# Global variability index for group of signals

Suppose I have a method that I can use to generate $$n_p$$ signals (we can intend them as realizations of an unknown not stationary discrete-time stochastic process). Modifying the method, I can obtain other $$n_p$$ signals.

I'd like to quantify the "variability" of the several group of $$n_p$$ generated signals.

The procedure I am using is the following: For each time step I have $$n_p$$ values and I compute the Interquartile Range (IQR). At this point I compute the average of the IQRs in order to obtain a single value that I can associate to that trial. Another approach is substituting the IQR with the range between 2.5th quantile and 97.5th quantile, but the procedure is the same.

Is this procedure correct? Does exist other more rigorous or correct way to quantify the global variability of the bunch of signals I have generated?

A related question is: variability among signals

• I'm not quite sure I understand what you're asking for correctly, but if you want to know how a change of an element in the parameter vector affects the output of a mapping, you're looking for the gradient of that mapping. This has, especially in optimization, a big meaning. Look for "backpropagation algorithms". If your mapping has multiple scalar outputs (in your case, it has $n_p$), then you'll get a vector of gradient vectors, i.e. a matrix ($n_{in}\times n_p$), where each entry tells you how the change in one input changes the value of one output. It's a common problem to find these – Marcus Müller Feb 27 at 12:39
• entries, and there's a wealth of methods to do that! The most commonly employed (probably) is gradient descent, or stochastic gradient descent, optimization of the parameters / weights of a network of nonlinear mappings of which calculating the local gradients is easy, optimizing the difference between the actual mapping's outputs and your network's outputs. – Marcus Müller Feb 27 at 12:44
• @MarcusMüller thank you for the answer but my question is more about the quantification of the "variability" of a bunch of signals than about the relation between parameters of my generating process and the obtained value. I described my idea but I don't know if it is the correct way to quantify "globally", in average, the variability of $n_p$ realizations of a stochastic process. – Linker Feb 27 at 17:17
• Your comment makes me think even more I got you correctly :) Point is: You have a stochastic process with a set of parameters, let's call them "inputs", which change the statistic properties of your process, which gives you a vector of statistic properties of dimension $n_p\times 1$, right? Then, let's call that statistics vector "output" And you wonder what the relation between the input. Does my comment make more sense now? – Marcus Müller Feb 27 at 17:58
• @MarcusMüller My question is: there exist a method/metric that I can use to quantify the variability inside the generated signals? The "metric" that I have used is the average of IQRs for all timesteps. There exist a more correct way? The relation to initial parameters I've mentioned in the question can be misleading and I've modified the question accordingly. – Linker Feb 28 at 15:56