Chapter 4.3. A partial list of VARbased 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 obtained in section 3.3. , with the exception of the renormalized PDC (rPDC). The rPDC requires estimating the inverse crosscovariance 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 lowlevel function est_mvtransfer()
.
Table 4. A partial list of VARbased spectral, coherence, and information flow / GC estimators implemented in SIFT.
Estimator  Formula  Primary Reference and Notes  
Spectral M.  Spectral Density Matrix 

(Brillinger, 2001) (f) is the spectrum for variable i. is the crossspectrum between variables i and j. 
Coherence Measures  Coherency 

(Brillinger, 2001) Complex quantity. Frequencydomain analog of the crosscorrelation. The magnitudesquared coherency is the coherence . The phase of the coherency can be used to infer laglead relationships, but, as with crosscorrelation, this should be treated with caution if the coherence is low, or if the system under observation may be closedloop. 
Imaginary Coherence (iCoh) 

(Nolte et al., 2004) The imaginary part of the coherency. This was proposed as a coupling measure invariant to linear instantaneous volumeconduction. only if the phase lag between i and j is nonzero, or equivalently,  
Partial Coherence (pCoh) 

(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, can regarded as the conditional coherence between i and j with respect to all other measured variables.  
Multiple Coherence 

(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) 

(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 magnitudesquared PDC is used. 
Generalized PDC (GPDC) 

(Baccalá and Sameshima, 2007) Modification of the PDC to account for severe imbalances in the variance of the innovations. Theoretically provides more robust smallsample estimates. As with PDC, the squaredmagnitude is typically used.  
Renormalized PDC (rPDC) 

(Schelter et al., 2009)
 
Directed Transfer Function Measures  Normalized Directed Transfer Function (DTF) 

(Kaminski and Blinowska, 1991; Kaminski et al., 2001)

FullFrequency DTF (ffDTF)  (Korzeniewska, 2003)
 
Direct DTF (dDTF)  (Korzeniewska, 2003)
 
GrangerGeweke  GrangerGeweke Causality (GGC)  (Geweke, 1982; Bressler et al., 2007)
