Algorithm for Estimating Number of Unique Monthly Visitors Is there a way to estimate the number of unique monthly visitors to a site based on a limited sample of one week of data? I have information about when a given user visited the site. This isn't as simple as just multiplying the number of unique visitors the first week by 4, due to the hotel problem. If 10 people visit your site the first week and the same people are the only visitors to your site the second, third, and fourth week, the total number of monthly unique visitors to your site is only 10.
I know you can use HLL to estimate the number of unique visitors to a site in O(1) space. I'm wondering if there's a similar approach to estimate how many unique visitors there will be after a month, preferably that also works in O(1) space.
After digging into this for a bit, I came across this paper which provides a solution. The paper gives an approach to estimating the number of new species that will be observed given an initial sampling period. It models observing a given species as a Poisson distribution. It gives the following estimator for the number of new species that will be discovered:
$$
\hat{\Psi}(t)=\sum_{k=1}^{k_{m a x}} N_{k} e^{-k}-\sum_{k=1}^{k_{m a x}} N_{k} e^{-k(1+t)}
$$
where $N_{k}$ is the number of species that were observed $k$ times and $t$ is the length of the second sample relative to the initial sample.