This is a post that I’ve been meaning to write for a while. Having worked on demand forecasting in the past, I was intrigued when I saw this paper posted on the arXiv pre-print archive from one of the research teams at Amazon.

Although it was obvious why Amazon would be interested in forecasting demand, I was intrigued that Amazon chose to use a state space model approach. About 6 months later I attended the ISBIS2018 conference, at which Lindsay Berry from Duke University presented this paper that also used a state-space model approach to modelling the demand for supermarket goods. I also subsequently became aware of this technical report from Ivan Svetunkov at Lancaster University’s Management School.

With three pre-prints on demand forecasting that all utilised a state-space modelling approach I thought it would be interesting to do a post summarizing the work from the Amazon team. I may get round to doing a further post on the other two pre-prints at a later date.

At this point it is worth explaining a bit about demand models in general. Demand models are statistical models that are usually built from 2-5yrs worth of historical sales data. A demand model enables us to forecast how many units of a product we will sell given the price of the product and a number of other variables. The models allow us to forecast a number of ‘what-if’ scenarios, e.g. what will happen to my sales if I reduce the product price by 20%? Ultimately, the demand model can enable us to determine what is the optimal price we should charge for a product depending on what business KPI we want to optimize. Some traditional approaches to demand modelling use a log-log model, with the log of the demand of an item being linear in the log of the price of the item. These models are of the Working-Leser type of demand models^{1,2}. For goods with large demand volumes, a log-log model form will be a reasonable assumption as the natural quantum of demand (a single unit) is much smaller than the typical level of demand and so we can consider the demand of such goods as effectively being continuous random variables.

The actual problem the Amazon team tackled was forecasting of intermittent demand, i.e. for what are commonly called, ‘slow-moving goods’, whose pattern of sales are bursty. We might typically only sell, on average, less than say 10 units a week for these kinds of products. There may be no sales for a week or two, followed by a burst of sales concentrated in a few days. Snyder *et al* give a good modern review of the problem of intermittent demand forecasting^{3}.

For such products, the traditional log-log type demand models can perform poorly, as we are dealing with products that sell only a few units per time period. However, there is no consensus approach to modelling such products, but this means it is an area ripe for novel and innovative methods. The paper by Seeger *et al* combines 3 interesting ideas,

- A multi-stage model – this means decomposing the modelling of demand into several models that cover different demand sizes. In this case separate models are constructed for when the expected demand is 0 units, 1 unit, and >1 unit.
- The combining of the multi-stage model with a state-space model. This has the effect of introducing exponential smoothing and hence some temporal continuity to the modelled demand.
- The use of a Kalman-filter approach to location of the mode when using a Laplace approximation to approximate a marginal posterior. This third innovation is the most technical, but, for me, also the most interesting.

The first of these innovations is not necessarily that much of a step-change. Other attempts to model slow-moving goods have also considered a mixture of distributions/processes to allow for the zero-inflation that one has in the weekly observed sales of a slow-moving good. Seeger *et al *use a three stage model, so that we have 3 latent functions,

, which is used in modelling the probability of zero sales at time point t

, which is used in modelling the probability of a single unit being sold at time point t

, which is used in modelling the distribution of units sold at time point t, given the number of units is greater than 1.

The second innovation is an interesting one. Whilst I had come across the use of self-excitation (Hawkes processes) to model the bursty behaviour of intermittent demand, I hadn’t seen temporal continuity enforced via a latent state contribution to the linear predictors of the mixture components. For demand greater than a single unit Seeger *et al* model the demand at time point t as following a Poisson distribution,

Here is a transfer function. The latent function depends upon a latent state and it is this latent state that is governed by a Kalman filter. Overall the latent process is,

The latent variables have to be integrated out to yield a marginal posterior distribution that can then be maximized to obtain parameter estimates for the parameters than control the innovation vectors .

It is the marginalization over that the third interesting technical innovation of Seeger *et al* is concerned with. The integration over is approximated using a Laplace approximation. The Laplace approximation simply replaces the exponent of the integrand by its second order Taylor expansion approximation, in order to approximate a complicated integration by an integration of a Gaussian. It is the simplest of a family of saddlepoint expansion techniques for obtaining asymptotic expansions of integrals (see for example the classic book by Wong).

The main task in a Laplace approximation is locating the maximum of the exponent of the integrand. Seeger *et al* do this via a Newton-Raphson procedure, i.e. expand the exponent to second order around the current estimate of the mode and then find the maximum of that second order approximation.

Consider a 1-dimensional example. Let be the function whose maximum, , we are trying to locate. If the expansion of around our current estimate of is,

The updated estimate of is then determined by maximizing the second order expansion above, and is given by,

The schematic below depicts how the iterative Newton-Raphson procedure locates the maximum of a one-dimensional function.

The multi-dimensional equivalent update rule when we are maximizing a function of a vector is,

where is the Hessian of evaluated at

As Seeger *et al* are marginalizing the posterior over , the Taylor expansion around any point is necessarily multi-variate, and so ordinarily, finding the maximum of that second order approximation would involve inverting the Hessian of the log-posterior evaluated at the current estimate of the mode. As the latent variables we are trying to marginalize over by doing the Laplace approximation are the innovations, and , this means that each step of the Newton-Raphson procedure would involve the inversion of a matrix, i.e. an operation for each Newton-Raphson step. However, Seeger *et al* point out that once we have replaced the log-posterior by a second-order approximation, finding the maximum of that approximation is equivalent to finding the posterior mean of a linear-Gaussian state-space model, and this can be done using Kalman smoothing. This means in each Newton-Raphson step we need only run a Kalman filter calculation, an calculation, rather than a Hessian inversion calculation which would be . When training on, say, 2 years of daily sales data with $T=730$, the speed-up will be significant. Seeger *et al* do point out that this trick of reducing the computation to one that scales linearly in is already known within the statistics literature^{4}, but not widely known within machine learning.

Seeger *et al* apply their methodology to a number of real-world scale datasets, for example to a ~40K item dataset with almost a year of historical data at the day-level. Overall run-times for the parameter learning are impressive (typically a few seconds for each of the separate demand model stages), though admittedly this is when running on a 150 node Spark cluster.

#### References

- Working, H. (1943) Statistical laws of family expenditure. J. Am. Statist. Ass., 38:43–56.
- Leser, C. E. V. (1963) Forms of Engel functions. Econometrica, 31, 694–703.
- Snyder, R., Ord, J., and Beaumont, A. (2012), Forecasting the intermittent demand for slow-moving inventories: A modelling approach. International Journal on Forecasting, 28:485-496
- Durbin, J. and Koopman, S. (2012), Time Series Analysis by State Space Methods. Oxford Statistical Sciences. Oxford University Press, 2nd Edition.