(^) power operator

(number ^ number) -> number, const autodiff

The power operation between two numbers. If the first number is negative, then the second number must be an integer, or the operation fails.

show summary "" with
  2 ^ 2
  2 ^ -2
  -2 ^ 3
  -2 ^ -3

(ranvar ^ number) -> ranvar

For a non-negative integer n, r ^ n is the distribution of the sum of n independent draws from r; r ^ 0 is dirac(0). Fractional exponents extend this operation through an approximate convolution root, as described below. This does not raise individual values to the power n.

Example:

show scalar "" with poisson(5) ^ 2

(ranvar ^ ranvar) -> ranvar

The convolution power between ranvars.

show scalar "" with poisson(5) ^ poisson(2)

Annex: convolution power

The convolution power is a relatively advanced mathematical operation. In supply chain, convolution power can be used to scale probabilistic demand forecasts up or down. Convolution power offers the possibility to perform linear-like numeric adjustments on probabilistic forecasts. Furthermore, convolution power can be interpreted as the probabilistic counterpart of the linear adjustments performed on “classic” forecasts - i.e. periodic forecasts regressed against the mean or the median.

Motivation

Probabilistic demand forecasts are particularly suitable for optimizing decisions while taking supply chain risks into account. However, unlike classic forecasts where the demand is expressed as a definite quantity associated with a specific period of time, probabilistic forecasts involve distributions of probabilities (referred to as ranvars, in the following).

While ranvars provide more insights about the future compared to single-point indicators, they are more complex to manipulate. Such manipulations may be required to reflect market evolutions that cannot be inferred from historical data. The convolution power is a mathematical operation that allows to scale a ranvar of probabilities in a pseudo-linear fashion.

For example, if a retailer knows that each promotion will bring a 100% increase in sales, then, all it takes to adjust a classic demand forecast - which ignores promotions - is to multiply the original number by 2. In the case of a probabilistic forecast ignoring promotions, it is not possible to multiply the ranvar by 2 in the naive sense because the sum of the ranvars needs to remain equal to 1.

Formal definition

In mathematics, the convolution power is the $n$-fold iteration of the convolution with itself. Thus, if $R$ is a function $\mathbb{Z} \to \mathbb{R}$ and if $n$ is a non-negative integer, the convolution power is defined by:

$$ R^{*n} = \underbrace{R * R * R * \cdots * R * R}_n,\quad R^{*0}=\delta_0 $$

where $*$ denotes the convolution operation and $\delta_0$ is the Dirac delta ranvar. The variable $n$ is referred to as the exponent.

If $R$ is the ranvar associated with the discrete random variable $X$ with $R(k)=\mathbf{P}[X=k]$, then the convolution power $R^{*n}$ can be interpreted as the ranvar of the sum of $n$ independent random variables all having the same individual ranvar $R$:

Fractional exponents

Fractional exponents allow adjustments such as a 50% promotional uplift, represented by r ^ 1.5. The target is a convolution root: for example, r ^ 0.5 should, when added to an independent copy of itself, approximately reconstruct r. More generally, for positive integers $p$ and $q$, raising $R^{*p/q}$ to the integer power $q$ should approximately reconstruct $R^{*p}$.

Not every probability distribution has the requested convolution root. Even when a root exists, the bucketed representation of a ranvar makes the result approximate. Fractional powers therefore do not guarantee exact reconstruction of the original distribution.

For suitable small, non-negative distributions, Envision attempts to compute and validate a convolution root. Otherwise, it uses a geometric scaling approximation based on transform, adjusted to preserve the intended mean. For an exponent with both an integer and a fractional part, the two results are combined by convolution.

For example, halving a Poisson demand horizon gives a distribution close to poisson(5) when starting from poisson(10):

r = poisson(10)
half = r ^ 0.5

show summary "Half demand horizon" a1d3 with
  mean(half) as "Mean"
  variance(half) as "Variance"

Both the mean and the variance are approximately 5.

Illustration: aerospace spare parts

Let’s consider an airline company that operates a homogeneous fleet of 100 aircrafts. The company needs to optimize its inventory of APUs (Auxiliary Power Units) which happen to be an expensive repairable component required by the aircraft. The demand for APUs has been forecast for the horizon of interest as a probabilistic demand forecast $D$

Now, this company has the opportunity to buy a small competitor operating 5 aircrafts that are homogeneous to our company’s own fleet. Through this competitor acquisition, the company gains extra aircrafts and extra passengers. If we assume that all aircrafts are statistically independent in their need for APUs, and if we assume that the competitor’s aircrafts have needs similar to those of the acquiring company, then, the total demand for APUs for the merged entity can be revised as $D^{*\frac{100 + 5}{100}}=D^{*1.05}$.

References

• Convolution, Wikipedia
• Convolution power, Wikipedia
• Negative binomial, Wikipedia
• Infinite divisibility, Wikipedia