# Probabilistic forecasting definition

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By Joannès Vermorel, October 2015

A probabilistic forecast represents an estimation of the respective probabilities for all the possible future outcomes of a random variable. In contrast to single-valued forecasts, such as median time-series forecasts or quantile forecasts, the probability forecast represents a probability density function. Probabilistic forecasts can be applied to numerous domains ranging from weather forecasting to sports betting, but they are especially useful for supply chain optimization.

## Application to supply chain optimization

In supply chain, the extreme events are driving the bulk of the costs from a statistical perspective. Surprisingly high demand levels cause stock-outs, while surprisingly low demand levels cause slow rotating inventory, and even dead inventory in the most extreme situations. Thus, correctly assessing the probabilities of such negative events taking place is very important in order to balance the resource allocations – mostly working capital through stocks – with the quality of service of the supply chain.

Single-valued forecasts tend to focus on mean or median situations which do not properly reflect the extreme situations described above. Indeed, the problem does not lie in the quality of the forecasts – which might be more or less accurate – but rather in the very definition of the forecasts themselves: a mean forecast does not address the question of extreme events. This issue is traditionally addressed in supply chain through safety stock analysis. However, this analysis usually relies on strong assumptions, such as a normal distribution of the demand, which are highly inaccurate when it comes to supply chain.

In contrast, probabilistic demand forecasts offer the possibility to implement prioritized ordering policies which outperform more classic ordering policies because they exploit the precise structure of the probabilities estimated for future demand.

## Probabilistic forecasts applied to time-series

Time-series is probably the most commonly found data model in supply chain. In this section, we formalize somewhat the notion of probabilistic forecasts in the context of time-series. Let $\mathbf{y}_t$ be the vector of past demand known at time $t$.

We can model future demand at time $t+h$ for each forecasting horizon $h$ as follows: $$y_{t+h}=g_h(\mathbf{y}_t)+\epsilon_{t+h}$$ Where:
• $g_h$ is a forecasting model specific of the horizon $h$
• $\epsilon_{t+h}$ denotes the model error

At this point, $g_h$ is still a single-valued forecasting model. We can adjust this definition into a probabilistic forecast by considering the following: $$Y_{t+h}=G_h(\mathbf{y}_t)$$ Where $G$ returns not a single value $y_{t+h}$, but a random variable $Y_{t+h}$ with an explicit density distribution, that is $P(y_{t+h}\leq y | \mathbf{y}_t)$. While this would go beyond the scope of the present entry, it can be noted that probabilistic forecasting can be reduced to the estimation of the cumulative distribution function.

## Practical representation of probabilistic forecasts

From a practical perspective, a probabilistic forecast $Y_{t+h}$ is usually represented as a histogram where each bin represents a range of future demand, and where the bin height represents the estimated probability that future demand will fall within the specific range associated to a bucket.

Unless some specific assumptions can be made about the probability distribution of the demand, histograms introduce a degree of numerical approximation when representing the demand. Indeed, many probability distributions are unbounded, with non-zero probabilities found for arbitrarily large values. Such distributions cannot be perfectly represented through finite histograms. Similarly, by construction, the histogram averages the fine structure of the distribution within each bucket.

While histograms require significantly more computing resources in order to be processed compared to single values (i.e. non-probabilistic forecasts), processing histograms for very large numbers of time-series (or more) is well within the range of capabilities of modern computing systems. In practice, histograms can be designed to contain a sufficient number of buckets to make sure that the numerical imprecisions introduced through the discretization of the demand is negligible compared to the uncertainty of the forecast itself.