Transcript CoherencePooya.ppt

Magnitude Squared
Coherence
Pooya Pakarian
Program in Neural Computation
Carnegie Mellon University
Magnitude Squared Coherence is the cosine squared of the angle between two
vectors. Each vector represents the Fourier transform of a given frequency
across several epochs of a time series signal. As is the output of Fourier
transform, the elements of these vectors are complex numbers. Calculating the
Magnitude Squared Coherence is like calculating the correlation coefficient
squared between two vectors of zero-mean stationary complex time series.
Coherence analysis between two signals
5 Hz, magnitude=1 volt, initial phase=-60° (w/r/t a cosine wave)
3 seconds
Cos(-60°)=0.5
magnitude modulation
Both amplitude and
phase modulation,
the negative
multiplier adds (or
subtracts) 180° to
(from) the phase.
Step 1: cut the signals to several snippets, conduct DFT on each.
DFT of 5 Hz:
magn=1 volt
phase=-60°
magn=1 volt
phase=-60°
magn=1 volt
phase=-60°
magn=2.48 volt
phase=-60°
magn=4.02 volt
phase=-60°
magn=2 volt
phase=+120°
Hint: the lower signal does have non-zero energy
in other frequencies as well, but in this example
we are only studying its 5 Hz component.
Hint: I chose the phase modulation in the lower signal
to occur at the start of its third snippet, but this is not
necessarily the case in real world practice.
Two complex vectors represent the DFT of 5 Hz of the two signals
Signal #1
Signal #2
magn=1 volt
phase=-60°
magn=2.48 volt
phase=-60°
magn=1 volt
phase=-60°
magn=4.02 volt
phase=-60°
magn=1 volt
phase=-60°
magn=2 volt
phase=+120°
An arbitrary example of maximum coherence (=1)
Signal #1
Signal #2
magn= 7 volt
phase=-60°
magn= 21 volt
phase=-40°
magn= 2 volt
phase=-30°
magn= 6 volt
phase=-10°
magn= 5 volt
phase=+90°
magn= 15 volt
phase=+110°
A consistent divisive relation between the magnitudes and a
consistent subtractive relation between the phases
Step 2: the numerator of coherence: inner product of the two complex vectors
each element of vector 1 multiplied in the conjugate
magnitudes multiplied
of their corresponding element in signal 2
Signal #1
Signal #2
magn=1 volt
phase=-60°
magn=2.48 volt
phase=-60°
magn=1 volt
phase=-60°
magn=4.02 volt
phase=-60°
magn=1 volt
phase=-60°
magn=2 volt
phase=+120°
1 2.48  e
1 4.02  e
1 2  e
 
i(  )
3 3
 
i(  )
3 3
 2
i(  )
3 3
Phases subtracted
magn=2.48
phase=0°
magn=4.02
phase=0°
magn=2 volt
phase=-180°
Vector
summation
The subtractive effect of the phase:
The magnitudes are always positive, so are their products. The inequality
of the phase difference, but, may force them to be subtracted from each
other so that even the numerator of coherence becomes zero.
magn=4.5
phase=0°
the numerator of coherence
Step 3: the denominator of coherence: the product of the sizes of complex vectors
Signal #1
Signal #2
magn=1 volt
phase=-60°
magn=2.48 volt
phase=-60°
magn=1 volt
phase=-60°
magn=4.02 volt
phase=-60°
magn=1 volt
phase=-60°
magn=2 volt
phase=+120°
12  12  12
The divisive effect of the magnitude:
Assume that all the phase differences are equal to zero, so in the numerator
we would have the inner product of two all positive vectors, that is always less
than the denominator calculated here or equal to it, but can never be zero.
2.482  4.022  22
Step 4: (magnitude squared) coherence= numerator times its
conjugate divided by the square of the product of the sizes
magn=4.5
phase=0°
12  12  12
magn=4.5
phase=-0°
2.482  4.022  22
2
= 0.5065
Note the conjugation of phase in the numerator. In this particular example that the phase
happened to be zero, its effect is not much apparent, but if the it is very helpful by forcing
the result to be a real number with zero imaginary part because the phase would be
subtracted from itself, and thus the result would be equal to the magnitude squared.
An example from the literature
Wang S, Aziz TZ, Stein JF, Bain PG, Liu X. 2006 Jul;117(7):1487-98. Clin Neurophysiol.
Physiological and harmonic components in neural and muscular coherence in Parkinsonian tremor.
As you see, the two signals are divided into “nd” segments. The key point is that, in the
numerator, the cross spectrum for each single frequency (represented by f above) is first
summed across segments and then its modulus is calculated and raised to the power of two.
In the denominator, in contrast, the complex numbers of auto-spectrum are first subject to
calculation of their modulus and being squared, and then the summation is performed. These
are complex numbers, therefore, the modulus of their sum can easily be much less
than the sum of their modulus, i.e. no surprise that the numerator becomes less than the
denominator, leading into a coherence less than “1”. Note that the segment-wise
multiplication and then summation in the numerator is like the inner product of two
multidimensional complex vectors..
Minor point: the modulus function in the denominator was not necessary, because the power of two is implemented by multiplication
of a complex number by its own conjugate that anyhow zeros out the phase and simply raises the magnitude to the power of two.
Further readings (the same method)
Srinivasan R, Winter WR, Ding J, Nunez PL.
EEG and MEG coherence: measures of functional connectivity at distinct spatial scales
of neocortical dynamics. J Neurosci Methods. 2007 Oct 15;166(1):41-52.
Pfurtscheller G, Andrew C
Event-Related changes of band power and coherence: methodology and interpretation.
J Clin Neurophysiol. 1999 Nov;16(6):512-9.
Farmer SF, Bremner FD, Halliday DM, Rosenberg JR, Stephens JA.
The frequency content of common synaptic inputs to motoneurones studied during
voluntary isometric contraction in man. J Physiol. 1993 Oct;470:127-55.
(This article, among others, cites Bloomfield 1976 for its method and also
mentions, as is also implied from the present series of slides, that smoothing of
the spectrograms during the process of coherence analysis is simply to reduce
the high frequency noise, it is not a genuine part of coherence analysis).
Halliday DM, Rosenberg JR, Amjad AM, Breeze P, Conway BA, Farmer SF.
A framework for the analysis of mixed time series/point process data--theory and
application to the study of physiological tremor, single motor unit discharges and
electromyograms. Prog Biophys Mol Biol. 1995;64(2-3):237-78.
Coherence and other analyses in the mixed time series/point process data!