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 obtained in section 3.3. , with the exception of the renormalized PDC (rPDC). The rPDC requires estimating the 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 |
|
(Brillinger, 2001) (f) is the spectrum for variable i. is the cross-spectrum between variables i and j. |
Coherence Measures | Coherency |
|
(Brillinger, 2001) Complex quantity. Frequency-domain analog of the cross-correlation. The magnitude-squared coherency is the coherence . 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) |
|
(Nolte et al., 2004) The imaginary part of the coherency. This was proposed as a coupling measure invariant to linear instantaneous volume-conduction. only if the phase lag between i and j is non-zero, 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 magnitude-squared 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 small-sample estimates. As with PDC, the squared-magnitude 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)
|
Full-Frequency DTF (ffDTF) | (Korzeniewska, 2003)
| ||
Direct DTF (dDTF) | (Korzeniewska, 2003)
| ||
Granger-Geweke | Granger-Geweke Causality (GGC) | (Geweke, 1982; Bressler et al., 2007)
|