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Variance prediction under systematic sampling with geometric probes

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

Published online by Cambridge University Press:  01 July 2016

Ximo Gual Arnau  and
Luis M. Cruz-Orive
Ximo Gual Arnau*
Affiliation:
Universitat Jaume I
Luis M. Cruz-Orive*
Affiliation:
University of Cantabria
*
∗ Postal address: Departament de Matematiques, Universitat Jaume I, Campus Riu Sec, E-12071 Castellón, Spain.
∗∗ 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]
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Abstract

In design stereology, and in the context of geometric sampling in general, the problem often arises of estimating the integral of a bounded non-random function over a bounded manifold D ⊂ ℝn by systematic sampling with geometric probes. Variance predictors, often based on Matheron's theory of regionalized variables, are available when the relevant function is sampled at the points of a grid intersecting D, but not when the dimension of the probes is greater than zero. For instance, the volume of a bounded object may be estimated using parallel systematic planes, which amounts to sampling on ℝ1 with systematic points, or using parallel systematic slabs of thickness t > 0, which amounts to sampling on ℝ1 with non-overlapping systematic segments of length t > 0. Useful variance predictors exist for the former case, but not for the latter. In this paper we set out a general scheme to predict estimation variances when the dimension of either D, or of the probes, is n. We make some progress when both dimensions are equal to n, and obtain explicit results for n = 1 (e.g. for systematic slice sampling). We check and illustrate our results for the volume estimators of ellipsoids and of rat lung.

Keywords

Cavalieri estimatorcovariogramellipsoidEuler-MacLaurin summation formulaFourier transformregularizationstereologysystematic samplingtest systemvariance prediction

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 30 , Issue 4 , December 1998 , pp. 889 - 903
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Research supported by the Fundació Caixa Castelló grant no. P1A-94-24 and the Dirección General de Investigación Cientifica y Técnica Project No. #PB94-1058.

References

Abramowitz, M. and Stegun, I. A. (1965). Handbook of Mathematical Functions. Dover, New York.Google Scholar
Burri, P. H. and Weibel, E. R. (1971). Morphometric estimation of pulmonary diffusion capacity. II. Effect of PO2 on the growing lung. Respir. Physiol. 11, 247–264.CrossRefGoogle ScholarPubMed
Cressie, N. A. C. (1991). Statistics for Spatial Data. Wiley, New York.Google Scholar
Cruz-Orive, L. M. (1985). Estimating volumes from systematic hyperplane sections. J. Appl. Prob. 22, 518–530.CrossRefGoogle Scholar
Cruz-Orive, L. M. (1989). On the precision of systematic sampling: a review of Matheron's transitive methods. J. Microsc. 153, 315–333.CrossRefGoogle Scholar
Cruz-Orive, L. M. (1990). On the empirical variance of a fractionator estimate. J. Microsc. 160, 89–95.CrossRefGoogle Scholar
Cruz-Orive, L. M. (1993). Systematic sampling in stereology. Bull. Int. Statist. Inst. Proc. 49th session, Florence 55, 451–468.Google Scholar
Cruz-Orive, L. M. (1997). Stereology of single objects. J. Microsc. 186, 93–107.CrossRefGoogle Scholar
Gual Arnau, X. and Cruz-Orive, L. M. (1996). Consistency in systematic sampling. Adv. Appl. Prob. (SGSA) 28, 982–992.CrossRefGoogle Scholar
Guelfand, I. M. and Chilov, G. E. (1962). Les Distributions. Dunod, Paris.Google Scholar
Gundersen, H. J. G. and Jensen, E. B. (1987). The efficiency of systematic sampling in stereology and its prediction. J. Microsc. 147, 229–263.CrossRefGoogle ScholarPubMed
Journel, A. G. and Huijbregts, Ch. J. (1978). Mining Geostatistics. Academic Press, London.Google Scholar
Matérn, B., (1986). Spatial Variation, 2nd edn. (Lecture Notes in Statistics 36) Springer, Berlin.CrossRefGoogle Scholar
Matheron, G. (1963). Principles of geostatistics. Econ. Geol. 58, 1246–1266.CrossRefGoogle Scholar
Matheron, G. (1965). Les Variables Régionalisées et Leur Estimation. Masson, Paris.Google Scholar
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
Matheron, G. (1973). The intrinsic random functions and their applications. Adv. Appl. Prob. 5, 439–468.CrossRefGoogle Scholar
Mattfeldt, T. (1987). Volume estimation of biological objects by systematic sections. J. Math. Biol. 25, 685–695.CrossRefGoogle ScholarPubMed
Miles, R. E. and Davy, P. J. (1977). On the choice of quadrats in stereology. J. Microsc. 110, 27–44.CrossRefGoogle Scholar
Ripley, B. D. (1981). Spatial Statistics. Wiley, New York.CrossRefGoogle Scholar
Ripley, B. D. (1988). Statistical Inference for Spatial Processes. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Roberts, N., Cruz-Orive, L. M., Reid, N., Brodie, D., Bourne, M. and Edwards, R. H. T. (1993). Unbiased estimation of human body composition by the Cavalieri method using magnetic resonance imaging. J. Microsc. 171, 239–253.CrossRefGoogle ScholarPubMed
Roberts, N., Garden, A. S., Cruz-Orive, L. M., Whitehouse, G. H. and Edwards, R. H. T. (1994). Estimation of fetal volume by MRI and stereology. Br. J. Radiol. 67, 1067–1077.CrossRefGoogle ScholarPubMed
Serra, J. (1982). Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
Thioulouse, J., Royet, J. P., Ploye, H. and Houllier, F. (1993). Evaluation of the precision of systematic sampling: nugget effect and covariogram modelling. J. Microsc. 172, 249–256.CrossRefGoogle Scholar
Weibel, E. R. (1970). An automatic sampling stage microscope for stereology. J. Microsc. 91, 1–18.CrossRefGoogle ScholarPubMed
Weibel, E. R. (1980). Stereological Methods. Vol. 2: Theoretical Foundations. Academic Press, London.Google Scholar