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# Chapter 14 ADAM for Intermittent Demand

So far we have discussed data that has regular occurrence. This is a characteristic of a well established product that is sold every observation. For example, daily sales of bread in a supermarket would have this regularity. However, there are time series, where non-zero values do not happen on every observation. In the context of demand forecasting, this is called “**Intermittent demand**”. The conventional example of such demand is monthly sales of jet engines: they will contain a lot of zeroes, when nobody buys the product and then all of a sudden several units, again followed by zeroes.

One of the simplest definitions of intermittent demand is that *it is the demand that happens at irregular frequency*. While at a first glance it might seem that it is an exotic problem, intermittent demand can be encountered in many areas, when the frequency of measurement is high enough. For example, daily sales of a specific type of tomatoes in a store might exhibit regular demand, but the same sales on hourly or minute frequency would exhibit intermittence. So, the problem is universal and might appear in almost any context.

Sometimes the term “count data” (or “integer-valued data”) is used in a similar context, but there is a difference between this term and intermittent data. “Count data” implies that demand can take integer values only and can be typically modelled via Poisson, Binomial or Negative Binomial distributions. It does not necessarily contain zeroes and does not explicitly allow demand to happen at random. If there are zeroes, then it is assumed that they are just one of the possible values of a distribution. In case of intermittent demand, we explicitly acknowledge that demand might not happen, but if it happens then the demand size will be greater than zero. Furthermore, intermittent demand does not necessarily need to be integer-valued. For example, daily energy consumption for charging electric vehicles would typically be intermittent (because the vehicle owners do not charge them every day), but the non-zero consumption will not be integer. Having said that, count distributions can be used in some cases of intermittent demand, but they do not necessarily always provide a good approximation of complex reality.

Before we move towards the proper discussion of the topic in context of ADAM, we should acknowledge that at the heart of what follows, there lies the following model (Croston 1972): \[\begin{equation} y_t = o_t z_t , \tag{14.1} \end{equation}\] where \(o_t\) is the demand occurrence variable, which can be either zero or one and has some probability of occurrence, \(z_t\) is the demand sizes captured by a model (for example, ADAM ETS) and \(y_t\) is the final observed demand. This model in context of intermittent demand was originally proposed by Croston (1972), but similar models (e.g. Hurdle and Zero Inflated Poisson) exist in other, non-forecasting related contexts.

In this chapter we will discuss the intermittent state space model that (14.1), both parts of which can be modelled via ADAM models, and we will see how they can be used, what they imply and how they connect to the convetional regular demand. If ETS model is used for \(z_t\) then (14.1) is called iETS. So, iETS(M,N,N) model refers to the intermittent state space model, where demand sizes are modelled via ETS(M,N,N). ETS can also be used for occurrence part of the model, so if the discussion is focused on demand occurrence part of the model (as in Subsection 14.1), we will use ``oETS’’ instead.

While ARIMA can be used in this context as well, it is not yet implemented for the occurrence part of the model. So we will focus the discussio on ADAM ETS. Furthermore, depending on how the occurrence part is modelled, these notations can be expanded to include references to specific parts of the occurrence part of the model. This is discussed in detail in Subsection 14.1.

### References

Croston, J D. 1972. “Forecasting and Stock Control for Intermittent Demands.” *Operational Research Quarterly (1970-1977)* 23 (3): 289. https://doi.org/10.2307/3007885.

Svetunkov, Ivan, and John Boylan. 2019a. “Multiplicative state-space models for intermittent time series.” Department of Management Science, Lancaster University. https://doi.org/10.13140/RG.2.2.35897.06242.