WWert

wWert is a hypothetical mathematical construct used in information theory and data analysis to denote a weighted value function. The term combines the German word Wert, meaning “value,” with a prefix intended to signal weighting. In this article, wWert is treated as an idealized operator that maps a data set and a weight assignment to a single scalar value.

Definition: Let x = (x1, ..., xn) be a vector of data points and w = (w1, ..., wn) a weight vector with sum_i w_i > 0. Choose a transformation g: R -> R, typically monotone and bounded where appropriate. The wWert of x under w is W_wert(x; w) = (sum_i w_i * g(x_i)) / (sum_i w_i). When g is the identity function, W_wert reduces to the standard weighted mean.

Properties: If g is non-decreasing and all weights w_i are nonnegative, W_wert is nonnegative. It is linear with respect to the weighting scheme in the sense of weighted averaging. If all weights are equal, W_wert equals the average of g(x_i). W_wert is continuous provided g is continuous.

Applications: Conceptually, wWert can be used to fuse heterogeneous signals in data integration, to produce composite scores in recommender systems, and as a pedagogical tool to illustrate how different transformations and weightings affect outcomes.

Limitations: The choice of g and the weight vector w strongly influence results; inappropriate selections can introduce bias. See also: weighted mean, aggregation function, distortion function.