# Combining Intermittent Demand and ARIMA

I have a time series dataset, where a customer may purchase fuel one week and not purchase again for 2-3 weeks. I need to forecast when a customer is likely to purchase and how much they will spend. As such, all the literature is pointing to using Croston's method for Forecasting Intermittent Demand, which I plan to do in SAS Forecast Studio.

However, my model for purchases needs to take in to account external factors e.g. price, weather and public holidays. Croston's method doesn't deal with external data and there appears to be no literature about combining 2 models.

My idea is that I could use Croston's method to forecast the interval between purchases, per customer. Then somehow combine this with an ARIMA forecast, which would give me an equation telling me how much that customer would spend.

Is this possible? Are there known techniques to do this a better way? How would I go about doing this?

TIA.

• Why do you think an ARIMA forecast will help here? – Skander H. Nov 16 '18 at 1:57
• I was thinking to predict overall spend with Arima, using external data such as weather, price and public holidays and then proportionally attribute that to customers who intermittently buy fuel. I was thinking to use Croston for the demand intervals and spend and then adjust the spend based on ARIMA's regression equation. I can find no literature where this problem has been solved before, so any suggestions are welcome!! – Mo van Praag Nov 16 '18 at 9:59

Is this possible?

In theory it be possible to combine Croston's and ARIMA. Croston's splits the time series into two component series (a demand interval series and a demand size series) and then uses exponential smoothing to forecast each series separately before recombining them into a rate of demand (or rate of sales) series. You could blend Croston's and ARIMA by using an ARIMA or ARMA model to forecast one or two of the components instead of exponential smoothing.

In practice it would be difficult though, since ARIMA requires a lot of data to be effective (otherwise exponential smoothing or a simple moving average works better). This means that since your data is intermittent, you would need to make up for that by having a very long series (keep in mind, by using Croston's you are effectively shortening your series - if you have 1 year of weekly data, i.e. 52 weeks, and you have a sales on average every two weeks, your series gets shortened to ~20 weeks which is very short for a time series to use ARIMA). So you need several years instead of just 2 or 3 years of weekly data. This is probably why you don't see the idea of mixing ARIMA and Croston's in the literature.

This problem is even more pronounced if you want to include external variables in your ARIMA model, because in Croston's you are rescaling your demand time series to a what is essentially a variable time scale. I don't see how can you rescale the external variables in such a way that it is synch with the component series that Croston's produces.

So yeah, combining Croston's and ARIMA would be very difficult to pull off.

Are there known techniques to do this a better way? How would I go about doing this?

Yes. For intermittent demand forecasting, you can use Negative Binomial distribution or Poisson distribution along with GLMs to model intermittent demand with causal factors.

• Thanks so much Alex! Looking in to Negative Binomial regression, it seems ideal for predicting low frequency events, along with causal factors. I just wonder if it is capable of predicting spend as well? Usually a value between $10 and$100 at each event. TIA. – Mo van Praag Nov 17 '18 at 21:49