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15
Indices of Dependence Between Types in Multivariate Point Patterns
 Scandinavian Journal of Statistics
, 1999
"... We propose new summary statistics quantifying several forms of dependence between types in a spatial pattern of points classified into distinct types. These statistics are the multivariate counterparts of the Jfunction for point processes of a single type, introduced in [18]. They are based on comp ..."
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Cited by 10 (3 self)
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We propose new summary statistics quantifying several forms of dependence between types in a spatial pattern of points classified into distinct types. These statistics are the multivariate counterparts of the Jfunction for point processes of a single type, introduced in [18]. They are based on comparing distances from a type i point to either the nearest type j point or to the nearest point in the pattern regardless of type to these distances seen from an arbitrary point in space. Information about the range of interaction can also be inferred. Our statistics can be computed explicitly for a range of wellknown multivariate point process models. Some applications to bivariate data sets are presented as well. Keywords & Phrases: ants' nests, beta cells, empty space function, hamster tumour, J  function, multitype point patterns, myrtle disease, nearestneighbour distance distribution function, point processes, random labelling, spatial interaction, spatial statistics. AMS Mathemat...
QuermassInteraction Processes: Conditions for Stability
"... We consider a class of random point and germgrain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let \Xi be the union of all grains. One can now construct new processes whose de ..."
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Cited by 6 (1 self)
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We consider a class of random point and germgrain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let \Xi be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear combination of quermass functionals of \Xi. If only the area functional is used, then the areainteraction point process is recovered. New point processes arise if we include the perimeter length functional, or the Euler functional (number of components minus number of holes). The main question addressed by the paper is that of when the resulting point process is welldefined: geometric arguments are used to establish conditions for the point process to be stable in the sense of Ruelle. Key words: areainteraction point process, Boolean model, germgrain model, Markov point process, Minkowski functional, quermass integral, semiMarkov random ...
KaplanMeier Type Estimators for Linear Contact Distributions
 Scand. J. Statist
, 1996
"... . The linear contact distribution function is shown to be continuously differentiable for any stationary random closed set, which implies the existence of a continuous density and hazard rate. Moreover, it is proved that the density is monotone decreasing. When the linear contact distribution functi ..."
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Cited by 6 (3 self)
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. The linear contact distribution function is shown to be continuously differentiable for any stationary random closed set, which implies the existence of a continuous density and hazard rate. Moreover, it is proved that the density is monotone decreasing. When the linear contact distribution function is estimated from observations in a bounded window, the distance to the set of interest from a fixed point in a given linear direction is rightcensored by its distance to the boundary of the window. We develop a KaplanMeier type estimator for the linear contact distribution function and hazard rate. We show that the new estimator has a ratiounbiasedness property and that it is an absolutely continuous distribution function. A CLT is derived for independent replications within a fixed observation window. The techniques are applied to the analysis of spatial patterns in acid milk. The feature of replication of the images and the CLT for the estimator give confidence bounds on the estimat...
TimeInvariance Estimating Equations
 Bernoulli
, 1995
"... We describe a general method for deriving estimators of the parameter of a statistical model, with particular relevance to highly structured stochastic systems such as spatial random processes and `graphical' conditional independence models. The method is based on representing the stochastic mo ..."
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Cited by 5 (0 self)
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We describe a general method for deriving estimators of the parameter of a statistical model, with particular relevance to highly structured stochastic systems such as spatial random processes and `graphical' conditional independence models. The method is based on representing the stochastic model as the equilibrium distribution of a Markov process Y = (Y t ; t ? 0) where the discrete or continuous `time' index t is to be understood as a fictional extra dimension added to the original setting. The parameter estimate b ` is obtained by equating to zero the generator of Y applied to a suitable statistic and evaluated at the data x. This produces an unbiased estimating equation for `. Natural special cases include the reduced sample estimator in survival analysis, the maximum pseudolikelihood estimator for random fields and for point processes, the TakacsFiksel method for point processes, `variational' estimators for random fields and multivariate distributions, and many standard esti...
First Contact Distributions For Spatial Patterns: Regularity And Estimation
 Advances in Applied Probability
, 2002
"... For applications in spatial statistics, an important property of a random set X in R k is its first contact distribution. This is the distribution of the distance from a fixed point 0 to the nearest point of X, where distance is measured using scalar dilations of a fixed test set B. We show that, ..."
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Cited by 5 (3 self)
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For applications in spatial statistics, an important property of a random set X in R k is its first contact distribution. This is the distribution of the distance from a fixed point 0 to the nearest point of X, where distance is measured using scalar dilations of a fixed test set B. We show that, if B is convex and contains a neighbourhood of 0, the first contact distribution function FB is absolutely continuous. We give two explicit representations of FB , and additional regularity conditions under which FB is continuously differentiable. A KaplanMeier estimator of FB is introduced and its basic properties examined. EMPTY SPACE FUNCTION; CONVEX TEST SETS; RANDOM CLOSED SETS; STOCHASTICGEOMETRY; SPATIAL STATISTICS; COAREA FORMULA; PRODUCT LIMIT ESTIMATES; HAZARD RATE AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60D05, 62H11 SECONDARY 62G05 1 Introduction The statistical analysis of a spatial pattern often involves treating the pattern as a realisation of a stationary random set X in R...
Structural Analysis of Sterol Distributions in the Plasma Membrane of Living Cells
, 2004
"... ABSTRACT: Although plasma membrane (PM) cholesterolrich andpoor domains have been isolated by subcellular fractionation, the realtime arrangement of cholesterol in such domains in living cells is still unclear. Therefore, dehydroergosterol (DHE), a naturally occurring fluorescent sterol, was inco ..."
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Cited by 3 (1 self)
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ABSTRACT: Although plasma membrane (PM) cholesterolrich andpoor domains have been isolated by subcellular fractionation, the realtime arrangement of cholesterol in such domains in living cells is still unclear. Therefore, dehydroergosterol (DHE), a naturally occurring fluorescent sterol, was incorporated into cultured Lcell fibroblasts. Two PM markers, the enhanced cyan fluorescent protein (ECFPMem) and 3′dioctadecyloxacarbocyanine perchlorate [DiOC18(3)], were used to distinguish DHE localized at the PM of living cells. Spatial enrichment of DHE in the PM of living cells was visualized in real time by multiphoton laser scanning microscopy (MPLSM). Quantitative models and imageprocessing techniques were developed for statistical analysis of the distribution of DHE within the PM. The PM was resolved from the cytoplasm in a twostep process, and a smooth trajectory reference of the PM was refined by statistical regression and momentsbased techniques. Thus, DHE intensities over the PM were measured following the major DHE intensity distributions. Spatial distributions of DHE within the PM were examined by a statistical inference technique, complete spatial randomness (CSR). For PM regions densely populated with DHE, the distributions of DHE exhibited statistical arrangements that were not spatial random (i.e., homogeneous Poisson process) or regular but, instead, exhibited strong cluster patterns. In effect, realtime MPLSM imaging data for the first time demonstrated that sterol enrichment occurred in clustered
The Empty Space Hazard of a Spatial Pattern
, 1994
"... This paper is concerned with the analysis of spatial patterns like those displayed in Figures 1 6, which may be point patterns, line segment or curvilinear patterns, or binary mosaics. The analysis usually begins with the computation of `exploratory' summary statistics such as the empty space ..."
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Cited by 3 (1 self)
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This paper is concerned with the analysis of spatial patterns like those displayed in Figures 1 6, which may be point patterns, line segment or curvilinear patterns, or binary mosaics. The analysis usually begins with the computation of `exploratory' summary statistics such as the empty space function F , the point pattern statistics G and K, and the spatial covariance function C [8, 11, 22, 28, 39, 43, 35, 36]. However, recent work on point patterns suggests that the densities of F; G and K are easier to interpret than the functions themselves, while
Estimating the J function without edge correction
 Statistica Neerlandica
, 2000
"... The interaction between points in a spatial point process can be measured by its empty space function F, its nearestneighbour distance distribution function G, and by combinations such as the Jfunction J = (1 − G)/(1 − F). The estimation of these functions is hampered by edge effects: the uncorrec ..."
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Cited by 2 (2 self)
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The interaction between points in a spatial point process can be measured by its empty space function F, its nearestneighbour distance distribution function G, and by combinations such as the Jfunction J = (1 − G)/(1 − F). The estimation of these functions is hampered by edge effects: the uncorrected, empirical distributions of distances observed in a bounded sampling window W give severely biased estimates of F and G. However, in this paper we show that the corresponding uncorrected estimator of the function J = (1 − G)/(1 − F) is approximately unbiased for the Poisson case, and is useful as a summary statistic. Specifically, consider the estimate ̂ JW of J computed from uncorrected estimates of F and G. The function JW(r), estimated by ̂ JW, possesses similar properties to the J function, for example JW(r) is identically 1 for Poisson processes. This enables direct interpretation of uncorrected estimates of J, something not possible with uncorrected estimates of either F, G or K. We propose a Monte Carlo test for complete spatial randomness based on testing whether JW(r) ≡ 1. Computer simulations suggest this test is at least as powerful as tests based on edge corrected estimators of J.
Nonparametric Estimation of the Chord Length Distribution
"... The distribution of the length of a typical chord of a stationary random set is an interesting feature of the set's whole distribution. We give a nonparametric estimator of the chord length distribution and prove its strong consistency. We report on a simulation experiment in which our estimato ..."
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The distribution of the length of a typical chord of a stationary random set is an interesting feature of the set's whole distribution. We give a nonparametric estimator of the chord length distribution and prove its strong consistency. We report on a simulation experiment in which our estimator compared favorably to a reduced sample estimator. Both estimators are illustrated by applying them to an image sample from a yoghurt ferment. We briefly discuss the closely related problem of estimation of the linear contact distribution. We show by a simulation experiment that a transformation of our estimator of the chord length distribution is more e#cient than a KaplanMeier type estimator of the linear contact distribution. Mathematics Subject Classification: 62M30 (Inference from spatial processes) .