Asymptotic Properties of Estimators for the Parameters of Spatial Inhomogeneous Poisson Point Processes

1994 ◽  
Vol 26 (01) ◽  
pp. 122-154 ◽  
Author(s):  
Stephen L. Rathbun ◽  
Noel Cressie
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Asymptotic Properties ◽  
Point Pattern ◽  
Spatial Point Pattern ◽  
Asymptotic Results ◽  
Borel Set ◽  
Inhomogeneous Poisson Process ◽  
Asymptotically Normal ◽  
Asymptotically Efficient

Consider a spatial point pattern realized from an inhomogeneous Poisson process on a bounded Borel set , with intensity function λ (s; θ), where . In this article, we show that the maximum likelihood estimator and the Bayes estimator are consistent, asymptotically normal, and asymptotically efficient as the sample region . These results extend asymptotic results of Kutoyants (1984), proved for an inhomogeneous Poisson process on [0, T] , where T →∞. They also formalize (and extend to the multiparameter case) results announced by Krickeberg (1982), for the spatial domain . Furthermore, a Cramér–Rao lower bound is found for any estimator of θ. The asymptotic properties of and are considered for modulated (Cox (1972)), and linear Poisson processes.

Asymptotic Properties of Estimators for the Parameters of Spatial Inhomogeneous Poisson Point Processes

1994 ◽  
Vol 26 (1) ◽  
pp. 122-154 ◽  
Author(s):  
Stephen L. Rathbun ◽  
Noel Cressie
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Asymptotic Properties ◽  
Point Pattern ◽  
Spatial Point Pattern ◽  
Asymptotic Results ◽  
Borel Set ◽  
Inhomogeneous Poisson Process ◽  
Asymptotically Normal ◽  
Asymptotically Efficient

Consider a spatial point pattern realized from an inhomogeneous Poisson process on a bounded Borel set , with intensity function λ (s; θ), where . In this article, we show that the maximum likelihood estimator and the Bayes estimator are consistent, asymptotically normal, and asymptotically efficient as the sample region . These results extend asymptotic results of Kutoyants (1984), proved for an inhomogeneous Poisson process on [0, T] , where T →∞. They also formalize (and extend to the multiparameter case) results announced by Krickeberg (1982), for the spatial domain . Furthermore, a Cramér–Rao lower bound is found for any estimator of θ. The asymptotic properties of and are considered for modulated (Cox (1972)), and linear Poisson processes.

THE NONSTATIONARY FRACTIONAL UNIT ROOT

1999 ◽  
Vol 15 (4) ◽  
pp. 549-582 ◽  
Author(s):  
Katsuto Tanaka
Keyword(s):  
Unit Root ◽  
Asymptotic Properties ◽  
Test Statistics ◽  
Finite Sample ◽  
Asymptotic Results ◽  
Simulation Experiments ◽  
Normality Assumption ◽  
Asymptotically Efficient ◽  
Uniformly Most Powerful ◽  
Fractional Unit

This paper deals with a scalar I(d) process {yj}, where the integration order d is any real number. Under this setting, we first explore asymptotic properties of various statistics associated with {yj}, assuming that d is known and is greater than or equal to ½. Note that {yj} becomes stationary when d < ½, whose case is not our concern here. It turns out that the case of d = ½ needs a separate treatment from d > ½. We then consider, under the normality assumption, testing and estimation for d, allowing for any value of d. The tests suggested here are asymptotically uniformly most powerful invariant, whereas the maximum likelihood estimator is asymptotically efficient. The asymptotic theory for these results will not assume normality. Unlike in the usual unit root problem based on autoregressive models, standard asymptotic results hold for test statistics and estimators, where d need not be restricted to d ≥ ½. Simulation experiments are conducted to examine the finite sample performance of both the tests and estimators.

On the MLE for a spatial point pattern

1996 ◽  
Vol 28 (2) ◽  
pp. 339-339
Author(s):  
Francisco Montes ◽  
Jorge Mateu
Keyword(s):  
Process Model ◽  
Exponential Family ◽  
Point Processes ◽  
Stochastic Methods ◽  
Point Pattern ◽  
Parameter Estimates ◽  
Spatial Point Pattern ◽  
Strauss Process ◽  
Markov Point Processes ◽  
Normalising Constant

Parameter estimation for a two-dimensional point pattern is difficult because most of the available stochastic models have intractable likelihoods ([2]). An exception is the class of Gibbs or Markov point processes ([1], [5]), where the likelihood typically forms an exponential family and is given explicitly up to a normalising constant. However, the latter is not known analytically, so parameter estimates must be based on approximations ([3], [6], [7]). In this paper we present comparisons amongst the different techniques available in the literature to obtain an approximation of the maximum likelihood estimate (MLE). Two stochastic methods are specifically illustrated: a Newton-Raphson algorithm ([7]) and the Robbins-Monro procedure ([8]). We use a very simple point process model, the Strauss process ([4]), to test and compare those approximations.

scholarly journals EXPLORATORY SPECTRAL ANALYSIS IN THREE-DIMENSIONAL SPATIAL POINT PATTERNS

2021 ◽  
Vol 39 (1) ◽  
pp. 177
Author(s):  
Edmary Silveira Barreto ARAÚJO ◽  
João Domingos SCALON ◽  
Lurimar Smera BATISTA
Keyword(s):  
Spectral Analysis ◽  
Point Processes ◽  
Three Dimensional ◽  
Dependence Structure ◽  
Point Pattern ◽  
Spatial Point Pattern ◽  
Space Domain ◽  
Point Patterns ◽  
Spatial Point ◽  
Spatial Point Patterns

A spatial point pattern is a collection of points irregularly located within a bounded area (2D) or space (3D) that have been generated by some form of stochastic mechanism. Examples of point patterns include locations of trees in a forest, of cases of a disease in a region, or of particles in a microscopic section of a composite material. Spatial Point pattern analysis is used mostly to determine the absence (completely spatial randomness) or presence (regularity and clustering) of spatial dependence structure of the locations. Methods based on the space domain are widely used for this purpose, while methods conducted in the frequency domain (spectral analysis) are still unknown to most researchers. Spectral analysis is a powerful tool to investigate spatial point patterns, since it does not assume any structural characteristics of the data (ex. isotropy), and uses only the autocovariance function, and its Fourier transform. There are some methods based on the spectral frameworks for analyzing 2D spatial point patterns. There is no such methods available for the 3D situation and, therefore, the aim of this work is to develop new methods based on spectral framework for the analysis of three-dimensional point patterns. The emphasis is on relating periodogram structure to the type of stochastic process which could have generated a 3D observed pattern. The results show that the methods based on spectral analysis developed in this work are able to identify patterns of three typical three-dimensional point processes, and can be used, concurrently, with analyzes in the space domain for a better characterization of spatial point patterns.

On Updating Algorithms and Inference for Stochastic Point Processes

1975 ◽  
Vol 12 (S1) ◽  
pp. 239-259 ◽  
Author(s):  
D. Vere-Jones
Keyword(s):  
Stochastic Process ◽  
Maximum Likelihood ◽  
Stochastic Approximation ◽  
Approximation Theory ◽  
Point Processes ◽  
Maximum Likelihood Estimates ◽  
Doubly Stochastic ◽  
Stochastic Point Processes ◽  
Asymptotically Efficient ◽  
Estimation And Filtering

This paper is an attempt to interpret and extend, in a more statistical setting, techniques developed by D. L. Snyder and others for estimation and filtering for doubly stochastic point processes. The approach is similar to the Kalman-Bucy approach in that the updating algorithms can be derived from a Bayesian argument, and lead ultimately to equations which are similar to those occurring in stochastic approximation theory. In this paper the estimates are derived from a general updating formula valid for any point process. It is shown that almost identical formulae arise from updating the maximum likelihood estimates, and on this basis it is suggested that in practical situations the sequence of estimates will be consistent and asymptotically efficient. Specific algorithms are derived for estimating the parameters in a doubly stochastic process in which the rate alternates between two levels.

scholarly journals Spatial point-pattern analysis as a powerful tool in identifying pattern-process relationships in plant ecology: an updated review

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Mariem Ben-Said
Keyword(s):  
Spatial Patterns ◽  
Pattern Analysis ◽  
Biotic Interactions ◽  
Plant Ecology ◽  
Point Pattern Analysis ◽  
Point Pattern ◽  
Summary Statistics ◽  
Spatial Point Pattern ◽  
Spatial Point ◽  
Spatial Point Pattern Analysis

Abstract Background Ecological processes such as seedling establishment, biotic interactions, and mortality can leave footprints on species spatial structure that can be detectable through spatial point-pattern analysis (SPPA). Being widely used in plant ecology, SPPA is increasingly carried out to describe biotic interactions and interpret pattern-process relationships. However, some aspects are still subjected to a non-negligible debate such as required sample size (in terms of the number of points and plot area), the link between the low number of points and frequently observed random (or independent) patterns, and relating patterns to processes. In this paper, an overview of SPPA is given based on rich and updated literature providing guidance for ecologists (especially beginners) on summary statistics, uni-/bi-/multivariate analysis, unmarked/marked analysis, types of marks, etc. Some ambiguities in SPPA are also discussed. Results SPPA has a long history in plant ecology and is based on a large set of summary statistics aiming to describe species spatial patterns. Several mechanisms known to be responsible for species spatial patterns are actually investigated in different biomes and for different species. Natural processes, plant environmental conditions, and human intervention are interrelated and are key drivers of plant spatial distribution. In spite of being not recommended, small sample sizes are more common in SPPA. In some areas, periodic forest inventories and permanent plots are scarce although they are key tools for spatial data availability and plant dynamic monitoring. Conclusion The spatial position of plants is an interesting source of information that helps to make hypotheses about processes responsible for plant spatial structures. Despite the continuous progress of SPPA, some ambiguities require further clarifications.

The Korteweg-de Vries equation and water waves. Solutions of the equation. Part 1

1973 ◽  
Vol 59 (4) ◽  
pp. 721-736 ◽  
Author(s):  
Harvey Segur
Keyword(s):  
Water Waves ◽  
Large Time ◽  
Vries Equation ◽  
Asymptotic Properties ◽  
Series Representation ◽  
Convergent Series ◽  
Asymptotic Results ◽  
De Vries ◽  
Method Of Solution

The method of solution of the Korteweg–de Vries equation outlined by Gardneret al.(1967) is exploited to solve the equation. A convergent series representation of the solution is obtained, and previously known aspects of the solution are related to this general form. Asymptotic properties of the solution, valid for large time, are examined. Several simple methods of obtaining approximate asymptotic results are considered.

Convergence to equilibrium in a traffic model with restricted passing

1979 ◽  
Vol 16 (4) ◽  
pp. 881-889 ◽  
Author(s):  
Hans Dieter Unkelbach
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Point Processes ◽  
Road Traffic ◽  
Poisson Processes ◽  
Traffic Model ◽  
Cluster Point ◽  
Convergence To Equilibrium ◽  
Constant Intensity

A road traffic model with restricted passing, formulated by Newell (1966), is described by conditional cluster point processes and analytically handled by generating functionals of point processes.The traffic distributions in either space or time are in equilibrium, if the fast cars form a Poisson process with constant intensity combined with Poisson-distributed queues behind the slow cars (Brill (1971)). It is shown that this state of equilibrium is stable, which means that this state will be reached asymptotically for general initial traffic distributions. Furthermore the queues behind the slow cars dissolve asymptotically like independent Poisson processes with diminishing rate, also independent of the process of non-queuing cars. To get these results limit theorems for conditional cluster point processes are formulated.

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