[Eeglablist] Units in output of timefreq - wavelet normalization

[Makoto Miyakoshi]

Dear Andreas, Niko, Norman, Mike, and the list,

What a detailed discussion!
Let me quickly confirm & summarize.

> The wavelets produced by dftfilt3 appear to always have an energy of
srate (thus they are *not* unit energy normalized?!):
[wavelet,cycles,freqresol,timeresol] = dftfilt3(F, ncycles, srate);
E = sum( abs( wavelet{ 1 } ) .^ 2 )
Consequently, to my understanding the correct „normfactor“ should be sqrt(
E ) or better sqrt( srate ).

I confirmed with your code that it's E ~= srate which is not normalized.
Yes, if it's all the same, it's better to be normalized using one of these
reasonable ways...

I discussed this normalization issue with Rey in Feb 2007 in SCCN as a
visitor (also asked the same question to Arno in August 2006 in email). It
was one of the most puzzling problems that I found that Herrmann's Morlet
equation was different from that of Tallon-Baudry's and Gruber's in the
normalization factor A; to make things worse, EEGLAB's timef()'s dftfilt()
was further different. Rey agreed with me, saying that he was puzzled too.
I was glad to know that I was not the one who was confused! Several months
before I met him, he had already written dftfilt3() to support the standard
Morlet wavelet in EEGLAB, for which I've been grateful.

Ok, let me confirm conclusions for the EEGLAB developers.

1) This is the bottom line: Let's not use the raw uV^2 outputs since these
changes due to non-interesting factors as Mike mentioned. Instead, let's
use dB changes against baseline etc.

2) If the raw uV^2 is absolutely necessary, there are probably two
reasonable ways to normalize the raw values. One is, in Andreas's sample
code, 'Analog to Gabor transform' in which A = 2.

3) The other good one is 'Unit energy' in which A =
sqrt(pnts./(sqrt(pi)*sigma_f). Probably, EEGLAB developers can implement
both as options and leave the choice to the users.

4) EEGLAB's explanation about dftfilt3() needs to be at least more
detailed, and potentially the default/optional normalization can be changes
to one of the above two, as long as it does not conflict with backword
compatibility.

Let me know if I'm wrong and/or missing something important.

As a list member, I'd like to express my deep gratitude for your depth of
knowledge and kindness to put so much effort on this discussion. I learned
a lot!

Makoto

On Thu, Aug 18, 2016 at 10:49 AM, Andreas Widmann <widmann at uni-leipzig.de>
wrote:

> P.S.: I forgot that I wanted to add to the final paragraph how to estimate
> the resulting amplitude of a sinusoid in the unit energy normalized wavelet
> transform: the energy of the wavelet with the corresponding frequency peak
> amplitude normalized to the RMS amplitude of the sinusoid, e.g. (for Niko’s
> example):
>
> mysig = 5 * sin(2*F*t*pi);
> [wavelet,cycles,freqresol,timeresol] = dftfilt3(F, ncycles, srate);
> temp = wavelet{1} /  max( abs( wavelet{1} ) ) * sqrt( mean( abs( mysig )
> .^ 2 ) );
> sum( abs( temp ) .^ 2 )
> sqrt( sum( abs( temp ) .^ 2 ) )
>
> Best,
> Andreas
>
> > Am 17.08.2016 um 21:07 schrieb Norman Forschack <forschack at cbs.mpg.de>:
> >
> > Dear Andreas,
> >
> > thanks a lot for your elaborate input! Below some further clarifications.
> >
> > First of all, I did not intend to further discuss question (1) as I
> found it quite convincing from your previous post that dftfilt3 is not
> computing results as we expect it to do.
> > But probably I should have put my initial question into other words
> (e.g. 'One question was' instead of 'The initial question was'), in order
> not to confuse things. So I am sorry for the mess!
> >
> > Coming to question (2) I do not fully agree here. Literally, Nico's
> questions were: 'what do values in tf represent and how should I
> "normalize" these values to match the amplitude of the input signal?' So, I
> was aiming to give some input on the second part. Below, using your code,
> Nico's signal is plotted together with both types of amplitude envelopes,
> i.e. gabor normalization and unit energy (see also picture attached). It's
> clear that gabor transform (i.e. maximal wavelet spectrum power)
> normalization closely matches the amplitude of the original signal, while
> the unit energy envelope shows much higher values. Unless there's a major
> bug in the code below (please check), I'd say gabor is more appropriate in
> this scenario and the normalization of a complex wavelet spectrum to
> maximal power is much more straight forward (see below).
> >
> > However, I totally agree that when it comes to EEG, things are
> different. As Makoto and Mike already pointed out before, we usually do not
> interpret raw EEG power, but rather refer to a baseline or another
> condition to compare with. In this case it shouldn't matter which wavelet
> normalization has been applied.
> >
> > The other points you mentioned, I found super super interesting and
> helpful and it would be great if you could comment on some emerging
> questions. Maybe it is also interesting for others to approach this
> intricate topic on a more detailed and basic level.
> >
> > You mentioned the gabor transform is rather suited for narrowband
> signals and EEG, however, is closer to white noise. But to my eyes EEG
> signals seem to be a lot more 'structured' e.g. when one thinks of the
> narrow spectral peaks of the rolandic rhythm in the alpha and beta band, or
> when a steady state evoked potential paradigm was applied.
> >
> > The gabor wavelet spectrum indeed seems to produce higher average
> amplitudes for higher frequencies, but the actual average values seem to
> vary much less than for the unit energy normalization, which then certainly
> does not produce 'flat' average amplitudes across frequencies. Do you think
> this is related to scaling issues?
> >
> > Unit energy normalization seems to increase frequency smearing for
> higher frequencies. Although one may apply a baseline normalization later,
> do you think this makes unit energy normalization less comparable to gabor
> normalization (i.e. contrary to what I've written above?)
> >
> > You said it doesn’t matter whether you normalize in the time or
> frequency domain. I'm confused now about this line of unit energy code:
> > A = sqrt( pnts ./ ( sqrt( pi ) * sigma_f ) );
> > Could you explain how sqrt(srate) normalization term transformed to the
> line above?
> > Also, when plotting the envelope based on this normalization, it yields
> an amplitude which is 25 times higher than in the original signal. When I
> normalized my complex wavelet with sqrt(srate)
> >> m =   exp(-t.^2/(2*s^2)).*exp(1i*2*pi*F.*t) ./ sqrt(srate);
> >
> > as you did before for the dft3filt wavelet
> >> wavelet_norm{ 1 } = wavelet{ 1 } / sqrt( srate );
> >
> > I got an envelope which was just 2.5 times higher than the original
> signal. So I guess there's an error in my code. could you elaborate on that
> please?
> >
> > I am very much looking forward to your comments, thanks a lot Andreas!
> > Best
> > Norman
> >
> > figure links:
> > .fig: https://www.dropbox.com/s/ktoffojfcxxu6vd/wavelet_
> normalization.fig?dl=0
> > .png: https://www.dropbox.com/s/v5i4d95y53t2woi/wavelet_
> normalization.png?dl=0
> >
> > clear
> > D = 4;       % total signal duration in seconds.
> > sigD = 1;    % duration of the test oscillation within the signal.
> > F = 10;      % frequency of the test oscillationin Hz.
> > P = .25;     % Phase of the test oscillation. 2 pi radians = 360 degrees
> > srate = 256; % sampling rate, i.e. N points per sec used to represent
> sine wave.
> > dt = 1/srate; % sampling period, i.e. for this e.g. points at 1 ms
> intervals
> > time = dt:dt:D; % time vector.
> >
> > sigpoints = length(time)/2 - (sigD*srate)/2:(length(time)/2 +
> (sigD*srate)/2)-1;
> > mysig = zeros(1,D*srate);
> > mysig(sigpoints) = sin(2*F*time(sigpoints)*pi+ 2*pi*P);
> >
> > figure; hold all
> > plot(time,mysig);
> >
> > cycles = 7;
> > pnts = D * srate;
> >
> > freqs = F * ( pnts / srate );
> > sigma_f = repmat( freqs / cycles, [ pnts 1 ] );
> > f = repmat( 0:pnts - 1, [ size( freqs, 2 ) 1 ] )' - repmat( freqs, [
> pnts 1 ] );
> > cmorlf = exp( -f .^ 2 ./ ( 2 * sigma_f .^ 2 ) );
> >
> > % Analog to Gabor transform
> > A = 2;
> > cmorlf_A = A .* cmorlf;
> >
> > tf = ifft( fft( mysig' ) .* cmorlf_A );
> >
> > plot( time, abs( tf ) .^ 2 , 'r-')
> >
> > % Unit energy
> >
> > A_unit = sqrt( pnts ./ ( sqrt( pi ) * sigma_f ) );
> > cmorlf_Au = A_unit .* cmorlf;
> >
> > tf_unit = ifft( fft( mysig' ) .* cmorlf_Au );
> >
> > plot( time, abs( tf_unit ) .^ 2, 'm:' )
> > legend({'Nico''s signal';'gabor';'unit energy'})
> >
> >
> >> ----- On Aug 17, 2016, at 3:05 PM, Andreas Widmann
> widmann at uni-leipzig.de wrote:
> >>
> >>> Dear Norman,
> >>>
> >>> sorry, I do not fully agree. First, I think it is important not to mix
> up the
> >>> two questions:
> >>>
> >>> (1) Does dftfilt3 provide the expected/documented results?
> >>> To my understanding this is not the case. I hope I do not miss
> anything obvious.
> >>>
> >>> vs. (2) What is the recommended/best/optimal wavelet normalization for
> EEG data
> >>> analysis?
> >>> I think there cannot be a generally valid recommendation and different
> solutions
> >>> are necessary for different applications. The normalization you
> suggested is
> >>> equivalent to the normalization of the Gabor transform (incorrectly
> abbreviated
> >>> to Gabor normalization in my previous post, sorry). This normalization
> is well
> >>> suited for narrowband signals and peaky spectra and the frequently
> used for
> >>> example in audio signal analysis. For broadband/noise signals,
> however, the
> >>> Gabor transform normalization overestimates high frequencies relative
> to low
> >>> frequencies.
> >>>
> >>> Please find below some code directly comparing both normalizations
> using a white
> >>> noise signal (btw. it doesn’t matter whether you normalize in the time
> or
> >>> frequency domain). In the global wavelet spectrum (middle column)
> power is
> >>> increasing with frequency for Gabor transform normalization (top row)
> but
> >>> „flat“ for unit energy normalization (bottom row).
> >>>
> >>> As EEG reflects a broadband and noisy signal I personally usually
> prefer to
> >>> apply energy normalization. As always, there are exceptions and other
> opinions
> >>> and special applications and ...
> >>>
> >>> Best,
> >>> Andreas
> >>>
> >>> fs = 256;
> >>> T = 4;
> >>> pnts = T * fs;
> >>> t = ( 0:pnts - 1 ) / fs;
> >>> cycles = 7;
> >>> F = 2:2:70;
> >>>
> >>> % White noise
> >>> rng( 0 )
> >>> signal = randn( fs * T, 1 );
> >>>
> >>> % Morlet wavelet
> >>> freqs = F * ( pnts / fs );
> >>> sigma_f = repmat( freqs / cycles, [ pnts 1 ] );
> >>> f = repmat( 0:pnts - 1, [ size( freqs, 2 ) 1 ] )' - repmat( freqs, [
> pnts 1 ] );
> >>> cmorlf = exp( -f .^ 2 ./ ( 2 * sigma_f .^ 2 ) );
> >>>
> >>> figure
> >>>
> >>> % Analog to Gabor transform
> >>> A = 2;
> >>> cmorlf = A .* cmorlf;
> >>>
> >>> tf = ifft( repmat( fft( signal ), [ 1 length( F ) ] ) .* cmorlf );
> >>>
> >>> h = subplot( 2, 3, 3 );
> >>> imagesc( t, F, abs( tf' ) .^ 2 )
> >>> set( h, 'YDir', 'normal' );
> >>> subplot( 2, 3, 2 );
> >>> plot( F, mean( abs( tf ) .^ 2 ) )
> >>> subplot( 2, 3, 1 )
> >>> plot( (0:pnts / 2) * fs / pnts, cmorlf( 1:pnts / 2 + 1, : ) )
> >>>
> >>> % Unit energy
> >>> A = sqrt( pnts ./ ( sqrt( pi ) * sigma_f ) );
> >>> cmorlf = A .* cmorlf;
> >>>
> >>> tf = ifft( repmat( fft( signal ), [ 1 length( F ) ] ) .* cmorlf );
> >>>
> >>> h = subplot( 2, 3, 6 );
> >>> imagesc( t, F, abs( tf' ) .^ 2 )
> >>> set( h, 'YDir', 'normal' );
> >>> subplot( 2, 3, 5 );
> >>> plot( F, mean( abs( tf ) .^ 2 ) )
> >>> subplot( 2, 3, 4 )
> >>> plot( (0:pnts / 2) * fs / pnts, cmorlf( 1:pnts / 2 + 1, : ) )
> >>>
> >>>> Am 15.08.2016 um 18:19 schrieb Norman Forschack <forschack at cbs.mpg.de
> >:
> >>>>
> >>>> Dear all,
> >>>>
> >>>> I'd like to contribute from the perspective of a discussion on Mike
> Cohen's
> >>>> Blog.
> >>>>
> >>>> The initial question was, how to obtain an amplitude envelope of a
> given signal
> >>>> which has in fact the same amplitude as the given signal, right?
> >>>> So coming from Nicos signal:
> >>>>
> >>>> clear all
> >>>> D = 4;       % total signal duration in seconds.
> >>>> sigD = 1;    % duration of the test oscillation within the signal.
> >>>> F = 10;      % frequency of the test oscillationin Hz.
> >>>> P = .25;     % Phase of the test oscillation. 2 pi radians = 360
> degrees
> >>>> srate = 256; % sampling rate, i.e. N points per sec used to represent
> sine wave.
> >>>> T = 1/srate; % sampling period, i.e. for this e.g. points at 1 ms
> intervals
> >>>> time = T:T:D; % time vector.
> >>>>
> >>>> sigpoints = length(time)/2 - (sigD*srate)/2:(length(time)/2 +
> (sigD*srate)/2)-1;
> >>>> mysig = zeros(1,D*srate);
> >>>> mysig(sigpoints) = sin(2*F*time(sigpoints)*pi+ 2*pi*P);
> >>>>
> >>>> one way to obtain equal amplitudes is to normalize the wavelet by its
> maximal
> >>>> value within the frequency domain:
> >>>>
> >>>> % some preparations
> >>>> mysig = mysig';
> >>>> ss = size(mysig);
> >>>> cycles = 4;
> >>>> dt = 1/srate;
> >>>> sf = F/cycles;
> >>>> s = 1./(2*pi*sf);
> >>>> t = (-4*s:dt:4*s)';
> >>>> nM = length(t);
> >>>> halfMsiz = (nM-1)/2;
> >>>> hz = linspace(0,srate/2,floor(nM/2)+1);
> >>>> Ly = ss(1)*ss(2)+nM-1;
> >>>> Ly2=pow2(nextpow2(ss(1)*ss(2)+nM-1));
> >>>>
> >>>> % fft of signal
> >>>> X=fft(reshape(mysig,ss(1)*ss(2),1), Ly2);
> >>>>
> >>>> % building morlet wavelet (without a normalization factor)
> >>>> m =   exp(-t.^2/(2*s^2)).*exp(1i*2*pi*F.*t);
> >>>> H = fft(m,Ly2); % fft of wavelet
> >>>>
> >>>> % normalize wavelet spectrum
> >>>> H = H./max(H);
> >>>>
> >>>> y =  ifft(X.*H,Ly2);
> >>>> y = y(floor(halfMsiz+1):Ly-ceil(halfMsiz));
> >>>> y_amp = 2* abs(y);
> >>>>
> >>>> figure; plot(time,mysig,'b',time,y_amp,'r')
> >>>>
> >>>> This seems to work for any combination of srate and cycles (except
> when number
> >>>> of cycles become large) because the signal spectrum is convolved by
> spectral
> >>>> wavelet values being maximally one.
> >>>>
> >>>> I have not fully worked my way through Andreas' example. It
> normalizes the
> >>>> wavelet in time, not in frequency domain as here. So it is probably
> not
> >>>> comparable.
> >>>> But doing time domain normalization within the lines above by just
> replacing the
> >>>> kernel formula:
> >>>>
> >>>> m =   exp(-t.^2/(2*s^2)).*exp(1i*2*pi*F.*t) ./ sqrt(srate); % unit
> energy
> >>>>
> >>>> and commenting out the max(H) normalization, however, yields an
> amplitude
> >>>> envelope which is 2.5 times larger than the original signal amplitude
> and
> >>>> increases when the number of wavelet cycles is increased. But as
> Andreas
> >>>> suggestion referred to the dftfilt3 output, the matter becomes more
> complicated
> >>>> as this function uses it's own normalization factor:
> >>>> A = 1./sqrt(s*sqrt(pi));
> >>>> and there are problably some more relevant differences (not even
> going into the
> >>>> timefreq function).
> >>>>
> >>>> In sum, this post may have fostered the general confusion (or at
> least mine) but
> >>>> for a more puristic approach to the matter of wavelet normalization,
> the lines
> >>>> above might be of some value (kudos to mike x cohen, of course).
> >>>>
> >>>> All the best
> >>>> Norman
> >>>>
> >>>>
> >>>> ----- On Aug 12, 2016, at 6:06 PM, Andreas Widmann
> widmann at uni-leipzig.de wrote:
> >>>>
> >>>>> Dear Niko,
> >>>>>
> >>>>> I’m puzzled by this difference since a long time too (and as you
> have written a
> >>>>> book chapter on WT actually I would have hoped you could help
> resolving this
> >>>>> issue ;).
> >>>>>
> >>>>> (Morlet) wavelet normalization always appeared somewhat arbitrary to
> me (as
> >>>>> signal amplitude will never be directly reflected across the whole
> TF plane for
> >>>>> peaky spectra/time courses). To my understanding the most common
> normalization
> >>>>> for wavelets is unit energy (and Gabor). The help text for timefreq
> states that
> >>>>> dftfilt3 is "exact Tallon Baudry“. TB (1998, JNeurosci) states that
> "Wavelets
> >>>>> were normalized so that their total energy was 1,…“.
> >>>>>
> >>>>> The wavelets produced by dftfilt3 appear to always have an energy of
> srate (thus
> >>>>> they are *not* unit energy normalized?!):
> >>>>> [wavelet,cycles,freqresol,timeresol] = dftfilt3(F, ncycles, srate);
> >>>>> E = sum( abs( wavelet{ 1 } ) .^ 2 )
> >>>>> Consequently, to my understanding the correct „normfactor“ should be
> sqrt( E )
> >>>>> or better sqrt( srate ).
> >>>>>
> >>>>> You might want to confirm by looking at the TF transform of the
> (real part of
> >>>>> the) wavelet itself
> >>>>> wavelet{ 1 } = wavelet{ 1 } / sqrt( srate ); % Unit energy normalize
> >>>>> mysig = zeros(1,D*srate);
> >>>>> delay = ceil( ( length( mysig ) - length( wavelet{ 1 } ) ) / 2 );
> >>>>> mysig( delay:delay + length(  wavelet{ 1 } ) - 1 )  = real( wavelet{
> 1 } ) * 2;
> >>>>> % Discard imag part
> >>>>> normfactor = sqrt( srate );
> >>>>> which should now have a peak amplitude of 1 (independent of sampling
> rate and
> >>>>> signal duration).
> >>>>>
> >>>>> As the issue appears to be not only in EEGLAB but also other
> implementations, I
> >>>>> always assumed my reasoning to be incorrect. Is it? What am I
> missing?
> >>>>>
> >>>>> Best,
> >>>>> Andreas
> >>>>>
> >>>>>> Am 10.08.2016 um 11:58 schrieb Niko Busch <niko.busch at gmail.com>:
> >>>>>>
> >>>>>> Dear Makoto (and everyone who replied to me personally regarding
> this post),
> >>>>>>
> >>>>>> thank you for your reply! I see that the result of the wavelet
> transform inside
> >>>>>> the timefreq function is dependent on the length of the signal,
> which in turn
> >>>>>> is dependent on the number of cycles and sampling rate. However,
> simply
> >>>>>> dividing by the length of the wavelet does not seem to be the
> solution either.
> >>>>>> I modified the code below by including a "normalization factor",
> which
> >>>>>> currently is simply the length of the wavelet. Dividing the wavelet
> transformed
> >>>>>> amplitudes by this factor gives the right order of magnitude, but
> the results
> >>>>>> are still quite off. By increasing the sampling rate or number of
> cycles, the
> >>>>>> results are even more off. I believe we are on the right track, but
> something
> >>>>>> is still missing. Do you have any ideas?
> >>>>>>
> >>>>>> Cheers,
> >>>>>> Niko
> >>>>>>
> >>>>>> %% Create sine wave
> >>>>>> clear all
> >>>>>> D = 4;       % total signal duration in seconds.
> >>>>>> sigD = 1;    % duration of the test oscillation within the signal.
> >>>>>> F = 10;      % frequency of the test oscillationin Hz.
> >>>>>> P = .25;     % Phase of the test oscillation. 2 pi radians = 360
> degrees
> >>>>>> srate = 256; % sampling rate, i.e. N points per sec used to
> represent sine wave.
> >>>>>> T = 1/srate; % sampling period, i.e. for this e.g. points at 1 ms
> intervals
> >>>>>> t = [T:T:D]; % time vector.
> >>>>>>
> >>>>>> sigpoints = length(t)/2 - (sigD*srate)/2:(length(t)/2 +
> (sigD*srate)/2)-1;
> >>>>>> mysig = zeros(1,D*srate);
> >>>>>> mysig(sigpoints) = sin(2*F*t(sigpoints)*pi+ 2*pi*P);
> >>>>>>
> >>>>>> %% TF computation
> >>>>>> ncycles = 4;
> >>>>>>
> >>>>>> [wavelet,cycles,freqresol,timeresol] = dftfilt3(F, ncycles, srate);
> >>>>>> normfactor = length(wavelet{1});
> >>>>>>
> >>>>>> [tf, outfreqs, outtimes] = timefreq(mysig', srate, ...
> >>>>>>   'cycles', ncycles, 'wletmethod', 'dftfilt3', 'freqscale',
> 'linear', ...
> >>>>>>   'freqs', F);
> >>>>>>
> >>>>>> %% Plot
> >>>>>> figure; hold all
> >>>>>> plot(t,mysig);
> >>>>>> plot(outtimes./1000,abs(tf)./normfactor)
> >>>>>> xlabel('Time (seconds)');
> >>>>>> ylabel('Amplitude');
> >>>>>> legend('input signal', 'wavelet result')
> >>>>>>
> >>>>>> _______________________________________________
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-- 
Makoto Miyakoshi
Swartz Center for Computational Neuroscience
Institute for Neural Computation, University of California San Diego
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