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Order Distance in Regular Point Patterns
Author(s) -
Miyagawa Masashi
Publication year - 2009
Publication title -
geographical analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.773
H-Index - 65
eISSN - 1538-4632
pISSN - 0016-7363
DOI - 10.1111/j.1538-4632.2009.00737.x
Subject(s) - k nearest neighbors algorithm , combinatorics , mathematics , hexagonal crystal system , point (geometry) , lattice (music) , minimum distance , square lattice , order (exchange) , hexagonal lattice , square (algebra) , closing (real estate) , nearest neighbour , geometry , discrete mathematics , statistical physics , computer science , physics , condensed matter physics , artificial intelligence , finance , antiferromagnetism , acoustics , law , political science , ising model , economics , chemistry , crystallography
This article examines the k th nearest neighbor distance for three regular point patterns: square, triangular, and hexagonal lattices. The probability density functions of the k th nearest distance and the average k th nearest distances are theoretically derived for k =1, 2, …, 7. As an application of the k th nearest distance, we consider a facility location problem with closing of facilities. The problem is to find the optimal regular pattern that minimizes the average distance to the nearest open facility. Assuming that facilities are closed independently and at random, we show that the triangular lattice is optimal if at least 68% of facilities are open by comparing the upper and lower bounds of the average distances.