- The Quantitative Supply Chain Manifesto
- The Lokad test of supply chain performance
- An overview of quantitative supply chain
- Generalized probabilistic forecasting
- Decision-driven optimization
- Economic drivers
- Data preparation
- The Supply Chain Scientist
- Timeline of a typical project
- Project deliverables
- Assessing success
- Antipatterns in supply chain

- Inventory forecasting
- Prioritized ordering report
- Old forecasting input file format
- Old forecasting output file format
- Choosing the service levels
- Managing your inventory settings
- The old Excel forecast report
- Using tags to improve accuracy
- Oddities in classic forecasts
- Oddities in quantile forecasts
- Stock-out's bias on quantile forecasts
- Daily, weekly and monthly aggregations

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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.

While distributions provide more insights about the future compared to single-point indicators, distributions 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 distribution 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 probabilistic forecasts (which also ignores promotions), it's not possible to multiply the distribution by 2 in the naive sense because the sum of the distribution needs to remain equal to 1 and represent the sum of the probabilities.

If $x$ is the probability density associated with the discrete random variable $X$ with $x(k)=\mathbf{P}[X=k]$, then the convolution power can be interpreted as a sum of random variables: $$ X^{*n} = \underbrace{X' + X' + X' + \cdots + X' + X'}_n $$ where all $X'$ are independent copies of the original random variable $X$.

For $a$, a non-negative real number, we re-define the convolution power as follows:

$$ x^{*a} = \mathcal{Z}^{-1} \Big\{ \mathcal{Z}\{x\}^a \Big\} $$ where $\mathcal{Z}$ is Z-transform of the discrete distribution $x$, defined as: $$ \mathcal{Z}\{x\} : z \to \sum_{k=-\infty}^{\infty} x[k] z^{-k} $$ and where $\mathcal{Z}\{x\}^a$ is the point-wise power over the Z-transform defined as: $$ \mathcal{Z}\{x\}^a : z \to \left( \sum_{k=-\infty}^{\infty} x[k] z^{-k} \right)^a $$ Finally, $\mathcal{Z}^{-1}$ is the inverse Z-transform $$ \mathcal{Z}^{-1} \{X(z) \}= \frac{1}{2 \pi j} \oint_{C} X(z) z^{n-1} dz $$ with $X(z) = \mathcal{Z}\{x\}(z)$, introduced for the sake of readability, and where $C$ is a counterclockwise closed path encircling the origin.

If $a$ is an integer, then the two definitions given above for the convolution power coincide.

In practice, the inverse Z-transform is not always defined. However, there are ways to generalize the notion of Z-transform inversion - somewhat similar to the notion of matrix pseudo inverse used in linear algebra. Details relating to Z-transform pseudo inverse go beyond of the scope of the present document.

Through this Z-transform pseudo inverse, the convolution power can be defined for all random variables of compact support, and for any non-negative real number used as the exponent.

`^*`

operator.
y := poisson(3) ^* 4.2 // fractional exponentThe script above illustrates how a Poisson distribution, obtained through the

`poisson()`

function, can be convoluted to the power of 4.2.For $x$ a function $\mathbb{Z} \to \mathbb{R}$ and $y$ a function $\mathbb{N} \to \mathbb{R}$, we can define the convolution power of $x$ by $y$ with: $$ x^{*y} = \sum_{k=0}^{\infty} y[k] x^{*k} $$ Envision also supports this alternative expression of the convolution power through the

`^*`

operator, as illustrated by the script below.
y := poisson(3) ^* exponential(0.05)The exponent is an exponential distribution obtained by using the

`exponential()`

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

- Convolution, Wikipedia
- Convolution power, Wikipedia
- Z-transform, Wikipedia
- Moore–Penrose pseudoinverse, Wikipedia