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Precision of area estimation: a numerical study
Author(s) -
Matérn Bertil
Publication year - 1989
Publication title -
journal of microscopy
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.569
H-Index - 111
eISSN - 1365-2818
pISSN - 0022-2720
DOI - 10.1111/j.1365-2818.1989.tb01477.x
Subject(s) - mathematics , sampling (signal processing) , plane (geometry) , line (geometry) , sample (material) , line segment , statistics , geometry , mathematical analysis , physics , optics , detector , thermodynamics
SUMMARY After listing some general formulae for sampling in n ‐dimensional space, the author considers the one‐dimensional case: the estimation of the length of a line segment by counting the number of points that happen to fall within the segment. If the points are equidistantly located, the variance of the estimate is a strictly periodic function of the length of the segment. This systematic sample has a higher efficiency than simple and stratified random samples of the same intensity. With some modifications, the results carry over to the two‐dimensional case: the estimation of the area of a plane figure by counting the number of sample points falling inside the figure. However, the strict periodicity of the variance in the one‐dimensional systematic case is replaced by a ‘Zitterbewegung’. The magnitude of this oscillation is seen to be very different for figures of different shapes. Some results are presented also for the estimation of areas by line transects, and for the estimation of volumes by aid of lattices of points in R 3 , and R 4 . Some comments are also given on the practical implications of the results for sampling in the plane.