# econometrics: weighting frequency over time

I want to measure a model like:

Score = arunning_sum_of_all_exercises + brunning_sum_of_identical_exercise + c*exercise_density_value_in_period_x --other variables --

Lets say I want to test progress in certain related exercises. The more I do each exercise, the better I will be in doing them. And the more exercises I do in a row are also important. If I practice over and over again in a short period of time, lets say n times , my score will rise at a faster pace compared to a longer period of time where I also have exercised n times. Therefore, the density of recent exercises plays a role.

How do I statistically measure this effect best? I thought of a geometric series where the time difference between the last x exercises has an diminishing effect on the variable measuring the density.

variable measuring density = exercise_density_value_in_period_x exercise_density_value_in_period_x = timediff(t)+(1-a)*timediff(t-1)+(1-a)^2*timediff(t-2).....

where t stands for the current exercise and t-n for the n-th exercise before t

Example:

exercise done at

Monday 12:00 (t-5)

-time inbetween: 5min-

Monday 12:05 (t-4)

-time inbetween: 5min-

Monday 12:10 (t-3)

-time inbetween: 10min-

Monday 12:20 (t-2)

-time inbetween: 360 min-

Monday 18:20 (t-1)

-time inbetween: 420 min-

Tuesday 01:20 (t)

The density for the first three exercises was the highest and then dropped. The result on Monday 12:10 is likely better than the result on Tuesday at 01:20, because I have done three exercises in a short period of time on Monday but more time has passed in the case of the exercise done on Tuesday 01:20. How much time lied between the first three exercises has a smaller effect on the score on Tuesday than the exercise right before (which is Monday 18:20). How do I measure this effect and how can I estimate it in statistical software like stata? Is OLS my way to go here?

Thanks for your help, I appreciate it if you took your time to read this