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.

Modelling Seismicity in California as a Spatio-Temporal Point Process Using inlabru: Insights for Earthquake Forecasting

2020 ◽  
Author(s):  
Mark Naylor ◽  
Kirsty Bayliss ◽  
Finn Lindgren ◽  
Francesco Serafini ◽  
Ian Main
Keyword(s):  
Point Process ◽  
Spatial Data ◽  
Point Processes ◽  
Earthquake Forecasting ◽  
Data Sets ◽  
Computationally Efficient ◽  
Data Types ◽  
Initial Application ◽  
Spatio Temporal ◽  
Diverse Data

<p>Many earthquake forecasting approaches have developed bespokes codes to model and forecast the spatio-temporal eveolution of seismicity. At the same time, the statistics community have been working on a range of point process modelling codes. For example, motivated by ecological applications, inlabru models spatio-temporal point processes as a log-Gaussian Cox Process and is implemented in R. Here we present an initial implementation of inlabru to model seismicity. This fully Bayesian approach is computationally efficient because it uses a nested Laplace approximation such that posteriors are assumed to be Gaussian so that their means and standard deviations can be deterministically estimated rather than having to be constructed through sampling. Further, building on existing packages in R to handle spatial data, it can construct covariate maprs from diverse data-types, such as fault maps, in an intutitive and simple manner.</p><p>Here we present an initial application to the California earthqauke catalogue to determine the relative performance of different data-sets for describing the spatio-temporal evolution of seismicity.</p>

scholarly journals Decomposition of Repulsive Clusters in Complex Point Processes with Heterogeneous Components

2019 ◽  
Vol 8 (8) ◽  
pp. 326
Author(s):  
Ci Song ◽  
Tao Pei
Keyword(s):  
Point Process ◽  
Probability Distribution Function ◽  
Large Scale ◽  
Point Processes ◽  
Small Scale ◽  
Simulation Experiments ◽  
Service Characteristics ◽  
Point Of Interest ◽  
Complex Point

The decomposition of a point process is useful for the analysis of spatial patterns and in the discovery of potential mechanisms of geographic phenomena. However, when a local repulsive cluster is present in a complex heterogeneous point process, the traditional solution, which is based on clustering, may be invalid for decomposition because a repulsive pattern is not subject to a specific probability distribution function and the effects of aggregative and repulsive components may be counterbalanced. To solve this problem, this paper proposes a method of decomposing repulsive clusters in complex point processes with multiple heterogeneous components. A repulsive cluster is defined as a set of repulsive density-connected points that are separated by a certain distance at a small scale and aggregated at a large scale simultaneously. The H-function is used to identify repulsive clusters by determining the repulsive distance and extracting repulsive points for further clustering. Through simulation experiments based on three datasets, the proposed method has been shown to effectively perform repulsive cluster decomposition in heterogeneous point processes. A case study of the point of interest (POI) dataset in Beijing also indicates that the method can identify meaningful repulsive clusters from types of POIs that represent different service characteristics of shops in different local regions.

scholarly journals MODELLING THE POSITION OF CELL PROFILES ALLOWING FOR BOTH INHOMOGENEITY AND INTERACTION

2011 ◽  
Vol 19 (3) ◽  
pp. 183 ◽  
Author(s):  
Linda Stougaard Nielsen
Keyword(s):  
Point Process ◽  
Mucous Membrane ◽  
Point Processes ◽  
Point Interaction ◽  
Process Models ◽  
Markov Point Processes ◽  
Point Process Models

It is of interest to consider models for point processes that allow for interaction between the points as well as for inhomogeneity in the intensity of the points. Markov point process models are very useful to describe point interaction and can also be used to describe inhomogeneity. A particular type of inhomogeneous Markov point processes obtained by transforming a homogeneous Markov point process will be considered. The position of cell proles in a 2D section of the mucous membrane in the stomach of a rat will be examined using this model.

scholarly journals INHOMOGENEITY IN SPATIAL COX POINT PROCESSES – LOCATION DEPENDENT THINNING IS NOT THE ONLY OPTION

2010 ◽  
Vol 29 (3) ◽  
pp. 133 ◽  
Author(s):  
Michaela Prokešová
Keyword(s):  
Point Process ◽  
Process Model ◽  
Point Processes ◽  
Homogeneous Model ◽  
Process Models ◽  
Point Patterns ◽  
Estimation Procedures ◽  
Markov Point Processes ◽  
Stationary Point Process ◽  
Point Process Models

In the literature on point processes the by far most popular option for introducing inhomogeneity into a point process model is the location dependent thinning (resulting in a second-order intensity-reweighted stationary point process). This produces a very tractable model and there are several fast estimation procedures available. Nevertheless, this model dilutes the interaction (or the geometrical structure) of the original homogeneous model in a special way. When concerning the Markov point processes several alternative inhomogeneous models were suggested and investigated in the literature. But it is not so for the Cox point processes, the canonical models for clustered point patterns. In the contribution we discuss several other options how to define inhomogeneous Cox point process models that result in point patterns with different types of geometric structure. We further investigate the possible parameter estimation procedures for such models.

scholarly journals Modeling Forest Tree Data Using Sequential Spatial Point Processes

2021 ◽  
Author(s):  
Adil Yazigi ◽  
Antti Penttinen ◽  
Anna-Kaisa Ylitalo ◽  
Matti Maltamo ◽  
Petteri Packalen ◽  
...  
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Process Models ◽  
Aerial Image ◽  
Tree Age ◽  
Data Set ◽  
Spatial Point Processes ◽  
Spatial Point ◽  
Forest Inventories

AbstractThe spatial structure of a forest stand is typically modeled by spatial point process models. Motivated by aerial forest inventories and forest dynamics in general, we propose a sequential spatial approach for modeling forest data. Such an approach is better justified than a static point process model in describing the long-term dependence among the spatial location of trees in a forest and the locations of detected trees in aerial forest inventories. Tree size can be used as a surrogate for the unknown tree age when determining the order in which trees have emerged or are observed on an aerial image. Sequential spatial point processes differ from spatial point processes in that the realizations are ordered sequences of spatial locations, thus allowing us to approximate the spatial dynamics of the phenomena under study. This feature is useful in interpreting the long-term dependence and spatial history of the locations of trees. For the application, we use a forest data set collected from the Kiihtelysvaara forest region in Eastern Finland.

A Computationally Efficient Method for Nonparametric Modeling of Neural Spiking Activity with Point Processes

2010 ◽  
Vol 22 (8) ◽  
pp. 2002-2030 ◽  
Author(s):  
Todd P. Coleman ◽  
Sridevi S. Sarma
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Intensity Function ◽  
Process Models ◽  
Conditional Intensity ◽  
Computationally Efficient ◽  
Neural Data ◽  
Neural Spiking ◽  
Point Process Models ◽  
Conditional Intensity Function

Point-process models have been shown to be useful in characterizing neural spiking activity as a function of extrinsic and intrinsic factors. Most point-process models of neural activity are parametric, as they are often efficiently computable. However, if the actual point process does not lie in the assumed parametric class of functions, misleading inferences can arise. Nonparametric methods are attractive due to fewer assumptions, but computation in general grows with the size of the data. We propose a computationally efficient method for nonparametric maximum likelihood estimation when the conditional intensity function, which characterizes the point process in its entirety, is assumed to be a Lipschitz continuous function but otherwise arbitrary. We show that by exploiting much structure, the problem becomes efficiently solvable. We next demonstrate a model selection procedure to estimate the Lipshitz parameter from data, akin to the minimum description length principle and demonstrate consistency of our estimator under appropriate assumptions. Finally, we illustrate the effectiveness of our method with simulated neural spiking data, goldfish retinal ganglion neural data, and activity recorded in CA1 hippocampal neurons from an awake behaving rat. For the simulated data set, our method uncovers a more compact representation of the conditional intensity function when it exists. For the goldfish and rat neural data sets, we show that our nonparametric method gives a superior absolute goodness-of-fit measure used for point processes than the most common parametric and splines-based approaches.

scholarly journals Transforming Spatial Point Processes into Poisson Processes Using Random Superposition

2012 ◽  
Vol 44 (1) ◽  
pp. 42-62 ◽  
Author(s):  
Jesper Møller ◽  
Kasper K. Berthelsen
Keyword(s):  
Model Checking ◽  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Intensity Function ◽  
Poisson Processes ◽  
Process Models ◽  
Death Process ◽  
Spatial Point Process ◽  
Spatial Point

Most finite spatial point process models specified by a density are locally stable, implying that the Papangelou intensity is bounded by some integrable function β defined on the space for the points of the process. It is possible to superpose a locally stable spatial point process X with a complementary spatial point process Y to obtain a Poisson process X ⋃ Y with intensity function β. Underlying this is a bivariate spatial birth-death process (Xt, Yt) which converges towards the distribution of (X, Y). We study the joint distribution of X and Y, and their marginal and conditional distributions. In particular, we introduce a fast and easy simulation procedure for Y conditional on X. This may be used for model checking: given a model for the Papangelou intensity of the original spatial point process, this model is used to generate the complementary process, and the resulting superposition is a Poisson process with intensity function β if and only if the true Papangelou intensity is used. Whether the superposition is actually such a Poisson process can easily be examined using well-known results and fast simulation procedures for Poisson processes. We illustrate this approach to model checking in the case of a Strauss process.

scholarly journals Transforming Spatial Point Processes into Poisson Processes Using Random Superposition

2012 ◽  
Vol 44 (01) ◽  
pp. 42-62 ◽  
Author(s):  
Jesper Møller ◽  
Kasper K. Berthelsen
Keyword(s):  
Model Checking ◽  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Intensity Function ◽  
Poisson Processes ◽  
Process Models ◽  
Death Process ◽  
Spatial Point Process ◽  
Spatial Point

Most finite spatial point process models specified by a density are locally stable, implying that the Papangelou intensity is bounded by some integrable function β defined on the space for the points of the process. It is possible to superpose a locally stable spatial point process X with a complementary spatial point process Y to obtain a Poisson process X ⋃ Y with intensity function β. Underlying this is a bivariate spatial birth-death process (X t , Y t ) which converges towards the distribution of (X, Y). We study the joint distribution of X and Y, and their marginal and conditional distributions. In particular, we introduce a fast and easy simulation procedure for Y conditional on X. This may be used for model checking: given a model for the Papangelou intensity of the original spatial point process, this model is used to generate the complementary process, and the resulting superposition is a Poisson process with intensity function β if and only if the true Papangelou intensity is used. Whether the superposition is actually such a Poisson process can easily be examined using well-known results and fast simulation procedures for Poisson processes. We illustrate this approach to model checking in the case of a Strauss process.

SEGMENTED POINT PROCESS MODELS FOR MAINTAINED SYSTEMS

2007 ◽  
Vol 14 (05) ◽  
pp. 431-458 ◽  
Author(s):  
A. SYAMSUNDAR ◽  
V. N. A. NAIKAN
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Single Point ◽  
Process Models ◽  
Parameter Estimates ◽  
Minimal Repair ◽  
Homogeneous Poisson Process ◽  
Stable Conditions ◽  
Point Process Models

A maintained system is generally modeled using point processes. The most common processes used are the renewal process and the non homogeneous Poisson process corresponding to maximal and minimal repair situations with homogeneous Poisson process being a special case of both. A general repair formulation with a factor indicating the degree of repair is introduced into the minimal repair model to form an Arithmetic Reduction of Intensity model. These processes are generally able to model maintained systems with a fair degree of accuracy when the system is operating under stable conditions. However whenever there is a change in the environment these models which are monotonic in nature are not able to accommodate this change. Such systems operating under different environments need to be modelled by segmented models with the system domain divided into segments at the points of changes in the environment. The individual segments can then be modeled by any of the above point process models and these can be combined to form a composite model. This paper proposes a statistical model of such an operating/maintenance environment. Its purpose is to quantify the impacts of changes in the environment on the failure intensities. Field data from an industrial-setting demonstrate that appropriate parameter estimates for such phenomena can be obtained and such models are shown to more accurately describe the maintained system in a changing environment than the single point process models usually used.

Sign in / Sign up

Export Citation Format

Share Document