# User-Specific Activity Level Along Weeks

I am assigning Low | Medium | High activity levels to users once a week.

At the end of an entire period,a user has been assign n weekly activity levels among {Low, Medium, High}.

Let k be the total number of users, then the data table would look like this:

User   Week_1   Week_2   ...   Week_n
User_1 Low      Low      ...   High
User_2 Medium   Low      ...   Low
.      .        .        .     .
.      .        .        .     .
.      .        .        .     .
User_k High     Medium   ...   Medium


How would you formulate a metric within [0,1] that quantifies the "stickiness" of user, that is, the variability of it's activity label over the entire period.

With such a metric, a user which is assigned always the same activity level (let's say, n times "High"), would have a 100% metric.

• Does the order matter? That is, would a sequence of High-Low-High-Low be equal to 0.5? So the same as High-High-Low-Low? – user2974951 Feb 12 at 9:26
• In the second example there is only one transition to different level, whereas in the first one there are three transitions to different levels so I would expect the metric of the second one to be higher than the first one. – Ronicho Feb 12 at 15:36
• In that case you need a measure that takes into effect the sequence of events, variance will not do that. – user2974951 Feb 13 at 14:19

I assume that Low-Medium or High-Medium combination are more stick than Low-High ones. That is, that are ordered so the "distance" from Low to High is bigger than to Medium. I also assume that the order does not matter: High High Low will have the same stickiness than High Low High

If this is it, 1. You can assign 1 to Low, 2 to Medium, 3 to High 2. calculate the variance for each row 3. 100% would be 0 variance, 0% would be the variance of the vector (1,3,1,3...) 4. Map your variances to the range given in 3.

Best!

• Many Thanks! It's a great solution. I was digging in the directions of "purity measures" (classification) which assumes to tag every user with an activity level (like the most frequent vote per user) and can be subject to mispecifications, hence my doubts. – Ronicho Feb 12 at 12:19