Download - An Introduction to Point Processes
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1. Definitions & examples
2. Conditional intensity & Papangelou intensity
3. Models
a) Renewal processes
b) Poisson processes
c) Cluster models
d) Inhibition models
An Introduction to Point Processes
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2Centroids of Los Angeles County wildfires, 1960-2000
Point process: a random point pattern.
Point pattern: a collection of points in some space.
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Aftershocks from global large earthquakes
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Epicenters & times of microearthquakes in Parkfield, CA
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Hollister, CA earthquakes: locations, times, & magnitudes
Marked point process: a random variable (mark) with each point.
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Los Angeles Wildfires: dates and sizes
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Time series:
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Time series: Palermo football rank vs. time
Marked point process: Hollister earthquake times & magnitudes
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Modern definition: A point process N is a Z+-valued random measure
N(a,b) = Number of points with times between a & b.
N(A) = Number of points in the set A.
Antiquated definition: a point process N(t) is a right-continuous, Z+-valued stochastic process:
--x-------x--------------x-----------------------x---x-x---------------
0 t T
N(t) = Number of points with times < t. Problem: does not extend readily to higher dimensions.
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More Definitions:
• -finite: finite number of pts in any bounded set.
• Simple: N({x}) = 0 or 1 for all x, almost surely. (No overlapping pts.)
• Orderly: N(t, t+ )/ ---->p 0, for each t.• Stationary: The joint distribution of {N(A1+u), …, N(Ak+u)} does not
depend on u.
Notation & Calculus:
• ∫A f(x) dN = ∑f(xi ), for xi in A.
•∫A dN = N(A) = # of points in A.
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Intensities (rates) and Compensators
-------------x-x-----------x----------- ----------x---x--------------x------0 t- t t+ T
• Consider the case where the points are observed in time only. N[t,u] = # of pts between times t and u.
• Overall rate: (t) = limt -> 0 E{N[t, t+t)} / t.
•Conditional intensity: (t) = limt -> 0 E{N[t, t+t) | Ht} / t, where Ht = history of N for all times before t.
•If N is orderly, then (t) = limt -> 0 P{N[t, t+t) > 0 | Ht} / t.
•Compensator: predictable process C(t) such that N-C is a martingale.If (x) exists, then ∫o
t (u) du = C(t).
•Papangelou intensity: p(t) = limt -> 0 E{N[t, t+t) | Pt} / t, where Pt = information on N for all times before and after t.
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Intensities (rates) and Compensators
-------------x-x-----------x----------- ----------x---x--------------x------0 t- t t+ T
These definitions extend to space and space-time:
Conditional intensity:
(t,x) = limt,x -> 0 E{N[t, t+t) x Bx,x | Ht} / tx,
where Ht = history of N for all times before t, and Bx,x is a ball around x of size x.
Compensator: ∫A (t,x) dt dx = C(A).
Papangelou intensity:
p(t,x) = limt,x -> 0 E{N[t, t+t) x Bx,x | Pt,x} / tx, where Pt,x = information on N for all times and locations except (t,x).
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Some Basic Properties of Intensities:
•Fact 1 (Uniqueness). If exists, then it determines the distribution of N. (Daley and Vere-Jones, 1988).
•Fact 2 (Existence). For any simple point process N, the compensator C exists and is unique. (Jacod, 1975) Typically we assume that exists, and use it to model N.
•Fact 3 (Kurtz Theorem). The avoidance probabilities, P{N(A)=0} for all measurable sets A, also uniquely determine the distribution of N.
•Fact 4 (Martingale Theorem). For any predictable process f(t),E ∫ f(t) dN = E ∫ f(t) (t) dt.
•Fact 5 (Georgii-Zessin-Nguyen Theorem). For any ex-visible process f(x),E ∫ f(x) dN = E ∫ f(x) p(x) dx.
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Some Important Point Process Models:
1) Renewal process. The inter-event times: t2 - t1, t3 - t2, t4 - t3, etc. are independent and identically distributed random variables. (Classical density estimation.)Ex.: Normal, exponential, power-law, Weibull, gamma, log-normal.
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2) Poisson process.
Fact 6: If N is orderly and does not depend on the history of the process, then N is a Poisson process:
N(A1), N(A2), … , N(Ak) are independent, and each has the Poisson dist.: P{N(A) = j} = [C(A)]j exp{-C(A)} / j!.Recall: C(A) = ∫A (x) dx.
a) Stationary (homogeneous) Poisson process: (x) = .
Fact 7: Equivalent to a renewal process with exponential inter-event times.
b) Inhomogeneous Poisson process: (x) = f(x),where f(x) is some fixed, deterministic function.
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The Poisson process is the limiting distribution in many important results:
Fact 8 (thinning; Westcott 1976): Suppose N is simple, stationary, & ergodic.
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Fact 9 (superposition; Palm 1943): Suppose N is simple & stationary.
Then Mk --> stationary Poisson.
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Fact 10 (translation; Vere-Jones 1968; Stone 1968): Suppose N is stationary.
Then Mk --> stationary Poisson.
For each point xi in N, move it to xi + yi, where {yi} are iid.Let Mk be the result of k such translations.
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Fact 11 (rescaling; Meyer 1971):
Suppose N is simple and has at most one point on any vertical line.
Rescale the y-coordinates: move each point (xi, yi) to (xi , ∫oyi (xi,y) dy).
Then the resulting process is stationary Poisson.
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3) Some cluster models.
a) Neyman-Scott process: clusters of points whose centers are formed from a stationary Poisson process. Typically each cluster consists of a fixed integer k of points which are placed uniformly and independently within a ball of radius r around each cluster’s center.
b) Cox-Matern process: cluster sizes are random: independent and identically distributed Poisson random variables.
c) Thomas process: cluster sizes are Poisson, and the points in each cluster are distributed independently and isotropically according to a Gaussian distribution.
d) Hawkes (self-exciting) process: “mothers” are formed from a stationary Poisson process, and each produces a cluster of “daughter” points, and each of them produces a cluster of further “daughter” points, etc. (t, x) = + ∑ g(t-ti, ||x-xi||).
ti < t
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4) Some inhibition models.
a) Matern (I) process: first generate points from a stationary Poisson process, and then if there are any pairs of points within distance d of each other, delete both of them.
b) Matern (II) process: generate a stationary Poisson process, then index the points j = 1,2,…,n at random, and then successively delete any point j if it is within distance d from any retained point with smaller index.
c) Simple Sequential Inhibition (SSI): Keep simulating points from a stationary Poisson process, deleting any if it is within distance d from any retained point, until exactly k points are kept.
d) Self-correcting process: Hawkes process where g can be negative: (t, x) = + ∑ g(t-ti, ||x-xi||).
ti < t
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Poisson (100) Poisson (50+50x+50y) Neyman-Scott(10,5,0.05) Cox-Matern(10,5,0.05)
Thomas (10,5,0.05) Matern I (200, 0.05) Matern II (200, 0.05) SSI (200, 0.05)
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In modeling a space-time marked point process, usually directly model (t,x,a).
For example, for Los Angeles County wildfires:
•Windspeed. Relative Humidity, Temperature, Precipitation, •Tapered Pareto size distribution f, smooth spatial background .
(t,x,a) = 1exp{2R(t) + 3W(t) + 4P(t)+ 5A(t;60)
+ 6T(t) + 7[8 - D(t)]2} (x) g(a).
Could also include fuel age, wind direction, interactions…
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r = 0.16(s
q m
)
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30(F)
(sq
m)
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In modeling a space-time marked point process, usually directly model (t,x,a).
For example, for Los Angeles County wildfires:
•Windspeed. Relative Humidity, Temperature, Precipitation, •Tapered Pareto size distribution f, smooth spatial background .
(t,x,a) = 1exp{2R(t) + 3W(t) + 4P(t)+ 5A(t;60)
+ 6T(t) + 7[8 - D(t)]2} (x) g(a).
Could also include fuel age, wind direction, interactions…
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In modeling a space-time marked point process, usually directly model (t,x,a).
For example, for Los Angeles County wildfires:
• Relative Humidity, Windspeed, Precipitation, Aggregated rainfall over previous 60 days, Temperature, Date • Tapered Pareto size distribution f, smooth spatial background .
(t,x,a) = 1exp{2R(t) + 3W(t) + 4P(t)+ 5A(t;60) + 6T(t) + 7[8 - D(t)]2} (x) g(a).
Could also include fuel age, wind direction, interactions…
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(Ogata 1998)
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Simulation.
1) Sequential.a) Renewal processes are easy to simulate: generate iid random variables z1, z2, … from the renewal distribution, and let t1=z1,t2= z1+ z2, t3= z1+z2+z3, etc.
b) Reverse Rescaling. In general, can simulate a Poisson process with rate 1, and move each point (ti, xi) to (ti , yi),
where xi = ∫oyi (ti,x) dx.
2) Thinning.If m = sup (t, x), first generate a Poisson process with rate m, and then keep each point (ti, xi) with probability (ti, xi)/m.
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Summary: • Point processes are random measures:
N(A) = # of points in A. • (t,x) = Expected rate around x, given history < time t.• Classical models are renewal & Poisson processes.• For Poisson processes, (t,x) is deterministic.• Poisson processes are limits in thinning, superposition, translation, and rescaling theorems.• Non-Poisson processes may have clustering (Neyman-Scott, Cox-Matern, Thomas, Hawkes) or inhibition (MaternI, MaternII, SSI, self-correcting).
Next time: How to estimate the parameters in these models, and how to tell how well a model fits….