Poisson Processes

14
Poisson Processes Shane G. Henderson http://people.orie.cornell.edu/~shane

TAGS:

description

Poisson Processes. Shane G. Henderson http://people.orie.cornell.edu/~shane. A Traditional Definition. What A re T hey For?. Times of customer arrivals (no scheduling and no groups). Locations, e.g., flaws on wafers, ambulance call locations , submarine locations. - PowerPoint PPT Presentation

Transcript of Poisson Processes

Page 1: Poisson Processes

Poisson ProcessesShane G. Henderson

http://people.orie.cornell.edu/~shane

http://people.orie.cornell.edu/~shane
Page 2: Poisson Processes

Shane G. Henderson

A Traditional Definition

2/15

Page 3: Poisson Processes

Shane G. Henderson

What Are They For?

3/15

Times of customer arrivals (no scheduling and no groups)

Locations, e.g., flaws on wafers, ambulance call locations,submarine locations

Ambulance call times and locations (3-D)

Page 4: Poisson Processes

Shane G. Henderson

“Palm-Khintchine Theorem”

User 1 ★ ★

User 2 ★ ★

User 3 ★

User 4 ★ ★

…

User n ★ ★

4/15

time

Page 5: Poisson Processes

Shane G. Henderson

A Point-Process Definition

5/15

Page 6: Poisson Processes

Shane G. Henderson

Poisson Point-Processes

6/15

Page 7: Poisson Processes

Shane G. Henderson

Superposition

7/15

Page 8: Poisson Processes

Shane G. Henderson

Transformations

8/15

Page 9: Poisson Processes

Shane G. Henderson

Inversion

9/15

t

Page 10: Poisson Processes

Shane G. Henderson

Marking

10/15

t

This “works” because oforder-statistic property

Page 11: Poisson Processes

Shane G. Henderson

Thinning

11/15

t

i.i.d.U(0,1)

Thinned points and retained points are in different regions,therefore independent“t” coordinates of retained points are a Poisson process, rate “t” coordinates of thinned points are too, rate

Page 12: Poisson Processes

Shane G. Henderson

More on Marking

• Suppose call times for an ambulance follow a Poisson process in time

• Mark each call with the call location (latitude, longitude)

• Resulting 3-D points are those of a Poisson process

12/15

Page 13: Poisson Processes

Shane G. Henderson

More on Marking

To generate Poisson processes in > 1 dimension, one way is to

– First generate their projection onto a lower dimensional structure (Poisson)

– Independently mark each point with the appropriate conditional distribution

Saltzman, Drew, Leemis, H. (2012). Simulating multivariate non-homogeneous Poisson processes using projections. TOMACS

13/15

Page 14: Poisson Processes

Shane G. Henderson

This View of Poisson Processes

• Is mathematically elegant• Is highly visual and therefore intuitive• Makes proving many results almost as

easy as falling off a log– Try proving thinned and retained points are

independent Poisson processes• Suggests other generation algorithms

14/15


APPM 7400 Lesson 11: Spatial Poisson Processes...The Spatial Poisson Process Consider a spatial con guration of points in the plane: Lesson 11: Spatial Poisson Processes Stochastic

APPM 7400 Lesson 11: Spatial Poisson Processes...The Spatial Poisson Process Consider a spatial con guration of points in the plane: Lesson 11: Spatial Poisson Processes Stochastic

POISSON PROCESSES - RLE at · PDF fileChapter 2 POISSON PROCESSES 2.1 Introduction A Poisson process is a simple and widely used stochastic process for modeling the times at which

POISSON PROCESSES - RLE at · PDF fileChapter 2 POISSON PROCESSES 2.1 Introduction A Poisson process is a simple and widely used stochastic process for modeling the times at which

Thinning spatial point processes into Poisson …frederic/papers/thinning6.pdfThinning spatial point processes into Poisson processes Jesper Møller1 and Frederic Paik Schoenberg2

Thinning spatial point processes into Poisson …frederic/papers/thinning6.pdfThinning spatial point processes into Poisson processes Jesper Møller1 and Frederic Paik Schoenberg2

Discrete Stochastic Processes, Chapter 2: Poisson Processes

Discrete Stochastic Processes, Chapter 2: Poisson Processes

Poisson Processes and Applications in Hockey · My honours project is on Poisson processes, which are named after the French mathe-matician Sim eon-Denis Poisson. The Poisson process

Poisson Processes and Applications in Hockey · My honours project is on Poisson processes, which are named after the French mathe-matician Sim eon-Denis Poisson. The Poisson process

Chapter 4: Stochastic Processes Poisson Processes and Markov Chains Presented by Vincent Buhr.

Chapter 4: Stochastic Processes Poisson Processes and Markov Chains Presented by Vincent Buhr.

Chapter 2. Poisson Processes ch.2...– Definition of renewal process – Definition of Poisson process • Properties of Poisson processes – Inter-arrival time distribution –

Chapter 2. Poisson Processes ch.2...– Definition of renewal process – Definition of Poisson process • Properties of Poisson processes – Inter-arrival time distribution –

STOCHASTIC PROCESSES AND CONTROL FOR JUMP …hanson/pub/SIREV07/sirev07iisc.pdfKey words. Jump-diffusions, Wiener processes, Poisson processes, random jump amplitudes, stochastic

STOCHASTIC PROCESSES AND CONTROL FOR JUMP …hanson/pub/SIREV07/sirev07iisc.pdfKey words. Jump-diffusions, Wiener processes, Poisson processes, random jump amplitudes, stochastic

Construction of Dependent Dirichlet Processes based on Poisson

Construction of Dependent Dirichlet Processes based on Poisson

Stochastic Processes: Examples - Stanford Universitystatweb.stanford.edu/~jtaylo/courses/stats116/simulation/processes.pdfStochastic Processes Stochastic Processes Poisson Process

Stochastic Processes: Examples - Stanford Universitystatweb.stanford.edu/~jtaylo/courses/stats116/simulation/processes.pdfStochastic Processes Stochastic Processes Poisson Process

18.440: Lecture 13 .1in Poisson processes - MIT Mathematicsmath.mit.edu/~sheffield/440/Lecture13.pdf · 18.440: Lecture 13 Poisson processes Scott She eld MIT 18.440 Lecture 13

18.440: Lecture 13 .1in Poisson processes - MIT Mathematicsmath.mit.edu/~sheffield/440/Lecture13.pdf · 18.440: Lecture 13 Poisson processes Scott She eld MIT 18.440 Lecture 13

POISSON PROCESSES - Research Laboratory of … · Poisson processes are the nicest, and in many ways most important, class of renewal pro-cesses. They are characterized by having

POISSON PROCESSES - Research Laboratory of … · Poisson processes are the nicest, and in many ways most important, class of renewal pro-cesses. They are characterized by having

Marked Point Processes - Purdue Universityhuang251/point2.pdf · Marked Poisson Point Process 0.2 0.4 0.6 0.8 1 1.2 Figure : A Simulated Example of Poisson Marked Poisson Processes

Marked Point Processes - Purdue Universityhuang251/point2.pdf · Marked Poisson Point Process 0.2 0.4 0.6 0.8 1 1.2 Figure : A Simulated Example of Poisson Marked Poisson Processes

Lecture 2: Everything you need to know to know about point processes Outline: basic ideas homogeneous (stationary) Poisson processes Poisson distribution.

Lecture 2: Everything you need to know to know about point processes Outline: basic ideas homogeneous (stationary) Poisson processes Poisson distribution.

Some notes on Poisson Processes from a web - Home Cc Umanitoba

Some notes on Poisson Processes from a web - Home Cc Umanitoba

TRANSFORMING SPATIAL POINT PROCESSES INTOpeople.math.aau.dk/~jm/KasperComplementaryProcessRR.pdf · note Poisson(S,β) the distribution of the Poisson process on S with intensity

TRANSFORMING SPATIAL POINT PROCESSES INTOpeople.math.aau.dk/~jm/KasperComplementaryProcessRR.pdf · note Poisson(S,β) the distribution of the Poisson process on S with intensity

Large deviations for Poisson driven processes in epidemiologycermics.enpc.fr/~lelievre/CEMRACS/slides/Kratz.pdf · Large deviations for Poisson driven processes ... Long-term behavior

Large deviations for Poisson driven processes in epidemiologycermics.enpc.fr/~lelievre/CEMRACS/slides/Kratz.pdf · Large deviations for Poisson driven processes ... Long-term behavior

Intensity estimation of non-homogeneous Poisson processes ...

Intensity estimation of non-homogeneous Poisson processes ...

Poisson Point Processes: Imaging, Tracking, and Sensing · Poisson Point Processes: Imaging, Tracking, and Sensing ... 154 6 Applications: Multiple Target Tracking ... Poisson Point

Poisson Point Processes: Imaging, Tracking, and Sensing · Poisson Point Processes: Imaging, Tracking, and Sensing ... 154 6 Applications: Multiple Target Tracking ... Poisson Point

Poisson representations of measure-valued processes

Poisson representations of measure-valued processes