1 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

2 Centroids of Los Angeles County wildfires, Point process: a random point pattern. Point pattern: a collection of points in some space.

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6 Aftershocks from global large earthquakes

7 Epicenters & times of microearthquakes in Parkfield, CA

8 Hollister, CA earthquakes: locations, times, & magnitudes Marked point process: a random variable (mark) with each point.

9 Los Angeles Wildfires: dates and sizes

10 Time series:

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12 Time series: Palermo football rank vs. time Marked point process: Hollister earthquake times & magnitudes

13 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 t T N(t) = Number of points with times < t. Problem: does not extend readily to higher dimensions.

14 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(A 1 +u), …, N(A k +u)} does not depend on u. Notation & Calculus: ∫ A f(x) dN = ∑f(x i ), for x i in A. ∫ A dN = N(A) = # of points in A.

15 Intensities (rates) and Compensators x-x x x---x x 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) = lim  t -> 0 E{N[t, t+  t)} /  t. Conditional intensity: (t) = lim  t -> 0 E{N[t, t+  t) | H t } /  t, where H t = history of N for all times before t. If N is orderly, then (t) = lim  t -> 0 P{N[t, t+  t) > 0 | H t } /  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) = lim  t -> 0 E{N[t, t+  t) | P t } /  t, where P t = information on N for all times before and after t.

16 Intensities (rates) and Compensators x-x x x---x x t- t t+ T These definitions extend to space and space-time: Conditional intensity: (t,x) = lim  t,  x -> 0 E{N[t, t+  t) x B x,  x | H t } /  t  x, where H t = history of N for all times before t, and B x,  x is a ball around x of size  x. Compensator: ∫ A (t,x) dt dx = C(A). Papangelou intensity: p (t,x) = lim  t,  x -> 0 E{N[t, t+  t) x B x,  x | P t,x } /  t  x, where P t,x = information on N for all times and locations except (t,x).

17 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.

18 Some Important Point Process Models: 1)Renewal process. The inter-event times: t 2 - t 1, t 3 - t 2, t 4 - t 3, etc. are independent and identically distributed random variables. (Classical density estimation.) Ex.: Normal, exponential, power-law, Weibull, gamma, log-normal.

19 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(A 1 ), N(A 2 ), …, N(A k ) 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.

20 The Poisson process is the limiting distribution in many important results: Fact 8 (thinning; Westcott 1976): Suppose N is simple, stationary, & ergodic.

21 Fact 9 (superposition; Palm 1943): Suppose N is simple & stationary. Then M k --> stationary Poisson.

22 Fact 10 (translation; Vere-Jones 1968; Stone 1968): Suppose N is stationary. Then M k --> stationary Poisson. For each point x i in N, move it to x i + y i, where {y i } are iid. Let M k be the result of k such translations.

23 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 (x i, y i ) to (x i, ∫ o y i (x i,y) dy). Then the resulting process is stationary Poisson.

24 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-t i, ||x-x i ||). t i < t

25 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-t i, ||x-x i ||). t i < t

26 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)

27 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) =  1 exp{  2 R(t) +  3 W(t) +  4 P(t)+  5 A(t;60) +  6 T(t) +  7 [  8 - D(t)] 2 }  (x) g(a). Could also include fuel age, wind direction, interactions…

28 r = 0.16 (sq m)

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30 (F) (sq m)

31 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) =  1 exp{  2 R(t) +  3 W(t) +  4 P(t)+  5 A(t;60) +  6 T(t) +  7 [  8 - D(t)] 2 }  (x) g(a). Could also include fuel age, wind direction, interactions…

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34 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) =  1 exp{  2 R(t) +  3 W(t) +  4 P(t)+  5 A(t;60) +  6 T(t) +  7 [  8 - D(t)] 2 }  (x) g(a). Could also include fuel age, wind direction, interactions…

35 (Ogata 1998)

36 Simulation. 1)Sequential. a) Renewal processes are easy to simulate: generate iid random variables z 1, z 2, … from the renewal distribution, and let t 1 =z 1, t 2 = z 1 + z 2, t 3 = z 1 +z 2 +z 3, etc. b) Reverse Rescaling. In general, can simulate a Poisson process with rate 1, and move each point (t i, x i ) to (t i, y i ), where x i = ∫ o y i (t i,x) dx. 2)Thinning. If m = sup (t, x), first generate a Poisson process with rate m, and then keep each point (t i, x i ) with probability (t i, x i )/m.

37 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….