Asymptotic variance of Newton–Cotes quadratures based on randomized sampling points

2020 ◽  
Vol 52 (4) ◽  
pp. 1284-1307
Author(s):  
Mads Stehr ◽  
Markus Kiderlen
Keyword(s):  
Point Process ◽  
Asymptotic Variance ◽  
Compact Object ◽  
Quadrature Rule ◽  
Randomized Sampling ◽  
Riemann Sum ◽  
Sampling Points ◽  
Stationary Point Process ◽  
The Mean ◽  
Point Distance

AbstractWe consider the problem of numerical integration when the sampling nodes form a stationary point process on the real line. In previous papers it was argued that a naïve Riemann sum approach can cause a severe variance inflation when the sampling points are not equidistant. We show that this inflation can be avoided using a higher-order Newton–Cotes quadrature rule which exploits smoothness properties of the integrand. Under mild assumptions, the resulting estimator is unbiased and its variance asymptotically obeys a power law as a function of the mean point distance. If the Newton–Cotes rule is of sufficiently high order, the exponent of this law turns out to only depend on the point process through its mean point distance. We illustrate our findings with the stereological estimation of the volume of a compact object, suggesting alternatives to the well-established Cavalieri estimator.

scholarly journals THE CAVALIERI ESTIMATOR WITH UNEQUAL SECTION SPACING REVISITED

2017 ◽  
Vol 36 (2) ◽  
pp. 135 ◽  
Author(s):  
Markus Kiderlen ◽  
Karl-Anton Dorph-Petersen
Keyword(s):  
Variance Reduction ◽  
Compact Object ◽  
Quadrature Rule ◽  
Special Situation ◽  
Randomized Sampling ◽  
Mild Conditions ◽  
Sampling Points ◽  
Random Variability ◽  
Simulation Results ◽  
Simpson's Rule

The Cavalieri method allows to estimate the volume of a compact object from area measurements in equidistant parallel planar sections. However, the spacing and thickness of sections can be quite irregular in applications. Recent publications have thus focused on the effect of random variability in section spacing, showing that the classical Cavalieri estimator is still unbiased when the stack of parallel planes is stationary, but that the existing variance approximations must be adjusted. The present paper considers the special situation, where the distances between consecutive section planes can be measured and thus where Cavalieri’s estimator can be replaced by a quadrature rule with randomized sampling points. We show that, under mild conditions, the trapezoid rule and Simpson’s rule lead to unbiased volume estimators and give simulation results that indicate that a considerable variance reduction compared to the generalized Cavalieri estimator can be achieved.

On the mean shape of particle processes

1997 ◽  
Vol 29 (04) ◽  
pp. 890-908 ◽  
Author(s):  
Wolfgang Weil
Keyword(s):  
Normal Distribution ◽  
Point Process ◽  
Stationary Point ◽  
Mean Shape ◽  
Stationary Point Process ◽  
The Mean

For a stationary point process X of sets in the convex ring in ℝ d , a relation is given between the mean particles of the section process X ∩ E (where E varies through the set of k-dimensional subspaces in ℝ d ) and a mean particle of X. In particular, it is shown that the mean bodies of all planar sections of X determine the Blaschke body of X and hence the mean normal distribution of X.

On the mean shape of particle processes

1997 ◽  
Vol 29 (4) ◽  
pp. 890-908 ◽  
Author(s):  
Wolfgang Weil
Keyword(s):  
Normal Distribution ◽  
Point Process ◽  
Stationary Point ◽  
Mean Shape ◽  
Stationary Point Process ◽  
The Mean

For a stationary point process X of sets in the convex ring in ℝd, a relation is given between the mean particles of the section process X ∩ E (where E varies through the set of k-dimensional subspaces in ℝd) and a mean particle of X. In particular, it is shown that the mean bodies of all planar sections of X determine the Blaschke body of X and hence the mean normal distribution of X.

scholarly journals Five-Year Enhanced Natural Attenuation of Historically Coal-Tar-Contaminated Soil: Analysis of Polycyclic Aromatic Hydrocarbon and Phenol Contents

2021 ◽  
Vol 18 (5) ◽  
pp. 2265
Author(s):  
Arkadiusz Telesiński ◽  
Anna Kiepas-Kokot
Keyword(s):  
Aromatic Hydrocarbons ◽  
Natural Attenuation ◽  
Soil Analysis ◽  
Coal Tar ◽  
Soil Samples ◽  
Lower Intensity ◽  
Sampling Points ◽  
Polycyclic Aromatic ◽  
Monoaromatic Hydrocarbons ◽  
The Mean

The objective of this study was to assess the soil pollution on an industrial wasteland, where coal-tar was processed in the period between 1880 and 1997, and subsequent to assess the decline in the content of phenols and polycyclic aromatic hydrocarbons (PAHs) during enhanced natural attenuation. The soil of the investigated area was formed from a layer of uncompacted fill. Twelve sampling points were established in the investigated area for collecting soil samples. A study conducted in 2015 did not reveal any increase in the content of heavy metals, monoaromatic hydrocarbons (BTEX), and cyanides. However, the content of PAHs and phenols was higher than the content permitted by Polish norms in force until 2016. In the case of PAHs, it was observed for individual compounds and their total contents. Among the various methods, enhanced natural attenuation was chosen for the remediation of investigated area. Repeated analyses of the contents of phenols and PAHs were conducted in 2020. The results of the analyses showed that enhanced natural attenuation has led to efficient degradation of the simplest substances—phenol and naphthalene. The content of these compounds in 2020 was not elevated compared to the standards for industrial wastelands. The three- and four-ring hydrocarbons were degraded at a lower intensity. Based on the mean decrease in content after 5-year enhanced natural attenuation, the compounds can be arranged in the following order: phenols > naphthalene > phenanthrene > fluoranthene > benzo(a)anthracene > chrysene > anthracene.

On projected thick sections of stationary point processes and Boolean models

1996 ◽  
Vol 28 (2) ◽  
pp. 335-335
Author(s):  
Markus Kiderlen
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Linear Subspace ◽  
Isotropic Case ◽  
Thick Section ◽  
Boolean Models ◽  
Dimensional Linear Subspace ◽  
Stationary Point Process ◽  
Stationary Point Processes

For a stationary point process X of convex particles in ℝd the projected thick section process X(L) on a q-dimensional linear subspace L is considered. Formulae connecting geometric functionals, e.g. the quermass densities of X and X(L), are presented. They generalize the classical results of Miles (1976) and Davy (1976) which hold only in the isotropic case.

scholarly journals Comparison of shortest sailing distance through random and regular sampling points

2004 ◽  
Vol 61 (1) ◽  
pp. 140-147 ◽  
Author(s):  
Alf Harbitz ◽  
Michael Pennington
Keyword(s):  
Random Sampling ◽  
Minimum Distance ◽  
Barents Sea ◽  
Pandalus Borealis ◽  
The Barents Sea ◽  
Theoretical Sampling ◽  
Sampling Points ◽  
Regular Sampling ◽  
The Mean ◽  
The Difference

Abstract The shortest sailing distance through n sampling points is calculated for simple theoretical sampling domains (square and circle) as well as for a rather irregular and concavely shaped real sampling domain in the Barents Sea. The sampling sites are either located at the nodes of a square grid (regular sampling) or they are randomly distributed. For n less than ten, the exact shortest sailing distance is derived. For larger n, a traveling salesman algorithm (simulated annealing) was applied, and its bias (distance from true minimum) was estimated based on a case where the true minimum distance was known. In general, the average minimum sailing distance based on random sampling was considerably shorter than for regular sampling, and the difference increased with sample size until an asymptotic value was reached at about n=60 for a square domain. For the sampling domain in the Barents Sea used for shrimp (Pandalus borealis) abundance surveys (n=118 stations), the cruise-track lengths based on random sampling were approximately normally distributed. The mean sailing distance was 18% shorter than the cruise track for regular sampling and the standard deviation equalled 2.6%.

On the weak convergence of a class of estimators of the variance-time curve of a weakly stationary point process

1977 ◽  
Vol 14 (01) ◽  
pp. 114-126 ◽  
Author(s):  
A. M. Liebetrau
Keyword(s):  
Weak Convergence ◽  
Gaussian Process ◽  
Point Process ◽  
Stationary Point ◽  
Stationary Gaussian Process ◽  
Time Curve ◽  
Second Moment ◽  
Stationary Point Process ◽  
Class Of Estimators

The second-moment structure of an estimator V*(t) of the variance-time curve V(t) of a weakly stationary point process is obtained in the case where the process is Poisson. This result is used to establish the weak convergence of a class of estimators of the form Tβ (V*(tTα ) – V(tTα )), 0 < α < 1, to a non-stationary Gaussian process. Similar results are shown to hold when α = 0 and in the case where V(tTα ) is replaced by a suitable estimator.

Some multivariate generalizations of results in univariate stationary point processes

1977 ◽  
Vol 14 (04) ◽  
pp. 748-757 ◽  
Author(s):  
Mark Berman
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Stationary Point Process ◽  
Stationary Point Processes

Some relationships are derived between the asynchronous and partially synchronous counting and interval processes associated with a multivariate stationary point process. A few examples are given to illustrate some of these relationships.

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