Convex combination

From Wikipedia, the free encyclopedia

A convex combination is a linear combination of data points (which can be vectors or scalars) where all coefficients are non-negative and sum up to 1. It is called "convex combination", since all possible convex combinations (given the base vectors) will be within the convex hull of the given datapoints. In fact, the set of all convex combinations constitutes the convex hull.

A special case is with only two data points, where the value of the new point (formed by the convex combination) will lie on a straight line between the two points.

[edit] Related constructions

Weighted means are functionally the same as convex combinations, but they use a different notation. The coefficients (weights) in a weighted mean are not required to sum to 1; instead the sum is explicitly divided from the linear combination.
Affine combinations are like convex combinations, but the coefficients are not required to be non-negative. Hence affine combinations are defined in vector spaces over any field.