I am looking for a similarity measure between series which exploits the time dimension (rather than just its "mapping" aspect), and was unsatisfied with everything I found.
First an example: assume I have daily data about expenditure of each of two indviduals, and I want to see if there is some imitation process by which when one spends more, the other does too - and possibly vice-versa - but maybe not on the same exact day.
So here are my requirements:
- the same event (e.g. a spike in spending) happening to both series on different days is "less similar" than if it happens on the same day
- ... but it is more similar if the two days are close than if they are distant
- all days have the same importance (with the possible exception of boundary effects)
- the measure is possibly linear, in the sense that distance doesn't change if I increase the same term of the two series by a same amount
- bonus points if the measure is elegant, intuitive, parameter free, and applicable to small series
Examples I have looked at:
- correlation, Spearman correlation, cosine similarity, Kendall rank correlation... don't satisfy 2.
- Dynamic Time Warping doesn't satisfy 1.
- absolute distance doesn't satisfy 2., and Euclidean distance doesn't even satisfy 4.
- I briefly fell in love with the Euclidean distance between cumulative sums... but it doesn't satisfy 3. (if I add 1 to the first term of a constant series it increases/decreases distance from another given series much more than if I add 1 to the last term)
- fitting functions and comparing parameters has problems (to the best of my knowledge) with 4. and 5.
I am currently considering a version of dtw which penalizes "misalignment", or (which might end up being the same thing) a version of the Euclidean distance between cumulative sums where the impact of each difference is somehow weighted depending on its position. But something inside me is telling "there must be a standard solution out there". Moreover, there is always a tradeoff between penalizing difference in levels and in timing, and I wonder whether there is a particularly meaningful choice to make.