STFT statistical analysis

Question

I am using the evolfft function in the RSEIS R package to do a STFT analysis of a signal.

The signal is one hour long and it was acquired during 3 different conditions, in particular 0-20' control, 20'-40' stimulus, 40'-60' after stimulus.

Visually, I see a change in the spectrogram during these 3 periods, with higher frequency and increased FFT power during the treatment, but I was wondering whether there was some kind of statistical analysis I could do to "put some numbers" on it.

Any suggestion?

EDIT: as suggested I will add an example of the data I'm dealing with

The treatment is between 20' and 40', as you can see it produces an increase in the power of the FFT over a fairly wide range of frequencies. I have 50-60 of these STFT for each experiment (for 10 total experiments). I can average the spectra for each experiments and still get a similar type of pattern. Now, my problem is how to correctly quantify the data I have and possibly do some stats to compare before, during and after the treatment.

Answer 1

I think the use of the spectrogram is visually interesting but not that obvious to exploit because of information redundency along frequencies. What we can see is that the changes between period are obvious. Also I would go back to the initial problem where you have for 3 different time periods indexed by $k=1,2,3$ a set of $n$ ($n=50$) signals of length $T>0$: $ i=1,\dots,n\; \; X^{k}_i\in \mathbb{R}^T$.

From this I would simply do a some sort of "Functional ANOVA" (or "multivariate ANOVA") :

$$X^{k}_i(t)=\mu_k+\beta_k(t)+\epsilon_{k,i}(t)$$

and test for difference in the mean i.e. test $\beta_1-\beta_2=0$ versus $\|\beta_1-\beta_2\|>\rho$.

You might be interested in this paper also this paper involves a different FANOVA modelling. The difficult point in your real case might be that all assumption that are made in these papers are false (homoscedasticity, or stationnarity, ...) and you might need to build a different "functional" test adapted to your problem.

Note that your idea of using multiscale analysis is not lost here because you can integrate it in the test (if I remember it is what is done in the first paper I mention).