Poisson Processes Shane G Henderson http people orie
- Slides: 14
Poisson Processes Shane G. Henderson http: //people. orie. cornell. edu/~shane
A Traditional Definition Shane G. Henderson 2/15
What Are They For? 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) Shane G. Henderson 3/15
“Palm-Khintchine Theorem” User 1 ★ User 2 ★ User 3 User 4 ★ ★ ★ … User n ★ ★ time Shane G. Henderson 4/15
A Point-Process Definition Shane G. Henderson 5/15
Poisson Point-Processes Shane G. Henderson 6/15
Superposition Shane G. Henderson 7/15
Transformations Shane G. Henderson 8/15
Inversion t Shane G. Henderson 9/15
Marking This “works” because of order-statistic property t Shane G. Henderson 10/15
Thinning 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 i. i. d. U(0, 1) t Shane G. Henderson 11/15
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 Shane G. Henderson 12/15
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 nonhomogeneous Poisson processes using projections. TOMACS Shane G. Henderson 13/15
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 Shane G. Henderson 14/15
