Bivariate stationary point processes, fundamental relations and first recurrence times

1972 ◽  
Vol 4 (2) ◽  
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.

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.

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.

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.

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.

On projected thick sections of stationary point processes and Boolean models

1996 ◽  
Vol 28 (02) ◽  
pp. 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 ℝ d 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.

Ergodicity and identifiability for random translations of stationary point processes

1975 ◽  
Vol 12 (04) ◽  
pp. 734-743
Author(s):  
Toshio Mori
Keyword(s):  
Point Process ◽  
Real Line ◽  
Stationary Point ◽  
Point Processes ◽  
Identification Problem ◽  
Original Process ◽  
The Real ◽  
Displacement Distribution ◽  
Stationary Point Process ◽  
Stationary Point Processes

A bivariate point process consisting of an original stationary point process and its random translation is considered. Westcott's method is applied to show that if the original point process is ergodic then the bivariate point process is also ergodic. This result is applied to an identification problem of the displacement distribution. It is shown that if the spectrum of the original process is the real line then the displacement distribution is identifiable from almost every sample realisation of the bivariate process.

Some multivariate generalizations of results in univariate stationary point processes

1977 ◽  
Vol 14 (4) ◽  
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.

Ergodicity and identifiability for random translations of stationary point processes

1975 ◽  
Vol 12 (4) ◽  
pp. 734-743 ◽  
Author(s):  
Toshio Mori
Keyword(s):  
Point Process ◽  
Real Line ◽  
Stationary Point ◽  
Point Processes ◽  
Identification Problem ◽  
Original Process ◽  
The Real ◽  
Displacement Distribution ◽  
Stationary Point Process ◽  
Stationary Point Processes

A bivariate point process consisting of an original stationary point process and its random translation is considered. Westcott's method is applied to show that if the original point process is ergodic then the bivariate point process is also ergodic. This result is applied to an identification problem of the displacement distribution. It is shown that if the spectrum of the original process is the real line then the displacement distribution is identifiable from almost every sample realisation of the bivariate process.

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