Biomedical Engineering Reference
In-Depth Information
coeffi cients for each signal pair, one at a time; thus, they are likely to be immune
from a shortage of computational resources. However, they cannot distinguish
between
direct
and
indirect
interactions, i.e., whether the information fl ow between
two signals is mediated by a third signal (Kus et al.
2004
). This can be overcome
by using multivariate measures, in which VAR coeffi cients from multiple (>2) sig-
nals are estimated simultaneously. However, signifi cantly more computational
resources will be required when the number of signals increases. Thus, dimension
reduction techniques, such as ICA, are essential for data mining with a large num-
ber of channels.
Granger-Geweke causality (GGC), or frequency-domain GC, is a commonly
used bivariate spectral effective connectivity measure (Geweke
1982
; Bressler
et al.
2007
). Thorough discussions of GGC in neurophysiological data can be
found in (Ding et al.
2006
; Scott and Kalaska
1997
). For multivariate effective con-
nectivity, we will focus primarily on three measures: partial directed coherence
(PDC) (Baccala and Sameshima
2001
), direct directed transfer function (dDTF)
(Korzeniewska et al.
2003
), and renormalized PDC (rPDC) (Schelter et al.
2009
).
The commonly used PDC and dDTF are multivariate extensions of the GC concept
and can be interpreted as frequency-domain conditional GC, in which all signals
are taken into account to disentangle direct and indirect infl uences among signal
pairs. rPDC is a scale-free measure that is independent of the units and variance of
the signals and can be used to assess the strength of direct infl uences (Schelter
et al.
2009
).
3.2.4
Statistical Signifi cance
The probability distributions of the activity features illustrated earlier are often not
known. Nonparametric statistical methods are usually employed to test their statisti-
cal signifi cance, to allow further inferences. This is particularly important and is
usually standard for computing the signifi cance level of connectivity measures.
Here, we will introduce some common numerical methods for computing statistical
signifi cance and discuss their reliability, particularly on connectivity measures.
Numerical Approaches for Statistical Signifi cance
Two methods are commonly used to obtain confi dence intervals and statistical sig-
nifi cance thresholds for activity features: bootstrap resampling (Efron and Tibshirani
1993
) and the leave-one-out method (LOOM) (Schlögl and Supp
2006
). Bootstrap
resampling is a numerical method that is used to obtain a signifi cant estimate by
generating surrogate data through randomly resampling the data with replacement.
This sampling with replacement is repeated many times (100 or 1,000) to approxi-
mate the true distribution. LOOM, which is based on jackknifi ng (Quenouille
1949
),
