Weighted Reduction Operations

It is often convenient to assign a weight $w_i$ to each value $a_i$. In this case, the basic reduction operations are redefined as follows.

count:

The number of values

$$\mathrm{count} := \sum_i w_i$$

sum:

The sum of the values

$$\mathrm{sum} := \sum_i w_i a_i$$

product:

The product of the values

$$\mathrm{product} := \exp \sum_i w_i \log a_i$$

sum2:

The sum of the squares of the values

$$\mathrm{sum2} := \sum_i w_i a_i^2$$

sumabs:

The sum of the absolute values

$$\mathrm{sum2} := \sum_i w_i \vert a_i\vert$$

sumabs2:

The sum of the squares of the absolute values

$$\mathrm{sumabs2} := \sum_i w_i \vert a_i\vert^2$$

min:

The minimum of the values

$$\mathrm{min} := \min_i w_i \ne 0: a_i$$

max:

The maximum of the values

$$\mathrm{max} := \max_i w_i \ne 0: a_i$$

maxabs:

The maximum of the absolute values

$$\mathrm{maxabs} := \max_i w_i \ne 0: \vert a_i\vert$$

The notation $\min_i w_i \ne 0: a_i$ means: ``The minimum of $a_i$ where $i$ runs over all values where $w_i \ne 0$''. The definition of the high-level reduction operations does not change when weights are present.