Abstract.
Investigations of spatial statistics, computed from lattice data in the plane, can lead to a special lattice point counting problem. The statistical goal is to expand the asymptotic expectation or large-sample bias of certain spatial covariance estimators, where this bias typically depends on the shape of a spatial sampling region. In particular, such bias expansions often require approximating a difference between two lattice point counts, where the counts correspond to a set of increasing domain (i.e., the sampling region) and an intersection of this set with a vector translate of itself. Non-trivially, the approximation error needs to be of smaller order than the spatial region’s perimeter length. For all convex regions in 2-dimensional Euclidean space and certain unions of convex sets, we show that a difference in areas can approximate a difference in lattice point counts to this required accuracy, even though area can poorly measure the lattice point count of any single set involved in the difference. When investigating large-sample properties of spatial estimators, this approximation result facilitates direct calculation of limiting bias, because, unlike counts, differences in areas are often tractable to compute even with non-rectangular regions. We illustrate the counting approximations with two statistical examples.
Similar content being viewed by others
References
Barvinok A I, Polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed, Math. Oper. Res. 19 (1994) 769–779
Barvinok A I and Pommersheim J, An algorithmic theory of lattice points in polyhedra, in: New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996–1997) (Math. Sci. Res. Inst. Publ., vol. 38. Cambridge Univ. Press, Cambridge) (1999) pp. 91–147
Cliff A D, and Ord J K, Spatial processes, models and application (London: Pion Limited) (1981)
Cressie N, Statistics for spatial data, 2nd edition (New York: Wiley) (1993)
De Loerab J A, Hemmeckeb R, Tauzera J and Yoshidab R, Effective lattice point counting in rational convex polytopes, J. Symbolic Computat. 38 (2004) 1273–1302
Dyer M, and Kannan R, On Barvinoks algorithm for counting lattice points in fixed dimension, Math. Oper. Res. 22 (1997) 545–549
Ehrhart E, Polynômes arithmétiques et methode des polyédres en combinatoire, International Series of Numerical Mathematics, 35 (Basel: Birkhäuser) (1977)
Fuentes M, Fixed-domain asymptotics for variograms using subsampling, Math. Geol. 33 (2001) 679–691
Fuller W, Introduction to statistical time series, 2nd edition (New York: Wiley) (1996)
Guyon X, Random fields on a network (Springer, New York) (1995)
Hall P, Horowitz J L and Jing B-Y, On blocking rules for the bootstrap with dependent data, Biometrika 82 (1995) 561–574
Huxley M N, Exponential sums and lattice points II, Proc. London Math. Soc. 66 (1993) 279–301
Huxley M N, Area, lattice points, and exponential sums (New York: Oxford University Press) (1996)
Kelly P J, and Weiss M L, Geometry and Convexity (New York: Wiley) (1979)
Krätzel E, Lattice points (Berlin: Deutscher Verlag Wiss.) (1988)
Künsch H R, The jackknife and the bootstrap for general stationary observations, Ann. Statist. 17 (1989) 1217–1261
Lenstra H W, Integer programming with a fixed number of variable, Math. Oper. Res. 8 (1983) 538–548
McAllister T B and Woods K M, The minimum period of the Ehrhart quasi-polynomial of a rational polytope, J. Combin. Theory A109 (2005) 345–352
Matheron G, The theory of regionalized random variables and its applications, Les Cahiers du Centre de Morphologie Mathematique, Fasc. 5, Centre de Geostatistique, Fontainebleau (1971)
Nordman D J and Lahiri S N, On optimal spatial subsample size for variance estimation, Ann. Statist. 32 (2004) 1981–2027
Pick G, Geometrishes zur Zahlentheorie, Sitzenber. Lotos 19 (1899) 311–319
Politis D N and Romano J P, Large sample confidence regions based on subsamples under minimal assumptions, Ann. Statist. 22 (1994) 2031–2050
Politis D N and White H, Automatic block-length selection for the dependent bootstrap, Economet. Revi. 23 (2004) 53–70
Politis D N, Romano J P and Wolf M, Subsampling (New York: Springer) (1999)
Possolo A, Subsampling a random field spatial statistics and imaging (ed.) A Possolo, IMS Lecture Notes Monograph Series 20 (CA: Institute of Mathematical Statistics, Hayward) (1991) pp. 286–294
Sherman M, Variance estimation for statistics computed from spatial lattice data, J. R. Stat. Soc. B58 (1996) 509–523
Sherman M and Carlstein E, Nonparametric estimation of the moments of a general statistic computed from spatial data, J. Amer. Statist. Assoc. 89 (1994) 496–500
Stanley R P, Enumerative combinatorics, vol I (California: Wadsworth, Belmont) (1986)
van der Corput J C, Über Gitterpunkte in der Ebene, Math. Ann. 81 (1920) 1–20
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
NORDMAN, D.J., LAHIRI, S.N. Bias expansion of spatial statistics and approximation of differenced lattice point counts. Proc Math Sci 121, 229–244 (2011). https://doi.org/10.1007/s12044-011-0024-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12044-011-0024-9