# Estimating the power in EEG frequency bands

Christian Hesse c.hesse at FCDONDERS.RU.NL
Thu Apr 19 10:58:23 CEST 2007

```Hi
>     A late follow-up to this topic. I have recentrly been musing
> over how to
> get a "clean" measure of the non-phase locked activity. I have tried
> subtracting the ERF out prior to time-frequency computation but this
> produces quite a bit of artifact...presumably since the single
> trial data
> will have considerable ;atency "jitter"

The ERF collapses two sources of "jitter"; in the latency of the
transient activity (if it exists) and the phase of ongoing
oscillatory activity.

>    The comments from Christian below make sense ( I think) why simply
> subtracting the two time-frequency power representaions is not
> valid. But I
> wonder would this subtractive approach be valid if one worked with the
> magnitude of the signal rather than power..omitting all the
> squaring operations?

Computing the magnitude is still a non-linear operation (square root
of a sum of squares, rectification, whatever ... ). The problem for
why this won't work either resides in averaging part: in the evoked
case you have a linear average followed by a non-linear operation,
and in the induced case you have an average of the non-linearly
transformed quantity. The "catch phrase" here is: the sum of the
squares is not the same as the square of the sum! (or the sum of the
rectified data is not the same as the rectified sum)

Hope this helps,
Christian

>   If this right theoretically, how to achieve this in Fieldtrip?.
> Would
> setting cfg.output = 'fourier then abs'ing the output work. My
> suspicion is
> no since the summing is being done first here. Alternatively, does
> one need
> to hack the code to return the magnitude.
>
> Thanks for your help on this and sorry for waking old threads :)
>
>    - Suresh
>
> On Fri, 23 Feb 2007 01:44:59 +0100, Christian Hesse
> <c.hesse at FCDONDERS.RU.NL> wrote:
>
>> One further comment (please see below):
>>
>>> Hi Thomas,
>>>> Following up on this conversation. It seems that the ‘induced
>>>> activity’ contains both phase-locked and non-phase-locked
>>>> activity, whereby the ‘evoked’ activity contains only phase-locked
>>>> activity. Is it then kosher to separate these components by linear
>>>> subtraction? For example, if we first compute the ‘induced’
>>>> activity by averaging power over individual trials, and from that
>>>> subtract the ‘evoked activity’ (calculated based on average
>>>> response) to get the induced activity without any phase-locked
>>>> activity?
>>>
>>> It is not correct to subtract because computing the induced and
>>> evoked power spectra involves squaring signal amplitudes (a non-
>>> linear operation), and hence, taking your terminology to refer to
>>> the instantaneous amplitudes of the signal components (this applies
>>> to any time-frequency tile)
>>>> Induced = Phase + Non-Phase
>>>>
>>>> And
>>>>
>>>> Evoked = Phase
>>>>
>>>> Then
>>>>
>>>> Non-Phase = Induced – Evoked
>>>>
>>>>
>>> what you actually get from spectral or time-frequency analysis is
>>> the power of your MEASURED signal
>>>
>>> Induced^2 = (Phase + Non-Phase)^2 = Phase^2 + 2*Phase*Non-Phase +
>>> Non-Phase^2
>>>
>>> Evoked^2 = Phase^2
>>>
>>> Then
>>>
>>> Induced^2 - Evoked^2 = 2*Phase*Non-Phase + Non-Phase^2  AND NOT Non-
>>> Phase^2
>>>
>> Note that the other crucial thing to consider here is that you are in
>> one case averaging power over trials over trials:
>>
>> E[ (Induced^2)  ] =  E[ (Phase + Non-Phase)^2 ] = E[ (Phase^2 +
>> 2*Phase*Non-Phase + Non-Phase^2) ] = E[ (Phase^2) ] E[ (Non-
>> Phase^2) ] + E[ 2*Phase*Non-Phase ]
>>
>> this is why taking the square root of sqrt(Induced^2) does not give
>> (Phase + Non-Phase) but sqrt(E[ (Phase+Non-Phase)^2 ]).
>>
>> in the evoked case you are taking the power of the average amplitude
>>
>> Evoked^2 = E[ Phase ]^2  (---> note the ^2 on the outside of the sum)
>>
>> so in subtracting you are actually assuming that E[Phase]^2 = E
>> [(Phase)^2] which is unlikely to be accurate the case in finite
>> samples.
>>
>> Hope I have not confused others (or myself) here.
>> Christian
>>
>>
>> ---------------------------------------------------------------------
>> -
>> Christian Hesse, PhD, MIEEE
>>
>> F.C. Donders Centre for Cognitive Neuroimaging
>> P.O. Box 9101
>> NL-6500 HB Nijmegen
>> The Netherlands
>>
>> Tel.: +31 (0)24 36 68293
>> Fax: +31 (0)24 36 10989
>>
>> Email: c.hesse at fcdonders.ru.nl
>> Web: www.fcdonders.ru.nl
>> ---------------------------------------------------------------------
>> -
>>
>>
>>
>>
>> Date:         Thu, 22 Feb 2007 15:52:56 +0100
> Sender:       FieldTrip discussion list <FIELDTRIP at NIC.SURFNET.NL>
> From:         Marcel Bastiaansen <Marcel.Bastiaansen at FCDONDERS.RU.NL>
> Subject:      Re: Estimating the power in EEG frequency bands
> Content-type: text/html; charset=ISO-8859-1
>
> Dear Thomas,
>
> This is indeed the approach that I have followed succesfully a
> couple of
> times (e.g. Bastiaansen et al., JOCN 2006), although the
> terminology that
> you are using is somewhat confusing. I (and I guess most people)
> would refer
> to induced activity as that part of the EEG that is non-phase-
> locked, so I
> would restate your equation to:
> induced = EEG - evoked.
>
> However, there is a drawback to this approach, since it assumes
> that the ERP
> is absolutely stationary over trials. This is not the case in
> reality (e.g.
> subjects' attentional level or other states may change from trial
> to trial,
> giving rise to variability in the single-trial ERPs). This means
> that by
> subtracting the average ERP, one may introduce frequency components
> in the
> residual EEG that were not present before. Klimesch, and Kalcher and
> Pfurtscheller, have come up with ways of scaling the average ERP so
> as to
> yield a best fit of the average with each single-trial ERP, but
> also that
> approach may be sub-optimal.
> My latest way around the problem is to run a TF analysis on the
> untreated
> EEG (containing both evoked and induced activity), and comparing
> this to a
> TF analysis of the subject-averaged ERPs (the evoked activity alone).
> Qualitative differences between the two analyses can now only be
> attributed
> to induced activity.
>
> Marcel
>
> Thomas Thesen wrote:
>>
>> Hi FieldTrippers,
>>
>>
>>
>> Following up on this conversation. It seems that the ‘induced
>> activity’
> contains both phase-locked and non-phase-locked activity, whereby the
> ‘evoked’ activity contains only phase-locked activity. Is it then
> kosher to
> separate these components by linear subtraction? For example, if we
> first
> compute the ‘induced’ activity by averaging power over individual
> trials,
> and from that subtract the ‘evoked activity’ (calculated based on
> average
> response) to get the induced activity without any phase-locked
> activity?
>>
>>
>>
>> So if
>>
>> Induced = Phase + Non-Phase
>>
>> And
>>
>> Evoked = Phase
>>
>> Then
>>
>> Non-Phase = Induced – Evoked
>>
>>
>>
>> Or does the fact that this is a linear operations on data that
>> have been
> constructed through a non-linear process render this somehow
> invalid? It has
>>
>>
>>
>> Cheers,
>>
>> Thomas
>

----------------------------------------------------------------------
Christian Hesse, PhD, MIEEE

F.C. Donders Centre for Cognitive Neuroimaging
P.O. Box 9101
NL-6500 HB Nijmegen
The Netherlands

Tel.: +31 (0)24 36 68293
Fax: +31 (0)24 36 10989

Email: c.hesse at fcdonders.ru.nl
Web: www.fcdonders.ru.nl
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