Chapter 4.3. A partial list of VAR-based spectral, coherence and GC estimators

Table 4 contains a list of the major spectral, coherence, and GC/information flow estimators currently implemented in SIFT. Each estimator can be derived from the quantities $S(f),A(f),H(f),\text{ and }\Sigma$ obtained in section 3.3. , with the exception of the renormalized PDC (rPDC). The rPDC requires estimating the $[(Mp)^2 \times (Mp)^2]$ inverse cross-covariance matrix of the VAR[p] process. SIFT achieves this using an efficient iterative algorithm proposed in (Barone, 1987) and based on the doubling algorithm of (Anderson and Moore, 1979). These estimators and more can be computing using the SIFT’s functions pop_est_mvarConnectivity() or the low-level function est_mvtransfer().

Table 4. A partial list of VAR-based spectral, coherence, and information flow / GC estimators implemented in SIFT.

	Estimator	Formula	Primary Reference and Notes
Spectral M.	Spectral Density Matrix	$S(f)=X(f)X(f)* \\ =H(f)\Sigma H(f)*$	(Brillinger, 2001) ${{S}_{ii}}$ (f) is the spectrum for variable i. $S_{ij}(f) = S_{ji}(f)^*$ is the cross-spectrum between variables i and j.
Coherence Measures	Coherency	$& {{C}_{ij}}(f)=\frac{{{S}_{ij}}(f)}{\sqrt{{{S}_{ii}}(f){{S}_{jj}}(f)}} \\ & 0\le {{\left\| {{C}_{ij}}(f) \right\|}^{2}}\le 1 \\$	(Brillinger, 2001) Complex quantity. Frequency-domain analog of the cross-correlation. The magnitude-squared coherency is the coherence $Coh_{ij}(f) = \|C_{ij}(f)\|^2$ . The phase of the coherency can be used to infer lag-lead relationships, but, as with cross-correlation, this should be treated with caution if the coherence is low, or if the system under observation may be closed-loop.
	Imaginary Coherence (iCoh)	$iCoh_{ij}}(f)=\operatorname{Im}({{C}_{ij}}(f))$	(Nolte et al., 2004) The imaginary part of the coherency. This was proposed as a coupling measure invariant to linear instantaneous volume-conduction. $iCoh_{ij}(f) > 0$ only if the phase lag between i and j is non-zero, or equivalently, $0<\text{angle}({{C}_{ij}}(f))<2\pi$
	Partial Coherence (pCoh)	$& {{P}_{ij}}(f)=\frac{{{{\hat{S}}}_{ij}}(f)}{\sqrt{{{{\hat{S}}}_{ii}}(f){{{\hat{S}}}_{jj}}(f)}} \\ & \hat{S}(f)=S{{(f)}^{-1}} \\ & 0\le {{\left\| {{P}_{ij}}(f) \right\|}^{2}}\le 1 \\$	(Brillinger, 2001) The partial coherence between i and j is the remaining coherence which cannot explained by a linear combination of coherence between i and j and other measured variables. Thus, ${{P}_{ij}}(f)$ can regarded as the conditional coherence between i and j with respect to all other measured variables.
	Multiple Coherence	${{G}_{i}}(f)=\sqrt{1-\frac{\det (S(f))}{{{S}_{ii}}(f){{\mathbf{M}}_{ii}}(f)}}$ ${{\mathbf{M}}_{ii}}(f)$ is the minor of S(f) obtained by removing the ith row and column of S(f) and returning the determinant.	(Brillinger, 2001) Univariate quantity which measures the total coherence of variable i with all other measured variables.
Partial Directed Coherence Measures	Normalized Partial Directed Coherence (PDC)	$& {{\pi }_{ij}}(f)=\frac{{{A}_{ij}}(f)}{\sqrt{\sum\nolimits_{k=1}^{M}{{{\left\| {{A}_{kj}}(f) \right\|}^{2}}}}} \\ & 0\le {{\left\| {{\pi }_{ij}}(f) \right\|}^{2}}\le 1 \\ & \underset{j=1}{\overset{M}{\mathop \sum }}\,{{\left\| {{\pi }_{ij}}(f) \right\|}^{2}}=1 \\$	(Baccalá and Sameshima, 2001) Complex measure which can be interpreted as the conditional granger causality from j to i normalized by the total amount of causal outflow from j. Generally, the magnitude-squared PDC ${{\left\| {{\pi }_{ij}}(f) \right\|}^{2}}$ is used.
	Generalized PDC (GPDC)	$& {{{\bar{\pi }}}_{ij}}(f)=\frac{\frac{1}{{{\Sigma }_{ii}}}{{A}_{ij}}(f)}{\sqrt{{{\sum\nolimits_{k=1}^{M}{\frac{1}{\Sigma _{kk}^{2}}\left\| {{A}_{kj}}(f) \right\|}}^{2}}}} \\ & 0\le {{\left\| {{{\bar{\pi }}}_{ij}}(f) \right\|}^{2}}\le 1 \\ & \sum\limits_{j=1}^{M}{{{\left\| {{{\bar{\pi }}}_{ij}}(f) \right\|}^{2}}}=1 \\$	(Baccalá and Sameshima, 2007) Modification of the PDC to account for severe imbalances in the variance of the innovations. Theoretically provides more robust small-sample estimates. As with PDC, the squared-magnitude ${{\left\| {{{\bar{\pi }}}_{ij}}(f) \right\|}^{2}}$ is typically used.
	Renormalized PDC (rPDC)	${{\lambda }_{ij}}(f)={{Q}_{ij}}(f)*{{V}_{ij}}{{(f)}^{-1}}{{Q}_{ij}}(f)$ where ${{Q}_{ij}}(f)=\left( \begin{matrix} \operatorname{Re}[{{A}_{ij}}(f)] \\ \operatorname{Im}[{{A}_{ij}}(f)] \\ \end{matrix} \right)$ and ${{V}_{ij}}(f)=\underset{k,l=1}{\overset{p}{\mathop \sum }}\,R_{jj}^{-1}(k,l){{\Sigma }_{ii}}Z(2\pi f,k,l)$ $\begin{align} & Z(\omega ,k,l) \\ & =\left( \begin{matrix} \cos (\omega k)\text{cos(}\omega l)\text{ }\cos (\omega k)\sin \text{(}\omega l) \\ \sin (\omega k)\text{cos(}\omega l)\text{ sin}(\omega k)\sin \text{(}\omega l) \\ \end{matrix} \right) \\ \end{align}$ R is the [ ${{(Mp)}^{2}}$ x ${{(Mp)}^{2}}$ ] covariance matrix of the VAR[p] process (Lütkepohl, 2006)	(Schelter et al., 2009) Modification of the PDC. Non-normalized PDC is renormalized by the inverse covariance matrix of the process to render a scale-free estimator (does not depend on the unit of measurement) and eliminate normalization by outflows and dependence of statistical significance on frequency. To our knowledge SIFT is the first publically available toolbox to implement this estimator.
Directed Transfer Function Measures	Normalized Directed Transfer Function (DTF)	$& {{\gamma }_{ij}}(f)=\frac{{{H}_{ij}}(f)}{\sqrt{\sum\nolimits_{k=1}^{M}{{{\left\| {{H}_{ik}}(f) \right\|}^{2}}}}} \\ & 0\le {{\left\| {{\gamma }_{ij}}(f) \right\|}^{2}}\le 1 \\ & \sum\nolimits_{j=1}^{M}{{{\left\| {{\gamma }_{ij}}(f) \right\|}^{2}}}=1 \\$	(Kaminski and Blinowska, 1991; Kaminski et al., 2001) Complex measure which can be interpreted as the total information flow from j to i normalized by the total amount of information inflow to i. Generally, the magnitude-squared DTF ${{\left\| {{\gamma }_{ij}}(f) \right\|}^{2}}$ is used and, in time-varying applications the DTF should not be normalized.
	Full-Frequency DTF (ffDTF)	$\eta _{ij}^{2}(f)=\frac{{{\left\| {{H}_{ij}}(f) \right\|}^{2}}}{\sum\nolimits_{f}{\sum\nolimits_{k=1}^{M}{{{\left\| {{H}_{ik}}(f) \right\|}^{2}}}}}$	(Korzeniewska, 2003) A different normalization of the DTF which eliminates the dependence of the denominator on frequency allowing more interpretable comparison of information flow at different frequencies.
	Direct DTF (dDTF)	$\delta _{ij}^{2}(f)=\eta _{ij}^{2}(f)P_{ij}^{2}(f)$	(Korzeniewska, 2003) The dDTF is the product of the ffDTF and the pCoh. Like the PDC, it can be interpreted as frequency-domain conditional GC.
Granger-Geweke	Granger-Geweke Causality (GGC)	${{F}_{ij}}(f)=\frac{\left( {{\Sigma }_{jj}}-(\Sigma _{ij}^{2}/{{\Sigma }_{ii}}) \right){{\left\| {{H}_{ij}}(f) \right\|}^{2}}}{{{S}_{ii}}(f)}$	(Geweke, 1982; Bressler et al., 2007) For bivariate models (M = 2), this is identical to Geweke’s 1982 formulation. However, it is not yet clear how this extends to multivariate models (M > 2).