összegbl

Összegbl is a hypothetical mathematical operator used in theoretical discussions of aggregation and data summarization. It serves as a parametric extension of the weighted sum, combining elements of a finite sequence through a block-based weighting scheme. Given a sequence (x1, x2, ..., xn), a block partition of indices into blocks B1, B2, ..., Bm is chosen with a block size parameter k. A block-weight function w assigns to each block a nonnegative weight b_j, and within each block the individual weights are distributed according to a subrule, such as equal distribution or a decreasing profile. The összegbl value is defined as S = sum_{j=1}^m b_j * (sum_{i in B_j} x_i) / |B_j|, optionally normalized to ensure sum b_j = 1. Different parameter choices produce variants; for example, choosing within-block equal weights yields a simple block-averaged sum, while nonuniform internal weights emphasize certain positions.

Etymology and naming: The name blends the Hungarian word összeg meaning "sum" with the abbreviation bl for "block-level" weighting. It first appeared in introductory texts and didactic papers as a fictional construct to illustrate aggregation concepts.

Properties: Összegbl is linear in the input sequence for a fixed block structure; it is not invariant to reordering within blocks and is sensitive to the chosen partition and weighting scheme. It can interpolate between a plain sum and a block-averaged statistic.

Applications: Used in teaching to demonstrate how partitioning an input affects aggregated results; in simulations of hierarchical data summarization; in simple feature engineering for machine learning.

See also: aggregation operator, weighted sum, block processing.