How to score/measure the change in a time series from t-x,...t-1 to t0 (Relative change)

Question

I think about having data such as:

Time   1   2   3   4   5   6   7   8   9 ... n

Set1  12  13  13  23  26  27  40  13  13
Set2   2   3   4   2   4   3  10  12  11 
.
.
.
Setn ...

All datasets are equivalent in terms of the meaning what a sudden (relative) change in values indicate. Means when in set1 is a change from 13 to 23 it indicates the same as when there is a change from 3 to 10 in set2. It only depends on the relative change of the actual t0 to the average of the past x periods.

My question is what is the best way to quantify such change? My goal is to have a value in each period for each dataset that indicates the change. This value should be comparable to the other datasets. Means when I have a change from 13 to 23 the value should approximately similar to a change from 3 to 10.

Would appreciate any advice!

Thanks and Best :)

Answer 1

To capture the relative effect i would go for x_t - x_t-1 / x_t-1 If you want the same result in your example x_t-1 = 13 x_t = 23 & z_t-1 = 3 z_t = 13 you could just go for the difference x_t - x_t-1

And if you want to put everything to respect of some stylized statistic, you could just change the denominator x_t-1 to t0 (t0 being median, mean, whatever you like)