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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

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….