Bigger Induced EEG than Total EEG problem

Hello everyone,
I am currently trying to get induced EEG by using the following equation :
Induced EEG = Total(Raw) EEG - trial averaged ERP

I got the result, but somehow in some trials Induced EEG power is bigger than the Total EEG power.
I think I got the code right, so I am completely lost on how to interpret the result.
Maybe I am misunderstanding how Induced EEG works, but shouldn’t Induced EEG power be smaller than the Total EEG power?

Any kind of insight is welcome!
Have a great day

YES

P.S. Following is the picture of my code and example of Induced power being bigger than Raw EEG power (around 1/3 of the trials show this pattern…)

Hi YES. Excellent observation. I know it seems weird that removing part of a signal can increase its spectral power. But it is possible and not that strange.

Keep in mind that the time-domain signals (ERP and single trials) contain energy from the entire spectrum (of course dominated by lower frequencies, but still the energy is distributed over a broad range). So the total energy can decrease while power in specific frequencies can increase. There is no requirement that the energy at each frequency must be smaller.

More importantly, the energy per frequency depends on the relative phases between the single trials and the ERP. So if the single trial and the ERP have opposing phases at some frequency (either because of noise or because of non-phase-locked signals), then subtracting the ERP will increase the amplitude on that trial. Here’s a quick simulation to illustrate the concept:

t=linspace(0,2*pi,30); plot(sin(t)-sin(t+pi))

Thank you so much Mike!

Thanks to your explanation I think I get why Induced EEG looks bigger than the Raw data.
I disregarded that ERP includes all the frequency spectrum from all the trials and because of that ERP and Total EEG could have different phase.

Have a great day!