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While having chronically data of population growth (registered users of a site), I want to compute a function that approximates future growth, based on past data. Also, what we ll be the distribution of that function? What is the distribution of interarrivals between consecutive registrations.

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You should check out the forecast package in R, specifically the ets() and auto.arima() functions. Both of these will fit a model to your historic data, and allow you to forecast into the future with confidence intervals. Note IrishStat's cautions about using a holdout sample to verify your forecasts as well as the particulars of ARMAX modeling (which auto.arima() can handle).

This will get you 80% of the way to a good forecast with little effort from you. To get 100% of the way there, you may need to spend more time developing your own arima model, rather than grabbing one out of the box.

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You are observing transactions of people becoming registered users. These transactions can be bucketed /grouped into time intervals or time buckets. Develop either a causative model or a mwmory +fixed effects model (ARMAX/Transfer Function). Ensure that this model has a Gaussian Error process to the best of your ability. Use this equation to forecast future buckets, Do not model the cumulated history as it can mask the behavior. After developing the forecasts for the next k intervals simply accumulate those forecasts and the history to date to privide an expectation for the future for total registrations. You might review How to do prediction from a linear regression? as I reflected on a cumulative box-office forecast for the movie "Alice". There is absolutely no need to have a model for these cumulative series as you can readily develop a model for the time buckets. Developing a model for the time between arrivals is not a subject that I can comment on, but I am sure others on the board will do so.

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