# Type of normalization [duplicate]

Suppose you have some vector $$[a_1, \ldots, a_n]$$ where the terms are non-negative.

How do you call the normalization method where you divide each element by the sum (to get a probability vector)? i.e. $$a_i \mapsto \frac{a_i}{\sum a_j}$$

## marked as duplicate by whuber♦Mar 3 at 22:08

• This is not necessarily a probability vector -- are the $a_i$ constrained to be positive? – Sycorax Mar 3 at 18:18
You could call $$a_i \mapsto \dfrac{a_i}{\sum\limits_j |a_j|}$$ a Manhattan or $$\ell_1$$-normalisation, effectively projecting each point onto the boundary of a unit hyperoctahedron or cross-polytope, in rather the same way that $$a_i \mapsto \dfrac{a_i}{\sqrt{\sum\limits_j a_j^2}}$$ could be called a Euclidean or $$\ell_2$$-normalisation projecting each point onto the boundary of a unit hypersphere.
As you say, if the $$a_i$$ are all non-negative and at least one is positive, then this is equivalent to $$a_i \mapsto \dfrac{a_i}{\sum\limits_j a_j}$$ and once normalised they add up to $$1$$ (you are effectively restricting yourself to a single orthant simplex facet of the unit hyperoctahedron) so these could represent a probability distribution. In a sense you could be starting with values proportional to the probabilities, and the normalisation turns them into actual probabilities.