Time Decay Formula for Conversion Attribution Question:

What is the Time Decay formula that web analytics packages use to distribute credit across the multiple sessions associated with a user before a conversion?

Context:

All web analytics packages like Google Analytics rely on the concept of:


USERS visitors, ie unique cookied browsers. Users have many...
SESSIONS visits, ie sets of pageviews within close time proximity.
CONVERSIONS  a success event such as a sign-up which can occur during a session.


These analytics packages allow you to distribute the credit of a single conversion across all the preceding sessions because each played a part in leading the user to conversion. One such model is Time Decay, which gives a diminishing share of credit to sessions further from the conversion.

Time Decay Visual: (1st session) ▁ ▂ ▃  ▅ ▆ (conversion)

more info on attribution models.

Sample Data:

I am assuming the data required would be:


n - the total number of sessions a user has before a conversion
t - the total duration of time from the user's first session to conversion
tbc - the individual time before conversion for each session 


So for three different sample users, data may look like this:


UserID  n   t   Session#    Timestamp   TBC Conversion in session?
User1   4   12  1           2015-10-01  12  No
User1   4   12  2           2015-10-06  7   No
User1   4   12  3           2015-10-11  2   No
User1   4   12  4           2015-10-13  0   YES
User2   3   4   1           2015-10-09  4   No
User2   3   4   2           2015-10-09  4   No
User2   3   4   3           2015-10-13  0   YES
User3   1   0   1           2015-10-14  0   YES
Here's one solution:

using the following data:

USER-LEVEL:


fss first session start (timestamp) - time of first pageview of
first session for that user
css converting session start (timestamp) - time of the first pageview of session where user converted
ns number of sessions (int) - total # of sessions
before conversion for user


SESSION-LEVEL:


ss session start (timestamp) - timestamp of first pageview in session


FORMULA: calculated for each session


FOR FIRST SESSION: Attribution Credit is just linear: (or really whatever you want to make it)  
ALL OTHER SESSIONS: 


First calculate the SUM of all session's "time before conversion":
Then Calculate the credit for each session:



it isn't perfect because it distributes credit based on the relative position of a session in the User's lifetime.  Ideally you have a preset halflife (like 7 days) so that comparison is more consistent.