Application of stochastic geometry to problems in plankton ecology

1992 ◽  
Vol 336 (1277) ◽  
pp. 225-237 ◽  
Keyword(s):  
Optical Sensors ◽  
Stochastic Geometry ◽  
Point Processes ◽  
Random Distribution ◽  
Interparticle Distance ◽  
Ecosystem Dynamics ◽  
Order Moment ◽  
Patch Structure ◽  
Covariance Functions ◽  
Plankton Ecology

The most fundamental linkages in ecosystem dynamics are trophodynamic. A trophodynamic theory requires a framework based upon inter-organism or interparticle distance, a metric important in its own right, and an essential component relating trophodynamics and the kinetic environment. It is typically assumed that interparticle distances are drawn from a random distribution, even though particles are known to be distributed in patches. Both random and patch-structure interparticle distance are analysed using the theory of stochastic geometry. Aspects of stochastic geometry - point processes and random closed sets (RCS) - useful for studying plankton ecology are presented. For point-process theory, the interparticle distances, random -distribution order statistics, transitions from random to patch structures, and second-order-moment functions are described. For RCS-theory, the volume fractions, contact distributions, and covariance functions are given. Applications of stochastic-geometry theory relate to nutrient flux among organisms, grazing, and coupling between turbulent flow and biological processes. The theory shows that particles are statistically closer than implied by the literature, substantially resolving the troublesome issues of autotroph-heterotroph nutrient exchange; that the microzone notion can be extended by RCS; that patch structure can substantially modify predator-prey encounter rates, even though the number of prey is fixed; and that interparticle distances and the RCS covariance function provide a fundamental coupling with physical processes. In addition to contributing to the understanding of plankton ecology, stochastic geometry is a potentially useful for improving the design of acoustic and optical sensors

scholarly journals Highly Local Model Calibration with a New GEDI LiDAR Asset on Google Earth Engine Reduces Landsat Forest Height Signal Saturation

2020 ◽  
Vol 12 (17) ◽  
pp. 2840 ◽  
Author(s):  
Sean P. Healey ◽  
Zhiqiang Yang ◽  
Noel Gorelick ◽  
Simon Ilyushchenko
Keyword(s):  
Optical Sensors ◽  
Forest Structure ◽  
Model Calibration ◽  
Google Earth ◽  
Ecosystem Dynamics ◽  
Ecosystem Functions ◽  
Calibration Data ◽  
Local Models ◽  
Google Earth Engine ◽  
Signal Saturation

While Landsat has proved to be effective for monitoring many elements of forest condition and change, the platform has well-documented limitations in measuring forest structure, the vertical distribution of the canopy. This is important because structure determines several key ecosystem functions, including: carbon storage; habitat suitability; and timber volume. Canopy structure is directly measured by LiDAR, and it should be possible to train Landsat structure models at a highly local scale with the dense, global sample of full waveform LiDAR observations collected by NASA’s Global Ecosystem Dynamics Investigation (GEDI). Local models are expected to perform better because: (a) such models may take advantage of localized correlations between structure and canopy surface reflectance; and (b) to the extent that models revert to the mean of the calibration data due to a lack of discrimination, local models will revert to a more representative mean. We tested Landsat-based relative height predictions using a new GEDI asset on Google Earth Engine, described here. Mean prediction error declined by 23% and important prediction biases at the extremes of the range of canopy height dropped as model calibration became more local, minimizing forest structure signal saturation commonly associated with Landsat and other passive optical sensors. Our results suggest that Landsat-based maps of structural variables such as height and biomass may substantially benefit from the kind of local calibration that GEDI’s dense sample of LiDAR data supports.

Stochastic geometry from the standpoint of integral geometry

1977 ◽  
Vol 9 (4) ◽  
pp. 792-823 ◽  
Author(s):  
R. V. Ambartzumian
Keyword(s):  
Integral Geometry ◽  
Stochastic Geometry ◽  
Point Processes ◽  
Convex Sets ◽  
Pattern Analysis ◽  
Marked Point ◽  
Marked Point Processes ◽  
Recent Developments ◽  
Euclidean Motion ◽  
Line Processes

This two-part paper surveys some recent developments in integral and stochastic geometry. Part I surveys applications of integral geometry to the theory of euclidean motion-invariant random fibrefields (a fibrefield is a collection of smooth arcs on the plane), involving marked point processes, Palm distribution theory and vertex pattern analysis. Part II develops the more sophisticated theory of Buffon sets in stochastic geometry and the characterisation of measures of lines, giving applications to problems concerning random triangles and colourings, line processes and fixed convex sets.

Analytical Expressions for Formal Kinetics

2012 ◽  
Vol 715-716 ◽  
pp. 971-976 ◽  
Author(s):  
Paulo Rangel Rios ◽  
Weslley L.S. Assis ◽  
Tatiana C. Salazar ◽  
Elena Villa
Keyword(s):  
Cellular Automata ◽  
Microstructural Evolution ◽  
Analytical Solutions ◽  
Stochastic Geometry ◽  
Poisson Point Process ◽  
Point Processes ◽  
Formal Kinetics ◽  
Bulk Nucleation ◽  
Inhomogeneous Poisson Point Process ◽  
Analytical Expressions

In recent papers Rios and Villa resorted to developments in stochastic geometry to revisit theclassical KJMA theory and generalize it for situations in which nuclei were located in space accordingto both homogeneous and inhomogeneous Poisson point processes as well as according to Materncluster process and surface and bulk nucleation in small specimens. Rigorous mathematical methodswere employed to ensure the reliability of the new expressions. These results are briefly described.Analytical expression for inhomogeneous Poisson point process nucleation gives very good agreementwith Cellular Automata simulations. Cellular Automata simulations complement the analyticalsolutions by showing the corresponding microstructural evolution. These new results considerablyexpand the range of situations for which analytical solutions are available.

scholarly journals A Cellular Network Model with Ginibre Configured Base Stations

2014 ◽  
Vol 46 (3) ◽  
pp. 832-845 ◽  
Author(s):  
Naoto Miyoshi ◽  
Tomoyuki Shirai
Keyword(s):  
Point Process ◽  
Coverage Probability ◽  
Stochastic Geometry ◽  
Point Processes ◽  
Spatial Configuration ◽  
Base Stations ◽  
Determinantal Point Processes ◽  
Geometry Model ◽  
Wireless Nodes ◽  
Ginibre Point Process

Stochastic geometry models for wireless communication networks have recently attracted much attention. This is because the performance of such networks critically depends on the spatial configuration of wireless nodes and the irregularity of the node configuration in a real network can be captured by a spatial point process. However, most analysis of such stochastic geometry models for wireless networks assumes, owing to its tractability, that the wireless nodes are deployed according to homogeneous Poisson point processes. This means that the wireless nodes are located independently of each other and their spatial correlation is ignored. In this work we propose a stochastic geometry model of cellular networks such that the wireless base stations are deployed according to the Ginibre point process. The Ginibre point process is one of the determinantal point processes and accounts for the repulsion between the base stations. For the proposed model, we derive a computable representation for the coverage probability—the probability that the signal-to-interference-plus-noise ratio (SINR) for a mobile user achieves a target threshold. To capture its qualitative property, we further investigate the asymptotics of the coverage probability as the SINR threshold becomes large in a special case. We also present the results of some numerical experiments.

scholarly journals Gaussian determinantal processes: A new model for directionality in data

2020 ◽  
Vol 117 (24) ◽  
pp. 13207-13213
Author(s):  
Subhroshekhar Ghosh ◽  
Philippe Rigollet
Keyword(s):  
Stochastic Geometry ◽  
Point Processes ◽  
Matrix Theory ◽  
Principal Component ◽  
Covariance Matrices ◽  
Negative Dependence ◽  
Determinantal Point Processes ◽  
Theoretical Investigations ◽  
Principal Directions ◽  
Parametric Modulation

Determinantal point processes (DPPs) have recently become popular tools for modeling the phenomenon of negative dependence, or repulsion, in data. However, our understanding of an analogue of a classical parametric statistical theory is rather limited for this class of models. In this work, we investigate a parametric family of Gaussian DPPs with a clearly interpretable effect of parametric modulation on the observed points. We show that parameter modulation impacts the observed points by introducing directionality in their repulsion structure, and the principal directions correspond to the directions of maximal (i.e., the most long-ranged) dependency. This model readily yields a viable alternative to principal component analysis (PCA) as a dimension reduction tool that favors directions along which the data are most spread out. This methodological contribution is complemented by a statistical analysis of a spiked model similar to that employed for covariance matrices as a framework to study PCA. These theoretical investigations unveil intriguing questions for further examination in random matrix theory, stochastic geometry, and related topics.

The disintegration of invariant measures

1976 ◽  
Vol 79 (2) ◽  
pp. 337-341 ◽  
Author(s):  
B. D. Ripley
Keyword(s):  
Integral Representation ◽  
Stochastic Geometry ◽  
Point Processes ◽  
Invariant Measures ◽  
Moment Measures ◽  
The Moment

Krickeberg (in (5) and (6)) showed that disintegration applied to invariant measures sometimes yields an integral representation which is useful in analysing the moment measures of point processes. His results, based on Bourbaki's disintegration theory, raised several questions. We refine the theory, using a more general disintegration theorem, and answer his questions by several examples. Finally we consider how far the enlarged theory is applicable in stochastic geometry.

Point processes on the complex plane with applications

2019 ◽  
Author(s):  
◽  
Weichao Wu
Keyword(s):  
Point Process ◽  
Complex Plane ◽  
Spatial Data ◽  
Point Processes ◽  
Dimensional Space ◽  
Gaussian Random Fields ◽  
Process Models ◽  
Covariance Functions ◽  
Complex Point ◽  
Log Linear

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] A point process is a random collection of points from a certain space, and point process models are widely used in areas dealing with spatial data. However, studies of point process theory in the past only focused on Euclidean spaces, and point processes on the complex plane have been rarely explored. In this thesis we introduce and study point processes on the complex plane. We present several important quantities of a complex point process (CPP) that investigate first and second order properties of the process. We further introduce the Poisson complex point process and model its intensity function using log-linear and mixture models in the corresponding 2-dimensional space. The methods are exemplified via applications to density approximation and time series analysis via the spectral density, as well as construction and estimation of covariance functions of Gaussian random fields.

scholarly journals The second-order analysis of stationary point processes

1976 ◽  
Vol 13 (2) ◽  
pp. 255-266 ◽  
Author(s):  
B. D. Ripley
Keyword(s):  
Stationary Point ◽  
Stochastic Geometry ◽  
Point Processes ◽  
Main Tool ◽  
Second Order ◽  
Spatial Point Processes ◽  
Order Analysis ◽  
Spatial Point ◽  
Moment Measures ◽  
Stationary Point Processes

This paper provides a rigorous foundation for the second-order analysis of stationary point processes on general spaces. It illuminates the results of Bartlett on spatial point processes, and covers the point processes of stochastic geometry, including the line and hyperplane processes of Davidson and Krickeberg. The main tool is the decomposition of moment measures pioneered by Krickeberg and Vere-Jones. Finally some practical aspects of the analysis of point processes are discussed.

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