One representative time series signal from multiple time series data

Question

I came across this basic problem in DSP which is not my area owf expertise but I consider this problem to be a common problem which others might be addressing.

I have multiple time series data and I need to find a time series data that is a representative of all those data. One method that I am applying is averaging the signals but I am sure there will be some other sophisticated methods to do the same.

What are the other methods that I can apply?

Thanks

Answer 1

Decision of representative is usually formulated as an optimization problem. The optimal representative minimizes sum of some dissimilarity measures from the observed signals. The average you indicated is optimal in the sense of minimization of Euclidean distance. If you want to minimize feet of perpendiculars, you can use the first principal component.

Now the problem is how to choose the dissimilarity measure. In signal processing, we often consider some probabilistic model and use the dissimilarity measure proportional to the negative log likelihood. This is often referred to as maximum likelihood estimation. For example, Euclidean distance corresponds to the negative log likelihood of mean parameter of Gaussian distribution. Even using the same Gaussian distribution, there can be various ways of modeling, and various dissimilarity measures are derived such as Mahalanobis distance and Itakura-Saito distance.