {\gamma _{ij} ( \omega )} \right|^2$ and $\pi _{ij}^{(\mathrm{b})} ( \omega )=\left| {\pi _{ij} ( \omega )} \right|^2$, thus proving property 8.

Properties 11 and 12 These properties state that the bPDC measures direct causality, while the bDC measures total causality, from one vector-valued process to another. First, we show that similar properties hold for the scalar-valued PDC and DC function. According to the definition of direct causality stated in Sect. 2.1, $y_{j}\rightarrow y_{i}$ when $A_{ij}(k)$ is nonzero for at least one value of $k (i\ne j)$; this entails, for some frequency $\omega ,\bar{A}_{i{j}} (\omega )=-A_{i{j}} (\omega )\ne 0$ and thus, according to (7), $\pi _{i{j}} (\omega )\ne 0$. As to the DC $\gamma _{i{j}} (\omega )$, it is possible to show that its numerator term $H_{i{j}}(\omega )$ can be expanded as a geometric series resulting in a sum of terms each one related to one of the (direct or indirect) transfer pathways connecting $y_{j}$ to $y_{i}$ (Eichler 2006). Therefore, $\gamma _{i{j}}(\omega )$ is nonzero whenever at least one path connecting $y_{j}$ to $y_{i}$ is significant, i.e., when $y_{j}\Rightarrow y_{i}$. These properties, when combined with the direct and total causality definitions stated for vector-valued processes, extend readily to the bPDC and bDC functions. Indeed, $Y_{j}\rightarrow Y_{i}$ entails that the matrix A $_{i{j}}(k)$, of dimension $M_{i}\times M_{j}$, is nonzero for at least one value of $k$, so that $\bar{{\mathbf{A}}}_{i{j}} (\omega )=-{{\bar{\mathbf{A}}}}_{i{j}} (\omega )\ne 0$ for some frequency $\omega$ (see (3)) and thus, recalling that $P_{j\rightarrow {i}}(\omega )$ is positive semidefinite, $\left| {{\mathbf{P}}_{j\rightarrow {i}} ( \omega )} \right|=\left| {{{\bar{\mathbf{A}}}}_{i{j}}^*( \omega ){{\varvec{\Sigma }}}_{m{m}}^{-1} {{\bar{\mathbf{A}}}}_{i{j}} ( \omega )} \right|>0$. This determines $\left| {{\mathbf{P}}_{j\rightarrow {-i}} ( \omega )} \right|<\left| {{\mathbf{P}}_{j{j}} ( \omega )} \right|$ and, according to (17), $\pi _{i{j}}^{{(\mathrm b)}} ( \omega )>0$. On the contrary, in the absence of direct causality from $Y_{j}$ to $Y_{i}$ we have A $_{ij}(k)=0$ for each $k$, so that ${ {\bar{\mathbf{A}}}}_{i{j}} (\omega )= 0,\,{ \mathbf{P}}_{j _\rightarrow { i}} (\omega )=0,\,{\mathbf{P}}_{j _\rightarrow {-i}} (\omega )={\mathbf{P}}_{j{j}} (\omega )$, and hence $\pi _{i{j}}^{(\mathrm{b})} ( \omega )=0$, thus completing the proof of property 11. Following a dual reasoning, property 12 is proved considering that total causality $Y_{j} \Rightarrow Y_{i}$ occurs when similar total causality relations are present between the constituent scalar processes of $Y_{j}$ and $Y_{i}$, so that the transfer matrix H $_{i{j}}(\omega )$ has at least one nonzero entry for some frequency $\omega$. Therefore, the presence of total causality entails $\vert {\mathbf{S}}_{i\vert {j}} (\omega )\vert =\vert {\mathbf{H}}_{i{j}} (\omega )\sum _{j{j}} {\mathbf{H}}_{i{j}} ^*(\omega )\vert >0$, in a way such that $\vert {\mathbf{S}}_{i\vert {-j}} (\omega )\vert <\vert {\mathbf{S}}_{i{i}} (\omega )\vert$ and, from (17), $\gamma _{i{j}}^{(\mathrm{b})} ( \omega )>0$. On the contrary, in the absence of total causality we have ${\mathbf{H}}_{i{j}} (\omega )=0$ for all $\omega$, ${\mathbf{S}}_{i\vert { j}} (\omega )=0,{\mathbf{S}}_{i\vert {-j}} (\omega )={\mathbf{S}}_{i{i}} (\omega )$ and hence $\gamma _{i{j}}^{(\mathrm{b})} ( \omega )=0$.

Properties 1–5 The properties of the proposed logarithmic frequency domain causality measures can be deduced from the corresponding properties of the non-logarithmic measures, recalling the relationships between the two types of measures derived in (15). The conditions $f_{j\rightarrow i}^{(\mathrm{m})} ( \omega )\ge 0$ and $g_{j\rightarrow i}^{(\mathrm{m})} ( \omega )\ge 0$ (properties 1 and 2) follow from (15) and from the fact that $0\le \gamma _{ij}^{(\mathrm{b})} ( \omega )\le 1$ and $0\le \pi _{ij}^{(\mathrm{b})} ( \omega )\le 1$; in particular, when ${\mathbf{H}}_{ij} (\omega )=0$ we have that $\gamma _{ij}^{(\mathrm{b})} ( \omega )=0$ (property 6) and thus $f_{j\rightarrow i}^{{(\mathrm{m})}} ( \omega )=\ln 1=0\,\,$. while, when ${{\bar{\mathbf{A}}}}_{ij} (\omega )=0$ we have that $\pi _{ij}^{(\mathrm{b})} ( \omega )=0$ (property 7) and thus $g_{j\rightarrow i}^{(\mathrm{m})} ( \omega )=\ln 1=0$. Properties 4 and 5 follow directly from properties 11 and 12 and from the fact that $f_{j\rightarrow i}^{(\mathrm{m})} \,$ is zero (respectively, nonzero) if and only if $\gamma _{ij}^{(\mathrm{b})} ( \omega )$ is zero (nonzero), while $g_{j\rightarrow i}^{(\mathrm{m})} \,$ is zero (nonzero) if and only if when $\pi _{ij}^{(\mathrm{b})} ( \omega )$ is zero (nonzero). Finally, property 3 derives from the observation that when only $M= 2$ vector processes are present (12a) reduces to (9), and thus (13a) reduces to (10) (i.e., $f_{j\rightarrow i}^{(\mathrm{m})} \,=f_{j\rightarrow i} )$.