# power law distribution of time events

Suppose you have the logs of a web server. In these logs you have tuples of this kind:

user1, timestamp1

user1, timestamp2

user1, timestamp3

user2, timestamp4

user1, timestamp5

...

These timestamps represent e.g. users' clicks. Now, user1 will visit the site multiple times (sessions) during the month, and you'll have bursts of clicks from each user during each session (supposing that when a user visits your site, he'll click on multiple pages).

Suppose you want to partition these burst of clicks in the sessions that generated them, but you don't have any additional source of information, only the list of timestamps. If you compute the distribution of intervals between two consequent clicks from the same user, you will obtain a power law distribution. Intuitively, you'd look for a "cut parameter", e.g. N seconds, where if timestamp_{i+1} - timestamp{i} > N, then you timestamp_{i+1} is the beginning of the new session.

The problem is that this power law distribution in reality is a mixture of two variables: X = "interval between two consequent clicks in the same session" and Y = "interval between the last click of the previous session and the first of the new one".

The question is, how to estimate this N, that divides the two distributions (with a bit of overlap, possibly) just by looking at the burst of clicks?

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Welcome to MO! You might get better responses to your question on a different site; for example a site on statistics in the stackexchange network. The focus here is current research in mathematics, and in practise the interests and expertise of most participants are even to the purer end of math. So, this question does not fit in too well. (I do not vote to close it, this is purely meant as suggestion and explanation.) – user9072 Nov 14 '12 at 17:13
thanks, will do that! – user28071 Nov 22 '12 at 0:50