Perinterval

Perinterval is a term used in mathematics and data analysis to denote a quantity assigned to each subinterval of a partition of a domain. It expresses local behavior by dividing a domain into consecutive, non-overlapping intervals and evaluating a quantity on each interval.

Formally, let [a,b] be partitioned by a = t_0 < t_1 < ... < t_n = b, with I_k = [t_{k-1}, t_k]. For a function f, a perinterval value can be the total over I_k, P_k = ∫_{I_k} f(x) dx, or the average per unit length, p_k = (1/|I_k|) ∫_{I_k} f(x) dx, where |I_k| = t_k - t_{k-1}. In discrete settings, one may take s_k = ∑_{x_j ∈ I_k} f(x_j) or other statistics.

Applications include finite element and finite-volume discretizations, histogram-based data analysis, time-series segmentation, and adaptive mesh refinement, where local properties guide reconstruction, refinement, or parallel computation. Perinterval methods are valued for local accuracy and modular computation, but results depend on the chosen partition, and mismatched bin sizes can obscure global structure.

See also interval analysis, binning, and piecewise constant approximation.