vmean

Vmean, short for vector mean, is a mathematical quantity that represents the average of a set of vectors. If a collection consists of n vectors v1, v2, ..., vn in R^d, the vector mean is the coordinate-wise average: vmean = (1/n) sum_{i=1}^n vi. In weighted datasets, with nonnegative weights w1, w2, ..., wn, the weighted vector mean is vmean = (sum w_i v_i) / (sum w_i).

In continuous contexts, the vector mean of a vector field v(x) over a domain D with finite measure is defined as vmean = (1/|D|) ∫_D v(x) dx, where |D| is the measure (volume) of D. This corresponds to averaging the field values across the domain. The concept generalizes the idea of a centroid or barycenter when the vectors represent positions or quantities attached to masses; with equal masses, the vector mean of position vectors yields the centroid, while mass-weighted means yield the barycenter.

Computation and implementation are straightforward for discrete data: sum the vectors (or their components) and divide by the number of samples, or divide by the sum of weights in the weighted case. In programming, many languages provide built-in support to compute the mean along a given axis, for example by summing across the desired dimension and dividing by the count.

Applications of vmean appear in statistics, physics, engineering, computer graphics, and data analysis, wherever a representative average of vector-valued data is required. Related concepts include the unweighted and weighted scalar mean, the centroid, and the barycenter.