[Eeglablist] eeglab causal filtering: impacts and applications?

[Andreas Widmann]

Hi Cedric,

in general most of your reasoning is correct and reasonable: Non-linear causal filters typically should only be used if the application explicitly requires this, for example is causality matters as in the detection of onset latencies (even if the problem is overestimated as it mainly affects ultra-sharp transients typically not observed in EEG/ERP), the analysis of small fast components before large slow components (e.g. if higher high-pass cutoff frequencies are required), or in the analysis of pre-stimulus activity, that is, your case.

There is, however, one problematic aspect in your reasoning which if resolved might possibly help you finding the right answer for you : You *cannot* delay correct a causal filter (linear or non-linear). This would make it automatically non-causal and break causality and there is no way around this.

Now you still have to decide whether to apply a linear or a non-linear filter. The difference between a linear causal and linear non-causal filter is exclusively the time axis. The output of the non-causal filter equals the delay corrected output of the causal filter. It is sufficient to change the EEG.times time axis. That is, if your signal of interest is further away from stimulus onset than the group delay you can simply use a linear non-causal filter.

% Example:
sig = [ 0 0 0 0 0 1 0 0 0 0 0 ]; % test signal (impulse)
b = [ 1 1 1 1 1 ] / 5; % some crude boxcar filter for demonstration purposes only, linear-phase, length = 5, order = 4, group delay = 2
fsig = filter( b, 1, sig ); % causal filter
plot( -5:5, [ sig; fsig ]', 'o' ) % the filtered impulse in the output does not start before the impulse in the input

fsig = filter( b, 1, [ sig 0 0 ] ) % padded causal filter
fsig = fsig( 3:end ); % delay correction by group delay, this is what makes the filter non-causal and zero-phase
plot( -5:5, [ sig; fsig ]', 'o' ) % the filtered impulse in the output starts before the impulse in the input BUT everything before x = -2 is unaffected

If the (group) delay is too large for your purpose you must reduce the delay and thus use a non-linear filter, e.g. minimum phase. Non-linear filters delay different frequencies by a different amount (and due to this difference you cannot easily delay correct the output of non-linear filters even if non-causality would be acceptable). Therefore, you must not interpret the shape of the waveform in ERPs as it is distorted to some extent and also not interpret cross-frequency relationships of amplitude or phase in ERSPs. But on the other hand you can be certain that your pre-stimulus data are not contaminated by post-stimulus activity.

> 2) Can I use the standard linear zero-phase filter for low-pass filtering after a causal filter as the paper showed that low-pass filtering using causal filters introduces large delays?
> My tests did not show any difference on the ERP plots compared to using only the causal filter without low-pass filtering but I haven't run more thorough tests.
No, I do not think that this is a valid strategy. The output of the non-linear high-pass in the first step (if I correctly understand your question) would only introduce a small delay. The non-causal effects of the zero-phase (delay corrected) low-pass will in most cases exceed the delay introduced by the high-pass. Thus, you have a non-linear *and* non-causal filter output.

% Check:
b_high = minphaserceps( firws( 10, 1/5, 'high' ) ); % minimum-phase non-linear high-pass
fsig_high = filter( b_high, 1, sig ); % causal filter
plot( -5:5, [ sig; fsig_high ]', 'o' ) % Causality preserved :)

fsig_high_low = filter( b, 1, [ fsig_high 0 0 ] ); % the low-pass from above
fsig_high_low = fsig_high_low( 3:13 ); % delay correction/zero-phase
plot( -5:5, [ sig; fsig_high_low ]', 'o' ) % Causality violated :(

> 3) the minimum-phase filter is called "beta" in eeglab GUI, can I confidently use it for my analysis and publication?
I didn’t notice or hear of any known errors in the last years. I would consider it safe for productive use.

> 4) If I need to apply a delay correction (for either causal filter), how can I do it?

You cannot without making the filter non-causal, see above.

Side note: I recommend not to use the legacy filter. It is broken. It would be relatively simple to use the new filter for causal linear filtering on the command line (just take the filter coefficients and use the regular MATLAB filter command; see above).

Hope this helps! Best,
Andreas

> Am 29.01.2021 um 22:05 schrieb Cedric Cannard <ccannard at protonmail.com>:
> 
> Hi all,
> 
> I am analyzing 64-channel EEG data of the pre-stimulus period [-1500 0] ms (3 conditions from 3 types of images being presented) and results will be obtained using LIMO on the ERP and ERSP data. I have been reading Andreas Widmann's papers on filtering (e.g. https://urldefense.com/v3/__https://www.frontiersin.org/articles/10.3389/fpsyg.2012.00233/full__;!!Mih3wA!VNo4GYqFILb305TK2eZbXRZWcQcgOnHzFo8Zfb510jeSOjaJCydWh45s-vE0funMJm7jiA$ ; https://urldefense.com/v3/__https://home.uni-leipzig.de/biocog/eprints/widmann_a2015jneuroscimeth250_34.pdf__;!!Mih3wA!VNo4GYqFILb305TK2eZbXRZWcQcgOnHzFo8Zfb510jeSOjaJCydWh45s-vE0fukqNL9gVw$ ), but I am still unsure I fully understand whether I should use the legacy causal FIR filter and apply a delay correction or the new minimum-phase causal filter instead. I also found this eeglablist discussion (https://sccn.ucsd.edu/pipermail/eeglablist/2013/006638.html) but it is from 2013 and there is now a beta minimum-phase causal filter available in EEGLAB, so
>  I wonder what the new recommendations would be.
> 
> I am pretty clear about using a non-linear/causal filter and not linear/non-causal filter to avoid "introducing non-causally smeared artificial or artificially enhanced components after high-pass filtering (Fig 7D; see Acunzo et al., 2012 for discussion), or smearing post-stimulus oscillations into the pre-stimulus interval leading to spurious interpretations of pre-stimulus phase (Zoefel and Heil, 2013)."
> However, regarding the causal filtering, I read: "zero-phase (non-causal) filters preserve peak latencies, while causal filters necessarily shift the signal in time. If a causal filter is needed, a non-linear minimum-phase filter should be considered as it introduces only the minimum possible delay at each frequency for a given magnitude response but distorting broadband or complex signals due to non-linearity (see Fig. 2). Causal high-pass minimum-phase (and other non-linear) filters introduce rather small delays (Fig. 2F and G) while causal low-pass (and band-pass and band-stop) filters introduce larger delays even with minimum-phase property (Fig. 2A and B), which is why they are not recommended in electrophysiology (Rousselet, 2012)."
> 
> --> This suggests I should use the minimum-phase for high-pass filtering (1 Hz) but not for low-pass filtering (50 Hz).
> But further in the paper the authors say: "However, as FIR filter coefficients necessarily must be symmetric (or antisymmetric) for the filter to have linear-phase characteristic ([Rabiner and Gold, 1975](https://urldefense.com/v3/__https://www.frontiersin.org/articles/10.3389/fpsyg.2012.00233/full*B6__;Iw!!Mih3wA!VNo4GYqFILb305TK2eZbXRZWcQcgOnHzFo8Zfb510jeSOjaJCydWh45s-vE0fukJiu_LIA$ ); [Ifeachor and Jervis, 2002](https://urldefense.com/v3/__https://www.frontiersin.org/articles/10.3389/fpsyg.2012.00233/full*B2__;Iw!!Mih3wA!VNo4GYqFILb305TK2eZbXRZWcQcgOnHzFo8Zfb510jeSOjaJCydWh45s-vE0ful963Bv-w$ )), this reduction of filter delay comes at the cost of a non-linear phase response and the introduction of a systematic delay in the signal (which can not easily be compensated due to non-linear phase). The recommendation for minimum-phase causal FIR filtering, thus, should be strictly limited to the detection of onset latencies and applications where causality is required for theoretical con
> siderations."
> 
> --> This suggests the minimum-phase filter not only introduces delays that are harder to compensate for compared to the causal FIR filter, but also that it distort complex and broadband signals? Does this mean the minimum-phase filtering might not necessarily be better than the legacy causal FIR filter for ERP/ERSP analyses?? My tests comparing the two during high-pass 1 Hz show extreme differences in the ERP plots for the same subject, obviously.
> 1) Which one should I use for ERP and ERSP analyses of the pre-stimulus period?
> 2) Can I use the standard linear zero-phase filter for low-pass filtering after a causal filter as the paper showed that low-pass filtering using causal filters introduces large delays?
> My tests did not show any difference on the ERP plots compared to using only the causal filter without low-pass filtering but I haven't run more thorough tests.
> 3) the minimum-phase filter is called "beta" in eeglab GUI, can I confidently use it for my analysis and publication?
> 4) If I need to apply a delay correction (for either causal filter), how can I do it?
> 
> Thanks in advance to anyone that could bring me more clarity on this topic.
> 
> Cedric
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