{\footnotesize\hfill[/jw/misc/roger/globdesc.tex --- \today]}

{\bf Hi, REG folks,}
\bigskip

the following is a brief rationale of my strategy of global assessment
of `multichannel data', i.~e.\ data recorded simultaneously from multiple
spatially distributed sites but measured on the same physical scale.
The background for the strategy presented here was the need to condense
information from several seconds of multichannel EEG data to few
physically meaningful state variables.  A copy of my original article
where the strategy was presented is enclosed; note, however, that there's
a lot of garbage not worth of reading (I had to explain to general audience
the meaning of simple mathematical operations, so the text is a compromise
between a tutorial and a research paper).  The following information should
be enough to implement computation of the global descriptors.
\medskip

Assume we have a series of $N$ vectors, each consisting of simultaneous
measurements at $K$ sites (e.~g.\ electrodes on the scalp or REGs on the
Earth surface): $\{ \vec{u}^{(0)}, \vec{u}^{(1)}, ..., \vec{u}^{(N-1)}\}$, sampled
in equidistant time steps $\Delta t = 1/f_{samp}$.  In other words,
$u^{(n)}_i$ is the value recorded at the $i$-th measuring site at time
$t=n \cdot \Delta t$.

Now, compute the following two mean values
\begin{eqnarray}		\label{EmZeroDef}
	m_0&=&\frac{1}{N} \sum_{n=0}^{N-1} \| \vec{u}^{(n)} \|^2	\\
	m_1&=&\frac{1}{N-1} \sum_{n=1}^{N-1}
	                \left\|\frac{\Delta\vec{u}^{(n)}}{\Delta t}\right\|^2
\end{eqnarray}
where $\| \vec{u} \|^2 \equiv \sum u_i^2$ is the squared
euclidean norm of vector $\vec{u}$, and
$\Delta\vec{u}^{(n)} \equiv \vec{u}^{(n)} - \vec{u}^{(n-1)}$
is the difference between the $n$-th and the previous data vector.

The two descriptors, $\Sigma$ und $\Phi$, are then defined as follows
\begin{eqnarray}
	\Sigma&=&\sqrt{\frac{m_0}{K}}	\\
	\Phi&=&\frac{1}{2\pi}\sqrt{\frac{m_1}{m_0}}
\end{eqnarray}
Obviously, $\Sigma^2$ is a measure of the global (averaged over space
and time) variance --- or `power' if the input data are electrical measurements
(EEG) --- while $\Phi$ is a measure of dominant frequency of the changes
(to get it more intuitively, imagine that the time series of vectors of
measurements draws a circular trajectory in the $K$-dimensional state space).
If the physical dimension of the measurements is $X$, then the dimension
of $\Sigma$ is $X$ as well, and the dimension of $\Phi$ is time$^{-1}$
(the $1/2\pi$ factor in the second equation above serves just to transform
the circular frequency [radians/time] to a wavenumber frequency 1/time.)

The third descriptor, $\Omega$, is a measure of spatial synchronization
among the $K$ measuring sites.  First, compute the covariance matrix
\begin{equation}
	{\bf C} = \frac{1}{N} \sum_{n=0}^{N-1} \vec{u}^{(n)} \vec{u}^{(n)}'
\end{equation}
that is
\begin{equation}
	c_{ij} = \frac{1}{N} \sum_{n=0}^{N-1} u^{(n)}_i u^{(n)}_j 
\end{equation}
Then compute the eigenvalues $\lambda_1,\dots,\lambda_K$ of the matrix
$\bf C$, and normalize the eigenvalues to a unit sum
\begin{equation}
	\lambda'_i = \frac{\lambda_i}{T}, \mbox{ where }
	                              T = \sum_{i=1}^K \lambda_i
\end{equation}
Note that the sum of eigenvalues $T$ is equal to the trace of the
covariance matrix tr {\bf C}$ = \sum c_{ii}$ and thus also to the
quantity $m_0$ defined above in (\ref{EmZeroDef}).  Finally, compute
\begin{equation}
	\log \Omega = - \sum_{i=1}^K \lambda'_i \log \lambda'_i
\end{equation}
In my article of 1995, $\Omega$ is defined as $\exp()$ of the
expression above; the (silly) idea behind was to scale $\Omega$ from
1 thru $K$, to make it more intuitive to physiologists who don't like
logarithms too much.  Well, forget it --- $\log \Omega$ is just fine,
attains values from 0 (maximal synchronization = minimal complexity)
through $\log K$ (minimal synchronization = maximal complexity).
\medskip 

Introducing the $\Omega$ complexity into EEG was no tremendous
intellectual achievement, of course; it is just one of many members
of a family of so-called {\it covariance complexities}, used in pattern
recognition business etc.  What was new was the use of those three
global descriptors together to characterize dynamical properties of
a spatially distributed systems; thus, the triples $(\Sigma, \Phi, \Omega)$
represent points in (macro)state space.  Physiologically similar states
project to subregions of this space, and changes in the working mode
of the brain manifest themselves as paths, making fine variations in
the global brain functional states to appear as filamentar or layered
manifolds.  For example, if a (putative) generator in the brain becomes
dominant over the other ongoing activity, the total variance $\Sigma$
increases {\it and} the $\Omega$ decreases as the activities recorded
from different locations become more and more correlated --- thus such
a change will display a different transition path than if the same goes
on in the brain, just with higher voltage.  The idea that emerged in
our talks with Roger was to try to apply this methodology to REGG data. 

Some questions arise, though; the following is a brief informal discussion,
no dogmatic opinion:
\begin{description}
  \item[What is the input data?]  With multichannel EEG, the primary data
	consists of a series of voltage measurements in $\mu$volts.
	With REGG data, we can use bitsums, relative frequencies of 1's,
	$Z$ values, chi-squares, whatever\dots\\
	IMHO the best candidates are relative frequencies of 1's, i.~e.\
	$r = \mbox{bitsum}/\mbox{bitcount}$, as these are invariant (provided
	the REG behaves constantly) against change of the bit count per time
	slice.  If time series are present in a form of $Z$ scores,
	these should be transformed back to relative frequencies.
  \item[Data centering.]  In the formulas above, data is tacitly assumed
	to be centered to zero baseline; the question is, whether we should
	center the data by subtracting the empirical mean value over given
	epoch, or subtracting the theoretical expected value; I~strongly
	prefer the latter option, i.~e.\ to subtract 0.5 from the observed
	relative frequencies; it is deviations from the theoretical baseline
	what we want to assess.
  \item[Sampling frequency.]  This is closely related to the abovementioned
	issue of bit counts per time slice and, on the other hand, to the
	expected {\it rate} of changes we want to assess.  Here I~have no
	clue; we'd touched the latter issue in a table talk in Halifax, but
	as far as I~remember without any definite conclusion.  I~guess the
	best way would be to sum up trials to combined time slices over
	a wide time scale, say, ranging from seconds to hours.\\
	(There was no problem with EEG: the sampling frequency is more or
	less enforced by the frequency content of brain's `humming' --
	I~have no idea at which frequency (if any) Gaia may be humming.)
  \item[Epoch length.]  While the previous comment was dealing with combining
	simultaneous REG data to a single data vector and the `base frequency'
	of the observed fluctuations, this one relates to the issue of
	{\it stationarity\/} of the fluctuations and also of {\it stability\/}
	of our estimates.\\
	With EEG, we were using epochs of length from a second to few seconds
	(brain's electrical activity is notoriously known to be non-stationary
	over longer time periods what regards its frequency contents as well
	as the spatial distribution of activities).  Using typical sampling
	frequencies of order of $10^2$ simultaneous measurements in a second,
	we had $N \approx 10^2 \div 10^3$.\\
	With REG data, I~have no {\it a priori\/} idea and think we have to
	experiment a bit again.  There must be a lower bound for $N$, anyway;
	the number of data vectors have to be large enough to allow for a good
	estimate of the covariance matrix {\bf C}, and thus for a reliable
	estimate of its eigenvalues.  I'm no expert in multivariate statistics,
	so I~can't tell the rule.  If anyone of you will want to elaborate
	this a bit, I'd be highly grateful!
  \item[Statistics.]  If there is no structure in the single REG data,
	the deviations of relative 1's frequencies $r$ from 0.5 will be
	normally distributed and uncorrelated in time (gaussian white noise).
	Confidence intervals should be specified for $\Sigma$ and $\Phi$,
	given the count of data vectors per epoch $N$ and the bit count
	per data point.  This should be an easy excercise in $\chi^2$
	statistics (admittedly, I~never did it as we {\it know\/} that EEG
	is no gaussian noise).\\
	If there is no structure in the multiple REG data, the theoretical
	covariance matrix {\bf C} is a diagonal matrix with equal values
	along the diagonal, thus the normalized spectrum of eigenvalues
	is uniform ($\lambda'_1 = \cdots \lambda'_K = \frac{1}{K}$) and
	the complexity attains its maximum $\log K$; however, we are dealing
	with sample estimates of {\bf C} based on a finite number $N$ of data
	vectors and, consequently, the distribution of the eigenvalues of the
	sample covariance matrix $\hat{\bf C}$ will not be exactly uniform.
	In other words, we have to expect {\it lower\/} empirical values of
	$\log \Omega$ even if these don't indicate a {\it significant}
	departure from `sphericity' of the data (no correlations between
	distant sites).  Would someone be so kind and derive the proper
	formula for a confidence interval for $\log \Omega$ with given
	$N$ and $K$?  I'd be much grateful again!
  \item[Presentation.]	The easy way are the three traces, $\Sigma$, $\Phi$
	and $\Omega$ along time axis, whatever our time resolution may be.
	However, as I~alluded to above, the strategy occured to be most
	useful with 3D displays of a `macro-state space' with coordinates
	$\Sigma$, $\Phi$ and $\Omega$, showing the state representations
	grouping and clustering.  This might be a useful approach even here;
	as for the axes, I~incline to $\log\Sigma$, $\log\Phi$, $\log\Omega$;
	as argued above, $\log\Omega$ is more `natural' than $\Omega$ by its
	original definition; for the former two, I~prefer logarithms just
	because transforming changes by multiplicative factors to equidistant
	shifts.
\end{description}
That's it\dots  I~think it's worth trying (and not abandoning too early, perhaps
we need a certain minimum number of REGs distributed around to observe any
reasonable patterns).  I'll be looking (well, with one eye, the lab business
takes a lot of time) forward to the results.
\bigskip

Jiri Wackermann {\tt <jw@igpp.de>}
