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14.2 Demand sizes part of the model

So far we have discussed the occurrence part \(o_t\) of the model and how to capture the probability of demand occurrence \(p_t\). But this is only a half of the intermittent state space model. The second one is the model for the demand sizes \(z_t\), which focuses on how many units of product will be sold if our customers decide to buy in a specific period of time. This can be modelled with any ADAM model, but has its own implications.

We start discussion with analysis of iETS(M,N,N)\(_F\) model, which can be formulate as: \[\begin{equation} \begin{aligned} & y_t = o_t z_t \\ & z_t = l_{z,t-1}(1 + \epsilon_{z,t}) \\ & l_{z,t} = l_{z,t-1}(1 + \alpha_{z} \epsilon_{z,t}) \\ & o_t \sim \text{Bernoulli}(p) \\ \end{aligned}, \tag{14.23} \end{equation}\] where the subscript \(z\) refers to the components and parameters of demand sizes. This model assumes that there is always a potential demand on the product which evolves over time, even when \(o_t=0\), we just do not always observe it. All the properties of this model have alread been discussed in chapter 7. The main challenge appears, when this model needs to be constructed and estimated, because \(z_t\) is not observable, when \(o_t=0\). In these situations the error term cannot be esimated, but according to the model it still exists, thus impacting the level of demand \(l_{z,t}\). In order to construct the model in the cases of no demand, we propose taking the conditional expectation for these periods, given the last available non-zero observations. This means that the model can be constructed using the following set of equations \[\begin{equation} \begin{aligned} & e_{z,t} = \frac{z_t - \hat{\mu}_{z,t}}{\hat{\mu}_{z,t}}, \text{ when } o_t=1 \\ & \hat{\mu}_{z,t} = \hat{l}_{z,t-1} \\ & \hat{l}_{z,t} = \left \lbrace \begin{aligned} & \hat{l}_{z,t-1} (1 + \hat{\alpha}_z e_t ), & \text{ when } o_t=1 \\ & \hat{l}_{z,t-1} , & \text{ when } o_t=0 \end{aligned} \right. \end{aligned}. \tag{14.24} \end{equation}\] This is only possible if \(\mathrm{E}(1+\epsilon_{z,t})=1\), which is an important assumption for multiplicative error models, discussed in Section 7.4. If this is violated, then the formula for the calculation of the level in (14.24) will become more complicated, involving the expectation of products of random variables.

In a similar way, we can construct more complicated models for the demand sizes. In a more general case this can be written as: \[\begin{equation} \begin{aligned} & e_{z,t} = \frac{z_t - \hat{\mu}_{z,t}}{\hat{\mu}_{z,t}}, \text{ when } o_t=1 \\ & \hat{\mathbf{v}}_{t} = \left \lbrace \begin{aligned} & f(\hat{\mathbf{v}}_{t-\boldsymbol{l}}) + g(\hat{\mathbf{v}}_{t-\boldsymbol{l}}) e_t, & \text{ when } o_t=1 \\ & f(\hat{\mathbf{v}}_{t-\boldsymbol{l}}) , & \text{ when } o_t=0 \end{aligned} \right. \end{aligned}, \tag{14.25} \end{equation}\] where all the functions and vectors have been defined for the original ADAM ETS model (6.1) in Section 6.

14.2.1 Additive vs multiplicative ETS for demand sizes

iETS supports any type of ETS model, including pure additive, pure multiplicative and mixed ones. But the selection of the appropriate model should be done based on the understanding of the problem. Typically, we expect demands to be non-negative: people want to buy our product, and usually the business does not want to buy from customers. In this case, we should use pure multiplicative models, as they will always produce meaningful results, as long as the assumption of positivity of \((1+\epsilon_{z,t})\) holds. This is important because the data would typically have low volume and the model might generate unreasonable (negative) pont and interval forecasts if a non-positive distribution is used (e.g. Normal). Thus, it is important to use either Inverse Gaussian, or Gamma, or Log Normal distribution for the error term of the demand sizes part of the model, when the volume of data is low and you expect the non-zero values to be strictly positive.

The main difficulty with pure multiplicative models arrises from the construction point of view - as discussed in Section 7.2 the point forecasts of such model in general do not correspond to the conditional expectations (the only exclusion is the ETS(M,N,N) model). At the same time, the construction of the model for demand sizes assumes that the conditional expectations are equal to point forecasts, when demand is not observed. If this is violated then (14.25) is no longer the correct way to construct the model. This problem becomes especially important for the models with multiplicative trend, where the conditional expectation might differ from point forecasts substantially. Still, point forecasts can be considered as proxies for the conditional expectation, especially when smoothing parameters are close to zero. In the boundary case with \(\alpha=0\) and \(\beta=0\) in ETS(M,M,N), the conditional expectation coincides with the point forecast. The higher the smoothing parameters are, the bigger descrepancy will be, implying that the model for the demand sizes is constructed incorrectly.

The pure additive models do not have the issue with the conditional expectation and thus can be constructed easily in case of intermittent demand. But as discussed earlier, they might violated the non-negativity assumption of the model. So, in practice they should be used with care.

14.2.2 Using ARIMA for demand sizes

Finally, ADAM ARIMA can be used for demand sizes as well, making iARIMA model. All the discussions in the previous subsection would apply to ARIMA as well, keeping in mind that ADAM ARIMA can be either pure additive or pure multiplicative. Given that the multiplicative ARIMA is formulated via logarithms, and still has the error term with the expectation of one, any ARIMA model can be used for the variable \(z_t\) and can be constructed via (14.25). This can also be used for the cases, when a pure multiplicative model with trend is needed, and there are difficulties with constructing ETS(M,M,N) (i.e. smoothing parameters are not close to zero). The relation between ARIMA and ETS might be useful in this case. For example, instead of constructing ETS(M,M,N) we can construct logARIMA(0,2,2) in this case, sidestepping the aforementioned problem.