Deltaadjacent

Deltaadjacent is a term used to describe a simple lag-based relation between elements in an ordered dataset. Two elements are deltaadjacent if the absolute difference between their positions equals a fixed nonnegative integer delta, typically delta > 0. In a finite sequence x1, x2, ..., xN, x_i and x_j are deltaadjacent when |i - j| = delta.

Formally, the set of deltaadjacent pairs for a given delta is E_delta = { (i, i+delta) | i = 1, 2, ..., N - delta }. If direction matters, one may consider the ordered pairs (i, i+delta) or (i+delta, i); for circular indexing, the relation extends to indices modulo N.

Deltaadjacency has applications in signal processing, time-series analysis, and graph representations. It captures lagged relationships such as the correlation between samples separated by a fixed time step, and can be used to define a delta-graph where vertices are positions and edges connect delta-separated positions. In pattern detection, deltaadjacent pairs help identify regular spacing or periodicity.

Related notions include adjacency (with delta = 1), lag, cross-correlation, and circular convolution. The concept is a straightforward way to formalize fixed-delay relationships in discrete data without requiring metric similarity beyond positional spacing.