Time series modeling the number of users of a mobile app

I want to model the number of users of an mobile app. This app has two kinds of users: free and paid. I thought of this autoregressive model:

$x_t = Ax_{t-1}$

with $x_t$ being a 4-dimensional vector, with quantities (all in period $t$) for the numbers of

• mobile users
• visitors of the app page
• free users
• paid users

I imagined $A$ as follows:

$A = \begin{bmatrix} a & 0 & b & c \\ d & e & 0 & 0 \\ 0 & 1-e & f & 0 \\ 0 & 0 & 1-(b+f) & g \end{bmatrix}$

Each parameter is the proportion of people in a group going to another group. Example: $f$ is the proportion of free users that remain free users in the next period; $d$ is the number of mobile users that visited the app page, and so on.

Problems I see with this model:

1. $b$ and $c$ are the proportion of users that leave the app. Not sure if it's nice to put them in the group of mobile users, or make another dimension for the group former users, that don't register back in the app.

2. The number of mobile users grow in time, so the way I thought to cope with this is by not demanding that $a$ and $d$ sum is 1. Not sure if it's valid.

So, the question is: how can I make a better model that I could feed the numbers after the launch of the app and get good estimates for these parameters, and good estimates for future numbers of users?

• Would be great to share experiences if we ever come to cope with our similar problems. Write to roland dot kofler at gmail if you want to share – Roland Kofler Jun 21 '12 at 7:35
• Did you ever receive any feedback on this or everyone left you dead in the water? – Ted Taylor of Life Aug 16 '16 at 1:26
• Dead in the water for four years, @TedTaylorofLife :) – Lucas Reis Aug 16 '16 at 3:28