θ-stationary point processes and their second-order analysis

1981 ◽  
Vol 18 (04) ◽  
pp. 864-878
Author(s):  
Karen Byth
Keyword(s):  
Point Process ◽  
Spatial Patterns ◽  
Stationary Point ◽  
Point Processes ◽  
Simulated Data ◽  
Second Order ◽  
Stationary Processes ◽  
Order Analysis ◽  
Stationary Point Processes ◽  
Point Of Reference

The concept of θ-stationarity for a simple second-order point process in R2 is introduced. This concept is closely related to that of isotropy. Some θ-stationary processes are defined. Techniques are given for simulating realisations of these processes. The second-order analysis of these processes which have an obvious point of reference or origin is considered. Methods are suggested for modelling spatial patterns which are realisations of such processes. These methods are illustrated using simulated data. The ideas are extended to multitype point processes.

θ-stationary point processes and their second-order analysis

1981 ◽  
Vol 18 (4) ◽  
pp. 864-878 ◽  
Author(s):  
Karen Byth
Keyword(s):  
Point Process ◽  
Spatial Patterns ◽  
Stationary Point ◽  
Point Processes ◽  
Simulated Data ◽  
Second Order ◽  
Stationary Processes ◽  
Order Analysis ◽  
Stationary Point Processes ◽  
Point Of Reference

The concept of θ-stationarity for a simple second-order point process in R2 is introduced. This concept is closely related to that of isotropy. Some θ-stationary processes are defined. Techniques are given for simulating realisations of these processes. The second-order analysis of these processes which have an obvious point of reference or origin is considered. Methods are suggested for modelling spatial patterns which are realisations of such processes. These methods are illustrated using simulated data. The ideas are extended to multitype point processes.

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.

scholarly journals The second-order analysis of stationary point processes

1976 ◽  
Vol 13 (02) ◽  
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.

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.

Bivariate stationary point processes, fundamental relations and first recurrence times

1972 ◽  
Vol 4 (02) ◽  
pp. 296-317 ◽  
Author(s):  
T. K. M. Wisniewski
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Counting Process ◽  
Derived Relations ◽  
Recurrence Times ◽  
Different Types ◽  
First Recurrence ◽  
Stationary Point Processes ◽  
Event Counting

Various types of time and event sampling of a stationary and orderly bivariate point process are considered. Fundamental relations between inter-event intervals and the event counting process are derived. Relations between first forward recurrence times and their moments for different types of sampling are obtained.

Selective interaction of a Poisson and renewal process: first-order stationary point results

1970 ◽  
Vol 7 (02) ◽  
pp. 359-372 ◽  
Author(s):  
A. J. Lawrance
Keyword(s):  
Stationary Point ◽  
Joint Distribution ◽  
Point Processes ◽  
First Order ◽  
Response Interval ◽  
Selective Interaction ◽  
Counting Distribution ◽  
Stationary Point Process ◽  
Equilibrium Process ◽  
Stationary Point Processes

The simple stationarity of a previously derived equilibrium process of responses in a renewal inhibited stationary point process is established by deriving the joint distribution of the number of responses in contiguous intervals in the process. For a renewal inhibited Poisson process the variancetime function of the process is obtained; the distribution of an arbitrary between-response interval and the synchronous counting distribution are also derived following analytic justification of the required results. These results strengthen earlier results in the theory of stationary point processes. Three other point processes arising from the interaction are briefly discussed.

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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