Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T15:56:49.709Z Has data issue: false hasContentIssue false
  • English
  • Français

Systematic sampling on the circle and on the sphere

Part of: Geometric probability and stochastic geometry Global differential geometry General convexity

Published online by Cambridge University Press:  19 February 2016

Ximo Gual-Arnau  and
Luis M. Cruz-Orive
Ximo Gual-Arnau*
Affiliation:
Universitat Jaume I, Castellón
Luis M. Cruz-Orive*
Affiliation:
Universidad de Cantabria
*
∗ Postal address: Departament de Matemàtiques, Universitat Jaume I, Campus Riu Sec, E-12071 Castellón, Spain. Email address: [email protected]
∗∗ Postal address: Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avda. Los Castros s/n, E-39005 Santander, Spain. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Useful approximations have been developed along the years to predict the precision of systematic sampling for measurable functions of a bounded support in ℝd. Recently, the theory of systematic sampling on ℝ has received a thrust. In geometric sampling, design based unbiased estimators exist, however, which imply systematic sampling on the circle (𝕊1) and the semicircle (ℍ1); the planimeter estimator of an area, or the Buffon-Steinhaus estimator of curve length in the plane constitute popular examples. Over the last two decades, many other estimators of geometric measures have been obtained which imply systematic sampling also on the sphere (𝕊2). In this paper we adapt the theory available for non-periodic functions of bounded support on ℝ to periodic functions, and thereby to 𝕊1 and ℍ1, and we obtain new estimators of the corresponding variance approximations. Further we consider - we believe for the first time - the problem of predicting the precision of systematic sampling in 𝕊2. The paper starts with a historical perspective, and ends with suggestions for further research.

Keywords

CovariogramEuler-MacLaurin summation formulaFourier seriesgeometric samplingnucleatorspherical harmonicsstereologytransitive theory

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry 53C65: Integral geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 3 , September 2000 , pp. 628 - 647
Copyright
Copyright © Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported by the Fundaci'o Caixa Castell' grant no. P1A-94-24 and the Direcci-on General de Ensenanza Superior (DGES) grant no. PM97-0043.

References

[1] Abramowitz, M. and Stegun, I. A. (1965). Handbook of Mathematical Functions. Dover, New York.Google Scholar
[2] Ayant, Y. and Borg, M. (1971). Fonctions spéciales a l'usage des étudiants en physique. Dunod, Paris.Google Scholar
[3] Baddeley, A. J., Gundersen, H. J. G. and Cruz-Orive, L. M. (1986). Estimation of surface area from vertical sections. J. Microsc. 142, 259–276.Google Scholar
[4] Bellhouse, D. R. (1988). Systematic sampling. In Sampling, eds Krishnaiah, P. R. and Rao, C. R.. Handbook of Statistics, Vol. 6. North-Holland, Amsterdam, pp. 125–145.Google Scholar
[5] Chadoeuf, J., Moran, C. and Goulard, M. (1998). A note on extension variances in R2 . Math. Geol. 30, 575–587.Google Scholar
[6] Cochran, W. G. (1977). Sampling Techniques, 3rd edn. John Wiley, New York.Google Scholar
[7] Cressie, N. A. C. (1991). Statistics for Spatial Data. John Wiley, New York.Google Scholar
[8] Cruz-Orive, L. M. (1987). Stereology: recent solutions to old problems and a glimpse into the future. Proc. ICS VII Caen 1987. Acta Stereologica 6/III, 3–18.Google Scholar
[9] Cruz-Orive, L. M. (1987). Particle number can be estimated using a disector of unknown thickness: the selector. J. Microsc. 145, 121–142.Google ScholarPubMed
[10] Cruz-Orive, L. M. (1989). On the precision of systematic sampling: a review of Matheron's transitive methods. J. Microsc. 153, 315–333.Google Scholar
[11] Cruz-Orive, L. M. (1989). Precision of systematic sampling on a step function. In Geobild '89, eds Hübler, A., Nagel, W., Ripley, B. D. and Werner, G.. Akademie-Verlag, Berlin, pp. 185–193.Google Scholar
[12] Cruz-Orive, L. M. (1993). Systematic sampling in stereology. Bull. Int. Statist. Inst., Proc. 49th Session, Florence 1993, 55, 451–468.Google Scholar
[13] Cruz-Orive, L. M. (1997). Stereology of single objects. J. Microsc. 186, 93–107.CrossRefGoogle Scholar
[14] Cruz-Orive, L. M. (1999). Precision of Cavalieri sections and slices with local errors. J. Microsc. 193, 182–198.CrossRefGoogle ScholarPubMed
[15] Cruz-Orive, L. M. and Howard, C. V. (1991). Estimating the length of a bounded curve in three dimensions using total vertical projections. J. Microsc. 163, 101–113.Google Scholar
[16] Cruz-Orive, L. M. and Roberts, N. (1993). Unbiased volume estimation with coaxial sections: an application to the human bladder. J. Microsc. 170, 25–33.CrossRefGoogle Scholar
[17] García-Fiñana, M. and Cruz-Orive, L. M. (1998). Explanation of apparent paradoxes in Cavalieri sampling. Acta Stereologica 17, 293–302.Google Scholar
[18] Goldberg, M. (1968). Divisors of a circle. Solution to problem 660, proposed by L. J. Upton. Math. Magazine 41, 46.Google Scholar
[19] Gradshteyn, I. S. and Ryzhik, I. M. (1994). Table of Integrals, Series and Products, 5th edn. Academic Press, New York.Google Scholar
[20] Gual Arnau, X. and Cruz-Orive, L. M. (1998). Variance prediction under systematic sampling with geometric probes. Adv. Appl. Prob. 30, 889–903.Google Scholar
[21] Gundersen, H. J. G. (1988). The nucleator. J. Microsc. 151, 3–21.Google Scholar
[22] Gundersen, H. J. G. and Jensen, E. B. (1987). The efficiency of systematic sampling in stereology and its prediction. J. Microsc. 147, 229–263.Google Scholar
[23] Gundersen, H. J. G., Jensen, E. B. V., Kiêu, K., and Nielsen, J. (1999). The efficiency of systematic sampling in stereology — reconsidered. J. Microsc. 193, 199–211.Google Scholar
[24] Hansen, M. H., Hurwitz, W. N. and Madow, W. G. (1953). Sample Survey Methods and Theory, Vols I and II. John Wiley, New York.Google Scholar
[25] Jeffrey, A. (1990). Linear Algebra and Ordinary Differential Equations. Blackwell, Oxford.Google Scholar
[26] Jensen, E. B. V. (1998). Local Stereology. World Scientific, Singapore.CrossRefGoogle Scholar
[27] Jensen, E. B. and Gundersen, H. J. G. (1989). Fundamental stereological formulae based on isotropically orientated probes through fixed points with applications to particle analysis. J. Microsc. 153, 249–267.Google Scholar
[28] Jensen, E. B. V. and Gundersen, H. J. G. (1993). The rotator. J. Microsc. 170, 35–44.Google Scholar
[29] Journel, A.G. and Huijbregts, Ch. J. (1978). Mining Geostatistics. Academic Press, London.Google Scholar
[30] Kellerer, A. M. (1989). Exact formulae for the precision of systematic sampling. J. Microsc. 153, 285–300.Google Scholar
[31] Kendall, D. G. (1948). On the number of lattice points inside a random oval. Quart. J. Math. 4, 178–189.Google Scholar
[32] Kendall, M. G. and Moran, P. A. P. (1963). Geometrical Probability. Griffin, London.Google Scholar
[33] Kiêu, K., (1997). Three Lectures on Systematic Geometric Sampling. Memoirs 13/1997, Department of Theoretical Statistics, University of Aarhus.Google Scholar
[34] Kiêu, K., Souchet, S. and Istas, J. (1999). Precision of systematic sampling and transitive methods. J. Statist. Plan. Inf. 77, 263–279.Google Scholar
[35] Lockwood, E. H. (1961). A Book of Curves. Cambridge University Press.Google Scholar
[36] Matérn, B., (1986). Spatial Variation, 2nd edn. Springer, Berlin.Google Scholar
[37] Matérn, B., (1989). Precision of area estimation: a numerical study. J. Microsc. 153, 269–284.Google Scholar
[38] Matheron, G. (1962). Traité de Geostatistique Appliquée, Tome I. Technip, Paris.Google Scholar
[39] Matheron, G. (1965). Les Variables Régionalisées et Leur Estimation. Masson, Paris.Google Scholar
[40] Matheron, G. (1971). The Theory of Regionalized Variables and Its Applications. Les Cahiers du Centre de Morphologie Mathématique de Fontainebleau, No. 5. École Nationale Supérieure des Mines de Paris, Fontainebleau.Google Scholar
[41] Moran, P. A. P. (1950). Numerical integration by systematic sampling. Proc. Camb. Phil. Soc. 46, 111–115.Google Scholar
[42] Moran, P. A. P. (1966). Measuring the length of a curve. Biometrika 53, 359–364.Google Scholar
[43] Murthy, M. N. (1967). Sampling Theory and Methods. Statistical Publishing Society, Calcutta.Google Scholar
[44] Ripley, B. D. (1981). Spatial Statistics. John Wiley, New York.Google Scholar
[45] Roberts, N., Howard, C. V., Cruz-Orive, L. M. and Edwards, R. H. T. (1991). The application of total vertical projections for the unbiased estimation of the length of blood vessels and other structures by magnetic resonance imaging. Magn. Res. Imaging 9, 917–925.Google Scholar
[46] Santaló, L. A. (1976). Integral Geometry and Geometrical Probability. Addison-Wesley, Reading, Massachusetts.Google Scholar
[47] Souchet, S. (1995). Précision de l'estimateur de Cavalieri. Rapport de stage, D.E.A. de statistiques et modèles aléatoires appliqués à la finance, Université Paris-VII. Laboratoire de Biométrie, INRA-Versailles.Google Scholar